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+{"text":"\\section{Introduction}\n\n\n\nAntimonene is a recent addition~\\cite{Zhang2015,Lei2016} to a growing family of elemental two-dimensional (2D) materials. This monolayer phase of antimony adopts a buckled honeycomb lattice ($D_{3d}$ point group), similar to those of \nsilicene~\\cite{Vogt2012} and germanene~\\cite{Zhang2016}, or more recently discovered blue phosphorene~\\cite{Gu2017}.\nStructurally, antimonene is identical to a single layer of bulk antimony, which possesses layered rhombohedral structure with a $D_{3d}$ point group.\nAntimonene was successfully obtained by various experimental techniques~\\cite{Ares2018}, including but not limited to, epitaxial growth~\\cite{Wu2017}, liquid- and solid-phase exfoliation~\\cite{Gibaja2016,Ares2016}.\nBased on both experimental observation and \\textit{ab initio} modeling~\\cite{Ares2016,Zhang2015}, antimonene is believed to be a material with remarkable stability in air and in water~\\cite{Ares2016,Wu2017}. \nThis alone makes antimonene an appealing candidate for various applications because instability at realistic conditions is one of the main factors limiting the application of 2D materials~\\cite{Morishita2015,Kuriakose2018}.\nFrom a fundamental point of view, characteristic feature of antimonene is a strong spin-orbit coupling (SOC)~\\cite{Rudenko2017}, with the intraatomic SOC constant $\\lambda_{\\text{SOC}}=0.34$ eV. A proper account of SOC is important for a correct description of the electronic structure and effective masses~\\cite{Rudenko2017}. Antimonene is expected to have high charge carrier mobility~\\cite{Pizzi2016} comparable to or exceeding that of the other 2D semiconductors. Besides, monolayer Sb is an indirect band semiconductor with a theoretically estimated band gap of 1.2 eV \\cite{Singh2016}. \nThese properties make antimonene suitable for application in electronic~\\cite{Pizzi2016} and optical devices~\\cite{Singh2016}, where thickness- and strain-tunable band gap could provide an additional control over the material's properties~\\cite{Zhao2015,Zhang2015}.\n\nSuperconductivity is not uncommon for two-dimensional materials~\\cite{Uchihashi2017}. It is observed in thin films~\\cite{Ozer2006}, atomic sheets~\\cite{Ludbrook2015}, surface atomic layers~\\cite{Ge2015} and other 2D structures of various chemical compositions.\nRecent experimental works~\\cite{Chapman2016,Ludbrook2015} revealed superconductivity in graphene laminates with the critical temperatures ${T_{\\text{c}}}$~in the range of 4--6~K.\nMagnetization measurements of intercalated black phosphorus reveal ${T_{\\text{c}}}$~$=3.8\\pm$0.1~K, which was shown to be the same for different intercalants~\\cite{Zhang2017}.\nAlso, a large number of experimental measurements of ${T_{\\text{c}}}$~were recently performed for doped 2D transition metal dichalcogenides: WS$_{2}$ and NbSe$_{2}$ were demonstrated to be superconducting below 3~K \\cite{Lu2018,Xi2015}, while MoS$_2$ was reported to have ${T_{\\text{c}}}$~in the range of 7--11~K \\cite{Lu2015,Ye1193}.\nSuperconductivity in two dimensions is especially interesting in view of the possibility of controlling the electronic structure of 2D materials by\nstrain, electric field, thickness, or substrate. Superconducting 2D materials also appear as an appealing testbed for studying interface phenomena and proximity effects. In particular, 2D materials can be implemented as a part of the Josephson junction~\\cite{Heersche2007,Yabuki2016}. \n\nComputational studies predicting the superconductivity in monolayer graphene~\\cite{Profeta2012,Margine2014} and phosphorene~\\cite{Shao2014} were preceding the experimental observations~\\cite{Ludbrook2015,Chapman2016,Zhang2017}.\nAlso, superconductivity at nonzero charge doping has already been predicted for recently proposed arsenene~\\cite{Kong2018} (monolayer As), as well as for silicene \\cite{Durajski2014}. Experimental data on superconductivity of these materials are not available yet.\nWhile theoretical studies lack realistic accounts of the experimental setup, they allow study of the underlying mechanisms of superconductivity, for example, the key contributions to electron-phonon coupling or important changes of electronic structure due to charge carrier doping~\\cite{Ge2013}.\nAdditionally, there is an opportunity to study modifications of the electronic structure and electron-phonon coupling properties including ${T_{\\text{c}}}$~in the presence of strain~\\cite{Shao2014} or other external factors.  Neither calculations nor measurements of ${T_{\\text{c}}}$~are available in the literature for doped antimonene.\n\nIn this paper, we present a systematic study of the electron-phonon coupling and conventional superconductivity in $n$- and $p$-doped antimonene. To this end, we use a combination of Density Functional Theory (DFT)~\\cite{Hohenberg1964,Kohn1965}  and Density Functional Perturbation Theory (DFPT)~\\cite{Gonze1997} in conjunction with the formalism of Maximally Localized Wannier Functions (MLWF)~\\cite{Giustino2007,Marzari2012}.\nBesides charge doping, we also consider the role of an electric field applied in the direction perpendicular to the atomic layer, which allows us to modify the band structure.\nConsidering the interplay of these effects is interesting for a number of reasons.\nFor example, the perpendicular electric field allows one to control the band gap and effective masses in silicene and germanene~\\cite{Ni2012,Acun2015}, as well is in few-layer phosphorene~\\cite{Rudenko2015,Kim723,Liu2015}, where it is possible to transform material from normal insulator to topological insulator or metal.\nBesides this, perpendicular electric field breaks inversion symmetry, which, in combination with strong SOC, induces spin-splitting of both valence and conduction bands~\\cite{Kormanyos2013}.  \nThis case is observed, for example, in experimental work~\\cite{Ye1193}, where setup combining liquid and solid gating is used to study superconductivity in MoS$_{2}$ flakes, and ionic liquid naturally creates the environment for emergence of perpendicular electric field.\nSince SOC in antimonene is strong, this situation is particularly interesting to study.\nFor these reasons, we study the dependence of the electron-phonon coupling on bias voltage.\n\nThe rest of the paper is organized as follows. We first present the theory and computational methods used to calculate the electron-phonon coupling and superconducting critical temperatures, and also discuss the relevant approximations (Sec.~II). We then give a description of the calculated electronic structure and phonon dispersion of antimonene (Sec.~III~A). Main results are presented in Sec.~III~B, where we discuss the obtained dependencies of ${T_{\\text{c}}}$~and electron-phonon coupling strength, as well as superconducting transition temperatures on charge carrier concentrations, with a detailed consideration of the most important cases. The effect of a perpendicular electric field (bias voltage) is analyzed and discussed in Sec.~III~C. \nIn Sec.~IV, we summarize our results and conclude the paper.\n\n\n\\section{Theoretical background and computational details}\n\n\\subsection{Electron-phonon coupling and Allen-Dynes-McMillan equation}\nIn this section, we give a short theoretical description of the electron-phonon coupling and its relation to superconductivity.\nMore detailed description of the theory from the viewpoint of \\textit{ab initio} calculations can be found in Ref.~\\onlinecite{Giustino2017}.\n\nThe interaction of electrons with phonons is described by the matrix elements\n\\begin{equation}\ng_{mn,\\nu}({\\bf k,q}) = \\bigg( \\frac{\\hbar}{2m_0\\omega_{{\\bf q}\\nu} }\n\\bigg)^{1\/2} M_{mn}^{\\nu}({\\bf k},{\\bf q}),\n\\label{eq:gmn}\n\\end{equation}\nwhere \n\\begin{equation}\t\t\nM_{mn}^{\\nu}({\\bf k},{\\bf q}) =  \n\\langle \\psi_{m{\\bf k+q}} | \\partial_{{\\bf q}\\nu}V | \\psi_{n{\\bf k}}\\rangle,\n\\label{eq:mmn}\n\\end{equation}\nand $\\psi_{n{\\bf k}}$ is the electronic Bloch function for band $n$ and wavevector ${\\bf k}$; $\\partial_{{\\bf q}\\nu}V$ is the derivative of the self-consistent potential associated with  the phonon of wavevector ${\\bf q}$, branch index $\\nu$; and frequency $\\omega_{{\\bf q}\\nu}$; and $m_{0}$ is the atomic mass. \n\nIt is useful to define a dimensionless representation of the electron-phonon coupling associated with the single phonon mode $\\nu$ and wavevector ${\\bf q}$ as an average over the Fermi surface:\n\n\\begin{equation}\n\\label{eq:lambda}\n\\begin{split}\n\\lambda_{{\\bf q}\\nu} = \n\\frac{1}{N_{\\rm F}\\omega_{{\\bf q}\\nu}}\\sum_{mn,{\\bf k}} \n\\mathrm{w}_{{\\bf k}} |g_{mn,\\nu}({\\bf k,q})|^2 \\\\ \n\\times\\delta(\\varepsilon_{n{\\bf k}}-\\varepsilon_F)\\delta(\\varepsilon_{m{\\bf k}+{\\bf q}}-\\varepsilon_F),\n\\end{split}\n\\end{equation}\t\t\t\t\t\t\t\nwhich is closely related to the Eliashberg electron-phonon spectral function $\\alpha^{2}F(\\omega)$\n\\begin{equation}\n\\alpha^{2}F( \\omega) = \\frac{1}{2} \\sum_{{\\bf q}\\nu} \n\\mathrm{w}_{\\bf q} \\omega_{{\\bf q}\\nu} \\lambda_{{\\bf q}\\nu}\n\\delta (\\omega - \\omega_{{\\bf q}\\nu}) \n, \n\\label{eq:a2f}\n\\end{equation}\t\t\t\t\t\t\t\t\t\t\t\nwhere $\\mathrm{w}_{\\bf k}$ ($\\mathrm{w}_{\\bf q}$) is the symmetry-dependent weight of ${\\bf k}$ (${\\bf q}$) points, $\\varepsilon_{n{\\bf k}} (\\varepsilon_{m{\\bf k+q}})$ is the electronic energy, and $N_{\\text{F}}$~ is the density of states (DOS) at the Fermi energy, $\\varepsilon_F$.\n\n\nBesides $\\lambda_{{\\bf q},\\nu}$, we also consider the nesting function $\\xi_{\\bf q}$ as defined in~\\cite{Bazhirov2010}:\n\\begin{equation}\n\\xi_{\\bf q}= \n\\sum_{mn,{\\bf k}} \n\\mathrm{w}_{{\\bf k}}\n\\delta(\\varepsilon_{n{\\bf k}}-\\varepsilon_F)\\delta(\\varepsilon_{m{\\bf k}+{\\bf q}}-\\varepsilon_F).\n\\label{eq:nesting}\n\\end{equation}\t\t\nWhile the nesting function is independent of electron-phonon coupling matrix elements, it provides the understanding of effects of Fermi surface topology on effective electron-phonon coupling. It is important to mention, that the singularities of nesting function, associated with such topological phenomena as Fermi surface nesting may result in a high electron susceptibility at specific \\textbf{q}, thus increasing the value of $\\lambda$. Additional effects can be related to softening of phonon frequencies near this \\textbf{q}-points, due to the giant Kohn anomaly~\\cite{Katsnelson1994}.\n\nThe critical temperature of the superconducting transition according to the McMillan equation~\\cite{McMillan1968}, modified by Allen and Dynes~\\cite{Allen1975} can be estimated as:\n\\begin{equation}\nT_{\\text{c}} = \n\\frac{\\omega_{\\text{log}}}{1.2}\n\\exp\\left(\n-\\frac{1.04(1+\\lambda)}\n{\\lambda-\\mu_{\\text{c}}^{*}(1+0.62\\lambda)}\n\\right), \n\\label{eq:ADMC}\n\\end{equation}\t\nwhich is based on the superconductivity theory of Migdal and Eliashberg~\\cite{Migdal1958,Eliashberg1960}. Although derivation of Eq.~(\\ref{eq:ADMC}) contains a number of approximations, it is a reasonable starting point for the estimation of~${T_{\\text{c}}}$.\nIn Eq.~(\\ref{eq:ADMC}), ${\\mu^{*}_{\\text{c}}}$~is the Morel-Anderson effective Coulomb potential~\\cite{Morel1962},\n\n\\begin{equation}\n\\omega_{\\text{log}}=\n\\exp\\left[\n\\frac{2}{\\lambda}\\int_{0}^{\\omega_{\\text{max}}}\nd\\omega \\frac{\\alpha^{2}F( \\omega)}{\\omega} \\log \\omega\n\\right]\n\\label{eq:wlog}\n\\end{equation}\nis the logarithmically averaged phonon frequency,\n\\begin{equation}\n\\lambda = \\sum_{{\\bf q}\\nu} \\mathrm{w}_{{\\bf q}} \\lambda_{{\\bf q}\\nu}\n\\label{eq:lambtot}\n\\end{equation}\nis the total Fermi energy-dependent electron-phonon coupling strength, which is also known as the mass-enhancement factor~\\cite{Peters2018}. In the present paper, ${\\mu^{*}_{\\text{c}}}$~is considered as a phenomenological parameter with the typical values in the range of 0.1--0.2 \\cite{McMillan1968,Allen1975,Giustino2017}.\nAmong 2D materials the Coulomb pseudopotential was estimated more rigorously for monolayer graphene in Ref.~\\onlinecite{Margine2014} (${\\mu^{*}_{\\text{c}}}$~$=0.16$) and Ref.~\\onlinecite{Profeta2012} (${\\mu^{*}_{\\text{c}}}$~$=0.1$),  {\\color{black} bilayer graphene (${\\mu^{*}_{\\text{c}}}$~$=0.155$) in Ref.~\\onlinecite{Margine2016}, as well as NbS$_2$ (${\\mu^{*}_{\\text{c}}}$~$=0.2$) in Ref.~\\onlinecite{Heil2017}.} We note that an accurate estimate of ${\\mu^{*}_{\\text{c}}}$~requires an account of dielectric screening, and, therefore, would be strongly influenced by the underlying substrate.\n\n\\subsection{Calculation details}\n\\label{subsec:methods}\n\nThe initial electronic structure and crystal structure optimization calculations were performed at the DFT level as implemented in the plane-wave {\\sc Quantum ESPRESSO} ({\\sc QE}) code~\\cite{Giannozzi2009,Giannozzi2017},\nusing fully relativistic norm conserving pseudopotentials.\nExchange and correlation were treated with the local density approximation (LDA).\nThe kinetic energy cutoff for plane waves was set to 90 Ry, the Brillouin zone was sampled with a (16$\\times$16) Monkhorst-Pack \\textbf{k}-point mesh~\\cite{Monkhorst1976}, and the electronic state occupancies were treated as fixed.\nThe crystal structure was fully relaxed with a threshold of \\pten{1}{-12}~eV for total energies and \\pten{1}{-12}~eV\/\\AA~ for forces.\nThe vacuum thickness of 30 \\AA~was used to avoid spurious interactions between the supercell periodic images in the direction perpendicular to the 2D plane.\nThe Brillouin zone for phonons within DFPT was sampled by a (16$\\times$16) \\textbf{q}-point mesh, and the self-consistency threshold of \\pten{1}{-16}~eV was used.\n\nElectronic structure, dynamical, matrices and electron-phonon matrix elements obtained from DFT and DFPT calculations were used as the initial data for Wannier interpolation within the MLWF formalism, as implemented in {\\sc EPW}~\\cite{Giustino2007,Ponce2016}.\nThe calculation of the electron-phonon-related properties was performed on dense grids of (432$\\times$432) \\textbf{k}- and (208$\\times$208) \\textbf{q}-points, which ensures the numerical convergence of the results presented in this work.\n\n\\subsection{Role of charge doping, out-of-plane acoustic phonons, and bias voltage}\n\\label{subsec:approx}\n\nIn this work, we simulate the charge carrier doping of antimonene using the rigid band shift approximation, that leaves the electronic structure and phonon dispersion unchanged.\nTypically, charge doping in 2D materials mainly affects the out-of-plane mode and optical phonons~\\cite{Margine2014,Kong2018,Shao2014}.\nAt the same time, out-of-plane acoustic mode (ZA) is not taken into account in our calculations. \nAlthough consideration of ZA mode may be important for fundamental understanding of the associated effects, it is of little practical interest for the superconductivity studies: Interaction of 2D materials with a substrate would suppress out-of-plane vibrations making the already small coupling of electrons with these modes negligible. Furthermore, a correct description of such effects is not trivial within the slab geometry.\nAn accurate description of electrons coupling with out-of-plane phonon modes is presented in Ref.~\\onlinecite{Sohier2017} for doped graphene, where it is also shown, that the contribution from ZA mode is negligible. \nOptical phonons, as we show below, have a minor effect on the electron-phonon coupling and~${T_{\\text{c}}}$. Therefore, the rigid band shift approximation is justified for the purpose of our study.\nThe charge carrier concentration $\\delta \\rho$ is thus chosen in accordance with the ground state DOS and the Fermi energy, $\\delta \\rho(\\varepsilon_F)=\\int_0^{\\varepsilon_F}d\\varepsilon \\, \\rho(\\varepsilon)$.\n\nWe consider both $n$- and $p$-doping cases. We limit ourselves to the concentrations less than~\\con{1}{15}.\nAlthough such charge carrier concentrations correspond to a heavy doping regime, they are not unrealistic. Electron concentrations of the order of~\\con{1}{15} are achievable in thin metallic films by means of electrochemical techniques~\\cite{Daghero2012}. \nSmaller electron and hole concentrations of the order of \\con{1}{14} can be achieved by liquid gating~\\cite{Efetov2010} and  solid-electrolyte gating~\\cite{Xu2017}.\n{\\color{black}At the same time, the presence of the strong Fermi-surface nesting~\\cite{Antropov1988,Vaks1989,Katsnelson1994}, as well as high charge carrier concentrations may lead to the loss of structural stability of the system.\nSince such instabilities cannot be detected within the rigid band approximation, we also consider lattice dynamics of antimonene using the jellium doping method, as implemented in the DFT code used.\nFor every doping case, relaxation of atomic positions is performed, while the lattice parameter of undoped antimonene was used to model the behavior of the system on a substrate.}\n\n\nTo simulate the effect of a perpendicular electric field, we consider a tight-binding Hamiltonian obtained in the MLWF basis, and add position-dependent bias voltage, yielding the following Hamiltonian\n\\begin{equation}\nH=\\sum_{ij}t_{ij}c_i^{\\dag}c_j+V_{\\text{b}}\/d\\sum_{i}z_ic_i^{\\dag}c_i,\n\\end{equation}\nwhere the sum runs over the real-space MLWF orbitals $i$ and $j$, $c_i^{\\dag}$,$c_i$ ($c_{j}$) are the creation and annihilation operators of electrons on the corresponding orbitals, $z_i$ is the $z$-component of the position operator of the orbital $i$, $t_{ij}$ is the corresponding hopping integral, $d$ is the buckling constant, and $V_{\\text{b}}$ is bias voltage applied to the upper and lower planes of antimonene. \nIt is worth mentioning that since the consideration of electric field is not performed in the self-consistent manner, as is done, for example, in~Ref.~\\onlinecite{Li2018}, additional screening arising due to the charge carrier doping is not taken into account in our calculations; thus, the bias effect is quantitatively overestimated.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[]{struct.pdf} \n\t\\caption{Crystal structure of hexagonal antimonene; red lines denote the hexagonal unit cell, used in calculations. Lattice parameter, buckling constant, and bond length, denoted as $a$, $d$, and $b$ respectively, are also given on the figure.}\n\t\\label{fig:structure}\n\n\\end{figure}\n\t\n\\section{Results and discussion}\n\\subsection{Electronic structure and phonon dispersion}\n\\label{subsec:dft}\n\n\nWe find the calculated relaxed lattice parameter of free-standing antimonene to be $a = 4.0$~\\AA~with the bond length $b = 2.82$~\\AA~and buckling constant $d = 1.6$~\\AA.\nThe structure and calculated parameters are illustrated in Fig.~\\ref{fig:structure}.\nThe found values are consistent with previously reported data~\\cite{Wang2015}.\nThe corresponding electronic band structure is given in Fig.~\\ref{fig:electrons}(a), from which one can see that antimonene is an indirect gap semiconductor with the gap width of 0.7~eV. The obtained value is less than 1.0 and 1.2~eV obtained in Refs.~\\onlinecite{Rudenko2017,Singh2016}, due to the differences in exchange-correlation functional, but agrees well with the value reported in~Ref.~\\onlinecite{Wang2015}. \nThe band structure and DOS [Fig.~\\ref{fig:electrons}(b)] display high degrees of electron-hole asymmetry even at small Fermi energies, unlike those of typical 2D materials including graphene and phosphorene.\n\n\n\n\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[]{{fig1_a_b.pdf}}\\vspace*{0.3cm}\n\t\\includegraphics[]{{fig1_c_f.pdf}} \n    \\caption{Electronic structure of antimonene. (a) Band structure plotted along high-symmetry directions of the Brillouin zone. (b) DOS in the relevant charge carrier concentration range. Horizontal lines on panels (a) and (b) mark the following charge carrier concentrations (from bottom to top): $N_{\\text{h}}=$~\\con{1}{15}, $N_{\\text{h}}=$~\\con{3.7}{14}, $N_{\\text{e}}=$~\\con{4}{14}, and $N_{\\text{e}}=$~\\con{1}{15}. The corresponding Fermi surface contours are given on subplots (c)--(f).}\n\t\\label{fig:electrons}\n   \n\\end{figure}\n\n\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[]{{fig2.pdf}}\n\t\\caption{Phonon specturm of antimonene. (a) Phonon dispersion plotted along high-symmetry directions of the Brillouin zone. (b) Phonon DOS.}\n\t\\label{fig:phonons}\n\n\\end{figure}\n\nAt negative Fermi energies (hole-doping) the Fermi surface is initially formed by one valence band. As the Fermi energy reduces, second and third occupied bands get involved, forming three concentric pockets [see Fig.~\\ref{fig:electrons}(c)].\nIn the \\textit{K}--$\\Gamma$ direction of the electronic structure one can see a flat region in the valence band, which gives rise to a van Hove singularity (VHS) in DOS ($\\varepsilon=-1.3$~eV in Fig.~\\ref{fig:electrons}). The corresponding\nconstant-energy contour is shown in Fig.~\\ref{fig:electrons}(d), which exhibits a hexagonal warping.\nThis topology opens up a possibility for the scattering between the parallel regions of the surface, known as the Fermi surface nesting~\\cite{Katsnelson1994,Landa2018}. In this case one can expect an increase of the electron-phonon coupling strength at the corresponding charge carrier concentrations ($N_{\\mathrm h}=~$\\con{3.7}{14}).\nFurther reduction of the Fermi energy leads to a widening of the contours \naccompanied by the bending of the hexagons, suppressing the nesting effect. \nThe topology of the Fermi surface at positive Fermi energies (electron-doping) is determined entirely by a single conduction band, yet involving multiple valleys.\nAt small energies the surface is formed by six closed droplet-shaped pockets, originating from the valleys centered along the $\\Gamma$--\\textit{M} direction.\nAdditional circle-shaped pockets appear as the Fermi energy reaches the valley at the \\textit{K} point [Fig.~\\ref{fig:electrons}(e)].\nFurther increase of the Fermi energy results in a VHS originating from the band bending around the \\textit{M} point.\n\n\n\nAt even higher conduction band fillings there is a distinctive change in the Fermi surface topology. Namely, previously closed pockets become connected, and an additional valley emerges around $\\Gamma$ [circle region in Fig.~\\ref{fig:electrons}(f)]. Similar to the valence band, one can see a hexagonal warping around the \\textit{K} point, which gives rise to a VHS ($\\varepsilon=1.2$ eV in Fig.~\\ref{fig:electrons}), corresponding to a heavy electron-doping with $N_{\\mathrm e}=~$\\con{1.0}{15}.\n\nCalculated phonon spectra is given on Fig.~\\ref{fig:phonons}.\nWhile the impact of SOC on the electronic structure is notable, it is significantly less prominent in the context of lattice dynamics.\nIn the long-wavelength limit, the frequency changes are nearly unnoticeable, introducing the difference below~1\\%. At higher $\\mathbf{q}$, particularly in vicinity of \\textit{K} high-symmetry point, the difference in acoustic phonon frequency reaches~5\\%.\nFrequencies of optical phonon modes, particularly ZO and LO, demonstrate comparable changes.\nOverall phonon spectra with SOC taken into account only slightly differs from that of without SOC (see the SM, Fig.~S1).\nIn comparison to other elemental 2D materials, antimonene has considerably lower phonon frequencies, which results, for example, in low thermal conductivity, as seen in~\\cite{SWang2016}.\nElastic constants can be estimated from the frequencies of long-wavelength phonons, using the expression $\\omega_\\nu=q\\sqrt{C_{ij}\/\\rho_{2D}}$, where $C_{ij}$ is the elastic constant, related to acoustic phonon mode $\\nu$, $\\rho_{2D}$ is the mass density of 2D material.\nOut-of-plane phonon mode ZA with quadratic dispersion at low ${\\mathbf q}$ is related to bending rigidity $\\kappa$ in the following way $\\omega_{ZA}=q^{2}\\sqrt{\\kappa\/\\rho_{2D}}$.\nThus, independent 2D elastic constants of antimonene would have the following values: $C_{11}=2.1$~eV\/\\AA$^{2}$, associated with LA phonon mode, and $C_{66}=0.8$~eV\/\\AA$^{2}$, associated with TA phonon mode, while $C_{12}=C_{11}-2C_{66}=0.5$~eV\/\\AA$^{2}$ and Young modulus $E = (C_{11}^{2}-C_{12}^2)\/C_{11} = 2.17~\\text{eV\/\\AA}^2$. The corresponding sound velocities are $v_{\\text{s},LA}=3.4$~km\/s and $v_{\\text{s},TA}=2.1$~km\/s. The bending rigidity associated with ZA phonons is $\\kappa=0.3$~eV.\nThese values are significantly smaller than the elastic moduli and speed of sound of graphene, as well as black and blue phosphorene~\\cite{DLiu2016,Zhu2014}.\nIndeed, in contrast to light elements, heavy antimony atoms suppress vibrations, leading to smaller frequencies.\nFinally, it is interesting to make a note of the role of anharmonic effects in antimonene. The characteristic cutoff wavevector below which anharmonic corrections become dominant is given by $q^{*}= \\sqrt{\n\\frac{3T_{\\text{R}}E}{16\\pi\\cdot\\kappa^2}\n}$.\nAt room temperature $T_{\\text{R}}=300$~K it can be estimated as $q^{*}=0.2~\\text{\\AA}^{-1}$, which is an order of magnitude larger than for black phosphorus~\\cite{Rudenko2016} and comparable with graphene~\\cite{Katsnelson2010}.\nIn the context of superconductivity, however, these effects are not relevant.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{{fig3.pdf}} \n\t\\caption{DOS [(a), (b)] and critical temperature [(c), (d)]  dependencies on charge carrier concentration for antimonene. Vertical dotted lines mark considered charge carrier concentrations. Multiple points for single $N_{\\text{h\/e}}$ on (c) and (d) represent ${T_{\\text{c}}}$~at different values of ${\\mu^{*}_{\\text{c}}}$~(0.1, 0.12,...0.2) from top to bottom. Red symbols and lines on panels (c) and (d) mark ${T_{\\text{c}}}$~values at ${\\mu^{*}_{\\text{c}}}$~$=0.1$. Dashed lines on bottom panels serve as a guide to the eye.}\n\t\\label{fig:dos_tc_bias0}\n    \\vspace*{-0.3cm}\n\\end{figure*}\t\n\n\\subsection{Superconductivity and electron-phonon coupling}\\label{subsec:main_res}\n\nCalculated values of ${T_{\\text{c}}}$~at various charge carrier concentrations are presented in Fig.~\\ref{fig:dos_tc_bias0}. \nAs can be seen, the concentrations resulting in the highest critical temperatures for both holes and electrons correlate with VHS of the corresponding charge carrier DOS. The highest value ${T_{\\text{c}}}$~$=17$ K is achieved for the hole-doping with $N_\\mathrm{h}=$\\con{3.7}{14}.\nIn the case of electron-doped antimonene, the highest value is comparable (${T_{\\text{c}}}$~$\\approx16$~K) yet it is observed at significantly higher charge carrier concentration $N_{\\text{e}}=$~\\con{1}{15}. In all considered cases, ${T_{\\text{c}}}$~decreases by 2--3 K at the maximum chosen ${\\mu^{*}_{\\text{c}}}$~$=0.2$.\n\nFor the sake of comparison with experimental results, let us consider reference experimental concentrations reported recently.\nIn the context of superconductivity of intercalated graphene laminates, chemical doping was utilized leading to ${T_{\\text{c}}}$~in the range 6--6.4~K at $N_{e}\\approx$~\\con{1}{14}~\\cite{Ludbrook2015,Chapman2016}.\nAt this concentration antimonene demonstrates slightly lower ${T_{\\text{c}}}$~right above the liquid helium temperatures (4.2~K). \nHowever, charge carrier concentrations achievable by the electrostatic doping~\\cite{Zhao2017,Efetov2010,Xu2017}   yield higher critical temperatures: 6.6--10.4~K at 2--\\pten{5}{14}~cm$^{-2}$ with a local maximum at~\\con{4.1}{14}.\nIn case of the hole-doping, ${T_{\\text{c}}}$~vanishes rapidly with increasing ${\\mu^{*}_{\\text{c}}}$~at charge carrier concentrations below \\con{2}{14}, mainly due to small DOS.\nIn the vicinity of VHS, ${T_{\\text{c}}}$~increases rapidly: At \\con{3}{14} we find ${T_{\\text{c}}}$~$=15.4$ K.\nOur estimation for ${T_{\\text{c}}}$~in antimonene is comparable with that of phosphorene, according to the calculations reported in Refs.~\\onlinecite{Shao2014,Ge2015}.\nAt $N_{\\text e}=$~\\con{7}{14} both materials yield ${T_{\\text{c}}}$~$\\approx10~$ K with phosphorene showing slightly higher values.\nAt smaller concentrations antimonene shows better results: At 1--\\con{4}{14} electron-doped phosphorene is predicted to have ${T_{\\text{c}}}$~in the range of 0.5--5~K, while antimonene exhibits ${T_{\\text{c}}}$~$=10$~K.\nApplication of strain to phosphorene can change the situation: ${T_{\\text{c}}}$~of phosphorene in this case can be tuned to exceed those of monolayer Sb in the aforementioned concentration range \\cite{Shao2014,Ge2015}.\nCalculated values of ${T_{\\text{c}}}$~for the other group V 2D material, namely arsenene, closely resemble those of antimonene, with a slight shift to lower concentrations~\\cite{Kong2018}.\nParticularly, at concentrations ranging from \\pten{0.9}{14} to \\con{3.7}{14} $n$-doped arsenene shows ${T_{\\text{c}}}$~in the range of 4.7--10.1~K, with the maximum value at \\con{2.8}{14}, which is just 1~K higher, than for antimonene at the same concentration.\nAs for phosphorene, a strain tuning of ${T_{\\text{c}}}$~is proposed for arsenene, which can increase these values.\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{{fig4.pdf}}\n\t\\caption{Contributions to total electron-phonon coupling ($\\lambda_\\mathrm{p}$) and critical temperature ($T_{\\mathrm{c}}$) as functions of hole (a) and electron (b) concentrations.\n\t\tGray pentagons denote critical temperatures at $\\mu^{*}$ = 0.1. Gray shaded region represents variability of ${T_{\\text{c}}}$~with respect to $\\mu^{*}$. Red triangles denote total electron phonon-coupling. Contributions to $\\lambda_{\\mathrm{p}}$ from acoustic and optical phonon modes are shown by green squares and blue circles, respectively.\n\t\tDashed horizontal line corresponds to $\\lambda=1.3$.\n        }\n\t\\label{fig:lambda_contrib_nza}\n\\end{figure*}\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]\n\t{{fig5.pdf}}\n\t\\caption{Electron-phonon coupling and nesting function of doped antimonene. Panels (a) and (c) show nesting function resolved in $\\bf q$-space for $N_{\\mathrm{h}}=$~\\con{3.7}{14} and $N_{\\mathrm{e}}=$~\\con{1}{15}, respectively. Heatmaps are given on the same scale for clarity, whereas actual maximum of $\\xi$ is marked by a yellow arrow in panel (c). Panel (b) shows the Fermi surface contour for $N_{\\mathrm{h}}=$~\\con{3.7}{14}, where arrows mark two distinct electron scattering channels discussed in the text. In dashed frame, panels (d)--(g) show electron-phonon coupling contributions from different phonon modes resolved in $\\bf q$-space. Green labels in panels (c) and (d) mark the Brillouin zone and high-symmetry directions. \n    }\n\t\\label{fig:lambda_single}\n    \\vspace*{-0.3cm}\n\\end{figure*}\n\nLet us analyze electron-phonon coupling in antimonene and its role in the superconducting properties in more details.\nData on the averaged and doping-dependent electron-phonon coupling $\\lambda$ [Eq.~(\\ref{eq:lambtot})] of antimonene is given in Fig.~\\ref{fig:lambda_contrib_nza} (red triangles).\nIt is worth mentioning that electron-phonon coupling strength of doped-antimonene is in general higher than that for other elemental monolayer materials discussed above. At comparable charge carrier concentrations, $\\lambda$ in the range of 0.5--1.4 was obtained for phosphorene~\\cite{Shao2014}, in the range of 0.76--1.27 for arsenene~\\cite{Kong2018}, while $\\lambda=0.6$ was measured experimentally for Li-doped graphene at $N_{\\text{e}}=$~\\con{1}{14}~\\cite{Ludbrook2015}.\nElectron-phonon coupling strength of antimonene exceeds these values already at $N_{\\text{e}}=$~\\con{1}{14}.\nDespite significantly higher values of $\\lambda$, this does not lead to a significant increase of ${T_{\\text{c}}}$~in comparison with the other materials discussed.\nIt can be explained by the difference in characteristic phonon frequencies for these systems (see Sec.~\\ref{subsec:dft}), and particularly smaller values of $\\omega_{\\text{log}}$ [Eq.~(\\ref{eq:wlog})].\nAs a result, in the context of superconductivity this effect compensates for higher $\\lambda$.\n{\\color{black}\nThe maximum value $\\lambda=5$ is observed at $N_{\\text{h}}=$~\\con{3.7}{14}.\nThus, the case of strong electronic nesting is observed for the hole-doped antimonene, which leads to a strong electron-phonon coupling for certain wave vectors, and to a higher $T_{\\text{c}}$.\nAt the same time, it has to be taken into account that the Fermi surface nesting, as well as Van Hove singularities in the electron energy spectrum leads to a general destabilization of the system~\\cite{Antropov1988,Vaks1989,Katsnelson1994}.\nThe most obvious manifestation of this effect is the loss of structural stability. \nWe discuss this aspect in Sec.~\\ref{subsec:stability} in details.\n}\n\nElectron-doped antimonene demonstrates considerably smaller $\\lambda$, yet larger than in the other elemental 2D materials: At $N_{\\text{e}}>$~\\con{3}{14} one has $\\lambda>1.3$, particularly $N_{\\text{e}}$=\\con{1}{15} and ~\\con{4}{14} give $\\lambda=1.8$ and $\\lambda=2.3$ respectively.\nWe note that in the original work where Eq.~(\\ref{eq:ADMC}) was derived~\\cite{Allen1975}, as well as in later works~\\cite{Kresin1987}, it is pointed out that Eq.~(\\ref{eq:ADMC}) underestimates ${T_{\\text{c}}}$~for materials with strong electron-phonon coupling, characterized by $\\lambda>1.3$.\nTherefore, the critical temperatures reported here should be considered as a lower limit.\n\nAs can be seen from Fig.~\\ref{fig:lambda_contrib_nza}, coupling with acoustic in-plane phonons (green squares) is the dominant contribution to $\\lambda$. \nThe coupling with optical phonons (blue circles) is small, but not negligible, at least in case of $p$-doping.\nWe now consider $\\lambda_{\\text{p}}$, contributions from individual phonon modes to the electron-phonon coupling at $N_{\\text{h}}=$~\\con{3.7}{14}, which are presented in Fig.~\\ref{fig:lambda_single}.\nThe main contribution is provided by longitudinal acoustic mode, while coupling with TA mode gives the second highest value.\nAs it was mentioned before, at $N_{\\text{h}}=$~\\con{3.7}{14} there is an indication of a strong Fermi surface nesting. \nTo further investigate these phenomena, we calculate the nesting function $\\xi$ [Eq.~(\\ref{eq:nesting})] shown in Fig.~\\ref{fig:lambda_single}(a).\nIndeed, $\\xi$ exhibits maxima in the $\\Gamma$--\\textit{K} direction of ${\\bf q}$, corresponding to the momentum transfer between parallel sections of the Fermi surface [dotted line in Fig.~\\ref{fig:lambda_single}(b)]. As can be seen from Fig.~\\ref{fig:lambda_single}(d), this mechanism provides a dominant contribution to the coupling with TA phonons. At the same time, one can also see another set of peaks in $\\xi_{\\bf q}$ along the $\\Gamma$--\\textit{M} direction. These peaks correspond to the momentum transfer between nonparallel parts of the Fermi surface [dashed line in Fig.~\\ref{fig:lambda_single}(b)]. This mechanism turns out to be more important for the coupling with LA phonons,\nwhich is shown in Fig.~\\ref{fig:lambda_single}(e), resulting in a higher value of $\\lambda_{\\text{LA}}$.\nThe electron-phonon coupling strengths for optical modes are two orders of magnitude smaller than those for the acoustic ones, with the exception of optical out-of-plane ZO mode [Figs.~\\ref{fig:lambda_single}(f)-\\ref{fig:lambda_single}(h)].\n\n\\begin{figure*}[ht]\n\t\\centering\n\t\\begin{minipage}[c]{8.6cm}\n\t\t\\centering\n\t\t\\includegraphics{{fig6_a.pdf}}\n\t\\end{minipage}\n\t\\begin{minipage}[c]{6.7cm}\n\t\t\\centering\n\t\t\\includegraphics{{fig6_b.pdf}}\n\t\\end{minipage}\n\t\\caption{Effect of applied bias voltage on the electronic structure of antimonene. \n\t\t(a) DOS at various values of bias voltage. \n\t\t(b) Comparison of the electronic band structure for ${V_{\\text{b}}}$=1.0~V (red line) and ${V_{\\text{b}}}$=0.0~V (gray line).\n\t}\n\t\\label{fig:dos_bias}\n\n\\end{figure*}\n\n\\begin{figure*}[ht]\n\t\\includegraphics[width=0.9\\textwidth]{fig7}\\vspace*{-0.2cm}\n\t\\caption{Critical temperature dependence on charge carrier concentration at bias voltage of 1~V. Vertical lines mark relevant hole (a) and electron (b) concentrations. Six points for a single $N_{\\text{h\/e}}$ represent different values of ${\\mu^{*}_{\\text{c}}}$~(0.1, 0.12,...0.2). Blue solid line represents ${T_{\\text{c}}}$~at ${\\mu^{*}_{\\text{c}}}$~$=0.1$ and ${V_{\\text{b}}}$~$=0.0$~V, and is given for comparison. Dashed lines serve as guide to the eye.} \n\t\\label{fig:dos_tc_bias1}\n\n\\end{figure*}\n\n\nTabulated values of $\\lambda_{\\text{p}}$ for different charge carrier concentrations is given in Supplemental Material (SM), Tables~S1 and~S2.\nIt is seen, that main contribution in case of electron-doping arises from acoustic phonons, for both $n$ and $p$-doping cases.\nHowever, for $N_{e}<$~\\con{2}{14} individual contributions of acoustic and optical phonons are of the same order.\nDistribution of the nesting function in ${\\bf q}$-space for $N_{\\text{e}}=$~\\con{1}{15} is more complex [Fig.~\\ref{fig:lambda_single}(b)] than for the hole-doping [Fig.~\\ref{fig:lambda_single}(a)]. This is because electron scattering now involves considerable intraband transitions.\nTherefore, it is not possible to determine specific scattering directions in this case. Here, the dominant contribution to $\\lambda$ arises from small ${\\bf q}$ (see SM, Fig.~S2), contrary to a short-wavelength character of the electron-phonon coupling at $N_{\\text{h}}=$~\\con{3.7}{14}. \n\n\n\n\n\\subsection{Effects of bias voltage}\n\n\\begin{figure*}[ht]\n\t\\begin{minipage}[c]{8.0cm}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{{fig8_a_c.pdf}} \n\t\\end{minipage}\\hspace{0.1cm}\n\t\\begin{minipage}[c]{8.0cm}\t\n\t\t\\includegraphics[width=\\textwidth]{{fig8_d_e_r.pdf}}\n\t\\end{minipage}\n\t\\caption{Dependence of critical temperature and electron-phonon coupling in doped antimonene on the bias voltage. Filled and open symbols correspond to $n$- and $p$-doping cases: (a) maximal critical temperature $T_{\\text{c}}^{\\text{max}}$; (b) charge carrier concentration $N_{\\text{e\/h}}$ corresponding to $T_{\\text{c}}^{\\text{max}}$; (c) total electron-phonon coupling strength $\\lambda$. (d) and (e) show variation of critical temperature and electron-phonon coupling strength with bias voltage at fixed concentrations  $N_{\\text{e\/h}}=$~\\con{2}{14}. }\n\t\\label{fig:tcmax}\n\\end{figure*}\n\nThe electronic bands of pristine antimonene are doubly degenerate with respect to spin~\\cite{Rudenko2017}, which is governed by the inversion symmetry.\nWhen an electric field is applied, the inversion symmetry is broken and the spin degeneracy is lifted, which gives rise to the band splitting~\\cite{Prishchenko2018}.\nThis can be clearly seen from Fig.~\\ref{fig:dos_bias}, where we show the effect of bias voltage on the electronic structure of antimonene.\nApart from the SOC related splitting, the band gap is enhanced by the field, unlike, for example, few-layer phosphorene~\\cite{Kim723,Liu2015,Rudenko2015}, where an opposite trend is observed.\nA small gap change of 0.1 eV is observed already at ${V_{\\text{b}}}$=1.0~V, while the highest considered bias voltage of 1.6 V results in a gap of around 30\\% larger than the original one. DOS changes accordingly, as one can see from Fig.~\\ref{fig:dos_bias}(a).\nTaking into account strong correlation of $\\lambda$ with DOS at the Fermi level, bias voltage is expected to have an effect on the superconducting critical temperatures.\nDOS at $n$- and $p$-doping behaves differently with the application of electric field. The most significant change of ${T_{\\text{c}}}$~is expected for $p$-doping at the concentrations corresponding to VHS at ${V_{\\text{b}}}$~$=$~0~V.\nBecause of the strong band splitting in the $\\Gamma$--\\textit{K} direction, the flat band at $\\varepsilon_F\\approx -1.3$~eV splits, resulting in two separate peaks in DOS. In contrast to the original VHS, the two resulting peaks have significantly smaller DOS, which becomes more clear at larger voltages.\nAt the same time, DOS at lower charge carrier concentrations slowly increases with ${V_{\\text{b}}}$, which is mostly observed for $p$-doping. The splitting of the VHS in the conduction band ($n$-doping) is less prominent.\n\n\n\nLet us now consider how ${T_{\\text{c}}}$~depends on the charge carrier concentration in the presence of ${V_{\\text{b}}}$~$=1.0$~V.\nAt this voltage one can clearly see qualitative changes of the carrier DOS [Fig.~\\ref{fig:dos_bias}(a)], i.e. the splitting of a peak at $\\varepsilon_F\\approx -1.3$~eV. \nAs can be seen from Fig.~\\ref{fig:dos_tc_bias1}(b), the change of ${T_{\\text{c}}}$~for the electron-doping is nearly negligible at concentrations $N_{\\text{e}}<$~\\con{3}{14}. At higher concentrations, ${T_{\\text{c}}}$~increase of the order of 1.5 K is observed, with the exception of the highest considered concentration, which results in $T_\\mathrm{c}$ being 1.5~K smaller than in the ${V_{\\text{b}}}$~$=0$~V case.\nMaximum achievable ${T_{\\text{c}}}$~is now 14.6 K for high electron concentration \\con{9.4}{14}, while hole-doping gives comparable values of 13.4 and 12.1~K at $N_{\\text{h}}$ of \\pten{2.7}{14}~and \\con{4.5}{14}, respectively. This corresponds to two new local maxima in DOS. Interestingly, ${T_{\\text{c}}}$~at these points are in good agreement with those observed at the same concentrations at ${V_{\\text{b}}}$$~=~$0~V.\n\nHigh holes concentration of~\\con{8.4}{14}~yields the critical temperature of 13~K, approximately 3~K higher than the highest considered concentration of~\\con{1}{15}~in the absence of the bias voltage. \nLower holes concentrations also yield higher ${T_{\\text{c}}}$~than in the absence of electric field: \\con{2}{14} now gives ${T_{\\text{c}}}$~of 3.2~K, which is nearly twice as large as it was at ${V_{\\text{b}}}$~$=0$~V.\nWhile the absolute value of ${T_{\\text{c}}}$~enhancement at this concentration is small, it represents a practically important trend: Increase of the bias voltage allows us to achieve higher values of ${T_{\\text{c}}}$~at lower concentrations.\n\n\nWe now turn to the dependencies of maximal critical temperature $T_{\\text{c}}^{\\text{max}}$ on the bias voltage [Fig.~\\ref{fig:tcmax}(a)] in the chosen charge carrier concentration range.\nAs expected, the observed trends are different for different doping types. In the case of $n$-doping, $T_{\\text{c}}^{\\text{max}}$ does not change significantly with bias, ranging from 14.6 to 16.4 K with the lowest value corresponding to ${V_{\\text{b}}}$~$=0.4$~V.   \nConcentrations required to achieve $T_{\\text{c}}^{\\text{max}}$ slowly decrease with bias voltage yet remain high: $N_{\\text{e}}=$~\\con{9}{14} at ${V_{\\text{b}}}$~$=1.6$~V. \nAs shown in Fig.~\\ref{fig:tcmax}(c), strong electron phonon coupling is observed for all considered concentrations with $\\lambda$ in the range of 2--3. At low bias voltages, $\\lambda$ decreases first until ${V_{\\text{b}}}$~$=0.4$ V and then increases in the rest of the range. The opposite trend is observed for $p$-doping.\nIn this case, $T_{\\text{c}}^{\\text{max}}$ decreases monotonously with ${V_{\\text{b}}}$~from 17 to 10~K.\nThe required hole concentration, on the contrary, increases, but does not exceed \\con{4.7}{14}, which is still significantly smaller than for electrons. In the presence of bias voltage, $\\lambda$ reduces significantly from 5 to 1.8, indicating a sufficiently strong electron-phonon coupling.\n\n\nCharge carrier concentrations presented in Fig.~\\ref{fig:tcmax}(b) are high, \nand thus may be difficult to achieve in practice. At the same time, \nbias voltage could increase DOS, as well as $\\lambda$ and ${T_{\\text{c}}}$~at moderate concentrations, i.e., below \\con{4}{14}.\nIn Figs.~\\ref{fig:tcmax}(d)-\\ref{fig:tcmax}(e), we consider $\\lambda$ and ${T_{\\text{c}}}$~at more realistic concentrations for both doping cases $N=$~\\con{2}{14}, which are typical values in the medium concentration range. In case of electron-doping, both $\\lambda$ and ${T_{\\text{c}}}$~increase monotonously with ${V_{\\text{b}}}$, yet for the highest bias voltage considered, ${T_{\\text{c}}}$~increases by only $\\approx$1~K.\nHole-doping yields $\\approx$2 K enhancement of ${T_{\\text{c}}}$~already at ${V_{\\text{b}}}$~$=1.0$~V, while $\\lambda$ reaches the value of 2.0.\nIn contrast to the case of electrons, $\\lambda$ and ${T_{\\text{c}}}$~exhibit a maximum at ${V_{\\text{b}}}$~$\\approx1.2$ V, after which one can see a gradual decrease of their values. This behavior can be attributed to a decreasing DOS, shown in Fig.~\\ref{fig:dos_bias}.\nAlthough in absolute values the effect of bias voltage is not large, it provides a possibility to\nincrease ${T_{\\text{c}}}$~and $\\lambda$ in antimonene by up to 20\\%.\n\n\n\n\n\t\\subsection{Dynamical stability of doped antimonene}\\label{subsec:stability}\n\t\n\n\tIn this section, we discuss the dynamical stability of doped antimonene.\n    To model the doping effect beyond the rigid band approximation, we use the jellium doping technique.\n    We are interested in studying the stability at doping concentrations corresponding to the strong Fermi-nesting, considered in the previous section, as well as to the highest charge carrier concentrations in case of $n$-doping. \n    In both cases the loss of structural stability would naturally provide the limiting factor for the applicability of used approximations.\n\nThe results of phonon spectra calculations are presented in Fig.~\\ref{fig:fig9}.\nFor brevity, we consider the $\\Gamma$--\\textit{M} direction only. Qualitatively similar effects are observed in the $\\Gamma$--\\textit{K} direction.\nLet us first consider the hole-doping case. At low enough doping, the system demonstrates the enhancement of acoustic phonon frequencies in the long-wavelength limit, which is especially evident for TA mode. At the same time, out-of-plane phonon mode ZA becomes nearly linear at low ${\\bf q}$.\n\tThese effects persist in the concentration range from~\\con{0.7}{14} to~\\con{1.8}{14}.\n\tAt~\\con{1.8}{14}, a pronounced softening of ZA mode in the $\\Gamma$--\\textit{M} direction is observed. At the concentration of~\\con{2.0}{14}, this mode becomes imaginary in a wide range of ${\\bf q}$, including the vicinity of high-symmetry \\textit{M} point. The TA mode also demonstrates pronounced softening.\n\tFurther imaginary frequencies appear for the TA mode at a slightly higher hole concentration of~\\con{2.1}{14} (+0.3 e\/cell).\n\tIt is important to distinguish out-of-plane mode related instability and in-plane instability.\n\tThe appearance of the former is reported for other doped 2D materials~\\cite{Margine2014,Kong2018} and is not considered as a sign of instability in the case of realistic applications, mainly due to the presence of the substrate in experimental setup. \n\tAt the same time, the presence of in-plane instability is an explicit indication of instability. \n\tTherefore, we conclude that antimonene at concentrations $N_{\\text{h}}>$~\\con{2}{14} is structurally unstable.\n\tIt is worth mentioning that to a certain extent strain can be used to stabilize the system~\\cite{Zeng2016}, even if in-plane instability appears in the phonon spectrum.\n\n\t\n\t\\begin{figure}\n\t\t\\includegraphics[]{{fig9.pdf}}\n\t\t\\caption{Phonon spectra of hole- and electron-doped antimonene (left and right columns respectively) in $\\Gamma$--M high-symmetry direction at various charge carrier concentrations, simulated using jellium doping technique. Dashed line corresponds to phonon spectra in undoped case.}\\label{fig:fig9}\n\t\\end{figure}\n\t\nOn the contrary, electron-doping would only lead to out-of-plane instability, even at the highest concentration considered. \nUnlike the $p$-doping case, electron-doped antimonene is stable at~\\con{2}{14} and higher. At lower concentration considered, i.e.,~\\con{0.7}{14}, only the ZA mode demonstrates imaginary frequencies at low ${\\bf q}$, while in-plane modes demonstrate stiffening. \nAt~\\con{2.1}{14}, however, softening of TA mode is observed, which further increases at higher concentrations\nThe doping effect on longitudinal modes is not as pronounced, with significant softening taking place only at highest charge carrier concentrations considered. This behavior is different from the hole-doping case, where the LA frequencies decrease significantly, especially away from the zone center.  \nThe reason for the higher stability of electron-doped antimonene can partially be associated with a more distributed electron nesting function [Fig.~\\ref{fig:lambda_contrib_nza}(a)], as well as $\\lambda$~(Fig.~S1 in Supplemental Materials).\n\n\t\n\tLet us now discuss possible effect of bias voltage on the stability and ${T_{\\text{c}}}$~in light of the results presented above. \n\tFirst of all, in the case of hole-doping, the effect of bias voltage on the electronic structure remains an important factor, since the increase of ${T_{\\text{c}}}$~is already observed at $N_{\\text{h}}$=~\\con{2}{14}, where the system remains stable.\n\tFurthermore, the splitting of the DOS peak in the presence of $V_{\\text{b}}$ allows us to assume that bias voltage may actually increase the stability of doped antimonene at $N_{\\text{h}}>$~\\con{2}{14} due to the modification of DOS and suppression of VHS (see Fig.~\\ref{fig:dos_bias}).\n    While the effect of bias voltage on the electronic structure of $n$-doped antimonene is less prominent, we expect the tendencies described for this case in the previous section to remain. \n\n\n\n\\FloatBarrier\n\\section{Summary and Conclusion}\n\nWe performed \\textit{ab initio} calculations of electron-phonon coupling for $n$- and $p$-doped antimonene using state-of-the-art computational techniques.\nWe estimated critical temperature of the superconducting transition at various charge carrier concentrations in a wide range from \\pten{5}{13} to \\con{1}{15}~using the Allen-Dynes-McMillan equation.\n{\\color{black}Doing so, we considered the dynamical stability of studied configurations.}\n\nDependence of the electron-phonon coupling strength from charge carrier concentration is essentially conditioned by the doping type and is mainly determined by the Fermi surface topologies.\nWe find that at hole concentrations below \\con{2}{14} and electron concentration below \\con{5}{13} superconductivity is not expected above 0.5~K. \nHigher charge carrier concentrations yield ${T_{\\text{c}}}$~in a wide range.\n{\\color{black}However, the increase of charge carrier concentration in the case of hole-doping is limited by the dynamical instability, which occurs for $N_{\\text{h}}>$\\con{2.0}{14}. This behavior can be explained by the presence of van Hove singularity close to dynamical stability limit in the density of hole states. Among the stable configurations, the maximum value of~${T_{\\text{c}}}$~is estimated to be 15~K, and corresponds to the electron-doping case. The hole-doping yields $T_{\\text{c}}^{\\text{max}}\\approx1.5$~K at the concentrations in the range of 1.8--\\con{2.0}{14}.}\nThese value are comparable or exceed those reported for other doped elemental monolayer materials.\nThe value of strong electron-phonon coupling strength as high as $\\lambda=2.3$ is predicted for the highest considered electron concentration of \\con{1}{15}.\nFor all studied charge carrier concentrations the main contribution to electron-phonon coupling arises from the interaction of carriers with in-plane acoustic phonons.\n\nWe also studied the effects of electric field applied in the direction perpendicular to the atomic sheet, on electron-phonon coupling and critical temperature of antimonene.\nIn the bias voltage range of 0.0--1.6~V, the electronic structure of antimonene undergoes spin-orbit-assisted bands splitting, as well as enhancement of the band gap width. \nThis effect allows us to increase the ${T_{\\text{c}}}$~for the stable hole-doped configurations; thus, in the vicinity of destabilization point at~\\con{2}{14} the maximum increase of ${T_{\\text{c}}}$~reaches 3.5 K at the highest considered bias voltage.\nIn contrast, the electron-doping case is less affected by the application of bias voltage, leading to a slight variation of the critical temperature and electron-phonon coupling over the whole concentration range, which potentially simplifies the application of gating-based doping methods for the system and making results more predictable.\nOverall, bias voltage allows us to control electron-phonon coupling as well as related properties in heavy-element 2D semiconductors, making them interesting objects for further studies.\n\n\n\n\n\n\\section{Acknowledgments}\n\nThis work is part of the research programme ``Two-dimensional semiconductor crystals'' with project number 14TWOD01, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO).\nThe calculations were preformed at computational cluster ``TCM'', Radboud University, Nijmegen, Nederlands and at computational cluster NUST ``MISIS'', Moscow, Russia. A.N.R. acknowledges  support from the Russian Science Foundation, Grant No. 17-72-20041.\n\n\\nocite{apsrev41Control}\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIn this paper, we are concerned with disturbed nonlinear reaction-diffusion\nequations of the form\n\\begin{align} \\label{eq:react-diffus-eq}\n\\begin{split}\n\\partial_t y(t,\\zeta) &= \\Delta y(t,\\zeta) + g(y(t,\\zeta)) + h(\\zeta) u(t) \\qquad (\\zeta \\in \\Omega) \\\\\ny(t,\\zeta) &= 0 \\qquad (\\zeta \\in \\partial \\Omega)\n\\end{split} \n\\end{align}\non a bounded domain $\\Omega \\subset \\mathbb{R}^d$\nwith smooth boundary $\\partial \\Omega$, \nwhere $g \\in C^1(\\mathbb{R},\\mathbb{R})$ and $h \\in L^2(\\Omega,\\mathbb{R})$ and the disturbance $u$ belongs to $\\mathcal{U} := L^{\\infty}([0,\\infty),\\mathbb{R})$. It is well-known~\\cite{Robinson} that the corresponding undisturbed equation\n\\begin{align} \\label{eq:react-diffus-eq, undisturbed}\n\\begin{split}\n\\partial_t y(t,\\zeta) &= \\Delta y(t,\\zeta) + g(y(t,\\zeta)) \\qquad (\\zeta \\in \\Omega) \\\\\ny(t,\\zeta) &= 0 \\qquad (\\zeta \\in \\partial \\Omega)\n\\end{split} \n\\end{align}\nhas a unique global attractor $\\Theta \\subset X := L^2(\\Omega,\\mathbb{R})$ under suitable growth and upper-boundedness conditions on the nonlinearity $g$ and its derivative $g'$ respectively. As usual, a global attractor for~\\eqref{eq:react-diffus-eq, undisturbed} is defined to be a compact subset of $X$ that is invariant and uniformly attractive for~\\eqref{eq:react-diffus-eq, undisturbed}. Also, it can be shown~\\cite{KaVa09} that the global attractor $\\Theta$ of~\\eqref{eq:react-diffus-eq, undisturbed} is a stable set for~\\eqref{eq:react-diffus-eq, undisturbed}. \n\\smallskip\n\nWhat we show in this paper is that the disturbed reaction-diffusion equations~\\eqref{eq:react-diffus-eq} are locally input-to-state stable w.r.t.~the global attractor $\\Theta$ of the undisturbed equation~\\eqref{eq:react-diffus-eq, undisturbed}.\nSo,\nwe show that there exist comparison functions $\\beta \\in \\mathcal{KL}$ and $\\gamma \\in \\mathcal{K}$ and radii $r_{0x}, r_{0u} > 0$ such that for every initial value $y_0 \\in X$ with $\\norm{y_0}_{\\Theta} \\le r_{0x}$ and every disturbance $u \\in \\mathcal{U}$ with $\\norm{u}_{\\infty} \\le r_{0u}$ the\nglobal weak solution $$[0,\\infty) \\ni t \\mapsto y(t,\\cdot) = y(t,y_0,u) \\in X$$\nof the boundary value problem~\\eqref{eq:react-diffus-eq}  \nwith initial condition $y(0,\\cdot) = y_0 \\in X$ satisfies the following estimate:\n\\begin{align} \\label{eq:lISS-estimate,intro}\n\\norm{y(t,y_0,u)}_{\\Theta} \\le \\beta(\\norm{y_0}_{\\Theta},t) + \\gamma(\\norm{u}_{\\infty}) \\qquad (t \\in [0,\\infty)). \n\\end{align}\nSee~\\cite{Mi16} for the analogous definition in the special case~$\\Theta = \\{0\\}$. \nIn the above relations, we use the standard notation\n\\begin{align} \\label{eq:norm-Theta,def}\n\\norm{x}_{\\Theta} := \\operatorname{dist}(x,\\Theta) := \\inf_{\\theta \\in \\Theta} \\norm{x-\\theta} \\qquad (x \\in X)\n\\end{align}\nand the standard definitions for the comparison function classes $\\mathcal{KL}$ and $\\mathcal{K}$, which are recalled in~\\eqref{eq:comparison-fct-classes} below. \nIn words,\nthe local input-to-state stability estimate~\\eqref{eq:lISS-estimate,intro} means\nthat  \n\\begin{itemize}\n\\item[(i)] the\ninvariant set $\\Theta$ for~\\eqref{eq:react-diffus-eq, undisturbed} is locally stable and\nattractive for the undisturbed system~\\eqref{eq:react-diffus-eq, undisturbed} and \n\\item[(ii)] these local stability and attractivity properties are affected only slightly in the presence of disturbances of small magnitude $\\norm{u}_{\\infty}$.\n\\end{itemize}\nIn order to achieve the estimate~\\eqref{eq:lISS-estimate,intro}, we will construct a suitable local input-to-state Lyapunov function $V$.\n\\smallskip\n\nAs far as we know, our result is the first (local) input-to-state stability result w.r.t.~attractors $\\Theta$ of concrete partial differential equation systems. All previous concrete pde results\nwe are aware of -- like those from~\\cite{DaMi13}, \\cite{JaNaPaSc16}, \\cite{JaSc18}, \\cite{KaKr16}, \\cite{KaKr17}, \\cite{MaPr11}, \\cite{MiKaKr17}, \\cite{ScZw18}, \\cite{TaPrTa17}, \\cite{ZhZh17a}, \\cite{ZhZh17b}, for instance --\nestablish input-to-state stability only w.r.t.~an equilibrium point $\\theta$, which without loss of generality is assumed to be $\\theta = 0$. In particular, all these previous results require their   nonlinearity $g$ to be such that $g(\\theta) = g(0) = 0$ and such that the undisturbed system has the singleton $\\Theta := \\{\\theta\\} = \\{0\\}$ as an attractor. \nWith our result, by contrast, we can treat much more general nonlinearities:\nwe can treat nonlinearities $g$ with $g(0) \\ne 0$ and, more importantly, nonlinearities $g$ for which the undisturbed system~\\eqref{eq:react-diffus-eq, undisturbed} has only a non-singleton attractor $\\Theta \\supsetneq \\{0\\}$. A simple example of such a nonlinearity is given by $g(r) := -r^3 + r$, which leads to the Chaffee--Infante equation.\nWe refer to~\\cite{KaKaVa15}, \\cite{GoKaKaPa14}, \\cite{GoKakaKh15} \\cite{DaKaRo17} for other interesting results about non-trivial global attractors of nonlinear, impulsive, or even multi-valued semigroups. \n\\smallskip\n\nIn the entire paper, we will use the following conventions and notations. As above, $X := L^2(\\Omega,\\mathbb{R})$ and $\\mathcal{U} := L^{\\infty}(\\mathbb{R}^+_0,\\mathbb{R})$ with $\\mathbb{R}^+_0 := [0,\\infty)$ and with the standard norm of $X$ being denoted simply by $\\norm{\\cdot} := \\norm{\\cdot}_{L^2(\\Omega)}$. As usual,\n\\begin{align*}\nB_r(x_0) = B_r^{X}(x_0), \\quad \\overline{B}_r(x_0) = \\overline{B}_r^{X}(x_0) \\quad \\text{and} \\quad B_r(u_0) = B_r^{\\mathcal{U}}(u_0), \\quad  \\overline{B}_r(u_0) = \\overline{B}_r^{\\mathcal{U}}(u_0)\n\\end{align*}\ndenote the open and closed balls in $X$ or $\\mathcal{U}$ of radius $r$ around $x_0 \\in X$ or $u_0 \\in \\mathcal{U}$ respectively. We will often use the notation~\\eqref{eq:norm-Theta,def} and\n\\begin{align*}\nB_r(\\Theta) := \\{x \\in X: \\norm{x}_{\\Theta} < r \\} \n\\qquad \\text{and} \\qquad\n\\overline{B}_r(\\Theta) := \\{x \\in X: \\norm{x}_{\\Theta} \\le r \\},\n\\end{align*}\nas well as the notation \n$\\operatorname{dist}(M,\\Theta) := \\sup_{x\\in M} \\norm{x}_{\\Theta}$\nfor subsets $M, \\Theta \\subset X$. Also, $\\mathcal{K}$, $\\mathcal{K}_{\\infty}$ and $\\mathcal{KL}$ will denote the following standard classes of comparison functions:\n\\begin{gather}\n\\mathcal{K} := \\{ \\gamma \\in C(\\mathbb{R}^+_0,\\mathbb{R}^+_0): \\gamma \\text{ strictly increasing with } \\gamma(0) = 0 \\} \\notag \\\\\n\\mathcal{K}_{\\infty} := \\{ \\gamma \\in \\mathcal{K}: \\gamma \\text{ unbounded} \\}  \\label{eq:comparison-fct-classes}\\\\\n\\mathcal{KL} := \\{ \\beta \\in C(\\mathbb{R}^+_0 \\times \\mathbb{R}^+_0,\\mathbb{R}^+_0): \\beta(\\cdot,t) \\in \\mathcal{K} \\text{ for } t \\ge 0 \\text{ and } \\beta(s,\\cdot) \\in \\mathcal{L} \\text{ for } s > 0 \\}, \\notag\n\\end{gather}\nwhere $\\mathcal{L} := \\{ \\gamma \\in C(\\mathbb{R}^+_0,\\mathbb{R}^+_0): \\gamma \\text{ strictly decreasing with } \\lim_{t\\to\\infty} \\gamma(t) = 0 \\}$. And finally, upper right-hand Dini derivatives will be denoted by\n\\begin{align*}\n\\overline{\\partial}_t^+ v(t) := \\varlimsup_{\\tau \\to 0+} \\frac{v(t+\\tau)-v(t)}{\\tau}.\n\\end{align*}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Some preliminaries}\n\nIn this section, we provide\nthe necessary preliminaries for our local input-to-state stability result. \nWe begin by recalling the definition of weak solutions of initial boundary value problems of the form \n\\begin{align} \\label{eq:ibvp}\n\\begin{split}\n\\partial_t y(t,\\zeta) &= \\Delta y(t,\\zeta) + g(y(t,\\zeta)) + h(\\zeta) u(t) \\qquad ((t,\\zeta) \\in [s,\\infty) \\times \\Omega) \\\\\ny(t,\\cdot)|_{\\partial \\Omega} &= 0 \\qquad \\text{and} \\qquad y(s,\\cdot) = y_s \\qquad (t \\in [s,\\infty)).\n\\end{split} \n\\end{align}\nIn fact, we will have to consider initial boundary value problems with more general inhomogeneities of the form\n\\begin{align} \\label{eq:ibvp-general}\n\\begin{split}\n\\partial_t y(t,\\zeta) &= \\Delta y(t,\\zeta) + \\overline{g}(y(t,\\zeta)) + \\overline{h}(t,\\zeta)  \\qquad ((t,\\zeta) \\in [s,\\infty) \\times \\Omega) \\\\\ny(t,\\cdot)|_{\\partial \\Omega} &= 0 \\qquad \\text{and} \\qquad y(s,\\cdot) = y_s \\qquad (t \\in [s,\\infty)),\n\\end{split} \n\\end{align}\nwhere $\\overline{g}$, $\\overline{h}$ satisfy the following conditions. \n\n\n\n\n\\begin{cond} \\label{cond:ol-g,ol-h}\n\\begin{itemize}\n\\item[(i)] $\\Omega$ is a bounded domain in $\\mathbb{R}^d$ for some $d \\in \\N$ with smooth boundary $\\partial \\Omega$ \nand, moreover, $p \\in [2,\\infty)$, $q \\in (1,2]$ are dual exponents: $1\/p + 1\/q = 1$\n\\item[(ii)] $\\overline{g} \\in C^1(\\mathbb{R},\\mathbb{R})$ and there exist constants $\\alpha_1, \\alpha_2, \\kappa, \\lambda \\in (0,\\infty)$ such that\n\\begin{align} \\label{eq:growth-cond-ol-g}\n-\\kappa - \\alpha_1 |r|^p \\le \\overline{g}(r)r \\le \\kappa - \\alpha_2 |r|^p\n\\qquad \\text{and} \\qquad\n\\overline{g}'(r) \\le \\lambda\n\\qquad (r \\in \\mathbb{R})\n\\end{align}\nand, moreover, $\\overline{h} \\in L^q_{\\mathrm{loc}}(\\mathbb{R}^+_0,L^q(\\Omega))$. \n\\end{itemize}\n\\end{cond}\n\n\nA bit more explicitly, the first two inequalities in~\\eqref{eq:growth-cond-ol-g} mean that $\\overline{g}|_{(0,\\infty)}$\nlies between $r \\mapsto -\\kappa\/|r| - \\alpha_1 |r|^{p-1}$ and $r \\mapsto \\kappa\/|r| - \\alpha_2 |r|^{p-1}$ \nand that $\\overline{g}|_{(-\\infty,0)}$ lies between $r \\mapsto -\\kappa\/|r| + \\alpha_2 |r|^{p-1}$ and $r \\mapsto \\kappa\/|r| + \\alpha_1 |r|^{p-1}$. \nA simple class of functions $\\overline{g}$ satisfying the three inequalities from~\\eqref{eq:growth-cond-ol-g}\nis given by the polynomials of odd degree with negative leading coefficient:\n\\begin{align*}\n\\overline{g}(r) = \\sum_{i=0}^{2m-1} c_i r^{i} \\qquad (r \\in \\mathbb{R})\n\\end{align*} \nwith $c_{2m-1} < 0$, where $m \\in \\N$. (Choose $p := 2m$.)\nIn particular, the nonlinearity of the Chaffee--Infante equation given by $\\overline{g}(r) := -r^3 + r$ falls into that class (Section~11.5 of~\\cite{Robinson}). \n\\smallskip \n\nSuppose that Condition~\\ref{cond:ol-g,ol-h} is satisfied and let $s \\in \\mathbb{R}^+_0$ and $y_s \\in X$. A function $y \\in C([s,\\infty),X)$ is\ncalled a \\emph{global weak solution of\n\\eqref{eq:ibvp-general}} iff $y(s) = y_s$ and for every $T \\in (s,\\infty)$ one has\n\\begin{align}\ny|_{[s,T]} \\in L^2([s,T],H^1_0(\\Omega)) \\cap L^p([s,T], L^p(\\Omega))\n\\end{align}\nand there exists a (then unique) $z \\in L^2([s,T],H^1_0(\\Omega)^*) + L^q([s,T], L^q(\\Omega))$ such that\n\\begin{align} \\label{eq:def-weak-sol}\n\\int_s^T \\big( z(t), \\phi(t)\\big) \\d t &= -\\int_s^T \\int_{\\Omega} \\nabla y(t)(\\zeta) \\cdot \\nabla \\phi(t)(\\zeta) \\d \\zeta \\d t + \\int_s^T \\int_{\\Omega} \\overline{g}\\big( y(t)(\\zeta) \\big) \\, \\phi(t)(\\zeta) \\d \\zeta \\d t  \\notag \\\\\n&\\quad + \\int_s^T \\int_{\\Omega} \\overline{h}(t)(\\zeta) \\, \\phi(t)(\\zeta)  \\d \\zeta \\d t \n\\end{align}\nfor every $\\phi \\in L^2([s,T],H^1_0(\\Omega)) \\cap L^p([s,T], L^p(\\Omega))$. See~\\cite{VaKa06} or~\\cite{KaVa09} and, for more background information, \\cite{ChVi96} or~\\cite{ChVi}. In this equation, $(\\cdot,\\cdot \\cdot)$ stands for the dual pairing of $H^1_0(\\Omega)^* + L^q(\\Omega)$ and $H^1_0(\\Omega) \\cap L^p(\\Omega)$, that is, \n\\begin{align} \\label{eq:dual-pairing}\n(z,\\phi) = (z_1,\\phi)_{H^1_0(\\Omega)^*,H^1_0(\\Omega)} + (z_2,\\phi)_{L^q(\\Omega),L^p(\\Omega)}\n\\end{align}\nfor every $z = z_1+z_2 \\in H^1_0(\\Omega)^* + L^q(\\Omega)$ and $\\phi \\in H^1_0(\\Omega) \\cap L^p(\\Omega)$, where $(\\cdot,\\cdot\\cdot)_{H^1_0(\\Omega)^*,H^1_0(\\Omega)}$ and $(\\cdot,\\cdot\\cdot)_{L^q(\\Omega),L^p(\\Omega)}$ denote the respective dual pairings. See~\\cite{BeLoe} (Theorem~2.7.1) and \\cite{DiUh} (Theorem~IV.1.1 and Corollary~III.2.13), for instance, to get that $H^1_0(\\Omega)^* + L^q(\\Omega)$, $H^1_0(\\Omega) \\cap L^p(\\Omega)$ and\n\\begin{align*}\nL^2([s,T],H^1_0(\\Omega)^*) + L^q([s,T], L^q(\\Omega)), \\quad L^2([s,T],H^1_0(\\Omega)) \\cap L^p([s,T], L^p(\\Omega))\n\\end{align*} \nare dual to each other. \nWe point out that if $y$ is a global weak solution to~\\eqref{eq:ibvp-general}, then for every $T \\in (s,\\infty)$ there is only one $z \\in L^2([s,T],H^1_0(\\Omega)^*) + L^q([s,T], L^q(\\Omega))$ satisfying~\\eqref{eq:def-weak-sol}. And this $z$ is called the \\emph{weak or generalized derivative} of $y|_{[s,T]}$. It is denoted by $\\partial_t y|_{[s,T]}$ or simply by $\\partial_t y$ in the following. \n\n\n\\begin{lm} \\label{lm:weak-sol-general}\nSuppose that Condition~\\ref{cond:ol-g,ol-h} is satisfied and let $s \\in \\mathbb{R}^+_0$ and $y_s \\in X$. Then the initial boundary value problem~\\eqref{eq:ibvp-general} has a unique global weak solution $y$ and, moreover, $t \\mapsto \\norm{y(t)}^2$ is absolutely continuous (hence differentiable almost everywhere) with\n\\begin{align}\n\\ddt \\norm{y(t)}^2 = 2 \\big( \\partial_t y(t),y(t) \\big) \n\\end{align} \nfor almost every $t \\in [s,\\infty)$, where $(\\cdot,\\cdot \\cdot)$ is the dual pairing from~\\eqref{eq:dual-pairing}. \n\\end{lm}\n\n\n\\begin{proof}\nIt is clear from the first two inequalities in~\\eqref{eq:growth-cond-ol-g} that\n\\begin{align} \\label{eq:estimate-g(r)r}\n|\\overline{g}(r)r| \\le \\kappa + \\alpha_1|r|^p \\qquad (r \\in \\mathbb{R}).\n\\end{align} \nSince $\\sup_{|r|\\le 1} |\\overline{g}(r)| < \\infty$ by the continuity of $\\overline{g}$, it follows from~\\eqref{eq:estimate-g(r)r} that for some constant $C_1 \\in (0,\\infty)$\n\\begin{align} \\label{eq:g(y)-in-Lq}\n|\\overline{g}(r)| \\le C_1 (1+|r|^{p-1}) \\qquad (r \\in \\mathbb{R})\n\\end{align}\nand therefore condition~(2) from~\\cite{VaKa06} is satisfied. Also, in view of the second and third inequalities in~\\eqref{eq:growth-cond-ol-g}, condition~(3) and condition~(4) from~\\cite{VaKa06} with $M=0$ are satisfied. \nConsequently, the assertions of the lemma follow from the remarks made in Section~2 (up to Remark~1) of~\\cite{VaKa06}. \n\\end{proof}\n\n\n\nWith this lemma at hand, it is easy to see that the initial boundary value problem~\\eqref{eq:ibvp} generates a semiprocess family $(S_u)_{u\\in\\mathcal{U}}$ on $X$ (Lemma~\\ref{lm:semiproc}).  A \\emph{semiprocess family on $X$} is a family of maps $S_u: \\Delta \\times X \\to X$ for every $u \\in \\mathcal{U}$ such that\n\\begin{gather}\nS_u(s,s,x) = x \\qquad \\text{and} \\qquad S_u\\big(t,s, S_u(s,r,x) \\big) = S(t,r,x) \\label{eq:def-semiproc-1}\\\\\nS_u(t+\\tau,s+\\tau,x) = S_{u(\\cdot+\\tau)}(t,s,x) \\label{eq:def-semiproc-2}\n\\end{gather} \nfor all $(t,s), (s,r) \\in \\Delta$, $\\tau \\in \\mathbb{R}^+_0$, $x \\in X$ and $u \\in \\mathcal{U}$, where we used the abbreviation $\\Delta := \\{(s,t) \\in \\mathbb{R}^+_0 \\times \\mathbb{R}^+_0: t \\ge s\\}$. See~\\cite{ChVi}, for instance, for more information on semiprocess families. \n\n\n\n\n\\begin{cond} \\label{cond:g,h}\n\\begin{itemize}\n\\item[(i)] $\\Omega$ is a bounded domain in $\\mathbb{R}^d$ for some $d \\in \\N$ with smooth boundary $\\partial \\Omega$ \nand, moreover, $p \\in [2,\\infty)$\n\\item[(ii)] $g \\in C^1(\\mathbb{R},\\mathbb{R})$ and there exist constants $\\alpha_1, \\alpha_2, \\kappa, \\lambda \\in (0,\\infty)$ such that\n\\begin{align} \\label{eq:cond-g}\n-\\kappa - \\alpha_1 |r|^p \\le g(r)r \\le \\kappa - \\alpha_2 |r|^p\n\\qquad \\text{and} \\qquad\ng'(r) \\le \\lambda\n\\qquad (r \\in \\mathbb{R})\n\\end{align}\nand, moreover, $h \\in X \\setminus \\{0\\}$.\n\\end{itemize}\n\\end{cond}\n\n\n\n\\begin{lm} \\label{lm:semiproc}\nSuppose that Condition~\\ref{cond:g,h} is satisfied. Then for every $s \\in \\mathbb{R}^+_0$ and every $(y_s,u) \\in X \\times \\mathcal{U}$ the initial boundary value problem~\\eqref{eq:ibvp} has a unique global weak solution $y(\\cdot,s,y_s,u)$. Additionally, $(S_u)_{u\\in\\mathcal{U}}$ defined by\n\\begin{align} \\label{eq:S_u-def}\nS_u(t,s,y_s) := y(t,s,y_s,u)\n\\end{align}\nis a semiprocess family on $X$. \n\\end{lm}\n\n\n\\begin{proof}\nIn order to see the unique global weak solvability, simply apply Lemma~\\ref{lm:weak-sol-general} with $\\overline{g}:=g$ and with $\\overline{h} \\in L^2_{\\mathrm{loc}}(\\mathbb{R}^+_0,L^2(\\Omega)) \\subset L^q_{\\mathrm{loc}}(\\mathbb{R}^+_0,L^q(\\Omega))$ defined by $\\overline{h}(t)(\\zeta) := h(\\zeta) u(t)$. \nIn order to see the semiprocess property, use the definition of weak solutions and the uniqueness statement from Lemma~\\ref{lm:weak-sol-general}. \n\\end{proof}\n\n\n\n\n\n\nIn the following, $(S_u)_{u\\in\\mathcal{U}}$ will always denote the semiprocess family from the previous lemma. Also, we will often refer to $(S_u)_{u\\in \\mathcal{U}}$ and $S_0$ as the disturbed and the undisturbed system, respectively. \nIn proving our local input-to-state stability result, the following estimates will play an important role. \n\n\\begin{lm} \\label{lm:semiproc-estimates}\nSuppose that Condition~\\ref{cond:g,h} is satisfied. Then\n\\begin{align}\n\\norm{S_0(t,0,y_{01})-S_0(t,0,y_{02})} &\\le \\e^{\\lambda t} \\norm{y_{01}-y_{02}} \\qquad (t \\in \\mathbb{R}^+_0) \\label{eq:semiproc-estimate-1}\\\\\n\\norm{S_u(t,0,y_0)-S_0(t,0,y_0)} &\\le 2 \\e^{2 \\lambda} \\norm{h} \\norm{u}_{\\infty} t \\qquad (t \\in [0,1]) \\label{eq:semiproc-estimate-2}\n\\end{align} \nfor all $y_0, y_{01}, y_{02} \\in X$ and all $u \\in \\mathcal{U}$.\n\\end{lm}\n\n\n\\begin{proof}\nAs a first step, we show that for every $y_{01},y_{02} \\in X$ and $u \\in \\mathcal{U}$ the function \n\\begin{align} \\label{eq:y-12-u}\ny_{12}^{u} := y_1^{u}-y_2^0 \\qquad \\text{with} \\qquad y_1^{u} := S_u(\\cdot,0,y_{01}) \\qquad \\text{and} \\qquad y_2^0 := S_0(\\cdot,0,y_{02})\n\\end{align}\nis a global weak solution of the initial boundary value problem\n\\begin{align} \\label{eq:semiproc-estimates-ibvp-general}\n\\begin{split}\n\\partial_t y(t,\\zeta) &= \\Delta y(t,\\zeta) + \\overline{g}(y(t,\\zeta)) + \\overline{h}(t,\\zeta)  \\qquad ((t,\\zeta) \\in [0,\\infty) \\times \\Omega) \\\\\ny(t,\\cdot)|_{\\partial \\Omega} &= 0 \\qquad \\text{and} \\qquad y(0,\\cdot) = y_{01}-y_{02} \\qquad (t \\in [0,\\infty)),\n\\end{split} \n\\end{align}\nwhere $\\overline{g} := g$ and $\\overline{h}(t)(\\zeta) := g(y_1^{u}(t)(\\zeta)) - g(y_2^{0}(t)(\\zeta)) - g(y_{12}^{u}(t)(\\zeta)) + h(\\zeta)u(t)$. \nSo, let $y_{01},y_{02} \\in X$ and $u \\in \\mathcal{U}$ and adopt the abbreviations from~\\eqref{eq:y-12-u}. It is not difficult -- using~\\eqref{eq:g(y)-in-Lq} and $q(p-1)=p$ -- to see from Condition~\\ref{cond:g,h} that with $\\overline{g}$, $\\overline{h}$ as defined above, Condition~\\ref{cond:ol-g,ol-h} is satisfied. \nSince $y_1^{u}$, $y_2^0$ are global weak solutions, we have $y_{12}^{u} \\in C(\\mathbb{R}^+_0,X)$ and for every $T \\in (0,\\infty)$ we have\n\\begin{align*}\ny_{12}^{u}|_{[0,T]} \\in L^2([0,T],H^1_0(\\Omega)) \\cap L^p([0,T], L^p(\\Omega))\n\\end{align*}\nand $\\partial_t y_{1}^{u}|_{[0,T]} - \\partial_t y_{2}^{0}|_{[0,T]} \\in L^2([0,T],H^1_0(\\Omega)^*) + L^q([0,T], L^q(\\Omega))$ as well as\n\\begin{align} \\label{eq:semiproc-estimates-step-1}\n\\int_0^T \\big( \\partial_t y_{1}^{u}(t) &- \\partial_t y_{2}^{0}(t), \\phi(t) \\big) \\d t \n= - \\int_0^T \\int_{\\Omega} \\nabla y_{12}^{u}(t)(\\zeta) \\cdot \\nabla \\phi(t)(\\zeta) \\d\\zeta \\d t \\notag \\\\\n&+  \\int_0^T \\int_{\\Omega} \\overline{g}\\big( y_{12}^{u}(t)(\\zeta) \\big) \\, \\phi(t)(\\zeta) \\d \\zeta \\d t  + \\int_0^T \\int_{\\Omega} \\overline{h}(t)(\\zeta) \\, \\phi(t)(\\zeta)  \\d \\zeta \\d t\n\\end{align}\nfor every $\\phi \\in L^2([0,T],H^1_0(\\Omega)) \\cap L^p([0,T], L^p(\\Omega))$. And therefore, $y_{12}^{u}$ is a weak solution of~\\eqref{eq:semiproc-estimates-ibvp-general}, as desired.\n\\smallskip\n\nAs a second step, we show that for every $y_{01},y_{02} \\in X$ and $u \\in \\mathcal{U}$ the function $y_{12}^{u}$ from~\\eqref{eq:y-12-u} satisfies the estimate\n\\begin{align} \\label{eq:semiproc-estimates-step-2}\n\\sup_{T \\in [0,t]} \\norm{y_{12}^{u}(T)}^2 \\le \\e^{2\\lambda t} \\Big( \\norm{y_{01}-y_{02}}^2 + 2\\norm{h}\\norm{u}_{\\infty} \\cdot t \\cdot \\sup_{T \\in [0,t]}  \\norm{y_{12}^{u}(T)} \\Big) \n\\end{align}\nfor every $t \\in \\mathbb{R}^+_0$. \nIndeed, by the first step and Lemma~\\ref{lm:weak-sol-general}, the function $t \\mapsto \\norm{y_{12}^{u}(t)}^2$ is absolutely continuous with \n\\begin{align*}\n\\ddt \\frac{\\norm{y_{12}^{u}(t)}^2}{2} = \\big( \\partial_t y_{12}^{u}(t), y_{12}^{u}(t) \\big) = \\big( \\partial_t y_{1}^{u}(t) - \\partial_t y_{2}^{0}(t), y_{12}^{u}(t) \\big) \n\\end{align*}\nfor almost every $t \\in \\mathbb{R}^+_0$. And therefore, by virtue of~\\eqref{eq:semiproc-estimates-step-1} with $\\phi := y_{12}^{u}$, we get\n\\begin{align} \\label{eq:semiproc-estimates-step-2,1}\n&\\frac{\\norm{y_{12}^{u}(T)}^2}{2} - \\frac{\\norm{y_{12}^{u}(0)}^2}{2}\n=\n\\int_0^T \\big( \\partial_t y_{1}^{u}(t) - \\partial_t y_{2}^{0}(t) , y_{12}^{u}(t) \\big)  \\d t \\notag \\\\\n&\\qquad \\le \n\\int_0^T \\int_{\\Omega} \\big( g(y_1^{u}(t)(\\zeta)) - g(y_2^{0}(t)(\\zeta)) \\big) y_{12}^{u}(t)(\\zeta) \\d\\zeta \\d t + \\int_0^T \\int_{\\Omega} h(\\zeta) u(t) y_{12}^{u}(t)(\\zeta) \\d\\zeta \\d t \\notag \\\\\n&\\qquad \\le \n\\lambda \\int_0^T \\norm{y_{12}^{u}(t)}^2 \\d t + \\norm{h}\\norm{u}_{\\infty} \\int_0^T \\norm{y_{12}^{u}(t)} \\d t\n\\end{align}\nfor every $T \\in (0,\\infty)$. In the last inequality, we used that $(g(r)-g(s)) (r-s) \\le \\lambda |r-s|^2$ for all $r,s \\in \\mathbb{R}$ due to~\\eqref{eq:cond-g}.\nSo, for every $t_0 \\in (0,\\infty)$, we obtain\n\\begin{align*}\n\\norm{y_{12}^{u}(T)}^2 \\le \\norm{y_{01}-y_{02}}^2 + 2\\norm{h}\\norm{u}_{\\infty} \\cdot t_0 \\cdot \\sup_{t\\in[0,t_0]} \\norm{y_{12}^{u}(t)} + 2 \\lambda \\int_0^T \\norm{y_{12}^{u}(t)}^2 \\d t\n\\end{align*}\nfor every $T \\in [0,t_0]$. And from this, in turn,\nthe claimed estimate~\\eqref{eq:semiproc-estimates-step-2} immediately follows by Gr\\\"onwall's lemma.  \n\\smallskip\n\nAs a third step, it is now easy to conclude the desired estimates~\\eqref{eq:semiproc-estimate-1} and~\\eqref{eq:semiproc-estimate-2} from the second step.\nIndeed, \\eqref{eq:semiproc-estimate-1} immediately follows from~\\eqref{eq:semiproc-estimates-step-2} with the special choice $u := 0 \\in \\mathcal{U}$ and~\\eqref{eq:semiproc-estimate-2} follows from~\\eqref{eq:semiproc-estimates-step-2} with the special choice $y_{01} = y_{02} := y_0 \\in X$.  \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe remark for later reference that our semiprocess family $(S_u)_{u\\in\\mathcal{U}}$, like any other semiprocess family~\\cite{ScKaDa19-wAG},\nsatisfies the following so-called cocycle property:\n\\begin{align}\nS_u(t+\\tau,0,x) = S_{u(\\cdot+\\tau)}\\big(t,0,S_u(\\tau,0,x)\\big)\n\\end{align}\nfor all $t, \\tau \\in \\mathbb{R}^+_0$, $x \\in X$ and $u \\in \\mathcal{U}$. (Just combine~\\eqref{eq:def-semiproc-1} and~\\eqref{eq:def-semiproc-2} to see this.) In particular, $S_0$ satisfies the following (nonlinear) semigroup property~\\cite{Miyadera}:\n\\begin{align} \\label{eq:S0-sgr}\nS_0(t+\\tau,0,x) = S_0\\big(t,0,S_0(\\tau,0,x)\\big) \\qquad (t,\\tau \\in \\mathbb{R}^+_0 \\text{ and } x \\in X).\n\\end{align} \n\nWe conclude this section with some remarks on the asymptotic behavior\nof this semigroup $S_0$ in terms of attractors~\\cite{Robinson}, \\cite{Temam}.\nA \\emph{global attractor} of $S_0$ is a compact subset $\\Theta$ of $X$ such that\n\\begin{itemize}\n\\item[(i)] $\\Theta$ is invariant under $S_0$, that is, $S_0(t,0,\\Theta) = \\Theta$ for every $t \\in \\mathbb{R}^+_0$\n\\item[(ii)] $\\Theta$ is uniformly attractive for $S_0$, that is, for every bounded subset $B \\subset X$ one has\n\\begin{align} \\label{eq:uniform-attractivity-def}\n\\operatorname{dist}\\big( S_0(t,0,B), \\Theta \\big) = \\sup_{x\\in B} \\norm{S_0(t,0,x)}_{\\Theta} \\longrightarrow 0 \\qquad (t \\to \\infty).\n\\end{align}\n\\end{itemize} \nIt directly follows from this definition that a global attractor of $S_0$ is minimal among all closed uniformly attractive sets of $S_0$ and maximal among all bounded invariant sets of $S_0$. And from this, in turn, it immediately follows that if $S_0$ has any global attractor then it is already unique. \n\n\n\n\\begin{lm} \\label{lm:S0-UGAS}\nSuppose that Condition~\\ref{cond:g,h} is satisfied. Then the undisturbed system $S_0$ has a unique global attractor $\\Theta$ and, moreover, $\\Theta$ is uniformly globally asymptotically stable for $S_0$, that is, there exists a comparison function $\\beta_0 \\in \\mathcal{KL}$ such that\n\\begin{align} \\label{eq:S0-UGAS}\n\\norm{S_0(t,0,x)}_{\\Theta} \\le \\beta_0(\\norm{x}_{\\Theta},t) \\qquad (t\\in \\mathbb{R}^+_0 \\text{ and } x \\in X). \n\\end{align}\n\\end{lm}\n\n\n\n\\begin{proof}\nIt is well-known that $S_0$ has a global attractor $\\Theta$ (by Theorem~11.4 of~\\cite{Robinson}, for instance) and that global attractors when existent are already unique (by the remarks preceding the lemma). \nSo, we have only\nto show that $\\Theta$ is uniformly globally asymptotically stable for $S_0$. And in order to do so, we will proceed in three steps, applying results from~\\cite{Mi17} to the system $S_0 = (S_u)_{u\\in\\mathcal{U}_0}$ with trivial disturbance space $\\mathcal{U}_0 := \\{0\\}$. (In this context, it should be noticed that by~\\eqref{eq:S0-sgr} and the continuity of weak solutions, $(S_u)_{u\\in\\mathcal{U}_0}$ is a forward-complete system in the sense of~\\cite{Mi17}, \\cite{MiWi18}, \\cite{Sc19-wISS}.)\n\\smallskip\n\nAs a first step, we show that $\\Theta$ is uniformly globally stable for $(S_u)_{u\\in\\mathcal{U}_0} = S_0$,\nthat is, there exists a comparison function $\\sigma_0 \\in \\mathcal{K}$ such that\n\\begin{align}\n\\norm{S_0(t,0,x)}_{\\Theta} \\le \\sigma_0(\\norm{x}_{\\Theta}) \\qquad (t \\in \\mathbb{R}^+_0)\n\\end{align}\nfor every $x \\in X$ (Definition~2.8 of~\\cite{Mi17}).\nIndeed, it immediately follows from the invariance of $\\Theta$ under $S_0$ and from the estimate~\\eqref{eq:semiproc-estimate-1} that for every $\\varepsilon > 0$ and every $T \\in (0,\\infty)$ there exists a $\\delta \\in (0,1]$ such that\n\\begin{align*}\n\\norm{S_0(t,0,x)}_{\\Theta} \\le \\inf_{\\theta \\in\\Theta} \\norm{S_0(t,0,x)-S_0(t,0,\\theta)} < \\varepsilon \\qquad (t \\in [0,T] \\text{ and } x \\in B_{\\delta}(\\Theta)).  \n\\end{align*}\nAnd from this and the uniform attractivity~\\eqref{eq:uniform-attractivity-def} of $\\Theta$ for $S_0$ (with $B := B_1(\\Theta)$), in turn, it follows that for every $\\varepsilon > 0$ there exists a $\\delta > 0$ such that\n\\begin{align} \\label{eq:Theta-ULS}\n\\norm{S_0(t,0,x)}_{\\Theta} < \\varepsilon \\qquad (t \\in \\mathbb{R}^+_0)\n\\end{align}\nfor every $x \\in B_{\\delta}(\\Theta)$. \nAlso, it is well-known that \n\\begin{align} \\label{eq:dissip-estimate-S0}\n\\norm{S_0(t,0,x)}^2 \\le \\e^{-2\\omega t} \\norm{x}^2 + \\frac{\\lambda |\\Omega|}{\\omega} \\qquad (t \\in \\mathbb{R}^+_0)\n\\end{align}\nfor all $x \\in X$, where $\\omega \\in (0,\\infty)$ is the smallest eigenvalue of $-\\Delta$, the negative Dirichlet Laplacian on $\\Omega$. (See the very last equation on page 286 of~\\cite{Robinson}, for instance.) Since $\\norm{S_0(t,0,x)}_{\\Theta} \\le \\norm{S_0(t,0,x)} + \\norm{\\Theta}$ and $\\norm{x} \\le \\norm{x}_{\\Theta} + \\norm{\\Theta}$ with $\\norm{\\Theta} := \\sup_{\\theta \\in \\Theta} \\norm{\\theta}$, it follows from~\\eqref{eq:dissip-estimate-S0}  that there exists a comparison function $\\sigma \\in \\mathcal{K}$ and a constant $c \\in (0,\\infty)$ such that\n\\begin{align} \\label{eq:Theta-Lagrange-stable}\n\\norm{S_0(t,0,x)}_{\\Theta} \\le \\sigma(\\norm{x}_{\\Theta}) + c \\qquad (t \\in \\mathbb{R}^+_0)\n\\end{align} \nfor every $x \\in X$. \nIn the terminology of~\\cite{Mi17}, the relations~\\eqref{eq:Theta-ULS} and~\\eqref{eq:Theta-Lagrange-stable} mean that $\\Theta$ is uniformly locally stable and Lagrange-stable for $(S_u)_{u\\in\\mathcal{U}_0}$, respectively. And therefore, $\\Theta$ is uniformly globally stable for $(S_u)_{u\\in\\mathcal{U}_0} = S_0$ by virtue of Remark~2.9 of~\\cite{Mi17}, as desired. \n\\smallskip\n\nAs a second step, we show\nthat $\\Theta$ is uniformly globally attractive for $(S_u)_{u\\in\\mathcal{U}_0} = S_0$, that is, for every $\\varepsilon >0$ and $r>0$ there exists a time $\\tau(\\varepsilon,r) \\in \\mathbb{R}^+_0$ such that \n\\begin{align} \\label{eq:Theta-UGATT}\n\\norm{S_0(t,0,x)}_{\\Theta} < \\varepsilon \\qquad (t \\ge \\tau(\\varepsilon,r))\n\\end{align} \nfor every $x \\in \\overline{B}_r(\\Theta)$ (Definition~2.8 of~\\cite{Mi17}). \nIndeed, this immediately follows from the uniform attractivity~\\eqref{eq:uniform-attractivity-def} of $\\Theta$ for $S_0$ with $B := \\overline{B}_r(\\Theta)$. \n\\smallskip\n\nAs a third step, we can now conclude the desired uniform global asymptotic stability of $\\Theta$ for $(S_u)_{u\\in\\mathcal{U}_0} = S_0$\nfrom Theorem~4.2 of~\\cite{Mi17} and the first two steps.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{A local input-to-state stability result}\n\n\n\nIn this section, we establish our local input-to-state stability result for the disturbed reaction-diffusion system~\\eqref{eq:react-diffus-eq}. We begin by showing that the undisturbed system~\\eqref{eq:react-diffus-eq, undisturbed} has a local Lyapunov function and, for that purpose, we will\nargue in a similar way as~\\cite{Henry} (Theorem~4.2.1). \n\n\n\\begin{lm} \\label{lm:L-fct}\nSuppose that Condition~\\ref{cond:g,h} is satisfied and let $\\Theta$ be the global attractor of the undisturbed system $S_0$.\nThen for every $r_0 > 0$ there exists a Lipschitz continuous function $V: \\overline{B}_{r_0}(\\Theta) \\to \\mathbb{R}^+_0$ with Lipschitz constant $1$ and comparison functions $\\underline{\\psi}, \\overline{\\psi}, \\alpha \\in \\mathcal{K}_{\\infty}$ such that\n\\begin{gather}\n\\underline{\\psi}(\\norm{x}_{\\Theta}) \\le V(x) \\le \\overline{\\psi}(\\norm{x}_{\\Theta}) \\qquad (x \\in \\overline{B}_{r_0}(\\Theta)) \\label{eq:V-coercive-Lfct}\\\\\n\\dot{V}_0(x) := \\varlimsup_{t\\to 0+} \\frac{1}{t}\\big( V(S_0(t,0,x))-V(x) \\big)  \\le -\\alpha(\\norm{x}_{\\Theta}) \\qquad (x \\in B_{r_0}(\\Theta)). \\label{eq:V-Dini-derivative}\n\\end{gather}\n\\end{lm}\n\n\n\n\\begin{proof}\nChoose an arbitrary $r_0 \\in (0,\\infty)$ and fix it for the rest of the proof. Also, choose $\\beta_0 \\in \\mathcal{KL}$ as in Lemma~\\ref{lm:S0-UGAS} and, for every $\\varepsilon > 0$,\nlet $T(\\varepsilon) = T_{r_0}(\\varepsilon)$ be a time such that\n\\begin{align} \\label{eq:Lfct-def-T(eps)}\n\\beta_0(r_0,t) \\le \\varepsilon \\qquad (t \\in [T(\\varepsilon),\\infty)).\n\\end{align} \nSet now, for every given $\\varepsilon > 0$,\n\\begin{align} \\label{eq:Lfct-def-V-eps}\nV^{\\varepsilon}(x) := \\e^{-(\\lambda+c_0)T(\\varepsilon)} \\sup_{t \\in [0,\\infty)} \\Big( \\e^{c_0 t} \\, \\eta_{\\varepsilon}\\big( \\norm{S_0(t,0,x)}_{\\Theta} \\big) \\Big)\n\\qquad (x \\in \\overline{B}_{r_0}(\\Theta)),\n\\end{align}\nwhere $c_0 \\in (0,\\infty)$ is an arbitrary constant (which is fixed throughout the proof) and $\\eta_{\\varepsilon}(r) := \\max\\{ 0,r-\\varepsilon\\}$ for every $r \\in \\mathbb{R}^+_0$. \nIn view of~\\eqref{eq:S0-UGAS} and~\\eqref{eq:Lfct-def-T(eps)}, the supremum in~\\eqref{eq:Lfct-def-V-eps} for $x \\in \\overline{B}_{r_0}(\\Theta)$ actually extends only over a compact interval, namely\n\\begin{align} \\label{eq:Lfct-V-eps-supremum-only-over-compact-interval}\nV^{\\varepsilon}(x) = \\e^{-(\\lambda+c_0)T(\\varepsilon)} \\sup_{t \\in [0,T(\\varepsilon)]} \\Big( \\e^{c_0 t} \\, \\eta_{\\varepsilon}\\big( \\norm{S_0(t,0,x)}_{\\Theta} \\big) \\Big)\n\\qquad (x \\in \\overline{B}_{r_0}(\\Theta)).\n\\end{align}\nIn particular, $V^{\\varepsilon}: \\overline{B}_{r_0}(\\Theta) \\to \\mathbb{R}^+_0$ is a well-defined map (with finite values) and\n\\begin{align} \\label{eq:Lfct-V-eps-le-beta0}\nV^{\\varepsilon}(x) \\le \\e^{-\\lambda T(\\varepsilon)}  \\sup_{t \\in [0,T(\\varepsilon)]} \\Big(  \\eta_{\\varepsilon}\\big( \\norm{S_0(t,0,x)}_{\\Theta} \\big) \\Big)\n\\le \\beta_0(\\norm{x}_{\\Theta},0) \n\\qquad (x \\in \\overline{B}_{r_0}(\\Theta))\n\\end{align}\nbecause $\\eta_{\\varepsilon}(r) \\le r$ for all $r \\in \\mathbb{R}^+_0$. \nSince, moreover,\n$|\\eta_{\\varepsilon}(r)-\\eta_{\\varepsilon}(s)| \\le |r-s|$ for all $r,s \\in \\mathbb{R}^+_0$, \nwe see from~\\eqref{eq:Lfct-V-eps-supremum-only-over-compact-interval} and~\\eqref{eq:semiproc-estimate-1} that \n\\begin{align} \\label{eq:Lfct-V-eps-Lipschitz-with-const-1}\n|V^{\\varepsilon}(x) &- V^{\\varepsilon}(y)| \n\\le \n\\e^{-(\\lambda+c_0)T(\\varepsilon)} \\sup_{t \\in [0,T(\\varepsilon)]} \\Big| \\e^{c_0 t} \\, \\eta_{\\varepsilon}\\big( \\norm{S_0(t,0,x)}_{\\Theta}\\big) - \\e^{c_0 t} \\, \\eta_{\\varepsilon}\\big( \\norm{S_0(t,0,y)}_{\\Theta} \\big) \\Big| \\notag \\\\\n&\\le \n\\e^{-\\lambda T(\\varepsilon)} \\sup_{t \\in [0,T(\\varepsilon)]} \\Big|  \\norm{S_0(t,0,x)}_{\\Theta} - \\norm{S_0(t,0,y)}_{\\Theta}  \\Big| \\\\\n&\\le \n\\e^{-\\lambda T(\\varepsilon)} \\sup_{t \\in [0,T(\\varepsilon)]} \\norm{ S_0(t,0,x) - S_0(t,0,y) }\n\\le \\norm{x-y}  \n\\qquad (x,y \\in \\overline{B}_{r_0}(\\Theta)).  \\notag\n\\end{align}\n(In the first inequality above, we used the elementary fact that $|\\sup_{t \\in I} a_t - \\sup_{t\\in I} b_t| \\le \\sup_{t\\in I} |a_t-b_t|$ for arbitrary bounded functions $t \\mapsto a_t, b_t$ on an arbitrary set $I$, and in the third inequality above, we used the elementary fact that $|\\norm{\\xi}_{\\Theta}-\\norm{\\eta}_{\\Theta}| \\le \\norm{\\xi-\\eta}$ for arbitrary $\\xi,\\eta \\in X$.) \nAdditionally, for every $x \\in B_{r_0}(\\Theta)$, we have $S_0(\\tau,0,x) \\in B_{r_0}(\\Theta)$ for $\\tau$ small enough and thus, by~\\eqref{eq:Lfct-def-V-eps} and the semigroup property~\\eqref{eq:S0-sgr},\n\\begin{align*}\nV^{\\varepsilon}(S_0(\\tau,0,x)) = \\e^{-(\\lambda+c_0)T(\\varepsilon)} \\sup_{t \\in [0,\\infty)} \\Big( \\e^{c_0 t} \\, \\eta_{\\varepsilon}\\big( \\norm{S_0(t+\\tau,0,x)}_{\\Theta} \\big) \\Big) \n\\le \\e^{-c_0 \\tau} V^{\\varepsilon}(x)\n\\end{align*}\nfor every $x \\in B_{r_0}(\\Theta)$ and all sufficiently small times $\\tau$. Consequently,\n\\begin{align} \\label{eq:Lfct-Dini-derivative-V-eps}\n\\dot{V}_0^{\\varepsilon}(x) = \\varlimsup_{\\tau \\to 0+} \\frac{1}{\\tau} \\big( V^{\\varepsilon}(S_0(\\tau,0,x)) - V^{\\varepsilon}(x) \\big) \\le -c_0 V^{\\varepsilon}(x)\n\\qquad (x \\in B_{r_0}(\\Theta)). \n\\end{align}\nWith the help of the auxiliary functions $V^{\\varepsilon}$, we can now construct a function $V: \\overline{B}_{r_0}(\\Theta) \\to \\mathbb{R}^+_0$ with the desired properties. Indeed, let \n\\begin{align} \\label{eq:Lfct-def-V}\nV(x) := \\sum_{k=1}^{\\infty} 2^{-k} V^{1\/k}(x) \\qquad (x \\in \\overline{B}_{r_0}(\\Theta)).\n\\end{align}\nWe then conclude from~\\eqref{eq:Lfct-V-eps-le-beta0}, \\eqref{eq:Lfct-V-eps-Lipschitz-with-const-1}, \\eqref{eq:Lfct-Dini-derivative-V-eps} that\n\\begin{gather}\nV(x) \\le \\beta_0(\\norm{x}_{\\Theta},0) \\qquad (x \\in \\overline{B}_{r_0}(\\Theta)), \\label{eq:Lfct-bounded-above-by-beta0}\\\\\n|V(x)-V(y)| \\le \\sum_{k=1}^{\\infty} 2^{-k} |V^{1\/k}(x)-V^{1\/k}(y)| \\le \\norm{x-y}\n\\qquad (x,y \\in \\overline{B}_{r_0}(\\Theta)),  \\label{eq:Lfct-Lipschitz-with-constant-1}\\\\\n\\dot{V}_0(x) \\le \\sum_{k=1}^{\\infty} 2^{-k} \\dot{V}_0^{1\/k}(x) \\le -c_0 V(x)  \\qquad (x \\in B_{r_0}(\\Theta)). \\label{eq:Lfct-Dini-derivative}\n\\end{gather}\nSince $\\sup_{t\\in[0,\\infty)} (\\e^{c_0 t} \\eta_{1\/k}(\\norm{S_0(t,0,x)}_{\\Theta})) \\ge \\eta_{1\/k}(\\norm{x}_{\\Theta})$ for all $x \\in X$, we also conclude from~\\eqref{eq:Lfct-def-V-eps} and~\\eqref{eq:Lfct-def-V} that\n\\begin{align} \\label{eq:Lfct-bounded-below}\nV(x) \\ge \\sum_{k=1}^{\\infty} 2^{-k} \\e^{-(\\lambda + c_0) T(1\/k)} \\eta_{1\/k}(\\norm{x}_{\\Theta}) \\qquad (x \\in \\overline{B}_{r_0}(\\Theta).\n\\end{align}\nIn view of these estimates, we now define the comparison functions $\\overline{\\psi}$, $\\underline{\\psi}$ and $\\alpha$  in the following way:\n\\begin{align*}\n\\overline{\\psi}(r) := \\beta_0(r,0) + r \\qquad \\text{and} \\qquad \\underline{\\psi}(r) := \\sum_{k=1}^{\\infty} 2^{-k} \\e^{-(\\lambda + c_0) T(1\/k)} \\eta_{1\/k}(r)\n\\end{align*}\nand $\\alpha(r) := c_0 \\underline{\\psi}(r)$ for $r \\in \\mathbb{R}^+_0$. It is easy to verify that $\\overline{\\psi}$, $\\underline{\\psi}$ and hence $\\alpha$ belong to $\\mathcal{K}_{\\infty}$. And, moreover, by virtue of~\\eqref{eq:Lfct-bounded-above-by-beta0}, \\eqref{eq:Lfct-Lipschitz-with-constant-1}, \\eqref{eq:Lfct-Dini-derivative}, \\eqref{eq:Lfct-bounded-below}, the desired estimates~\\eqref{eq:V-coercive-Lfct} and~\\eqref{eq:V-Dini-derivative} follow. \n\\end{proof}\n\n\n\nIt should be noticed that the functions $V, \\underline{\\psi}, \\alpha$ constructed in the proof above all depend on the chosen radius $r_0 \\in (0,\\infty)$ because these functions are defined in terms of the times $T(\\varepsilon) = T_{r_0}(\\varepsilon)$ from~\\eqref{eq:Lfct-def-T(eps)}. \nWith the next lemma, we show that the local Lyapunov function $V$ for the undisturbed system is also a local input-to-state Lyapunov function\nfor the disturbed system w.r.t.~$\\Theta$. (See~\\cite{DaMi13} for the definition of local input-to-state Lyapunov functions w.r.t.~an equilibrium point.) \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{lm} \\label{lm:lISS-L-fct}\nSuppose that Condition~\\ref{cond:g,h} is satisfied and let $\\Theta$ be the global attractor of the undisturbed system $S_0$. Also, let $r_0 > 0$ and let $V: \\overline{B}_{r_0}(\\Theta) \\to \\mathbb{R}_0^+$ be chosen as in the previous lemma. Then there exist comparison functions $\\alpha, \\sigma \\in \\mathcal{K}$ such that\nfor every $u \\in \\mathcal{U}$\n\\begin{align*}\n\\dot{V}_u(x) := \\varlimsup_{t\\to0+} \\frac{1}{t}\\big( V(S_u(t,0,x))-V(x) \\big) \\le -\\alpha(\\norm{x}_{\\Theta}) + \\sigma(\\norm{u}_{\\infty})\n\\qquad (x \\in B_{r_0}(\\Theta)).\n\\end{align*}\n\\end{lm}\n\n\n\\begin{proof}\nChoose $\\alpha = \\alpha_{r_0} \\in \\mathcal{K}_{\\infty}$ as in Lemma~\\ref{lm:L-fct} and define $\\sigma \\in \\mathcal{K}_{\\infty}$\nby $\\sigma(r) := 2\\e^{2\\lambda} \\norm{h} r$ for all $r \\in \\mathbb{R}^+_0$. \nWe then see from Lemma~\\ref{lm:L-fct} and from \\eqref{eq:semiproc-estimate-2}\nthat for every $x \\in B_{r_0}(\\Theta)$ and every $u \\in \\mathcal{U}$\n\\begin{align}\n\\dot{V}_u(x) \\le \\varlimsup_{t\\to 0+} \\frac{1}{t}\\big( V(S_0(t,0,x))-V(x) \\big) + \\varlimsup_{t\\to 0+} \\frac{1}{t}\\big( V(S_u(t,0,x))-V(S_0(t,0,x)) \\big) \\notag \\\\\n\\le -\\alpha(\\norm{x}_{\\Theta}) + \\varlimsup_{t\\to 0+} \\frac{1}{t} \\norm{ S_u(t,0,x) - S_0(t,0,x) } \n\\le -\\alpha(\\norm{x}_{\\Theta}) + \\sigma(\\norm{u}_{\\infty}),\n\\end{align}\nas desired. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nWith these lemmas at hand, we can now establish the local input-to-state stability of the disturbed reaction-diffusion system~\\eqref{eq:react-diffus-eq} w.r.t.~the global attractor of the undisturbed system~\\eqref{eq:react-diffus-eq, undisturbed}. \nIt is an open question -- left to future research -- whether this result can actually be extended to a semi-global\ninput-to-state stability\nresult. See the remarks after the proof for a discussion of the obstacles to such an extension. \n\n\n\\begin{thm} \\label{thm:lISS}\nSuppose that Condition~\\ref{cond:g,h} is satisfied and let $\\Theta$ be the global attractor of the undisturbed system $S_0$. Then the disturbed system $(S_u)_{u\\in\\mathcal{U}}$ is locally input-to-state stable w.r.t.~$\\Theta$, that is, there exist comparison functions $\\beta \\in \\mathcal{KL}$ and $\\gamma \\in \\mathcal{K}$ and radii $r_{0x}, r_{0u} > 0$ such that\n\\begin{align}\n\\norm{S_u(t,0,x_0)}_{\\Theta} \\le \\beta(\\norm{x_0}_{\\Theta},t) + \\gamma(\\norm{u}_{\\infty}) \\qquad (t \\in \\mathbb{R}^+_0)\n\\end{align}\nfor all $(x_0,u) \\in X \\times \\mathcal{U}$ with $\\norm{x_0}_{\\Theta} \\le r_{0x}$ and  $\\norm{u}_{\\infty} \\le r_{0u}$.\n\\end{thm}\n\n\n\\begin{proof}\nChoose an arbitrary $r_0 \\in (0,\\infty)$ and fix it for the entire proof. Also,  take $V = V_{r_0}$ and $\\underline{\\psi} = \\underline{\\psi}_{r_0}$, $\\overline{\\psi}$ as in Lemma~\\ref{lm:L-fct}.\nIt then immediately follows from Lemma~\\ref{lm:lISS-L-fct} that there exist comparison functions $\\alpha = \\alpha_{r_0} \\in \\mathcal{K}$ and $\\chi = \\chi_{r_0} \\in \\mathcal{K}$ such that for all $(x_0,u) \\in X \\times \\mathcal{U}$ with $r_0 \\ge \\norm{x_0}_{\\Theta} \\ge \\chi(\\norm{u}_{\\infty})$ one has\n\\begin{align} \\label{eq:lISS-L-fct,implicative}\n\\dot{V}_u(x_0) \\le -\\alpha(\\norm{x_0}_{\\Theta}). \n\\end{align}\n(Simply choose $\\chi(r) := \\alpha_0^{-1}(2\\sigma_0(r))$ and $\\alpha(r) := \\alpha_0(r)\/2$, where $\\alpha_0, \\sigma_0 \\in \\mathcal{K}_{\\infty}$ are as in Lemma~\\ref{lm:lISS-L-fct}.)\nAccording to the comparison lemma from~\\cite{MiIt16} (Corollary~1), we can then choose a comparison function $\\overline{\\beta} = \\overline{\\beta}_{\\alpha\\circ \\overline{\\psi}^{-1}}$ in such a way that for every $T \\in (0,\\infty]$ and every function $v \\in C([0,T),\\mathbb{R}^+_0)$ with\n\\begin{align*}\n\\overline{\\partial}_t^+ v(t) \\le -(\\alpha \\circ \\overline{\\psi}^{-1})(v(t)) \\qquad (t \\in [0,T))\n\\end{align*} \none has $v(t) \\le \\overline{\\beta}(v(0),t)$ for all $t\\in[0,T)$. \nWe now define\n\\begin{align} \\label{eq:def-beta-gamma}\n\\beta(r,t) := \\underline{\\psi}^{-1}\\big( \\overline{\\beta}(\\overline{\\psi}(r),t) \\big) \n\\qquad \\text{and} \\qquad\n\\gamma(r) := \\underline{\\psi}^{-1}\\big( \\overline{\\psi}(\\chi(r)) \\big)\n\\end{align} \nfor $r,t \\in \\mathbb{R}^+_0$ and choose $r_{0x}, r_{0u} \\in (0,\\infty)$ so small that\n\\begin{align} \\label{eq:def-r0x-and-r0u}\nr_{0x} < r_0 \\qquad \\text{and} \\qquad \\beta(r_{0x},0) < r_0 \n\\qquad \\text{and} \\qquad\n\\gamma(r_{0u}) < r_0. \n\\end{align}\nAlso, we will write\n\\begin{align} \\label{eq:M_u-def}\nM_u := \\big\\{ x \\in \\overline{B}_{r_0}(\\Theta): V(x) \\le \\overline{\\psi}(\\chi(\\norm{u}_{\\infty})) \\big\\}\n\\end{align}\nfor $u \\in \\mathcal{U}$.\nClearly, $\\beta \\in \\mathcal{KL}$, $\\gamma \\in \\mathcal{K}$ and $M_u$ is closed for every $u \\in \\mathcal{U}$. Additionally, for every $u \\in \\overline{B}_{r_{0u}}(0)$ we have by~\\eqref{eq:def-r0x-and-r0u} that\n\\begin{align} \\label{eq:M_u-contained-in-better-sets}\nM_u \\subset \\big\\{ x \\in \\overline{B}_{r_0}(\\Theta): \\norm{x}_{\\Theta} \\le \\gamma(\\norm{u}_{\\infty}) \\big\\} \\subset B_{r_0}(\\Theta).\n\\end{align}\n\nAfter these preliminary considerations, we now prove that\n\\begin{align} \\label{eq:lISS-estimate}\n\\norm{S_u(t,0,x_0)}_{\\Theta} \\le \\beta(\\norm{x_0}_{\\Theta},t) + \\gamma(\\norm{u}_{\\infty}) \\qquad (t \\in \\mathbb{R}^+_0)\n\\end{align}\nfor all $(x_0,u) \\in \\overline{B}_{r_{0x}}(\\Theta) \\times \\overline{B}_{r_{0u}}(0)$ and thus obtain the desired local input-to-state stability. So, let $(x_0,u) \\in \\overline{B}_{r_{0x}}(\\Theta) \\times \\overline{B}_{r_{0u}}(0)$ be fixed for the rest of the proof.\nWe will distinguish two cases in the following, namely the case where $x_0 \\in M_u$ (part (i) of the proof) and the case where $x_0 \\notin M_u$ (part (ii) of the proof). \n\\smallskip\n\n(i) Suppose we are in the case $x_0 \\in M_u$. In order to establish~\\eqref{eq:lISS-estimate} in that case,\nwe will show -- in two steps -- that for every $t_0 \\in [0,\\infty)$ one has\n\\begin{align} \\label{eq:M_u-invariant}\nS_u(t,t_0,M_u) \\in M_u \\qquad (t \\in [t_0,\\infty)). \n\\end{align}\nSo, let $t_0 \\in [0,\\infty)$ and $x_{t_0} \\in M_u$\nand \n\\begin{align} \\label{eq:def-T-case(i)}\nT := \\sup \\big\\{ T' \\in (t_0,\\infty): \\norm{x(t)}_{\\Theta} < r_0 \\text{ for all } t \\in [t_0,T') \\big\\}, \n\\end{align}\nwhere we use the abbreviation $x(t) := S_u(t,t_0,x_{t_0})$. \nSince $x(t_0) = x_{t_0} \\in M_u$ and thus $\\norm{x(t_0)}_{\\Theta} < r_0$ by~\\eqref{eq:M_u-contained-in-better-sets}, we observe\nthat $T \\in (t_0,\\infty]$ and that\n\\begin{align} \\label{eq:norm-initially-less-than-r0,case(i)}\n\\norm{x(t)}_{\\Theta} < r_0 \\qquad (t \\in [t_0,T)).\n\\end{align}\n\nAs a first step, we show that $x(t) \\in M_u$ at least for all $[t_0,T)$. \nAssuming the contrary, we find a $t \\in [t_0,T)$ and an $\\varepsilon > 0$ such that $V(x(t)) > \\overline{\\psi}(\\chi(\\norm{u}_{\\infty})) + \\varepsilon$. Since $x(t_0) = x_{t_0} \\in M_u$ and thus $V(x(t_0)) \\le \\overline{\\psi}(\\chi(\\norm{u}_{\\infty})) + \\varepsilon$,\nwe observe\nthat\n\\begin{align}\nt_1 := \\inf \\big\\{ t \\in [t_0,T): V(x(t)) > \\overline{\\psi}(\\chi(\\norm{u}_{\\infty})) + \\varepsilon \\big\\}\n\\end{align}\nbelongs to the interval $(t_0,T)$ and, moreover, $V(x(t_1)) = \\overline{\\psi}(\\chi(\\norm{u}_{\\infty})) + \\varepsilon$. So,\n\\begin{align*}\n\\overline{\\psi}(\\norm{x(t_1)}_{\\Theta}) \\ge V(x(t_1)) > \\overline{\\psi}(\\chi(\\norm{u}_{\\infty})) \\ge \\overline{\\psi}\\big(\\chi(\\norm{u(\\cdot+t_1)}_{\\infty})\\big)\n\\end{align*} \nand therefore we get by virtue of~\\eqref{eq:lISS-L-fct,implicative} that\n\\begin{align}\n\\varlimsup_{t\\to 0+} \\frac{1}{t}\\Big( V(x(t_1+t))-V(x(t_1)) \\Big) \n&= \\varlimsup_{t\\to 0+} \\frac{1}{t}\\Big( V\\big(S_{u(\\cdot + t_1)}(t,0,x(t_1))\\big)-V(x(t_1)) \\Big)  \\notag \\\\\n&= \\dot{V}_{u(\\cdot + t_1)}(x(t_1)) \\le -\\alpha(\\norm{x(t_1)}_{\\Theta}) < 0.\n\\end{align}\nConsequently, there exists a $\\delta >0$ such that $V(x(t_1+t)) \\le V(x(t_1)) = \\overline{\\psi}(\\chi(\\norm{u}_{\\infty})) + \\varepsilon$ for all $t \\in [0,\\delta)$. Contradiction to the definition of $t_1$!\n\\smallskip\n\nAs a second step, we show that $T = \\infty$. Indeed, assuming $T < \\infty$, we would get by the first step and continuity\nthat even $x(T) \\in M_u$ and thus $\\norm{x(T)}_{\\Theta} < r_0$ by~\\eqref{eq:M_u-contained-in-better-sets}. And from this, in turn, it would follow again by continuity that $\\norm{x(t)}_{\\Theta} < r_0$ for all $t \\in [T,T+\\delta)$ with some $\\delta > 0$. In conjunction with~\\eqref{eq:norm-initially-less-than-r0,case(i)}, this would yield a contradiction to the definition~\\eqref{eq:def-T-case(i)} of $T$!\n\\smallskip\n\nCombining now the first and the second step, we finally obtain the desired invariance~\\eqref{eq:M_u-invariant}, which clearly implies~\\eqref{eq:lISS-estimate} in the case $x_0 \\in M_u$.\n\\smallskip\n\n(ii) Suppose we are in the case $x_0 \\notin M_u$. In order to establish~\\eqref{eq:lISS-estimate} in that case,\nwe will show -- in three steps -- that for some $t_0 \\in (0,\\infty]$ one has\n\\begin{align} \n\\norm{S_u(t,0,x_0)}_{\\Theta} &\\le \\beta(\\norm{x_0}_{\\Theta},t) \\qquad (t\\in [0,t_0)) \\label{eq:part-(ii),1} \\\\\n\\norm{S_u(t,0,x_0)}_{\\Theta} &\\le \\gamma(\\norm{u}_{\\infty}) \\qquad (t \\in (t_0,\\infty)). \\label{eq:part-(ii),2}\n\\end{align}\nIndeed, let $t_0 := \\inf \\{t \\in \\mathbb{R}^+_0: x(t) \\in M_u\\}$\nand \n\\begin{align} \\label{eq:def-T-case(ii)}\nT := \\sup \\big\\{ T' \\in (0,t_0): \\norm{x(t)}_{\\Theta} < r_0 \\text{ for all } t \\in [0,T') \\big\\},\n\\end{align}\nwhere we use the abbreviation $x(t) := S_u(t,0,x_0)$. (In view of the standard convention $\\inf \\emptyset := \\infty$, we have $t_0 = \\infty$ in case $x(t) \\notin M_u$ for all $t \\in \\mathbb{R}^+_0$.) Since $x(0) = x_0 \\in (X\\setminus M_u) \\cap \\overline{B}_{r_{0x}}(\\Theta)$ and thus $\\norm{x(0)}_{\\Theta} < r_0$ by~\\eqref{eq:def-r0x-and-r0u}, we observe that $t_0 \\in (0,\\infty]$ and $T \\in (0,t_0]$ and that\n\\begin{align} \\label{eq:norm-initially-less-than-r0,case(ii)}\nx(t) \\notin M_u \\qquad (t \\in [0,t_0)) \\qquad \\text{and} \\qquad \\norm{x(t)}_{\\Theta} < r_0 \\qquad (t \\in [0,T)). \n\\end{align} \n\nAs a first step, we show that $\\norm{x(t)}_{\\Theta} \\le \\beta(\\norm{x_0}_{\\Theta},t)$ at least for all $t \\in [0,T)$. \nIndeed, in view of~(\\ref{eq:norm-initially-less-than-r0,case(ii)}.a) and (\\ref{eq:norm-initially-less-than-r0,case(ii)}.b) we have\n\\begin{align*}\n\\overline{\\psi}(\\norm{x(t)}_{\\Theta}) \\ge V(x(t)) > \\overline{\\psi}(\\chi(\\norm{u}_{\\infty})) \\ge \\overline{\\psi}\\big(\\chi(\\norm{u(\\cdot+t)}_{\\infty})\\big)\n\\qquad (t \\in [0,T))\n\\end{align*}\nand therefore we get by virtue of~\\eqref{eq:lISS-L-fct,implicative} that\n\\begin{align}\n\\overline{\\partial}_t^+ V(x(t)) &= \\varlimsup_{\\tau\\to 0+} \\frac{1}{\\tau}\\Big( V(x(t+\\tau))-V(x(t)) \\Big) \\notag \\\\\n&= \\varlimsup_{\\tau\\to 0+} \\frac{1}{\\tau}\\Big( V\\big(S_{u(\\cdot + t)}(\\tau,0,x(t))\\big)-V(x(t)) \\Big) = \\dot{V}_{u(\\cdot + t)}(x(t)) \\notag \\\\\n&\\le -\\alpha(\\norm{x(t)}_{\\Theta}) \\le -\\big(\\alpha \\circ \\overline{\\psi}^{-1}\\big)\\big(V(x(t))\\big)\n\\qquad (t \\in [0,T)).\n\\end{align}\nConsequently, by our choice of $\\overline{\\beta}$ we see that\n\\begin{align*}\nV(x(t)) \\le \\overline{\\beta}(V(x(0)),t) \\qquad (t \\in [0,T)).\n\\end{align*}\nIn view of~(\\ref{eq:norm-initially-less-than-r0,case(ii)}.b) and our definition~\\eqref{eq:def-beta-gamma} of $\\beta$, the assertion of the first step is then clear. \n\\smallskip\n\nAs a second step, we show that $T = t_0$. \nIndeed, assuming $T < t_0$, we would get by the first step and continuity that even $\\norm{x(T)}_{\\Theta} \\le \\beta(\\norm{x_0}_{\\Theta},T) \\le \\beta(r_{0x},0)$ and thus $\\norm{x(T)}_{\\Theta} < r_0$ by~\\eqref{eq:def-r0x-and-r0u}. And from this, in turn, it would follow that $\\norm{x(t)}_{\\Theta} < r_0$ for all $t \\in [T,T+\\delta)$ with some $\\delta > 0$. In conjunction with~(\\ref{eq:norm-initially-less-than-r0,case(ii)}.b), this would yield a contradiction to the definition~\\eqref{eq:def-T-case(ii)} of $T$!\n\\smallskip\n\nAs a third step, we show that $\\norm{x(t)}_{\\Theta} \\le \\gamma(\\norm{u}_{\\infty})$ for all $t \\in [t_0,\\infty)$. \nWe can assume $t_0 < \\infty$\nbecause in the case $t_0 = \\infty$ the assertion is empty. So, by\nthe definition of $t_0$ it then follows that $x(t_0) \\in M_u$ and therefore by virtue of~\\eqref{eq:M_u-invariant} \n\\begin{align*}\nx(t) = S_u(t,0,x_0) = S_u(t,t_0,x(t_0)) \\in M_u \\qquad (t\\in[t_0,\\infty)).\n\\end{align*} \nIn view of~\\eqref{eq:M_u-contained-in-better-sets}, the assertion of the third step is then clear.\n\\smallskip\n\nCombining now the first, second and third step, we finally obtain the desired estimates~\\eqref{eq:part-(ii),1} and~\\eqref{eq:part-(ii),2}, which clearly imply~\\eqref{eq:lISS-estimate} in the case $x_0 \\notin M_u$.\n\\end{proof}\n\n\n\n\n\n\n\nAn inspection of the above proof shows that we actually proved a bit more than local input-to-state stability, namely we have: for every $r_0 > 0$ there exist $\\beta \\in \\mathcal{KL}$ and $\\gamma \\in \\mathcal{K}$ and $r_{0x}, r_{0u} > 0$ such that the estimate~\\eqref{eq:lISS-estimate} holds true for all $\\norm{x}_{\\Theta} \\le r_{0x}$ and $\\norm{u}_{\\infty} \\le r_{0u}$. So, if by choosing $r_0$ large enough, we could also ensure that $r_{0x}$ and $r_{0u}$ with~\\eqref{eq:def-r0x-and-r0u} can be chosen arbitrarily large, we would even have semi-global input-to-state stability.\nYet, this is not so clear because the functions $\\beta = \\beta_{r_0}$ and $\\gamma = \\gamma_{r_0}$ from~\\eqref{eq:def-r0x-and-r0u} which determine our choice of $r_{0x}$ and $r_{0u}$ depend on $r_0$ themselves (basically because $V = V_{r_0}$ and $\\underline{\\psi} = \\underline{\\psi}_{r_0}$ depend on $r_0$ as was pointed out after Lemma~\\ref{lm:L-fct}).\nWe therefore leave the question of semi-global input-to-state stability to future research. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\n\nS. Dashkovskiy and O. Kapustyan are partially supported by the German Research Foundation (DFG) and the State Fund for Fundamental Research of Ukraine (SFFRU) through the joint German-Ukrainian grant ``Stability and robustness of attractors of nonlinear infinite-dimensional systems with respect to disturbances'' (DA 767\/12-1).\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{small}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\n\nWe consider the existence and properties of radially symmetric weak solutions to the following system of differential equations:\n\n\\begin{empheq}[left=\\empheqlbrace]{align}\n&\\frac {\\partial F} {\\partial t}(t, p)=n(t) I_3(F(t))(p)\\qquad t>0,\\; p\\in \\mathbb{R}^3, \\label{PA}\\\\\n&n'(t)=-n(t)\\int _{ \\mathbb{R}^3 } I_3(F(t))(p)dp \\qquad t>0,\\label{PB}\n\\end{empheq}\nwhere\n\\begin{align}\n&I_3(F(t))(p)=\\!\\!\\iint _{(\\mathbb{R}^3)^2}\\!\\!\\big[R(p, p_1, p_2)\\!-\\!R(p_1, p, p_2)\\!-\\!R(p_2, p_1, p) \\big]dp_1dp_2, \\label{E1BCD} \\\\\n&R(p, p_1, p_2)\\,=\\left[\\delta (|p|^2-|p_1|^2-|p_2|^2)  \\delta (p-p_1-p_2)\\right]\\,\\times \\nonumber \\\\\n&\\hskip 2.5cm \\times\\left[ F_1F_2(1+F)-(1+F_1)(1+F_2)F \\right],  \\label{S1EA4BC}\n\\end{align}\nand we denote $F=F(t,p)$ and $F_{\\ell}=F(t,p_{\\ell})$ for $\\ell=1, 2$.\n\nThe system (\\ref{PA}), (\\ref{PB}) is motivated by the mathematical description of a weakly interacting dilute gas of bosons. Given such a gas at equilibrium, if its temperature is below the so-called critical temperature $T_c$, a macroscopic density of bosons, called a condensate, appears at the lowest quantum state (cf.\\cite{lieb}). A description of the system of particles out of equilibrium at zero temperature has also been rigorously obtained  (\\cite{ESY}). The system (\\ref{PA}), (\\ref{PB})  is more directly related to a gas out of equilibrium and at non zero temperature. The equations that, in the physic's literature, describe a gas in such a situation have not been the object of a mathematical proof; they have rather been deduced on the basis of physical arguments  (cf. \\cite{Zoller}, \\cite{GNZB, ZNG}, \\cite{Stoof2} for example). \nWe are particularly interested in the kinetic description of the interaction between the condensate and the particles in the dilute gas, when most of the particles are still in the gas, and so when the system is at a temperature close to  $T_c$.  \n\n\\subsection{The Nordheim equation}\n\nThe kinetic equation consistently used to describe the evolution of the  distribution function for a spatially homogeneous, weakly interacting dilute gas of bosons of momentum $p_1$ is\n\\begin{align}\n&\\frac{\\partial F}{\\partial t}(t, p_1)=I_4(F(t))(p_1)\\qquad  t>0,\\; p_1\\in \\mathbb{R}^3, \\label{S1E0a}\n\\end{align}\nwhere\n\\begin{align}\n&I_4(F(t))(p_1)=\\iiint_{\\left(\\mathbb{R}^{3}\\right)  ^{3}}q(F)  d\\nu (p_2, p_3, p_4),  \\label{S1Eb}\\\\\n&q(F)= F_{3}F_{4}( 1+ F_{1})(1+F_{2}) -F_{1}F_{2}(1+F_{3})(1+F_{4}), \\label{S1Eq}\\\\\n&d\\nu (p_1, p_2, p_3)=2a^2 \\pi ^{-3} \\delta\\left(  p_{1}+p_{2}-p_{3}-p_{4}\\right)\\times \\nonumber \\\\\n&\\hskip 2.5cm \\delta \\left(  E(p_1)+ E(p_2)- E(p_3)- E(p_4)\\right) dp_{2}dp_{3}dp_{4}.\n\\end{align}\nsometimes called Nordheim equation (\\cite{Nordheim1}), (cf. for example  \\cite{Zoller}, \\cite{GNZB}, \\cite{Stoof2}). We are assuming that the particles have mass $m=1\/2$ and $E(p)$ denotes  the energy of a particle of momentum $p$.  The constant $a$ is the scattering length that parametrizes the Fermi pseudopotential  of scattering. In the absence of condensate, the energy of the particles is taken to be $E(p)=|p|^2$.  \n\nFor a condensed Bose gas, it is necessary to include  the collisions involving the condensate. A kinetic equation is derived in \\cite{Eckern} and \\cite{Kirkpatrick} describing such processes.  More recently, \\cite{ZNG}  extended the treatment to a trapped Bose gas by including Hartree-Fock\ncorrections to the energy of the excitations, and  have derived coupled kinetic equations for the\ndistribution functions of the normal and superfluid components. Later on the results where generalized  to low temperatures in \\cite{IG}  using  the Bogoliubov-Popov approximation to describe the energy particle. The system is as follows\n\\begin{empheq}[left=\\empheqlbrace]{align}\n&\\frac {\\partial F} {\\partial t}(t, p)=I_4(F(t))(p)+ 32 a^2 n(t) \\widetilde I_3(F(t))(p)\\quad t>0,\\; p\\in \\mathbb{R}^3,  \\label{S1EA1}\\\\\n& n'(t)=-n(t)\\int  _{ \\mathbb{R}^3 } \\widetilde I_3(F(t))(p)dp\\qquad t>0. \\label{S1EA1B}\n\\end{empheq}\n(cf. \\cite{Eckern}, \\cite{GNZB}, \\cite{Kirkpatrick} for a deduction based on physic's arguments). The term $I_4(F)$ is exactly as in (\\ref{S1Eb}) and   the constant $32 a^2$ comes from the approximation of  the transition probability:\n$|\\mathcal M(p, p_1, p_2)|^2\\approx 32 a^2 n(t)$. The integral collision $\\widetilde I_3$ is given by an expression similar to (\\ref{E1BCD}), (\\ref{S1EA4BC})\n but where the corresponding terms $\\widetilde R(p, p_1, p_2)$ are as follows,\n \\begin{align}\n&\\widetilde R(p, p_1, p_2)\\,=\\left[\\delta (E(p)-E(p_1)-E(p_2))  \\delta (p-p_1-p_2)\\right]\\,\\times \\nonumber \\\\\n&\\hskip 1cm \\times\\left[ F(p_1)F(p_2)(1+F(p))-(1+F(p_1))(1+F(p_2))F(p) \\right].   \\label{S1EBog}\n\\end{align}\nIn presence of a condensate, the energy $E(t, p)$ of the particles at time $t$ is now taken as $E(t, p)=\\sqrt{|p|^4+16a\\,n(t)|p|^2}$,  where $n(t)$ is the condensate density (\\cite{Chiara}, \\cite{GNZB}). \nOnce equation (\\ref{S1EA1}) has been obtained, the equation (\\ref{S1EA1B}) is just what is needed in order to ensure that the total number of particles $n(t)+\\int  _{ \\mathbb{R}^3 }F(t, p)dp$ in the system is constant in time.  \n\nWe are particularly interested in a situation where most of the particles are in the gas, and the condensate density $n$ is very small. The energy of the particles is then usually approximated as $E(t, p)\\approx |p|^2+4 a\\pi n(t)$ (cf.\\cite{GNZB}).\nIn all what follows we need  the strongest simplification  $E(t, p)\\approx |p|^2$ to have the collision integral $I_3$ in (\\ref{E1BCD}).\n \nMoreover, in the problem (\\ref{PA}), (\\ref{PB})  only the term that in the equation  (\\ref{S1EA1})  describes the interactions involving one particle of the condensate has been kept. The term $I_4$, the same as in equation (\\ref{S1E0a}), that only considers interactions between particles in the gas, has been dropped. The term $I_4$ has been studied with detail to prove the existence of solutions to the Nordheim equation (\\ref{S1E0a}) and describe some of their properties. The problem (\\ref{PA}), (\\ref{PB}) only takes into account the collision processes involving a particle of the condensate. \n\nSince we are only concerned with radial solutions $(F,n)$ of (\\ref{PA}), (\\ref{PB}), a very natural independent variable is  $x=|p|^2$. But  this introduces a jacobian and then, the most suitable quantity is not always $f(x)=F(p)$ but may be sometimes  $\\sqrt x f(x)$.\n\n\n\\subsection{The term $I_4$ and the Nordheim equation}\n\nThe local existence of bounded solutions for Nordheim equation (\\ref{S1E0a}) was proved in \\cite{BE}. Global existence of bounded solutions has been proved in \\cite{Lu5} for bounded and suitably small initial data. The existence of radially symmetric weak solutions was first proved in \\cite{Lu1} for all initial data $f_0$ in the space of nonnegative radially symmetric measures on $[0, \\infty)$. \n\nFor radially symmetric solutions $F(p)=f(x)$, $x=|p|^2$, the expression of the Nordheim equation simplifies because it is possible to perform the angular variables in the collision integral. After rescaling the time variable $t$ (in order to absorb some constants), the Nordheim equation reads:\n\n\\begin{align}\n&\\frac{\\partial f}{\\partial t}(t, x_1)= J_4(f(t))(x_1),\\qquad t>0,\\;x_1\\geq0, \\label{S1E1}\n\\end{align}\nwhere\n\\begin{align}\n&J_4(f)(x_1)=\\iint _{[0,\\infty)^2}\\hskip -0.4cm  \\frac{w(x_1, x_2, x_3)}{\\sqrt{x_1}} q(f)(x_1,x_2,x_3)dx_{2}dx_{3}, \\label{S1E2}\\\\ \n&q(f)=(1+  f_{1})(1+f_{2})  f_{3}f_{4}-( 1+ f_{3})(1+f_{4})  f_{1}f_{2},  \\label{S1E3}\\\\\n&w(x_1,x_2,x_3)=\\min\\left\\{\\sqrt{x_{1}},\\sqrt{x_{2}},\\sqrt{x_{3}},\\sqrt{x_{4}}\\right\\},\\,\\,x_{4}=(x_{1}+x_{2}-x_{3})_+. \\label{E2'}\n\\end{align}\n\nThe factor $\\frac{w}{\\sqrt{x_1}}$ in the collision integral comes from the angular integration of the Dirac's delta of the energies $|p _{ \\ell }|^2$.\n\nIf we denote $\\mathscr M _+([0, \\infty))$ the space of positive and finite Radon measures on $[0, \\infty)$, and define for all $\\alpha \\in \\mathbb{R}$\n\\begin{align}\n\\label{S1E17}\n&\\mathscr{M}^{\\alpha} _+([0, \\infty))=\\left\\{G\\in  \\mathscr{M}_+([0, \\infty)):\\; M_{\\alpha}(G)<\\infty\\right\\},\\\\\n&M_{\\alpha}(G)=\\int_{[0,\\infty)}x^{\\alpha}G(x)dx\\qquad\\text{(moment of order $\\alpha$)},\\label{S1E17'}\n\\end{align}\nthe definition of weak solution introduced in \\cite{Lu1} is the following.\n\n\\begin{definition}[Weak radial solutions of (\\ref{S1E0a})]\n\\label{S1D0}\nLet $G$ be a  map from $[0, \\infty)$ into $\\mathscr M_+^1([0, \\infty))$ and consider $f$ defined as $\\sqrt xf(t)=G(t)$. We say that $f$ is a weak radial solution of (\\ref{S1E0a}) if $G$ satisfies:\n\\begin{align}\n&\\forall t>0:\\,\\,\\,G(t)\\in \\mathscr M_+^1([0, \\infty)), \\label{S1ED1}\\\\\n&\\forall  T>0:\\,\\,\\,\\sup _{ 0\\le t<T } \\int  _{ [0, \\infty) }(1+x)G(t, x)dx<\\infty, \\label{S1ED2} \\\\\n& \\forall \\,\\varphi \\in C^{1, 1}_b([0, \\infty)): \\,\\, \\int  _{ [0, \\infty) } \\varphi (x)G(t , x) dx\\in C^1([0, \\infty)),  \\label{S1ED3} \n\\\\\n&\\frac {d} {dt} \\int_{\\left[  0,\\infty\\right)}\\varphi(t,x)G(t,x)dx =\\mathcal{Q}_4(\\varphi,G(t)), \\label{S1E4}\\\\\n&\\mathcal Q_4(\\varphi,G)=\\iiint_{\\left[  0,\\infty\\right)^3} \\frac{G_{1}G_{2}G_{3}\n}{\\sqrt{x_{1}x_{2}x_{3}}}w\\Delta\\varphi \\;dx_1dx_2dx_3+ \\nonumber \\\\\n&\\hskip 2cm +\\frac{1}{2}\\iiint_{\\left[  0,\\infty\\right)^3} \\frac{G_{1}G_{2}}{\\sqrt{x_{1}x_{2}}}w \\Delta\\varphi \\;dx_1dx_2dx_3  \\label{S1E5}\\\\\n&\\Delta\\varphi (x_1, x_2, x_3)=\\varphi (x_4)+\\varphi (x_3)-\\varphi (x_2)-\\varphi (x_1),\\label{S2E1}\\\\\n&w(x_1,x_2,x_3)=\\min\\{\\sqrt{x_1}, \\sqrt{x_2}, \\sqrt{x_3}, \\sqrt{x_4}\\},\\,\\,\\,x_4=(x_1+x_2-x_3)_+. \\label{S1E6'}\n\\end{align}\n\\end{definition}\n\nFor all initial data $f_0$ such that $G_0=\\sqrt x f_0\\in \\mathscr M_+^1([0, \\infty))$, the existence of a weak solution was proved in \\cite{Lu1}.\nThe moments of order zero and one of $G$ where shown to be constant in time. \nIt was shown in \\cite{Lu3} that  a definition equivalent to Definition \\ref{S1D0} would be to impose $\\varphi (0)=0$ to the test functions in Definition \\ref{S1D0} and impose the conservation of mass on $G(t)$ for all $t>0$. Further properties of the solutions, such as the gain of moments, asymptotic behavior,   where obtained in a series of articles  \\cite{Lu1,Lu2, Lu3, Lu4}\n\nIt is proved in Proposition \\ref{S1P0} below that if the measure $G$ is written as $G(t)=n(t)\\delta _0+g(t)$, where\n$n(t)=G(t,\\{0\\})$, then for all $\\varphi \\in C^{1,1}_b([0, \\infty))$ the term $\\mathcal Q_4(\\varphi,G)$ may be decomposed as follows:\n\\begin{align}\n\\mathcal Q_4(\\varphi,G(t))=\\mathscr Q _4(\\varphi,g(t))+n(t)\\mathscr{Q}_3(\\varphi,g(t)),\\label{S1E13B}\n\\end{align}\nwhere\n\\begin{align}\n&\\mathscr Q_4(\\varphi,g)=\\iiint_{(0,\\infty)^3}\\frac{g_{1}g_{2}g_{3}\n}{\\sqrt{x_{1}x_{2}x_{3}}}w \\Delta\\varphi \\;dx_1dx_2dx_3 \\nonumber \\\\\n&\\hskip 2cm +\\frac{1}{2}\\iiint_{(0,\\infty)^3}\\frac{g_{1}g_{2}}{\\sqrt{x_{1}x_{2}}}w \\Delta\\varphi \\;dx_1dx_2dx_3, \\label{S1E15}\\\\\n&\\mathscr Q _3(\\varphi,g)=\\mathscr{Q}_3^{(2)}(\\varphi,g)-\\mathscr{Q}_3^{(1)}(\\varphi,g),\\label{S1E1Q3}\\\\\n&\\mathscr{Q}_3^{(2)}(\\varphi,g)=\\iint_{(0, \\infty)^2} \\frac {\\Lambda(\\varphi)(x, y)} {\\sqrt{x y}}g(x)g(y)dxdy,\\label{S1E1Q32}\\\\\n&\\mathscr{Q}_3^{(1)}(\\varphi,g)=\\int_{ (0, \\infty)}\\frac {\\mathcal{L}_0(\\varphi)(x)} {\\sqrt x}g(x)dx, \\label{S1E1Q31}\\\\\n&\\Lambda(\\varphi)(x, y)=\\varphi (x+y)+\\varphi (|x-y|)-2\\varphi (\\max\\{x, y\\}), \\label{S1E154}\\\\\n&\\mathcal {L}_0(\\varphi )(x)=x\\big(\\varphi (0)+\\varphi (x)\\big)-2\\int _0^x \\varphi (y)dy. \\label{S1E155}\n\\end{align}\n\n\nIt was also proved in \\cite{Lu1} that as $t\\to \\infty$, the measure $G$ converges in the weak sense of measures  to one of the measures:\n\\begin{eqnarray}\n\\label{BED}\nG _{ \\beta, \\mu, C} =\\frac {\\sqrt x} {e^{\\beta x-\\mu  }-1}+C\\delta _0,\\,\\,\\,\\beta >0,\\,\\,\\mu  \\le 0, \\,\\,\\,C\\ge 0\n\\end{eqnarray}\nwhere the constants $C$ and $\\mu  $ are such that $C\\mu  =0$.  \n\nWhen $C=0$ and $\\mu \\le 0$, the function  $F_{ \\beta , \\mu , 0  }(p)=|p|^{-1} G _{ \\beta , \\mu , 0  }(|p|^2)$ is an equilibrium of the Nordheim equation (\\ref{S1E0a}) because\n$q(F _{ \\beta , \\mu , 0 })d\\nu \\equiv 0$. When $C>0$ and $\\mu =0$, then $F _{ \\beta , 0, C  }(p)=|p|^{-1} G _{ \\beta , 0, C  }(|p|^2)$ is an equilibria of (\\ref{S1EA1}) because\n$q(f _{ \\beta , 0 , 0 }) \\equiv 0$ and  $R(p, p', p'')\\equiv 0$ for all $(p, p', p'')\\in (\\mathbb{R}^3)^3$ for  $f _{ \\beta , 0 , 0} $, where  $R(p, p', p'')$ is defined in  (\\ref{S1EA4BC}).  It was proved in \\cite{Lu1} that $F_{ \\beta , \\mu, C  }$ is a weak solution of the Nordheim equation  (\\ref{S1E1}) if and only if $\\mu C=0$. \n\nOn the other hand, it was proved in \\cite{EV1} that, given any  $N>0$, $E>0$ there exists initial data $f_{0}\\in L^{\\infty}\\left( \n\\mathbb{R}_+;\\left( 1+x\\right) ^{\\gamma}\\right) $ with $\\gamma >3$, satisfying\n$$\n\\int_{\\mathbb{R}^{+}}f_{0}(x)\\sqrt{x }dx=N,\\qquad\\int_{\\mathbb{R}^{+}}f_{0}(x)\\sqrt{x^{3}}dx=E,\n$$\nand such that there exists a global weak solution $f$  and  positive times  $0<T_{\\ast}<T^*$ such that:\n\\begin{eqnarray}\n\\label{S1EC1}\n\\sup_{0< t\\leq T_{\\ast}}\\left\\Vert f\\left( t,\\cdot\\right) \\right\\Vert\n_{L^{\\infty}\\left( \\mathbb{R}^{+}\\right) }<\\infty,\\,\\,\\,\\,\\,\\sup _{T_*<t\\le T^* }\\int_{\\left\\{ 0\\right\\} }\\sqrt{x}f\\left( t,x\\right)\ndx>0.\n\\end{eqnarray}\nProperty (\\ref{S1EC1}) shows that the  solution $G=\\sqrt x f$ of  (\\ref{S1ED1})--(\\ref{S1E6'}) is a bounded function on the time interval $[0, T^*)$ and a Dirac mass is formed at the origin at some time $T_0$ between $T_*$ and $T^*$. After that time $T_0$, the solution $G$ is such that $G(t, \\{0\\})>0$. \n\nIn the simplified description of the physical system of particles that we are using, where only the radial density  $G$ of particles of momentum $p$  is considered, the description of the physical Bose-Einstein condensate can just be given by a  Dirac measure at the origin. \n\nNotwithstanding the similarity of these two phenomena, the extent to which the first one is a truthful mathematical description of the second is not clear.  Nevertheless, we refer to the term $n(t)\\delta _0$  that appears in finite time in some of the weak solutions of the Nordheim equation as ``condensate'', with some abuse of language.\n\n\\subsection{The term $I_3$ in radial variables.}\n\nThe results briefly presented in the previous sub Section describe some of the properties of the weak solutions to the Nordheim equation in terms of the measure $G$.  In particular, the weak convergence of $G$ to the measures defined in (\\ref{BED}) shows what is the limit of  $G(t, \\{0\\})$ as $t\\to \\infty$.   \nTo understand better  the dynamics of $G(t, \\{0\\})\\delta _0$ and its interaction with $G(t)-G(t, \\{0\\})\\delta _0$ it seems suitable to  write  $G(t)=G(t, \\{0\\})\\delta _0+g(t)$ and consider the system (\\ref{S1EA1}), (\\ref{S1EA1B}).  \n\nFor radially symmetric functions $F(p)=f(x)$, $x=|p^2|$, the  system (\\ref{PA}), (\\ref{PB}) reads, after a suitable time rescaling to absorb some constants:\n\\begin{empheq}[left=\\empheqlbrace]{align}\n&\\frac {\\partial f} {\\partial t}(t, x)=\\frac {n(t)} {\\sqrt x} J_3(f(t))(x)\\qquad t>0,\\;x>0, \\label{PR}\\\\\n&n'(t)=-n(t)\\int _0^\\infty J_3(f(t))(x)dx\\qquad t>0, \\label{PR19}\n\\end{empheq}  \nwhere\n\\begin{align}\n&J_3(f)(x)=\\int_0^x \\Big(f(x-y)f(y)-f(x)\\big[1+f(x-y)+f(y)\\big]\\Big)dy +\\nonumber\\\\\n&\\hskip 1cm +2\\int_x^\\infty \\Big(f(y)\\big[1+f(y-x)+f(x)\\big]-f(y-x) f(x)\\Big) dy.\\label{PR2}\n\\end{align}\n(cf. \\cite{ST1} and \\cite{Svis} for the isotropic system that also contains the term $J_4(f)$, that comes from $I_4$ in  (\\ref{S1EA1})).\nNotice that\n\\begin{align}\n&\\int _0^\\infty J_3(f(t))(x)dx\\nonumber\\\\\n&=\\int _0^\\infty\\int _0^\\infty \\Big(f(t, x)f(t, y)- f(t, x+y)\\big[1+f(t, x)+f(t, y)\\big]\\Big)dxdy \\label{S1E9}\n\\end{align}\nwhenever the integral in the right hand side is finite, for example, if\n$f\\in L^1\\big(\\mathbb{R}_+, (1+x)dx\\big)$. In that case we also have,\n\\begin{equation}\n\\int _0^\\infty J_3(f(t))(x)dx=M _1(f(t)).\n\\end{equation}\n\nThe factor $x^{-1\/2}$ in the right hand side  of (\\ref{PR}) comes from the angular integration of the Dirac's measure of energies of $I_3$, just as the \n$\\frac{w}{\\sqrt{x_1}}$ term of (\\ref{S1E2}) in $I_4$.  But since $\\frac{w}{\\sqrt{x_1}}$ is a bounded function,\nit appears that the operator $I_3$ is more singular than $I_4$ for small values of $x$. \n\nIf we denote  $F(t, p)=f(t, |p|^2)=|p|^{-1}g(t, |p|^2)$ and $x=|p^2|$, from the original motivation of the Nordheim equation  it is very natural to expect\n$$\n\\int  _{ \\mathbb{R}^3 }F(t, p)dp= 2\\pi\\int _0^\\infty f(t, x)\\sqrt x dx= 2\\pi\\int_0^\\infty g(t, x)dx<\\infty, \n$$\n(that corresponds to the  number of particles in the normal fluid), and\n$$\n\\int  _{ \\mathbb{R}^3 }F(t, p)|p|^2dp= 2\\pi\\int _0^\\infty f(t, x)x^{3\/2}dx=2\\pi\\int_0^\\infty g(t, x)xdx<\\infty,\n$$\n(corresponding to the total energy in the system).  But there is no particular reason to expect \n$$\n\\int  _{ \\mathbb{R}^3 }F(t, p)\\frac {dp} {|p|}= 2\\pi\\int _0^\\infty f(t, x)dx=2\\pi\\int_0^\\infty g(t, x)\\frac {dx} {\\sqrt x}<\\infty.\n$$\nWithout that last condition, the convergence of the integrals in the term $I_3(F(t))$ (cf. (\\ref{E1BCD}), (\\ref{S1EA4BC})), or in (\\ref{PR}), (\\ref{PR2}), is delicate.\nThat difficulty is usually avoided using a suitable weak formulation. \n\nIf we suppose that  $f=x^{-1\/2}g\\in L^1\\big(\\mathbb{R}_+, (1+x)dx\\big)$, and multiply the equation (\\ref{PR}) by $\\sqrt x\\, \\varphi $, we obtain by Fubini's Theorem, \n\\begin{equation}\n\\label{g11}\n\\frac {d} {dt}\\int_{ [0, \\infty) } \\varphi (x) g(t, x)dx=n(t)\\widetilde{\\mathscr Q}_3(\\varphi,g(t))\\quad\\forall  \\varphi \\in C^1_b([0, \\infty)),\n\\end{equation}\nwhere\n\\begin{align}\n&\\widetilde{\\mathscr Q}_3(\\varphi,g)=\\mathscr{Q}_3^{(2)}(\\varphi,g)-\\widetilde{\\mathscr Q}_3^{(1)}(\\varphi,g),\\label{S1EB2}\\\\\n&\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi,g)=\\int_{ (0, \\infty)} \\frac {\\mathcal{L}(\\varphi )(x)} {\\sqrt x}g(x)dx, \\label{S1E20R}\\\\\n&\\mathcal{L}(\\varphi)(x)=x\\varphi (x)-2\\int _0^x \\varphi (y)dy. \\label{S1E21R}\n\\end{align}\nNotice that, by (\\ref{S1E1Q3}),\n\\begin{align}\n\\mathscr{Q}_3(\\varphi,g)=\\widetilde{\\mathscr{Q}}_3(\\varphi,g)-\\varphi(0)M_{1\/2}(g).\\label{S1EB1}\n\\end{align}\n\nA natural weak formulation for $G=n(t)\\delta _0+g$ is then obtained by adding (\\ref{PR19}) to (\\ref{g11}). \nWe  then define a weak radially symmetric solution of the Problem (\\ref{PA}), (\\ref{PB}) as follows.\n\\begin{definition}[Weak radial solution of (\\ref{PA}), (\\ref{PB})]\n\\label{S1D1}\nConsider a map $G:[0, T) \\to \\mathscr M_+^1([0, \\infty))$  for some $T\\in (0, \\infty]$, that we decompose as follows:\n$$\n\\forall t\\in [0, T):\\,\\,\\,\\,\\,G(t)=n(t)\\delta _0+g(t), \\,\\,\\,\\, \\hbox{where}\\,\\,\\,\\, n(t)=G(t, \\{0\\});\n$$\nand  define  $ F(t, p)=|p|^{-1}g(t, |p|^2)$ for all $t>0$ and $p\\in \\mathbb{R}^3$. We say that $(F, n)$ is a weak radial solution of (\\ref{PA}), (\\ref{PB}) on $(0, T)$ if:\n\n\\begin{align}\n&\\forall T'\\in (0, T]:\\qquad\\sup _{ 0\\le t<T' } \\int  _{ [0, \\infty) }(1+x)G(t, x)dx<\\infty, \\label{S1ED2S} \\\\\n& \\forall \\varphi \\in C^{1}_b([0, \\infty)): \\quad t\\mapsto \\int  _{ [0, \\infty) } \\varphi (x)G(t, x) dx \\in W^{1,\\infty } _{ loc }([0, T)),  \\label{S1ED3S}\n\\end{align}\nand for a.e. $t\\in (0, T)$\n\\begin{equation}\n\\frac {d} {dt}\\int_{ [0, \\infty) } \\varphi (x) G(t, x)dx=n(t) \\mathscr{Q}_3(\\varphi,g(t))\\quad\\forall  \\varphi \\in C^1_b([0, \\infty)), \\label{S1E16}\\\\\n\\end{equation}\nwhere $\\mathscr Q _3(\\varphi,g)$ is defined by (\\ref{S1E1Q3})-(\\ref{S1E1Q31}).\n\\end{definition}\n\nWe  show in Proposition \\ref{S1P0} that the Definition \\ref{S1D1} substantially  coincides with the Definition \\ref{S1D0} of radial weak solution of (\\ref{S1E0a}) when the term $\\mathscr Q_4(\\varphi,g)$ in (\\ref{S1E13}) is dropped (cf. Remark \\ref{SXR1}). As a  consequence, the measures $f _{ \\beta , 0, C  }(p)$\ndefined above are weak radial solutions of (\\ref{PA}), (\\ref{PB}) (cf. Proposition \\ref{equilibria}).\n\n\n\n\\subsection{Main results}\nThe existence of weak radial solutions for the Cauchy problem associated with the system (\\ref{PA}), (\\ref{PB}) is given in the following Theorem.\n\n\\begin{theorem}[Existence result]\n\\label{S1T1}\nSuppose that $G_0\\in \\mathscr{M}^1_+([0,\\infty))$ satisfies $G_0(\\{0\\})>0$, and define $F_0(p)=|p|^{-1}g_0(|p|^2)$, where $g_0=G_0-G_0(\\{0\\})\\delta _0$. Then, there exists  a weak radial solution $(F, n)$ of  (\\ref{PA}), (\\ref{PB}) on $(0, \\infty)$ such that $F(t, p)=|p|^{-1}g(t, |p|^2)$, where $G=n\\delta _0+g$  satisfies:\n\\begin{eqnarray}\n\\label{S1T1E0}\nG\\in C\\big([0,\\infty),\\mathscr{M}_+^1([0,\\infty))\\big),\\quad G(0)=G_0\n\\end{eqnarray} \nand:\n\\begin{enumerate}[(i)]\n\\item$G$ conserves the total number of particles $N$ and energy $E$:\n\\begin{align}\n&M_0(G(t))=M_0(G_0)=N\\qquad\\forall t\\geq 0,\\label{S1E210}\\\\\n&M_1(G(t))=M_1(G_0)=E\\qquad\\forall t\\geq 0.\\label{S1E220}\n\\end{align}\n\n\\item For all $\\alpha \\geq 3$,  if $M_\\alpha (G_0)<\\infty$, then $G\\in C \\big((0, \\infty), \\mathscr{M}_+^{\\alpha}([0,\\infty))\\big)$ and \n\\begin{flalign}\n\\label{S1E23}\n&M_{\\alpha}(G(t))\\le \\left(M_{\\alpha}(G_0)^{\\frac{2}{\\alpha-1}}\n+\\alpha2^{\\alpha-1}E^{\\frac{\\alpha+1}{\\alpha-1}}\\tau(t)\\right)^{\\frac{\\alpha-1}{2}}\\quad\\forall t>0,&\\\\\n&\\text{where}\\quad\\tau (t)=\\int _0^t G(s, \\{0\\})ds.& \\label{E657tau}\n\\end{flalign}\n\n\\item For all $\\alpha \\geq3$,\n\\begin{equation}\n\\label{S5Ealphahh}\n\\quad M_{\\alpha}(G(t))\\leq C(\\alpha,E)\\left(\\frac{1}{1-e^{-\\gamma(\\alpha,E)\\tau(t)}}\\right)^{2(\\alpha-1)}\\quad\\forall t >0,\n\\end{equation}\nwhere $\\tau(t)$ is given by (\\ref{E657tau}), and the constants $C(\\alpha,E)$ and $\\gamma (\\alpha,E)$ are defined in Theorem \\ref{S5T5R}.\n\n\\item If $\\alpha \\in (1, 3]$ and\n\\begin{align}\nE> C(\\alpha)N^{5\/3},\\label{PRO112}\n\\end{align}\n\\begin{flalign}\n&\\text{where}\\qquad\n\\label{PRO115}\nC(\\alpha)=\n\\begin{cases}\n\\Big(\\frac{(2^{\\alpha}-2)(\\alpha+1)}{(\\alpha-1)}\\Big)^{\\frac{2}{3}}&\\text{if}\\quad\\alpha\\in(1,2],\\\\\n\\big(\\alpha(\\alpha+1)\\big)^{\\frac{2}{3}}&\\text{if}\\quad\\alpha\\in(2,3],\n\\end{cases}\n&\n\\end{flalign}\nthen $M_{\\alpha}(G(t))$ is a decreasing function on $(0,\\infty)$.\n\\end{enumerate}\n\\end{theorem}\n\nThe next result is a property satisfied by all  the weak radial solutions of  (\\ref{PA}), (\\ref{PB}).\n\\begin{theorem}\n\\label{S1Treg}\nLet $G_0$ be  as in Theorem \\ref{S1T1}, and $G$ a weak radial solution of  (\\ref{PA}), (\\ref{PB}).\nThen for all $T>0$, $R>0$ and $\\alpha\\in\\left(-\\frac{1}{2},\\infty\\right)$,\n\\begin{align}\n&\\int_0^TG(t,\\{0\\})\\int_{(0,R]}x^{\\alpha}G(t,x) dxdt\\leq \\nonumber \\\\\n&\\,\\,\\, \\le \\frac{2R^{\\frac{1}{2}+\\alpha}}{1-\\left(\\frac{2}{3}\\right)^{\\frac{1}{2}+\\alpha}}\\left(\\int_0^T\\!\\! G(t,\\{0\\})dt\\right)^{\\frac{1}{2}}\\left(\\frac{\\sqrt{E}}{2}\\int_0^T\\!\\!G(t,\\{0\\})dt+\\sqrt{N}\\right).\\label{MNEG2}\n\\end{align}\n\\end{theorem}\nThe only\npossible algebraic behavior for such a measure $G$ near the origin is then $x^{-1\/2}$.\n\\begin{remark}\n\\label{S1R1}\nThe functions $F _{ \\beta , 0, C }$ defined above are weak radial solutions of (\\ref{PA}), (\\ref{PB}) for all $\\beta >0$ and $C\\ge 0$ (cf. Proposition \\ref{equilibria}). Since\n\\begin{eqnarray*}\n\\int  _{ (0, \\infty) } x^{\\alpha}  G _{ \\beta , 0, C}\\;dx<\\infty\\quad\\Longleftrightarrow\\quad\\alpha >-1\/2,\n\\end{eqnarray*}\nthe estimate (\\ref{MNEG2}) can not hold for all radial weak solutions if $\\alpha\\leq -1\/2$.\n\\end{remark}\n\nIn the next two results we describe the evolution of the measure at the origin $n(t)=G(t,\\{0\\})$ by taking the limit $\\varepsilon\\to 0$ in the weak formulation \n(\\ref{S1E16}) for test functions $\\varphi_{\\varepsilon}$ as follows:\n\\begin{remark}\n\\label{TEST}\nGiven $\\varphi\\in C^1_b([0,\\infty))$ nonnegative, convex, with $\\varphi(0)=1$ and \n$\\lim_{x\\to\\infty}\\sqrt{x}\\varphi(x)=0$, denote $\\varphi_{\\varepsilon}(x)=\\varphi(x\/\\varepsilon)$ for $\\varepsilon>0$. Notice that for any \n$G\\in\\mathscr{M}_+([0,\\infty))$,\n\\begin{align}\n\\label{TEST2}\nG(\\{0\\})=\\lim_{\\varepsilon\\to 0}\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)dG(x).\n\\end{align}\nThe standard example is $\\varphi_{\\varepsilon}(x)=(1-x\/\\varepsilon)^2_+$.\n\\end{remark}\n\n\\begin{theorem}\n\\label{THn01}\nLet $G$ be the solution of (\\ref{S1E16}) obtained in Theorem \\ref{S1T1}, with initial data $G_0\\in \\mathscr{M}^1_+([0,\\infty))$ such that $N=M_0(G_0)>0$, $E=M_1(G_0)>0$ and $G_0(\\{0\\})>0$. Denote $G(t)=n(t)\\delta _0+g(t)$, with\n$n(t)=G(t, \\{0\\})$. Then $n$ is right continuous and a.e. differentiable on $[0,\\infty)$. Moreover, there exists a positive measure $\\mu$ on $[0,\\infty)$ whose cumulative distribution function is given by\n\\begin{align}\n\\label{ZE01}\n\\mu((0,t])=\\lim_{\\varepsilon\\to 0}\\int_{0}^{t}n(s)\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},g(s))ds\n\\end{align}\nfor any $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST},\nand  such that:\n\\begin{align}\n\\label{ZE02}\nn(t)-n(0)+\\int_{0}^{t}n(s)M_{1\/2}(g(s))ds=\\mu((0,t])\\qquad\\forall t>0.\n\\end{align}\n\\end{theorem}\n\n\\begin{theorem}\n\\label{MU1}\nLet $G$ and $\\mu$ be as in Theorem \\ref{THn01}. Then\n\\begin{align}\n\\label{ZE00}\n0<\\mu((0,t]))<\\infty\\qquad\\forall t>0.\n\\end{align}\n\\end{theorem}\nThe measure $\\mu$ in (\\ref{ZE01}) depends on the atomic part of  $g$, and on the behaviour of $g$ at the origin (it seems to be actually related with its moment of order $-1\/2$ c.f. Proposition \\ref{LM-1\/2} and Remark \\ref{HH}). This measure $\\mu $ appears as a source term in the equation (\\ref{ZE02}) for $n$.\nGiven  the function $n$, the equation (\\ref{PR}) satisfied by $g$ on $(0, \\infty)$ has also a natural weak formulation by itself. In terms of $g(t)$, where $g(t)=G(t)-G(t, \\{0\\})\\delta _0$ and $\\sqrt x f(t, x)=g(t, x)$ it reads \n\n\\begin{eqnarray}\n\\label{2E765}\n\\frac{d}{dt}\\int_{[0,\\infty)} \\!\\!\\!\\!\\!\\varphi(x)g(t,x)dx=n(t)\\mathscr{Q}_3(\\varphi,g(t)),\\;\\forall \\varphi\\in C_b^1([0, \\infty)),\\,\\varphi (0)=0.\n\\end{eqnarray}\n\nIn the next result we describe the relation between a weak solution $(F,n)$ of (\\ref{PA}), (\\ref{PB}), where $F(t, p)=|p|^{-1}g(t, |p|^2) $, $G(t)=n(t)\\delta _0+g(t)$, $n(t)=G(t, \\{0\\})$, and a pair $(g, n)$ where $g$ is a weak radial solution of the equation (\\ref{PA}) and $n$ satisfies (\\ref{PB}). \n\\begin{theorem}\n\\label{EQUIV}\nSuppose that $G\\in C\\big([0,\\infty),\\mathscr{M}_+([0,\\infty))\\big)$ is such that $G(0)=G_0\\in \\mathscr{M}^1_+([0,\\infty))$ with $G_0(\\{0\\})>0$, and denote  $G(t)=n(t)\\delta _0+g(t)$ with $n(t)=G(t, \\{0\\})$.\\\\\n(i) If $(F, n)$  is a weak radial solution of (\\ref{PA}), (\\ref{PB}) and $F(t, p)=|p|^{-1}g(t, |p|^2)$, then $n$ is given by (\\ref{ZE02}), (\\ref{ZE01}),\nand $g$ satisfies (\\ref{2E765}) for a.e. $t>0$.\\\\\n(ii) On the other hand, if  $g$ satisfies (\\ref{S1ED2S}), (\\ref{S1ED3S}) and (\\ref{2E765}) for some nonnegative bounded function $n$,  \nthen the limit in (\\ref{ZE01}) exists. If $n$ also satisfies\n\\begin{equation}\n\\label{ene1}\nn(t)=n(0)+\\lim_{\\varepsilon\\to 0}\\int_0^t n(s)\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},g(s))ds-\\int_0^t n(s)M_{1\/2}(g(s))ds\n\\end{equation}\nand $F(t, p)=|p|^{-1}g(t, |p|^2)$, then $(F, n)$ is a weak radial solution of (\\ref{PA}), (\\ref{PB}).\n\\end{theorem}\n\nIf in the Definition \\ref{S1D1} only test functions satisfying $\\varphi (0)=0$ are taken, it becomes necessary to introduce some other condition to the system. Otherwise the system would be reduced to  find $g$ satisfying (\\ref{S1ED3S})--(\\ref{S1E16})  for a given function $n(t)$ and  for test functions such that  $\\varphi (0)=0$. If we impose just the conservation of mass, we prove below (Corollary \\ref{S1C31}) that we recover a solution that satisfies the Definition \\ref{S1D1}.\n\n\\begin{corollary}\n\\label{S1C31}\nIf $g$ satisfies (\\ref{S1ED2S}), (\\ref{S1ED3S}) and (\\ref{2E765}) for some nonnegative bounded function $n=n(t)$ such that \n\\begin{equation}\n\\label{MBC}\nn(t)+\\int  _{ (0, \\infty) }g(t, x)dx= constant\n\\end{equation}\nand $F(t, p)=|p|^{-1}g(t, |p|^2)$, then $(F, n)$ is a weak radial solution of (\\ref{PA}), (\\ref{PB}).\n\\end{corollary}\n\nIn our last  result we show that, under some sufficient conditions, the condensate density $n(t)$ tends to zero as $t\\to\\infty$, fast enough to be  integrable.\n\\begin{theorem} \n\\label{S1T5}\nSuppose that  $G_0\\in \\mathscr M_+^1([0, \\infty))$  satisfies $G_0(\\{0\\})>0$ and let $(F,n)$ be the weak radial solution of (\\ref{PA}), (\\ref{PB}) obtained in Theorem \\ref{S1T1}. Let us call $N=M_0(G_0)$ and $E=M_1(G_0)$. If condition (\\ref{PRO112}), (\\ref{PRO115}) hold for some $\\alpha \\in (1, 3]$,\nthen, for all $t_0>0$,\n\\begin{equation}\n\\label{S1ET5B}\n\\int _{ t_0 }^{\\infty} n(t)dt\\leq M_{\\alpha}(G(t_0)) C(N,E,\\alpha)\n\\end{equation}\nfor some explicit constant $C(N,E,\\alpha)$ given in (\\ref{PRO116}), and\n\\begin{equation}\n\\label{S1ET5C}\n\\lim_{t\\to \\infty }n(t)=0.\n\\end{equation}\n\\end{theorem}\n\n\\begin{remark}\n\\label{S1R2}\nThe quantity $E\/N^{5\/3}$ has a very precise  interpretation in physical terms.  Suppose that $T$ is the temperature of  a system of  particles at equilibrium with total number of particles $N$ and total energy $E$. And denote $T_c$ the critical temperature, that is the temperature at which the ground state of the system becomes macroscopically occupied. Then:\n\\begin{eqnarray*}\n\\frac {E} {N^{5\/3}} = b\\, \\frac {T} {T_c},\\qquad\\hbox{where}\\qquad\nb=\\frac {3}{(2\\pi )^{\\frac {1} {3}}}\\frac  {\\zeta (5\/2)}{\\zeta (3\/2)^{5\/3}}.\n\\end{eqnarray*}\nand  condition  (\\ref{PRO112}) implies\n$$\n\\frac {T} {T_c}=\\frac {1} {b}\\,\\frac {E} {N^{5\/3}}> \\frac {C(\\alpha )} {b}.\n$$\nThe function $C(\\alpha )\/b$ is continuous and strictly increasing on $[1, 3]$ and its limit as $\\alpha \\to 1^+$ is $\\log(16)^{2\/3}\/b\\approx 4.48403$. Condition (\\ref{PRO112}) means that, when at equilibrium, the  system of particles would be at a temperature  clearly above the critical temperature. Anyway,  the solution $F$ of the problem (\\ref{PA}), (\\ref{PB}) may be far from any real distribution of particles of the original system of particles.\n\\end{remark}\n\n\\subsection{Some arguments of the proofs.}\nIt is very natural to make the following change of variables in  problem (\\ref{S1E16}). Given $G(t)=n(t)\\delta _0+g(t)$, where $n(t)=G(t, \\{0\\})$, we define\n\\begin{eqnarray}\n\\label{S1E45}\n&&H(\\tau)=G(t),\\qquad\\text{where}\\qquad\\tau =\\int _0^t n(s) ds. \\label{S1E45b}\n\\end{eqnarray}\nIn terms of $H$, (\\ref{S1E16}) reads\n\\begin{equation}\n\\frac {d} {d\\tau }\\int_{ [0, \\infty) } \\varphi (x)H(\\tau , x)dx= \\mathscr Q _3(\\varphi,H(\\tau))\\quad\\forall \\varphi \\in C^1_b([0, \\infty)). \\label{S1E16Ha}\n\\end{equation}\nTo obtain a measure $H$ that satisfies (\\ref{S1E16Ha}), we first find $h$ satisfying\n\\begin{align}\n\\frac {d} {d\\tau }\\int_{ [0, \\infty) } \\varphi (x)h(\\tau , x)dx= \\widetilde{\\mathscr Q}_3(\\varphi,h(\\tau))\n\\quad\\forall \\varphi \\in C^1_b([0, \\infty)),\\label{S1E16ha}\n\\end{align}\nwhere $\\widetilde{\\mathscr Q}_3$ is given in (\\ref{S1EB2})--(\\ref{S1E21R}).\nThen we define $H$ as\n\\begin{eqnarray}\n\\label{S1EdecompH}\nH(\\tau)=h(\\tau)-\\left(\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma\\right)\\delta_0.\n\\end{eqnarray}\nBy (\\ref{S1EB1}), the measure  $H$ will satisfy (\\ref{S1E16Ha}).\n\nAs it will be seen in Section \\ref{existenceH}, all the arguments are much simpler and clear in the equation for $H$ than in the equation for $G$. In particular, the measure $\\lambda$, that corresponds to the measure $\\mu$ of Theorem \\ref{THn01}, appears as the Lebesgue-Stieltjes measure associated to $m(\\tau )=h(\\tau , \\{0\\})$.\n\nThe proofs of Theorem \\ref{S1T1} and Theorem \\ref{S1Treg} make great use of the change of variables (\\ref{S1E45b}).\nSeveral of our arguments will need the measure $h(\\tau )$ to satisfy only one inequality in (\\ref{S1E16ha}). This requires the following:\n\\begin{definition}\n\\label{SUPER}\nA  function $h:[0,\\infty)\\to\\mathscr{M}_+([0,\\infty))$ is said to be a super solution of (\\ref{S1E16ha}) if\n\\begin{empheq}[left=\\empheqlbrace]{align}\n&\\forall  \\varphi \\in C^1_b([0, \\infty))\\;\\hbox{nonnegative, convex and decreasing} :\\nonumber\\\\\n&\\frac {d} {d\\tau }\\int_{ [0, \\infty) } \\varphi (x)h(\\tau,x)dx\\ge \\mathscr{Q}_3^{(2)}(\\varphi,h(\\tau))\\qquad a.e.\\;\\tau>0.\\label{S1E16haB}\n\\end{empheq}\n\\end{definition}\nThe operator $\\mathscr Q_3^{(2)}$ is considered in \\cite{KIER} and \\cite{AV1}, where a problem similar to (\\ref{S1E16ha}) is studied, with\n$\\widetilde{\\mathscr{Q}}_3$  replaced by $\\mathscr Q_3^{(2)}$ and for which, the property of instantaneous condensation is proved. We extend this result to the solutions $h$ of the problem (\\ref{S1E16ha}) with the whole $\\widetilde{\\mathscr{Q}}_3$, using similar arguments (monotonicity,  convexity of  test functions) and taking care of the linear term.\n \nTheorem \\ref{S1T1} is deduced from the corresponding existence result of $h$, that is proved using very classical arguments: regularization of the problem,  fixed point, a priori estimates and passage to the limit. Then, the delicate point is to invert the change of variables (\\ref{S1E45b}) in order to obtain a global in time nonnegative solution $G$.\n\nThe Plan of the article is the following. In Section \\ref{model} we prove Proposition \\ref{S1P0}. Section \\ref{existenceH} is devoted to the proof of the existence of the measure $H$. In Section \\ref{SectionC} we obtain several properties of $h(\\tau , \\{0\\})$. In Section \\ref{sectionG} we prove Theorem \\ref{S1T1} (existence for the measure $G$) and Theorem \\ref{S1Treg}. The contents of Section \\ref{SectionK} are the\nproofs of Theorem \\ref{THn01}, Theorem \\ref{MU1}, Theorem \\ref{EQUIV} and Corollary \\ref{S1C31}. Finally in Section \\ref{SectionD} we prove Theorem \\ref{S1T5}. Several technical results are presented in an Appendix.\n\n\\section{On weak formulations.}\n\\label{model}\n\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\n\nWe  deduce first a  detailed  expression of  the weak formulation of (\\ref{S1E1}) for a radial measure $G$.\n\\begin{proposition}\n\\label{S1P0}\nLet $G$ satisfy (\\ref{S1ED1})--(\\ref{S1E6'}) for some $T>0$, and write \n$G(t)=n(t)\\delta_0+g(t)$, where $n(t)=G(t,\\{0\\})$. Then, for all $\\varphi \\in C^{1,1}_b([0, \\infty))$ and for all $t\\in (0, T)$:\n\\begin{align}\n&\\frac {d} {dt}\\int _{ [0, \\infty) } \\varphi (x)G(t, x)dx=\\mathscr{Q}_4(\\varphi,g(t))+n(t)\\mathscr{Q}_3(\\varphi,g(t)),\\label{S1E13}\n\\end{align}\nwhere $\\mathscr Q_4(\\varphi,g)$ and $\\mathscr Q _3(\\varphi,g)$ are defined in (\\ref{S1E15})--(\\ref{S1E155}).\n\\end{proposition}\n\n\n\\begin{remark}\n\\label{SXR1}\nIf the term $\\mathscr Q_4(\\varphi,g)$ in (\\ref{S1E13}) is dropped, we recover the equation (\\ref{S1E16})  that defines a radial weak solution of (\\ref{PA}), (\\ref{PB}).\n\\end{remark}\n\n\\begin{proof}\n[\\upshape\\bfseries{Proof of Proposition \\ref{S1P0}}]\nWe may rewrite $\\mathcal{Q}_4(\\varphi,G)$ in (\\ref{S1E5}) as \n\\begin{align*}\n\\mathcal{Q}_4(\\varphi,G)&=\\iiint_{[0,\\infty)^3}\\Phi_{\\varphi}\\;dG_1dG_2dG_3\n+\\frac{1}{2}\\iiint_{[0,\\infty)^3}\\sqrt{x_3}\\Phi_{\\varphi}\\;dG_1dG_2dx_3,\n\\end{align*}\nwhere $\\Phi_{\\varphi}$ is as in Lemma \\ref{S2L1}, and we have used notation $dG$ instead of $Gdx$. Then we decompose \n$[0,\\infty)^3=(0,\\infty)^3\\cup A\\cup P$, where, for $\\{i,j,k\\}=\\{1,2,3\\}$,\n\\begin{align*}\n&A=\\{(x_1,x_2,x_3)\\in\\partial[0,\\infty)^3\\;:\\; x_i=x_j=0,\\; x_k> 0\\}\\cup\\{(0,0,0)\\},\\\\\n&P=\\{(x_1,x_2,x_3)\\in\\partial[0,\\infty)^3\\;:\\; x_i=0,\\; (x_j,x_k)\\in (0,\\infty)^2\\}.\n\\end{align*}\nLet $\\varphi\\in C^{1.1}_b([0,\\infty)$. By (\\ref{S2E2}) in Lemma \\ref{representation of Deltavarphi} and the definition (\\ref{S2E3}) of $W$, it follows that $\\Phi_{\\varphi}\\equiv 0$ on $A$. Hence, recalling the definition (\\ref{S1E15}) of $\\mathscr{Q}_4(\\varphi,g)$ and the definition of $\\Phi_{\\varphi}$ in Lemma \\ref{representation of Deltavarphi}, we have\n\\begin{align}\n\\label{999C}\n\\mathcal{Q}_4(\\varphi,G)=\\mathscr{Q}_4(\\varphi,g)+\\iiint_{P}\\Phi_{\\varphi}\\;dG_1dG_2dG_3\n+\\frac{1}{2}\\iiint_{P}\\sqrt{x_3}\\Phi_{\\varphi}\\;dG_1dG_2dx_3.\n\\end{align}\nWe now study the integral over $P$ for the cubic and the quadratic terms in (\\ref{999C}).\\\\\n(a) The cubic term.\nSince $\\Phi_{\\varphi}$ is symmetric in the $x_1$, $x_2$ variables, and $\\Phi_{\\varphi}$ is uniformly continuous on $[0,\\infty)^3$ by Lemma\n\\ref{S2L1}, then\n\\begin{align}\n\\iiint_{P}\\Phi_{\\varphi}\\;dG_1dG_2dG_3=&2 \\iiint_{\\{x_2=0,\\;x_1>0,\\,x_3>0\\}}\\Phi_{\\varphi}\\;dG_1dG_2dG_3\\nonumber\\\\\n&+\\iiint_{\\{x_3=0,\\;x_1>0,\\,x_2>0\\}}\\Phi_{\\varphi}\\;dG_1 dG_2 dG_3\\nonumber\\\\\n=&2G(t,\\{0\\})\\iint_{(0,\\infty)^2}\\Phi_{\\varphi}(x_1,0,x_3)\\;dG_1dG_3\\nonumber\\\\\n&+G(t,\\{0\\})\\iint_{(0,\\infty)^2}\\Phi_{\\varphi}(x_1,x_2,0)\\;dG_1 dG_2.\\label{mid step}\n\\end{align}\nUsing now the definition of $\\Phi_{\\varphi}$, we have\n\\begin{align}\n\\label{line 1}\n&2\\iint_{(0,\\infty)^2}\\Phi_{\\varphi}(x_1,0,x_3)\\;dG_1dG_3\\\\\n&=2\\iint_{\\{x_1>x_3>0\\}}\\big[\\varphi(x_1-x_3)+\\varphi(x_3)-\\varphi(0)-\\varphi(x_1)\\big] \\frac{dG_1dG_3}{\\sqrt{x_1x_3}}\\nonumber\\\\\n&=\\iint\\limits_{(0,\\infty)^2}\\big[\\varphi(|x_1-x_3|)+\\varphi(\\min\\{x_1,x_3\\})-\\varphi(0)-\\varphi(\\max\\{x_1,x_3\\})\\big] \\frac{dG_1dG_3}{\\sqrt{x_1x_3}}.\\nonumber\n\\end{align}\nand\n\\begin{align}\n\\label{line 2}\n&\\iint_{(0,\\infty)^2}\\Phi_{\\varphi}(x_1,x_2,0)\\,dG_1 dG_2\\\\\n&=\\iint\\limits_{(0,\\infty)^2}\\big[\\varphi(x_1+x_2)+\\varphi(0)-\\varphi(\\min\\{x_1,x_2\\})-\\varphi(\\max\\{x_1,x_2\\})\\big]\\frac{dG_1 dG_2}{\\sqrt{x_1x_2}}.\\nonumber\n\\end{align}\nNotice in (\\ref{line 1}) that $\\varphi(|x_1-x_3|)+\\varphi(\\min\\{x_1,x_3\\})-\\varphi(0)-\\varphi(\\max\\{x_1,x_3\\})=0$ on the diagonal $\\{x_1=x_3>0\\}$.\nThen, using (\\ref{line 1}) (changing the labels $x_3$ by $x_2$) and (\\ref{line 2}) in (\\ref{mid step}), and recalling the definition (\\ref{S1E154}) of $\\Lambda(\\varphi)$, we obtain\n\\begin{align}\n\\label{999A}\n\\iiint_{P}&\\Phi_\\varphi\\;dG_1dG_2dG_3=G(t,\\{0\\})\\iint_{(0,\\infty)^2}\\frac{\\Lambda(\\varphi)(x_1,x_2)}{\\sqrt{x_1x_2}}dG_1 dG_2.\n\\end{align}\n(b) The quadratic term.\nAgain, by the symmetry of $\\Phi_{\\varphi}$ in $x_1$, $x_2$, and the continuity of $\\Phi_{\\varphi}$ on $[0,\\infty)^3$, we obtain\n\\begin{align}\n\\label{999B}\n\\frac{1}{2}\\iiint_{P}\\sqrt{x_3}\\,\\Phi_{\\varphi}\\;dG_1dG_2d x_3\n&=\\iiint_{\\{x_2=0,\\;x_1>0,\\,x_3>0\\}}\\sqrt{x_3}\\,\\Phi_{\\varphi} \\,dG_1dG_2d x_3\\nonumber\\\\\n&=G(t,\\{0\\})\\iint_{(0,\\infty)^2}\\sqrt{x_3}\\,\\Phi_{\\varphi}(x_1,0,x_3)\\,dG_1d x_3\\nonumber\\\\\n&=G(t,\\{0\\})\\iint_{\\{x_1>x_3>0\\}}\\frac{\\Delta\\varphi(x_1,0,x_3)}{\\sqrt{x_1}}dG_1d x_3\\nonumber\\\\\n&=-G(t,\\{0\\})\\int_{(0,\\infty)}\\frac{\\mathcal{L}_0(\\varphi)(x_1)}{\\sqrt{x_1}}dG_1,\n\\end{align}\nwhere $\\mathcal{L}_0(\\varphi)$ is given in (\\ref{S1E155}).\nUsing (\\ref{999A}) and (\\ref{999B}) in (\\ref{999C}), the result follows.\n\\end{proof}\n\n\n\n\\begin{proposition}\n\\label{equilibria}\nFor all $C> 0$ and all $\\beta >0$, the measure $f _{ \\beta , 0, C }$ is a radial weak solutions of (\\ref{PA}),(\\ref{PB}).\n\\end{proposition}\n\n\\begin{proof}\nBy Proposition \\ref{S1P0},\n\\begin{eqnarray*}\n\\mathcal Q_4(\\varphi,G _{ \\beta , 0, C })=\\mathscr{Q}_4(\\varphi,G _{ \\beta , 0, 0 })+C\\mathscr{Q}_3(\\varphi,G _{ \\beta , 0, 0 }).\n\\end{eqnarray*}\nWe already know by Theorem  5 of \\cite{Lu1} that\n$\\mathcal{Q}_4(\\varphi,G _{ \\beta , 0, C })=0$ for all  $\\varphi \\in  C^{1,1}([0, \\infty))$. \nSince $\\mathscr{Q}_4(\\varphi,G _{ \\beta , 0, 0 })\\equiv \\mathcal{Q}_4(\\varphi,G _{ \\beta , 0, 0 })$, we deduce\n$\\mathscr{Q}_4(\\varphi,G _{ \\beta , 0, 0 })=0$ for all $\\varphi \\in  C^{1,1}([0, \\infty))$. Then, since $C>0$,\n$$\\mathscr{Q}_3(\\varphi,G _{ \\beta , 0, 0 })=0\\quad\\forall \\varphi \\in  C^{1,1}([0, \\infty)).$$\n\\end{proof}\n\n\\section{Existence of solutions $H$ to (\\ref{S1E16Ha})}\n\\label{existenceH}\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\n\nThe main result of this Section is the following,\n\\begin{theorem}\n\\label{S5T5R}\nLet $h_0\\in\\mathscr{M}^1_+([0,\\infty))$ with $N=M_0(h_0)>0$ and $E=M_1(h_0)>0$. Then, there exists $h\\in C\\big((0,\\infty), \\mathscr{M}_+^\\alpha([0,\\infty))\\big)$ for any $\\alpha\\geq1$,  \nthat satisfies the following properties: for all $\\varphi\\in C^1_b([0,\\infty)$\n\\begin{align}\n\\label{lip loc h}\n(i)\\quad&  \\tau\\mapsto\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)dx\\in W^{1,\\infty} _{loc}([0,\\infty)),\\\\\n\\label{AUXW}  \n(ii)\\quad&\\frac{d}{d \\tau}\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)\\,dx=\\widetilde{\\mathscr{Q}}_3(\\varphi,h(\\tau))\\quad a.e.\\,\\tau>0,\\\\\n(iii)\\quad&  h(0)=h_0,\\\\\n\\label{MMI}\n(iv)\\quad& M_0(h(\\tau))\\le\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg)^2\\quad\\forall\\tau \\ge 0,\\\\\n \\label{EE}\n(v)\\quad&M_1(h(\\tau))=E\\quad\\forall\\tau \\ge 0,\\\\\n(vi)\\quad& \\text{For all }\\alpha \\geq 3,\\,\\,\\hbox{if}\\,\\,M_{\\alpha}(h_0)<\\infty,\\,\\,\\hbox{then}  \\nonumber\\\\\n\\label{MAh}\n&\\quad M_{\\alpha}(h(\\tau))\n\\leq \\left(M_{\\alpha}(h_0)^{\\frac{2}{\\alpha-1}}+\\alpha2^{\\alpha-1}E^{\\frac{\\alpha+1}{\\alpha-1}}\\tau\\right)^{\\frac{\\alpha-1}{2}}\\quad\\forall\\tau\\geq 0,\\\\\n\\label{S5Ealpha }\n(vii)\\quad& M_{\\alpha}(h(\\tau))\\leq C(\\alpha,E)\\left(\\frac{1}{1-e^{-\\gamma(\\alpha,E)\\tau}}\\right)^{2(\\alpha-1)}\\,\\,\\forall \\alpha\\geq 3,\\;\\forall\\tau>0,\n\\end{align}\nwhere $C=C(\\alpha,E)$ is the unique positive root of the algebraic equation\n\\begin{equation}\n\\label{algebraic equation}\n2^{\\alpha-2}(\\alpha+1)E^{\\frac{2\\alpha+3}{2(\\alpha-1)}}(1+C)=C^{\\frac{2\\alpha-1}{2(\\alpha-1)}},\n\\end{equation}\nand $\\gamma=\\gamma(\\alpha,E)$:\n\\begin{equation}\n\\label{gamma}\n\\gamma=\\frac{1}{2(\\alpha+1)}\\left(\\frac{C}{E}\\right)^{\\frac{1}{2(\\alpha-1)}}.\n\\end{equation}\n\n\\end{theorem}\nThe proof of Theorem \\ref{S5T5R} is in three steps. In the first, a regularized problem is solved (Theorem \\ref{Ex1T2}). Then, using an approximation argument, a solution is obtained that satisfies (\\ref{lip loc h})--(\\ref{MAh}) but not yet (\\ref{S5Ealpha }) (Theorem \\ref{Ex1T1}). The Theorem  \\ref{S5T5R} is proved with  a second approximation argument on the initial data.\n\nAs a Corollary, we obtain the measure $H$ (not necessarily positive). \n\n\\begin{corollary}\n\\label{S5C52R}\nSuppose that $h_0\\in\\mathscr{M}^1_+([0,\\infty))$ with $N=M_0(h_0)>0$ and $E=M_1(h_0)>0$, consider $h$ given by Theorem \\ref{S5T5R},\nand define, for $\\tau\\geq 0$\n\\begin{align}\n\\label{DEFH}\nH(\\tau)=h(\\tau)-\\left(\\int_0^\\tau M_{1\/2}(h(\\sigma))d\\sigma\\right)\\delta_0.\n\\end{align}\nThen $H\\in C\\big([0,\\infty),\\mathscr{M}^1([0,\\infty))\\big)$ and for all $\\tau\\in[0,\\infty)$ and $\\varphi\\in C^1_b([0,\\infty))$:\n\\begin{align}\n\\label{lip loc H}\n(i)\\quad&\\tau\\mapsto\\int_{[0,\\infty)}\\varphi(x)H(\\tau,x)dx\\in W^{1,\\infty} _{loc}([0,\\infty)),\\\\\n(ii)\\quad&\n\\label{AUXWH}\n\\frac{d}{d \\tau}\\int_{[0,\\infty)}\\varphi(x)H(\\tau,x)\\,dx=\\mathscr{Q}_3(\\varphi,H(\\tau))\\quad a.e.\\,\\tau>0,\\\\\n\\label{S5IDh}\n(iii)\\quad& H(0)=h_0,\\\\\n(iv)\\quad& \\label{MMH}M_0(H(\\tau))=N\\quad\\forall \\tau \\ge 0,\\\\\n(v)\\quad& \\label{EEH} M_1(H(\\tau))=E\\quad\\forall \\tau \\ge 0,\\\\\n(vi)\\quad& \\forall \\alpha \\geq3, \\,\\,\\hbox{if}\\,\\,  M_{\\alpha}(h_0)<\\infty\\,\\hbox{ then, for all}\\,\\tau >0, \\nonumber\\\\\n&\\label{MAH} \\hskip 1.5cm  M_{\\alpha}(H(\\tau))\\leq \\left(M_{\\alpha}(h_0)^{\\frac{2}{\\alpha-1}}+\\alpha2^{\\alpha-1}E^{\\frac{\\alpha+1}{\\alpha-1}}\\tau\\right)^{\\frac{\\alpha-1}{2}},\\\\\n(vii)\\quad&\\label{S5EalphaR }\nM_{\\alpha}(H(\\tau))\\leq C(\\alpha,E)\\left(\\frac{1}{1-e^{-\\gamma(\\alpha,E)\\tau}}\\right)^{2(\\alpha-1)},\\,\\,\\forall \\alpha\\geq 3,\n\\end{align}\nwhere the constants $C(\\alpha,E)$ and $\\gamma (\\alpha,E)$ are defined in Theorem \\ref{S5T5R}.\n\\end{corollary}\n\n\\begin{remark}\nUnder the hypothesis that all the moments of the initial data $h_0$  are bounded it is easy to obtain the estimate (\\ref{S5Ealpha }) using the weak formulation  (\\ref{AUXW}). However, it is not so easy using the regularized weak formulation (\\ref{REGWEAK}) below. For that reason, we first want to obtain a solution $h$ satisfying (\\ref{AUXW}) with an initial data with bounded moments of all order.\n\\end{remark}\n\n\\subsection{A first result.}\n\\begin{theorem}\n\\label{Ex1T1}\nFor any $h_0\\in\\mathscr{M}^1_+([0,\\infty))$ with $N=M_0(h_0)$ and $E=M_1(h_0)$, there exists\n$h\\in C\\big([0,\\infty), \\mathscr{M}_+^1 ([0,\\infty))\\big)$ that satisfies (\\ref{lip loc h})--(\\ref{MAh}).\n\\end{theorem}\n\nThe proof of  Theorem \\ref{Ex1T1} is made in two steps. We first solve a regularised version of (\\ref{AUXW}). Then, in a second step, we use an approximation argument. More precisely, we consider the following cutoff:\n\\begin{cutoff}\n\\label{cut-off}\nFor every $n\\in \\mathbb{N}$ let $\\phi _n\\in  C_c([0,\\infty))$ be such that $\\operatorname{supp}\\phi_n=[0,n+1]$,\n$\\phi _n(x)\\le x^{-1\/2}$ for all $x>0$ and $\\phi_n(x)=x^{-1\/2}$ for all $x\\in\\left(\\frac {1} {n}, n \\right)$,\nin such a way that:\n\\begin{align}\n\\forall x>0\\,\\,\\,\\,\\,\\lim _{ n\\to \\infty }\\phi _n(x)=\\frac {1} {\\sqrt x}.\n\\end{align}\n\\end{cutoff}\n\n\\subsection{Regularised problem}\nWe now solve in Theorem \\ref{Ex1T2} a regularised version of (\\ref{AUXW}) with the operator $\\widetilde{\\mathscr{Q}}_{3,n}$ defined in \n(\\ref{Aq3tilden})--(\\ref{Q3n1w}). The solution $h_n$ is obtained as a mild solution to the equation\n\\begin{align}\n\\label{Ex1EApn}\n\\frac {\\partial h_n} {\\partial\\tau}(\\tau,x)=J_{3,n}(h_n(\\tau))(x),\n\\end{align} \nwhere $J_{3,n}$ is defined in (\\ref{A1E32})-(\\ref{A1E35}), and corresponds to a regularised version of the term $J_3$ defined in (\\ref{PR2}). Namely,\n$J_{3,n}(h)=J_3(h\\phi_n)$, where $\\phi_n$ is as in Cutoff \\ref{cut-off}.\n\\begin{theorem}\n\\label{Ex1T2}\nFor any $n\\in\\mathbb{N}$ and any nonnegative function  $h_0\\in C_c([0,\\infty))$,\nthere exists a unique nonnegative function $h_n\\in C\\big([0,\\infty), L^{\\infty}(\\mathbb{R}_+)\\cap L^1_{x}(\\mathbb{R}_+)\\big)$ such that for all $\\tau\\in[0,\\infty)$ \nand all $\\varphi\\in  L^1_{loc}(\\mathbb{R}_+)$:\n\\begin{align}\n&\\tau\\mapsto\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)dx\\in W^{1,\\infty}_{ loc  }([0,\\infty)) \\label{S5E765}\\\\\n&\\frac{d}{d\\tau}\\int_0^{\\infty}\\varphi(x)h_n(\\tau,x)dx=\\widetilde{\\mathscr{Q}}_{3,n}(\\varphi,h_n(\\tau)).\\label{REGWEAK}\\\\\n&h_n(0,x)=h_0(x)\\label{datan}\n\\end{align}\nMoreover, if we denote by $N=M_0(h_0)$ and $E=M_1(h_0)$, then for every $\\tau\\in[0,\\infty)$ and $\\alpha\\geq 3$:\n\\begin{align}\n&M_0(h_n(\\tau))\\leq\\bigg(\\frac{E}{2}\\tau+\\sqrt{N}\\bigg)^2, \\label{mass inequality} \\\\\n&M_1(h_n(\\tau))=E, \\label{conservation of energy}\\\\\n&M_{\\alpha}(h_n(\\tau))\\leq\\left(M_{\\alpha}(h_0)^{\\frac{2}{\\alpha-1}}+\\alpha 2^{\\alpha-1}E^{\\frac{\\alpha+1}{\\alpha-1}}\\tau\n\\right)^{\\frac{\\alpha-1}{2}}. \\label{MAhn}\n\\end{align}\nFurthermore, there exist two positive constants $C _{ 1, n }$ and $C _{ 2, n }$ depending on $n$ and $\\|h_0\\| _{ L^\\infty\\cap L^1_x }$ such that for all $\\tau >0$:\n\\begin{equation}\n\\label{a priori sup norm}\\|h_n(\\tau)\\|_{\\infty}\\leq C _{ 1, n }e^{C _{ 2, n }(\\tau^2+\\tau)}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nUsing\n(\\ref{A1E32}) we write equation (\\ref{Ex1EApn}) as\n\\begin{eqnarray}\n\\label{Ex1EApn2}\n\\frac {\\partial h_n} {\\partial \\tau }+h_nA_n(h_n)=K_n(h_n)+L_n(h_n),\n\\end{eqnarray}\nand the solution $h_n$ is obtained as a fixed point of the operator:\n\\begin{align}\nR_n(h_n)(\\tau,x)=&h_0(x)S_n(0, \\tau; x)\\nonumber \\\\\n&+\\int_0^\\tau S_n(\\sigma , \\tau; x)\\big(K_n(h_n)(\\sigma,x)+L_n(h_n)(\\sigma,x)\\big)d\\sigma, \\label{Ex1E1}\\\\\nS_n(\\sigma,\\tau; x)=&e^{-\\int_\\sigma ^\\tau A_n(h_n)(\\sigma,x)d\\sigma}\\label{Ex1E4}\n\\end{align} \non\n\\begin{align}\nB(T):=\\Big\\{h\\in C\\big([0,T],L^{\\infty}&(\\mathbb{R}_+)\\cap L^1_x(\\mathbb{R}_+)\\big): h\\geq 0\\quad\\text{and} \\nonumber\\\\\n&\\sup_{\\tau\\in[0,T]}\\|h(\\tau)\\|_{L^{\\infty}\\cap L^1_x}\\leq 2\\|h_0\\|_{L^{\\infty}\\cap L^1_x}\\Big\\}. \\label{Ex1E2}\n\\end{align}\nLet us show first that $R_n$ sends $B(T)$ into itself. Let $r_0:=\\|h_0\\|_{L^\\infty\\cap L^1_x}$ and for an arbitrary $T>0$, let $h\\in  B(T)$. \nBy Proposition \\ref{well defined operators}  with $\\rho (x)=x$, \n\\begin{align*}\n&R_n(h)(\\tau,x)\\geq 0\\qquad\\forall\\tau\\in[0,T],\\;\\forall x\\in\\mathbb{R}_+,\\\\\n&R_n(h)\\in C\\big([0,T],L^{\\infty}(\\mathbb{R}_+)\\cap L^1_x(\\mathbb{R}_+)\\big).\n\\end{align*}\nMoreover, using (\\ref{SAE100}) and  (\\ref{bound L}):\n\\begin{align*}\n\\sup _{ \\tau \\in [0,T] }\\|R_n(h)(\\tau)\\|_{L^{\\infty}\\cap L^1_x}\n\\leq r_0+T\\,C(n)(4r_0^2+2r_0).\n\\end{align*}\nIf $T$ satisfies:\n\\begin{eqnarray}\nT\\le \\frac {1} {C(n)(4r_0+2)}\n\\label{Ex1E257}\n\\end{eqnarray}\nthen \n$R_n(h)\\in B(T)$. \n\n\nTo prove that $R_n$ is a contraction, let $h_1\\in B(T)$, $h_2\\in B(T) $ and write:\n\n\\begin{align*}\n&\\big|R_n(h_1)(\\tau,x)-R_n(h_2)(\\tau,x)\\big|\\le h_0(x)\\left|  S_1(0, \\tau ; x)-S_2(0, \\tau ; x)\\right| +\\\\\n&+\\int_0^{\\tau}\\left|  S_1(\\sigma , \\tau ; x)-S_2(\\sigma , \\tau ; x) \\right|\n\\big(K_n(h_1)(\\sigma,x)+L_n(h_1)(\\sigma,x)\\big)d\\sigma\\\\\n&+\\int_0^{\\tau}\\big|K_n(h_1)(\\sigma,x)-K_n(h_2)(\\sigma,x)\\big|d\\sigma\\\\\n&+\\int_0^{\\tau} \\big|L_n(h_1)(\\sigma,x)-L_n(h_2)(\\sigma,x)\\big|d\\sigma.\n\\end{align*}\nBy (\\ref{SaE121}), for all $\\sigma \\ge 0$ and $\\tau \\ge0$\n\\begin{align}\n\\left|  S_1(\\sigma , \\tau ; x)-S_2(\\sigma , \\tau ; x) \\right| & \\le   \\int_0^\\tau |A_n(h_1)(\\sigma,x)-A_n(h_2)(\\sigma,x)|d\\sigma \\nonumber \\\\\n& \\le C(n)\\,\\tau  \\sup_{\\tau\\in[0,T]}\\|h_1(\\tau)-h_2(\\tau)\\|_{\\infty}.   \\label{Ex1E258}\n\\end{align}\nUsing now (\\ref{Ex1E258}) and (\\ref{SAE100})--(\\ref{SaE121}), we deduce:\n\\begin{align*}\n&\\|R_n(h_1)(\\tau)-R_n(h_2)(\\tau)\\| _{ L^\\infty \\cap L^1_x }\\le  C_1\n\\sup_{\\tau\\in[0,T]}\\|h_1(\\tau)-h_2(\\tau)\\|_{\\infty},\\\\\n&C_1\\equiv C_1(n, T, r_0)=C(n)T\\left(1+ 3r_0+2Tr_0(1+2r_0)\\right).\n\\end{align*}\nIf (\\ref{Ex1E257}) holds and\n\\begin{align*}\nC(n)T\\left(1+ 3r_0+2Tr_0(1+2r_0)\\right)<1,\n\\end{align*}\n$R_n$ will be a contraction from $B(T)$ into itself. This is achieved, for example, as soon as:\n\\begin{align*}\nT<\\min \\left\\{ \\frac {1} {2r_0(1+2r_0)}, \\frac {1} {2C(n) (1+2r_0)}\\right\\}=\\kappa _{ r_0 }.\n\\end{align*}\nThe fixed point $h_n$  of $R_n$ in $B(T)$  is then a  mild solution of  (\\ref{Ex1EApn}), that can be extended to a maximal interval of existence $[0, T _{ n, \\max })$. \n\nWe claim now that $h_n$ satisfies (\\ref{S5E765}), (\\ref{REGWEAK}).\nSince $h_n$ is a mild solution of (\\ref{Ex1EApn}):\n\\begin{equation}\n\\label{Ex1mildE}\nh_n(\\tau , x)=h_0(x)S_n(0, \\tau; x)+\\int_0^\\tau S_n(\\sigma , \\tau; x)\\big(K_n(h_n)(\\sigma,x)+L_n(h_n)(\\sigma,x)\\big)d\\sigma \n\\end{equation}\nWe multiply this equation by $\\varphi\\in L^1_{loc}(\\mathbb{R}_+)$ and integrate on $(0, \\infty)$:\n\\begin{align*}\n\\int _0^\\infty & h_n(\\tau , x)\\varphi ( x)dx=\\int _0^\\infty h_0(x)S_n(0, \\tau ; x) \\varphi (x)dx+\\\\\n&+\\int _0^\\tau \\int _0^\\infty S_n(\\sigma , \\tau; x)\\big(K_n(h_n)(\\sigma,x)+L_n(h_n)(\\sigma,x)\\big)\\varphi (x)dx d\\sigma.\n\\end{align*}\nUsing Lemma \\ref{well defined operators} and $h_0\\in C_c([0,\\infty))$, it follows that the integrals above are well define.\nIt also follows from Lemma \\ref{well defined operators} and (\\ref{Ex1E4}) that $\\tau \\mapsto \\int _0^\\infty h_n(\\tau , x)\\varphi (x)dx$ is  locally Lipschitz on $(0, T _{ n, \\max })$, and:\n\\begin{align*}\n\\frac {d} {dt}\\int _0^\\infty &h_n(\\tau , x)\\varphi (x)dx= \\int _0^\\infty h_0(x)(S_n(0, \\tau ; x)) _{ \\tau  } \\varphi  (x)dx+\\\\\n&+\\int _0^\\infty\\big(K_n(h_n)(\\tau ,x)+L_n(h_n)(\\tau ,x)\\big)\\varphi (x) dx+\\\\\n&+\\int _0^\\tau \\int _0^\\infty (S_n(\\sigma , \\tau; x))_\\tau \\big(K_n(h_n)(\\sigma,x)+L_n(h_n)(\\sigma,x)\\big)\\varphi (x)dx d\\sigma.\n\\end{align*}\nWe use now that $(S_n(\\sigma , \\tau; x))_\\tau=-A_n(h_n)(\\tau , x)S_n(\\sigma , \\tau; x)$\n and the identity (\\ref{Ex1mildE}) to deduce:\n\\begin{align*}\n\\frac {d} {dt}\\int _0^\\infty h_n(\\tau , x)\\varphi (x)dx=&\\int _0^\\infty\\!\\!\\big(K_n(h_n)(\\tau ,x)+L_n(h_n)(\\tau, x)\\big)\\varphi (x) dx- \\nonumber\\\\\n&-\\int _0^\\infty A_n(h_n)h_n(\\tau , x)\\varphi (\\tau , x)dx,\n\\end{align*}\nthat is (\\ref{REGWEAK}).\n\nSuppose now that $T _{ n, \\max }<\\infty$ and \n$$\n\\sup_{\\tau \\in [0, T _{ n, \\max })}\\|h_n(\\tau)\\|_{L^{\\infty}\\cap L^1_x}<\\infty.\n$$\nThen there is an increasing sequence \n$\\tau_j\\rightarrow T_{n,\\max}$  as $j\\rightarrow\\infty$  and $L>0$ such that\n$$\n\\sup_{j }\\|h_n(\\tau_j)\\|_{L^{\\infty}\\cap L^1_x}\\le L<\\infty.\n$$\n\nFix $\\delta >0$ such that $\n\\delta <\\kappa _{ r_0+1 }$.\nStarting with the initial value $h(\\tau _j)$ we have a mild solution $h_j$ defined on $[0, \\delta]$. Gluing together $h$ with $h_j$ we obtain a mild solution on\n$[0, t_j+\\delta]$. For $j$ large enough, $t_j+\\delta >T _{ n, \\max }$, and this is a contradiction. Therefore, either $T _{ n, \\max }=\\infty$ or, if \n$T _{ n, \\max }=\\infty$, then $\\limsup \\|h_n(\\tau)\\|_{L^{\\infty}\\cap L^1_x}=\\infty$ as $\\tau\\to T_{n,\\max}$.\n \nLet us prove now the estimates (\\ref{mass inequality}), (\\ref{conservation of energy}) and (\\ref{a priori sup norm}), first for all $\\tau\\in (0, T _{ n, \\max })$. Then,  the property $T _{ n, \\max }=\\infty$ will follow. We start proving (\\ref{conservation of energy}). To this end we use (\\ref{REGWEAK}) with $\\varphi=x$. Since in that case \n$\\Lambda(\\varphi)(x,y)=0$ and $\\mathcal{L}(\\varphi)(x)=0$, \n(\\ref{conservation of energy}) is immediate. To prove (\\ref {mass inequality}), we use  (\\ref{REGWEAK})  with $\\varphi =1$. \nThen, $\\Lambda(\\varphi)(x,y)=0$ and $\\mathcal{L}(\\varphi)(x)=-x$\n and then, using $\\phi _n\\le x^{-1\/2}$, H\\\"older inequality and  (\\ref{conservation of energy}): \n\\begin{align*}\n\\frac{d}{d\\tau}\\left(\\int_0^\\infty h_n(\\tau,x)d x\\right)^{1\/2}\\leq\\frac{\\sqrt{E}}{2},\n\\end{align*}\nfrom where (\\ref {mass inequality}) follows. \n\nIn order to prove \\eqref{a priori sup norm} we use \\eqref{mass inequality}:\n\\begin{align*}\n\\|K_n(h_n)(\\sigma)\\|_{\\infty}&\\leq\\|\\phi_n\\|_{\\infty}^2\\|h_n(\\sigma)\\|_1\\|h_n(\\sigma)\\|_{\\infty}\\\\\n&\\leq\\|\\phi_n\\|_{\\infty}^2\\bigg(\\frac{\\sqrt{E}}{2}\\sigma+\\sqrt{N}\\bigg)^2\\|h_n(\\sigma)\\|_{\\infty},\n\\end{align*}\nwhich combined with the estimate $\\|L_n(h_n)(\\sigma)\\|_{\\infty}\\leq 2\\|\\phi_n\\|_1\\|h_n(\\sigma)\\|_{\\infty}$, gives \n\\begin{align*}\n\\|h_n(\\tau)\\|_{\\infty}&\\leq\\|h_0\\|_{\\infty}\n+\\int_0^{\\tau}\\big(\\|K_n(h_n)(\\sigma)\\|_{\\infty}+\\|L_n(h_n)(\\sigma)\\|_{\\infty}\\big)d\\sigma\\\\\n&\\leq \\|h_0\\|_{\\infty}+C(n,h_0)\\int_0^{\\tau}(\\sigma^2+1)\\|h_n(\\sigma)\\|_{\\infty}d\\sigma.\n\\end{align*}\nwhere\n\\begin{eqnarray*}\nC(n,h_0)=\\max\\left\\{\\|\\phi _n\\|_1\\|\\phi _n\\| ^2_{ \\infty } \\|h_0\\|_1,\\, \\frac {\\|\\phi _n\\| ^2_{ \\infty }} {4}\\|h_0\\| _{ L^1_x }\\right\\}.\n\\end{eqnarray*}\nThen \\eqref{a priori sup norm} follows from Gronwall's inequality.\n\nFor the proof of (\\ref{MAhn}) we use (\\ref{REGWEAK}) with $\\varphi(x)=x^{\\alpha}$ for $\\alpha\\geq 3$:\n\\begin{eqnarray}\n\\label{S5E902}\n\\frac {d} {d\\tau }M_{\\alpha}(h_n(\\tau))=\\widetilde{\\mathscr{Q}}_{3,n}(\\varphi,h_n(\\tau)).\n\\end{eqnarray}\nSince:\n\\begin{align}\n\\label{MAL}\n\\mathcal{L}(\\varphi)(x)=\\left(\\frac{\\alpha-1}{\\alpha+1}\\right)x^{\\alpha+1}\\geq 0,\n\\end{align}\nwe have,\n\\begin{eqnarray*}\n\\frac{d}{d\\tau}M_{\\alpha}(h_n(\\tau))\\leq \n 2\\int_0^{\\infty}\\!\\!\\!\\!\\int_0^x \\Lambda(\\varphi)(x,y)\\phi_n(x)\\phi_n(y)h_n(\\tau,x)h_n(\\tau,x) dydx.\n\\end{eqnarray*}\nThen, we write\n$\\Lambda(\\varphi)(x,y)=x^{\\alpha}\\big((1+z)^{\\alpha}+(1-z)^{\\alpha}-2\\big),$\nwhere $z=y\/x$, and by Taylor's expansion around $z=0$:\n\\begin{align*}\nu(z)\\leq\\frac{\\|u''\\|_{\\infty}}{2}z^2\\leq \\alpha(\\alpha-1)2^{\\alpha-3}z^2.\n\\end{align*}\nHence for all $0\\leq y\\leq x$:\n\\begin{align}\n\\label{MAQ}\n\\Lambda(\\varphi)(x,y)&\\leq C_{\\alpha}x^{\\alpha-2}y^2,\\qquad\\text{where}\\qquad C_{\\alpha}=\\alpha(\\alpha-1)2^{\\alpha-3},\n\\end{align}\nand then, using $\\phi_n(x)\\phi_n(y)\\leq y^{-1}$ and (\\ref{conservation of energy}),\n\\begin{align*}\n\\frac{d}{d\\tau}M_{\\alpha}(h_n(\\tau) \\leq 2C_{\\alpha}M_{\\alpha-2}(h_n(\\tau))E.\n\\end{align*}\nSince  by Holder's inequality and (\\ref{conservation of energy})\n\\begin{align*}\nM_{\\alpha-2}(h_n(\\tau))\\leq E^{\\frac{2}{\\alpha-1}}M_{\\alpha}(h_n(\\tau))^{\\frac{\\alpha-3}{\\alpha-1}},\n\\end{align*} \nwe deduce\n\\begin{align*}\n\\frac{d}{d\\tau}\\left(M_{\\alpha}(h_n(\\tau))^{\\frac{2}{\\alpha-1}}\\right)\\leq\\frac{4C_{\\alpha}}{\\alpha-1}E^{\\frac{\\alpha+1}{\\alpha-1}},\n\\end{align*}\nand (\\ref{MAhn}) follows.\n\\end{proof}\n\\subsection{Proof of Theorem \\ref{Ex1T1}.}\n\nThe solution $h$ whose existence is claimed in Theorem \\ref{Ex1T1}  is obtained as the limit of a subsequence of solutions  $(h_n) _{ n\\in \\mathbb{N} }$  to the regularized problems obtained in Theorem \\ref{Ex1T2}. We first prove the following Lemma.\n\n\\begin{lemma}\n\\label{precomp}\nLet $h_0\\in C_c([0,\\infty))$ be nonnegative with $N=M_0(h_0)>0$ and $E=M_1(h_0)>0$, and consider $(h_n)_{ n\\in \\mathbb{N} }$ the sequence of functions  given by Theorem \\ref{Ex1T2}. Then for every $\\tau\\in[0,\\infty)$ there exists a subsequence, still denoted $(h_n(\\tau))_{n\\in\\mathbb{N}}$, and a measure $h(\\tau)\\in\\mathscr{M}^1_+([0,\\infty))$ such that, as $n\\to \\infty$,  \n$h_n(\\tau)$ converges to $h(\\tau)$ in the following sense:\n\\begin{align}\n\\label{growth condition}\n&\\forall\\varphi \\in C([0, \\infty));\\;\\exists\\theta\\in [0, 1): \\quad \\sup _{ x\\ge 0} \\frac {\\varphi (x)} {1+x^\\theta}<\\infty,\\\\\n\\label{sense of convergence}\n&\\lim _{ n\\to \\infty }\\int_{[0,\\infty)}\\varphi(x)h_{n}(\\tau,x)d x=\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)d x.\n\\end{align}\nMoreover, for every $\\tau\\in[0,\\infty)$:\n\\begin{align}\n\\label{pre conservation laws for widetildeG}\n&M_0(h(\\tau))\\leq\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg)^2,\\\\\n\\label{pre energy}\n&M_1(h(\\tau))\\leq E.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nLet us prove first the convergence for a subsequence of $(h_n(\\tau))_{n\\in\\mathbb{N}}$. For every $\\tau\\geq 0$ we have by (\\ref{mass inequality}) that \n\\begin{align*}\n\\sup_{n\\in\\mathbb{N}}\\int_0^\\infty h_n(\\tau,x)d x\\leq \\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg)^2.\n\\end{align*}\nTherefore, there exists a subsequence, still denoted $(h_n(\\tau))_{n\\in\\mathbb{N}}$, and a measure $h(\\tau)$ such that \n $(h_n(\\tau))_{n\\in\\mathbb{N}}$  converges to $h(\\tau)$ in the weak* topology of $\\mathscr{M}([0,\\infty))$, as $n\\to \\infty$: \n\\begin{equation}\n\\label{weak* 1}\n\\lim _{ n\\to \\infty  }\\int_{[0,\\infty)}\\!\\!\\!\\varphi(x)h_{n}(\\tau,x)d x=\n\\int_{[0,\\infty)}\\!\\!\\varphi(x)h(\\tau,x)d x,\\,\\,\n\\forall\\varphi\\in C_0([0,\\infty)).\n\\end{equation}\nSince for all $n\\in\\mathbb{N}$, $h_n(\\tau)$ is nonnegative, then $h(\\tau)$ is a positive measure. Also by weak* convergence and \\eqref{mass inequality} we deduce that $h(\\tau)$ is a finite measure:\n\\begin{align}\n\\label{weak mass}\n\\int_{[0,\\infty)}h(\\tau,x)d x\\leq\\liminf_{n\\rightarrow\\infty}\\int_0^\\infty h_{n}(\\tau,x)d x\\leq\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg)^2.\n\\end{align}\nMoreover, by (\\ref{conservation of energy}) we also have that the sequence $(h_n(\\tau))_{n\\in\\mathbb{N}}$ is bounded in $L^1_x(\\mathbb{R}_+)$. \nHence there exists a subsequence (not relabelled) that converges to a measure \n$\\nu(\\tau)$ in the weak* topology of $\\mathscr{M}([0,\\infty))$, i.e., such that\n\\begin{equation}\n\\label{weak* 2}\n\\lim _{ n\\to \\infty  }\\int_0^\\infty\\varphi(x)\\,x\\,h_{n}(\\tau,x)d x=\\int_{[0,\\infty)}\\!\\!\\!\\varphi(x)\\nu(\\tau,x)dx,\\,\\forall\\varphi\\in C_0([0,\\infty)).\n\\end{equation}\nAgain, since $h_n(\\tau)$ is nonnegative for all $n\\in\\mathbb{N}$ then $\\nu(\\tau)$ is a positive measure. Also by weak* convergence and \\eqref{conservation of energy} we have\n\\begin{align}\n\\label{weak energy}\n\\int_{[0,\\infty)}\\nu(\\tau,x)d x\\leq\\liminf_{n\\rightarrow\\infty}\\int_0^\\infty\\,x\\,h_{n}(\\tau,x)d x=E.\n\\end{align}\nLet us show now that $\\nu(\\tau)=x\\,h(\\tau)$. This will follow from \n\\begin{align}\n\\label{equality measures}\n\\forall \\varphi\\in C_0([0,\\infty)):  \\int_{[0,\\infty)}\\varphi(x)\\nu(\\tau,x)d x=\\int_{[0,\\infty)}\\varphi(x)\\,x\\,h(\\tau,x)d x\n\\end{align}\n In a first step we show that (\\ref{equality measures}) holds for $\\varphi\\in C_c([0,\\infty))$ and then we use a density argument. \nLet $\\varepsilon>0$ and \n$\\varphi\\in C_c([0,\\infty))$. Using \\eqref{weak* 2} with test function $\\varphi$, and \\eqref{weak* 1} with test function $x\\varphi(x)$, we deduce that \n\\begin{align*}\n\\bigg|\\int_{[0,\\infty)}&\\varphi(x)\\nu(\\tau,x)d x-\\int_{[0,\\infty)}\\varphi(x)\\,x\\,h(\\tau,x)d x\\bigg |\\\\\n&\\leq \\bigg|\\int_0^\\infty\\varphi(x)\\nu(\\tau,x)d x-\\int_{[0,\\infty)}\\varphi(x)\\,x\\,h_n(\\tau,x)d x\\bigg|\\\\\n&+\\bigg|\\int_0^\\infty\\varphi(x)\\,x\\,h_n(\\tau,x)d x-\\int_{[0,\\infty)}\\varphi(x)\\,x\\,h(\\tau,x)d x\\bigg|<\\varepsilon\n\\end{align*}\nfor $n$ large enough. Hence \\eqref{equality measures} holds for all $\\varphi\\in C_c([0,\\infty))$. Now let $\\varphi\\in C_0([0,\\infty))$ and consider a sequence \n$(\\varphi_k)_{k\\in\\mathbb{N}}\\subset C_c([0,\\infty))$ such that \\\\$\\|\\varphi_k-\\varphi\\|_{\\infty}\\rightarrow0$ as $k\\rightarrow\\infty$. \nUsing \\eqref{equality measures} with $\\varphi_k$ and the bounds \\eqref{weak mass} and \\eqref{weak energy}, we deduce that\n\n\\begin{align*}\n\\bigg|\\int_{[0,\\infty)}&\\varphi(x)\t\\nu(\\tau,x)d x-\\int_{[0,\\infty)}\\varphi(x)\\,x\\,h(\\tau,x)d x \\bigg|\\\\\n&\\leq\\int_{[0,\\infty)}\\big|\\varphi(x)-\\varphi_k(x)\\big|\\nu(\\tau,x)d x\\\\\n&+\\bigg|\\int_{[0,\\infty)}\\varphi_k(x)\\nu(\\tau,x)d x-\\int_{[0,\\infty)}\\varphi_k(x)\\,x\\,h(\\tau,x)d x \\bigg|\\\\\n&+\\int_{[0,\\infty)}\\big|\\varphi_k(x)-\\varphi(x)\\big|\\,x\\,h(\\tau,x)d x<\\varepsilon\n\\end{align*}\nfor $k$ large enough. Therefore \\eqref{equality measures} holds for all $\\varphi \\in C_0([0,\\infty))$, i.e., $\\nu(\\tau)=x\\,h(\\tau)$. Hence we rewrite \\eqref{weak* 2} as \n\\begin{align}\n\\lim _{ n\\to \\infty }\\int_0^{\\infty}\\varphi(x)x\\,h_n(\\tau,x)d x =\\int_{[0,\\infty)} \\!\\!\\!\\!\\!\\!\\varphi(x)x\\,h(\\tau,x)dx, \\,\\,\\forall\\varphi\\in C_0([0,\\infty)).\n\\end{align}\nLet us show now (\\ref{growth condition}), (\\ref{sense of convergence}).\nLet  then  $\\varphi\\in C([0, \\infty))$  be any nonnegative test function that satisfies (\\ref{growth condition}). We denote $(\\zeta _j) _{ j\\in \\mathbb{N} }$ a sequence of nonnegative and nonincreasing functions  of  $C_c^\\infty([0, \\infty))$ such that:\n\\begin{equation*}\n\\zeta_j(x)= 1\\,\\,\\hbox{if}\\,\\,\\,x\\in [0, j),\\qquad\n\\zeta_j(x)= 0\\,\\,\\,\\hbox{if}\\,\\,x>j+1,\n\\end{equation*}\nand  define $\\varphi_j=\\varphi\\,\\zeta_j$. Then for every $n,\\,j\\in\\mathbb{N}$:\n\\begin{align}\n\\label{est 1. lemma convergence for tau fixed}\n\\bigg|\\int_0^{\\infty}&\\varphi(x)h_n(\\tau,x)d x-\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)d x \\bigg|\\\\\n&\\leq\\int_0^{\\infty}\\big|\\varphi(x)-\\varphi_j(x)\\big|h_n(\\tau,x)d x\\nonumber\\\\\n&+\\bigg|\\int_0^{\\infty}\\varphi_j(x)h_n(\\tau,x)d x-\\int_{[0,\\infty)}\\varphi_j(x)h(\\tau,x)d x \\bigg|\\nonumber\\\\\n&+\\int_{[0,\\infty)}\\big|\\varphi_j(x)-\\varphi(x)\\big|h(\\tau,x)d x\\nonumber\n\\end{align}\nSince $\\varphi_j\\in C_0([0,\\infty))$, using \\eqref{weak* 2}, the second term in the right hand side of \n\\eqref{est 1. lemma convergence for tau fixed} converges to zero as $n\\rightarrow\\infty$ for every $j\\in\\mathbb{N}$. The first and the third term in the right hand side of \\eqref{est 1. lemma convergence for tau fixed} are treated in the same way, using that $\\varphi _j(x)=\\varphi (x)$ for all $x\\in [0, j)$. For instance, in the first term:\n\\begin{align*}\n\\int_0^{\\infty}\\big|\\varphi(x)&-\\varphi_j(x)\\big|h_n(\\tau,x)d x\n=\\int_j^{\\infty}\\big|\\varphi(x)-\\varphi_j(x)\\big|h_n(\\tau,x)d x\\\\\n&\\leq 2\\int_j^{\\infty}|\\varphi(x)| h_n(\\tau,x)d x \\leq 2C\\int_j^{\\infty}(1+x^{\\theta})h_n(\\tau,x)d x\\\\\n&\\leq 2C\\left(\\frac{1+j^{\\theta}}{j}\\right)\\int_j^{\\infty}x\\,h_n(\\tau,x)d x\\leq 2C\\left(\\frac{1+j^{\\theta}}{j}\\right)E.\n\\end{align*}\nTherefore this term is small provided $j$ is large enough. In conclusion, the difference in \\eqref{est 1. lemma convergence for tau fixed} is less than $\\varepsilon$ for $n$ sufficiently large, i.e., \\eqref{sense of convergence} holds.\n\\end{proof}\n\n\n\\begin{remark}\n\\label{remark. weak* topology generated by a distance}\nThe so-called narrow topology $\\sigma(\\mathscr{M}([0,\\infty)),C_b([0,\\infty)))$ on $\\mathscr{M}_+([0,\\infty))$ is generated by the metric\n$d(\\mu,\\nu)=\\|\\mu-\\nu\\|_0$, where\n\\begin{align*}\n\\|\\mu\\|_0=\\sup\\left\\{\\int_{[0,\\infty)}\\varphi d\\mu:\\varphi\\in\\text{Lip}_1([0,\\infty)),\\;\\|\\varphi\\|_{\\infty}\\leq 1\\right\\},\n\\end{align*}\n(cf. \\cite{BOG} Theorem 8.3.2).\n\\end{remark}\n\nUsing this Remark, Lemma \\ref{precomp} and the Arzel\\`{a}-Ascoli's Theorem we prove now the following:\n\\begin{proposition}\n\\label{equicont}\nLet $h_0$ and $(h_n) _{ n\\in \\mathbb{N} }$ be as in Lemma \\ref{precomp}.\nThen there exist a subsequence (not relabelled) and $h\\in C\\big([0,\\infty),\\mathscr{M}_+([0,\\infty))\\big)$ such that \n\\begin{align}\n\\label{S5E90}\nh_n\\xrightarrow[n\\rightarrow\\infty]{}h\\quad\\text{in}\\quad C\\big([0,\\infty),\\mathscr{M}_+([0,\\infty))\\big).\n\\end{align}\nMoreover, if we denote by $N=M_0(h_0)$ and $E=M_1(h_0)$, then for all $\\tau\\geq 0$ \n\\begin{align}\n\\label{MASS IN}\n&M_0(h(\\tau))\\leq\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg)^2,\\\\\n\\label{EN IN}\n&M_1(h(\\tau))\\leq E,\n\\end{align}\nand for all $\\varphi\\in C([0,\\infty))$ satisfying the growth condition (\\ref{growth condition}):\n\\begin{equation}\n\\label{sense of convergence 2}\n\\lim _{ n\\to \\infty  }\\int_0^{\\infty}\\varphi(x)h_{n}(\\tau,x)d x=\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)d x.\n\\end{equation}\n\\end{proposition}\n\n\n\\begin{proof}[\\upshape\\bfseries{Proof of Proposition \\ref{equicont}}]\nBy Lemma \\ref{precomp} the sequence \n$(h_n(\\tau))_{n\\in\\mathbb{N}}$ is relatively compact in $\\mathscr{M}([0,\\infty))$ for every $\\tau\\in[0,\\infty)$. Let us show now that $(h_n)_{n\\in\\mathbb{N}}$ is also equicontinuous. To this end let $\\tau_2\\geq\\tau_1\\geq 0$, and consider $\\varphi$ as in Remark \\ref{remark. weak* topology generated by a distance}, i.e., \n$\\varphi\\in\\text{Lip}([0,\\infty))$ with Lipschitz constant $\\text{Lip}(\\varphi)\\leq 1$, and $\\|\\varphi\\|_{\\infty}\\leq 1$.\nThen, using $\\phi_n(x)\\leq x^{-1\/2}$, (\\ref{lemma regularity 1}) and (\\ref{lemma regularity 4}) in Lemma \\ref{lemma regularity}, we have\n\\begin{align}\n&\\bigg|\\int_0^{\\infty}\\varphi(x)h_n(\\tau_1,x)d x-\\int_0^{\\infty}\\varphi(x)h_n(\\tau_2,x)d x\\bigg|\\nonumber \\\\\n&\\leq\\int_{\\tau_1}^{\\tau_2}\\big|\\widetilde{\\mathscr{Q}}_{3,n}(\\varphi,h_n(\\sigma))\\big|d\\sigma\\leq 2\\int_{\\tau_1}^{\\tau_2}\\bigg(\\int_0^{\\infty}h_n(\\sigma,x)d x\\bigg)^2d\\sigma\\nonumber\\\\\n&+4\\int_{\\tau_1}^{\\tau_2}\\int_0^{\\infty}\\sqrt{x}\\,h_n(\\sigma,x)d xd\\sigma.\\label{equicontinuity 1}\n\\end{align}\nUsing H\\\"{o}lder's inequality and the estimates (\\ref{mass inequality}) and (\\ref{conservation of energy}) in (\\ref{equicontinuity 1}), it follows that \n\\begin{align*}\n&\\bigg|\\int_0^{\\infty}\\varphi(x)h_n(\\tau_1,x)d x-\\int_0^{\\infty}\\varphi(x)h_n(\\tau_2,x)d x\\bigg|\\\\\n&\\leq 2\\int_{\\tau_1}^{\\tau_2}\\bigg(\\frac{\\sqrt{E}}{2}\\sigma+\\sqrt{N}\\bigg)^4d\\sigma\n+4\\sqrt{E}\\int_{\\tau_1}^{\\tau_2}\\bigg(\\frac{\\sqrt{E}}{2}\\sigma+\\sqrt{N}\\bigg)d\\sigma\\quad\\forall n\\in\\mathbb{N}.\\nonumber\n\\end{align*}\nWe then deduce using Remark \\ref{remark. weak* topology generated by a distance} that $(h_n)_{n\\in\\mathbb{N}}$ is equicontinuous.\nIt then follows from Arzel\\`{a}-Ascoli's Theorem (cf. for example \\cite{Roy})  that there exists $h\\in C\\big([0,\\infty),\\mathscr{M}_+([0,\\infty))\\big)$ such that\n$h_n\\rightarrow h$ in $C\\big([0,T], \\mathscr{M}_+([0,\\infty))\\big)$, for every $T>0$, as $n\\rightarrow\\infty$. \n\nThe estimates \n(\\ref{MASS IN}), (\\ref{EN IN}) and the convergence (\\ref{sense of convergence 2}) are deduced in the same way as in the Proof of  Lemma \\ref{precomp}.\n\\end{proof}\n\n\n\\begin{proof}[\\upshape\\bfseries{Proof of Theorem \\ref{Ex1T1}}]\nBy Corollary \\ref{APD1}, there exists a sequence of nonnegative function $(h_{0,n})_{n\\in\\mathbb{N}}\\in C_c([0,\\infty))$ that approximate $h_0$ in the weak* topology of the space $C_b([0,\\infty))^*$.\nLet then  $(h_n) _{ n\\in \\mathbb{N} }\\subset  C\\big([0,\\infty),\\mathscr{M}_+([0,\\infty))\\big)$ be the sequence of solutions to (\\ref{S5E765}), (\\ref{REGWEAK})\nobtained by Theorem \\ref{Ex1T2} with the initial data $h_{0,n}$.  By Proposition \\ref{equicont}  there exists a subsequence, still denoted $(h_n) _{ n\\in \\mathbb{N} }$, and $h\\in  C\\big([0,\\infty), \\mathscr{M}_+([0,\\infty))\\big)$ such that\n$h_n$ converges to $h$ in the topology of $C\\big([0,\\infty),\\mathscr{M}_+([0,\\infty))\\big)$. \n\nBy (\\ref{REGWEAK}) and (\\ref{datan}), for all  $\\varphi \\in C^1_b([0, \\infty))$ and $\\tau >0$:\n\\begin{equation}\n\\label{S5E456}\n\\int_0^{\\infty}\\varphi (x)h_n(\\tau , x)dx-\\int_0^{\\infty}\\varphi (x)h _{ 0, n }( x)dx=\\int _0^\\tau \\widetilde{\\mathscr{Q}}_{3,n}(\\varphi,h_n(\\sigma))d\\sigma. \n\\end{equation}\nBy construction, for every $\\varphi\\in C_b^1([0,\\infty))$ and every $\\tau\\in[0,\\infty)$:\n\\begin{align}\n\\label{LIM55}\n\\lim _{ n\\to \\infty  }\\int_0^{\\infty}\\varphi(x)h_n(\\tau,x)d x=\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)d x.\n\\end{align}\nWe prove now the convergence of the linear term: for all $\\varphi\\in C_b^1([0,\\infty))$ and $\\tau\\in[0,\\infty)$\n\\begin{align}\n\\label{linear limit 3}\n\\lim_{n\\to\\infty}\\widetilde{\\mathscr{Q}}_{3,n}^{(1)}(\\varphi, h_n(\\tau))=\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi, h_n(\\tau)).\n\\end{align}\nBy definition:\n\\begin{align}\n&\\Big|\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi,h(\\tau))- \\widetilde{\\mathscr{Q}}_{3,n}^{(1)}(\\varphi,h_n(\\tau))\\Big|\\nonumber \\\\\n&\\leq  \\bigg|\\int_0^{\\infty}\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}h(\\tau,x)d x-\\int_0^{\\infty}\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}h_n(\\tau,x)d x \\bigg|\\nonumber\\\\\n&+\\int_0^{\\infty}\\bigg|\\mathcal{L}(\\varphi)(x)\\phi_n(x)-\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}\\bigg|h_n(\\tau,x)d x. \\label{linear limit 1}\n\\end{align}\nFrom Lemma \\ref{lemma regularity} (iii) and (\\ref{sense of convergence 2}):\n\\begin{equation}\n\\lim _{ n\\to \\infty } \\bigg|\\int_0^{\\infty}\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}h(\\tau,x)d x-\\int_0^{\\infty}\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}h_n(\\tau,x)d x \\bigg|=0\n\\end{equation}\nFor the second term  in the right hand side of (\\ref {linear limit 1}) we split the integral $\\int _0^\\infty$ in two:  $\\int_0^R$ and  $\\int_R^{\\infty}$ for   $R>0$,  \nand apply (\\ref{lemma regularity 4}). We obtain:\n\\begin{align}\n\\label{linear limit 2}\n\\int_0^{\\infty}&\\bigg|\\mathcal{L}(\\varphi)(x)\\phi_n(x)-\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}\\bigg|h_n(\\tau,x)d x  \\\\\n&\\leq \\bigg\\|\\mathcal{L}(\\varphi)(x)\\phi_n(x)-\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}\\bigg\\|_{C([0,R])}\\int_0^R h_n(\\tau,x)d x+\\nonumber\\\\\n&\\hskip 4.5cm +4\\|\\varphi\\|_{\\infty}\\int_R^{\\infty}\\sqrt{x}\\;h_n(\\tau,x)d x\\nonumber.\n\\end{align}\nBy (\\ref{conservation of energy}), for any $\\varepsilon >0$ and $R> (E\/\\varepsilon )^2$:\n\\begin{align*}\n\\int_R^{\\infty}\\sqrt{x}\\;h_n(\\tau,x)d x \\leq\\frac{E}{\\sqrt{R}}<\\varepsilon \\qquad\\forall n\\in\\mathbb{N}.\n\\end{align*}\nThen by Lemma \\ref{regularised operators converge uniformly} and (\\ref{mass inequality}), the part on $[0,R]$ converges to zero as $n\\to\\infty$. \nSince $R>0$ is arbitrary we finally deduce that (\\ref{linear limit 2}) converges to zero as $n\\to\\infty$.\nTherefore (\\ref{linear limit 3}) holds.\n\nLet us prove now the convergence of the quadratic term: for all $\\varphi\\in C_b^1([0,\\infty))$ and all $\\tau\\in[0,\\infty)$:\n\\begin{align}\n\\label{quadratic limit 2}\n\\lim_{n\\to\\infty}\\mathscr{Q}_{3,n}^{(2)}(\\varphi, h_n(\\tau))=\\mathscr{Q}_3^{(2)}(\\varphi, h_n(\\tau)).\n\\end{align}\nAs before\n\\begin{align}\n\\label{quadratic limit 1}\n&\\bigg|\\mathscr{Q}_{3}^{(2)}(\\varphi,h(\\tau))-\\mathscr{Q}_{3,n}^{(2)}(\\varphi,h_n(\\tau))\\bigg|\\\\\n&\\leq\\bigg|\\mathscr{Q}_{3}^{(2)}(\\varphi,h(\\tau))-\\int_0^{\\infty}\\!\\!\\!\\!\\int_0^{\\infty}\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}h_n(\\tau,x)h_n(\\tau,y)d xd y\\bigg|\\nonumber\\\\\n&+\\int_0^{\\infty}\\!\\!\\!\\!\\int_0^{\\infty}\\bigg|\\Lambda(\\varphi)(x,y)\\phi_n(x)\\phi_n(y)-\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}\\bigg|h_n(\\tau,x)h_n(\\tau,y)d xd y\\nonumber.\n\\end{align}\nIt follows from Lemma \\ref{lemma regularity} (ii) and (\\ref{sense of convergence 2})\nthat the first term in the right hand side above converges to zero as $n\\rightarrow\\infty$.\nFor the second term we proceed as before. For any $R>0$ we split the double integral:\n\\begin{align*}\n&\\int_0^{\\infty}\\!\\!\\!\\!\\int_0^{\\infty}\\bigg|\\Lambda(\\varphi)(x,y)\\phi_n(x)\\phi_n(y)-\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}\\bigg|h_n(\\tau,x)h_n(\\tau,y)d xd y\\\\\n&\\leq \\bigg\\|\\Lambda(\\varphi)(x,y)\\phi_n(x)\\phi_n(y)-\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}\\bigg\\|_{C([0,R]^2)}\\left(\\int_0^R h_n(\\tau,x)dx\\right)^2\\nonumber\\\\\n&+\\iint_{(0,\\infty)^2\\setminus (0,R)^2}\\bigg|\\Lambda(\\varphi)(x,y)\\phi_n(x)\\phi_n(y)-\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}\\bigg|h_n(\\tau,x)h_n(\\tau,y)d xd y\\\\\n&=I_1+I_2.\n\\end{align*}\nBy Lemma \\ref{regularised operators converge uniformly} and (\\ref{mass inequality}), $I_1$ converges to zero as \n$n\\to\\infty$.\nFor the term $I_2$ we use (\\ref{lemma regularity 1}) in Lemma \\ref{lemma regularity} and the estimates (\\ref{conservation of energy}) and (\\ref{mass inequality}):\n\\begin{align*}\n&\\int_R^{\\infty}\\!\\!\\!\\!\\int_R^{\\infty}\\bigg|\\Lambda(\\varphi)(x,y)\\phi_n(x)\\phi_n(y)-\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}\\bigg|\nh_n(\\tau,x)h_n(\\tau,y)d xd y\\\\\n&\\leq 4\\|\\varphi'\\|_{\\infty}\\bigg(\\int_R^{\\infty}h_n(\\tau,x)d x\\bigg)^2\n\\leq \\frac{4\\|\\varphi'\\|_{\\infty}E^2}{R^2}\\qquad\\forall n\\in\\mathbb{N}\n\\end{align*}\nand\n\\begin{align*}\n&2\\int_R^{\\infty}\\!\\!\\!\\int_0^R\\bigg|\\Lambda(\\varphi)(x)\\phi_n(x)\\phi_n(y)-\\frac{\\Lambda(\\varphi)(x)}{\\sqrt{xy}}\\bigg|\nh_n(\\tau,x)h_n(\\tau,y)d xd y\\\\\n&\\leq4\\|\\varphi'\\|_{\\infty}\\int_R^\\infty\\int_0^Rh_n(\\tau,x)h_n(\\tau,y)d xd y\n\\leq \\frac{4 \\|\\varphi'\\|_{\\infty}E}{R}\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg)^2.\n\\end{align*}\nSince $R>0$ is arbitrary we deduce that $I_2$ also converges to zero as $n\\to\\infty$.\nWe then conclude that (\\ref{quadratic limit 2}) holds.\n\nCombining (\\ref{linear limit 3}) and (\\ref{quadratic limit 2}) it follows that for all $\\varphi\\in C_b^1([0,\\infty))$ and all $\\tau\\in[0,\\infty)$: \n\\begin{align}\n\\label{LIMQ3N}\n\\lim _{ n\\to \\infty  }\\widetilde{\\mathscr{Q}}_{3,n}(\\varphi,h_n(\\tau))=\\widetilde{\\mathscr{Q}}_3(\\varphi,h(\\tau)).\n\\end{align}\nMoreover, using $\\phi_n(x)\\leq x^{-1\/2}$, (\\ref{lemma regularity 1}) and (\\ref{lemma regularity 4}) in Lemma \\ref{lemma regularity}, and the estimates\n(\\ref{mass inequality}) and (\\ref{conservation of energy}), we have for all $\\varphi\\in C_b^1([0,\\infty))$, all $\\tau\\in[0,\\infty)$ and all $n\\in\\mathbb{N}$:\n\n\\begin{align*}\n&\\Big|\\widetilde{\\mathscr{Q}}_{3,n}(\\varphi,h_n(\\tau))\\Big|\\leq\\\\\n&\\leq 2\\|\\varphi'\\|_{\\infty}\\bigg(\\int_0^{\\infty}h_n(\\tau,x)dx\\bigg)^2+4\\|\\varphi\\|_{\\infty}\\int_0^{\\infty}\\sqrt{x}\\,h_n(\\tau,x)dx\\\\\n&\\leq2\\|\\varphi'\\|_{\\infty}\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg)^4+4\\|\\varphi\\|_{\\infty}\\sqrt{E}\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg).\n\\end{align*}\nBy  (\\ref{LIMQ3N}) and the dominated convergence Theorem:\n\\begin{align}\n\\label{CQ3I}\n\\lim _{ n\\to \\infty  }\\int_0^{\\tau}\\widetilde{\\mathscr{Q}}_{3,n}(\\varphi,h_n(\\sigma))d\\sigma=\\int_0^{\\tau}\\widetilde{\\mathscr{Q}}_3(\\varphi,h(\\sigma))d\\sigma.\n\\end{align}\nUsing now (\\ref{LIM55}) and (\\ref{CQ3I}), we may pass to the limit as $n\\to \\infty$ in (\\ref{S5E456}) for all $\\varphi\\in C_b^1([0,\\infty))$ and all $\\tau\\in[0,\\infty)$ to obtain:\n\\begin{equation}\n\\label{S5E963}\n\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)dx=\\int_{[0,\\infty)}\\varphi(x)h_0(x)dx+\\int_0^{\\tau}\\widetilde{\\mathscr{Q}}_3(\\varphi,h(\\sigma))d\\sigma.\n\\end{equation}\nThe map $\\tau\\mapsto\\int_{[0,\\infty)}\\varphi(x)h(\\tau,x)dx$ is then locally Lipschitz on $[0,\\infty)$, and  $h$ satisfies  (\\ref{lip loc h}), (\\ref{AUXW}) for all\n$\\varphi\\in C_b^1([0,\\infty))$ and for a.e. $\\tau\\in[0,\\infty)$. It also follows from (\\ref{S5E963}) that $h(0)=h_0$ in $\\mathscr M_+$.\n\nThe property (\\ref{MMI}) follows from (\\ref{MASS IN}). The  conservation of energy (\\ref{EE}) is obtained as follows. We already know by (\\ref{EN IN}) that $M_1(h(\\tau ))\\le E$. On the other hand, let  $\\varphi_k\\in C^1_b([0,\\infty))$ be a concave test function such that $\\varphi_k(x)=x$ for $x\\in[0,k)$ and $\\varphi_k(x)=k+1$ for $x\\geq k+2$. Notice that there exists a positive constant $C$ such that\n\\begin{align}\n\\label{SUP9}\n\\sup_{k\\in\\mathbb{N}}\\|\\varphi_k'\\|_{\\infty}\\leq C.\n\\end{align}\nBy Remark \\ref{concave-negativity}, $\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi_k,h)\\leq 0$ and $\\mathscr{Q}_3^{(2)}(\\varphi_k,h)\\leq 0$ for all $k\\in\\mathbb{N}$,\nand then, from (\\ref{S5E963}):\n\\begin{align}\n\\label{INW1}\n\\int_{[0,\\infty)}\\!\\!\\!\\varphi_k(x)h(\\tau,x)dx\\geq\\int_{[0,\\infty)}\\!\\!\\!\\varphi_k(x)h_0(x)dx+\\int_0^{\\tau}\\mathscr{Q}_{3}^{(2)}(\\varphi_k,h(\\sigma))d\\sigma.\n\\end{align}\nWe now prove that for all $\\tau\\in[0,\\infty)$:\n\\begin{align}\n\\label{DOMQ32}\n\\lim _{ k\\to \\infty }\\int_0^{\\tau}\\mathscr{Q}_{3}^{(2)}(\\varphi_k,h(\\sigma))d\\sigma =0.\n\\end{align}\nNotice that $\\Lambda(\\varphi_k)(x,y)\\to 0$ as $k\\to\\infty$, since $\\varphi_k(x)\\to x$. Then, using (\\ref{lemma regularity 1}) in Lemma \\ref{lemma regularity}, (\\ref{SUP9}) and (\\ref{mass inequality}), we deduce for all $\\tau\\in[0,\\infty)$ and $\\sigma \\in (0, \\tau )$:\n\\begin{align}\n&\\lim _{ k\\to \\infty }\\mathscr{Q}_{3}^{(2)}(\\varphi_k,h(\\sigma))=0\\\\\n&\\Big| \\mathscr{Q}_{3}^{(2)}(\\varphi_k,h(\\tau))\\Big|\\leq 2C\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg)^4\\qquad\\forall k\\in\\mathbb{N}.\n\\end{align}\nand  (\\ref{DOMQ32}) follows from the dominated convergence Theorem. We take now limits in (\\ref{INW1}) as $k\\to \\infty$. By (\\ref{DOMQ32}) and the monotone convergence Theorem we obtain that $M_1(h(\\tau ))\\ge E$ and then $M_1(h(\\tau ))=E$ for all $\\tau >0$.\n\nWe assume now that $M_\\alpha (h_0)<\\infty $ for some $\\alpha \\geq 3$ and prove (\\ref{MAh}). By  (\\ref{MAhn}) and \n(\\ref{APD56}) in Corollary \\ref{APD1}:\n\\begin{align}\nM_{\\alpha}(h(\\tau))&\\leq  \\liminf_{ n\\to \\infty } \\left(M_{\\alpha}(h _{ 0, n })^{\\frac{2}{\\alpha-1}}+\\alpha 2^{\\alpha-1}M_1(h_{0,n})^{\\frac{\\alpha+1}{\\alpha-1}}\\tau\n\\right)^{\\frac{\\alpha-1}{2}}\\\\\n&\\leq\\left(M_{\\alpha}(h _{ 0})^{\\frac{2}{\\alpha-1}}+\\alpha 2^{\\alpha-1}E^{\\frac{\\alpha+1}{\\alpha-1}}\\tau\n\\right)^{\\frac{\\alpha-1}{2}}.\n\\end{align}\n\\end{proof}\n\n\\subsection{ Proof of Theorem \\ref{S5T5R}}\n\\begin{proof}[\\upshape\\bfseries{Proof of Theorem \\ref{S5T5R}}] \nConsider again the sequence of initial data $h _{ 0, n }$  used in the proof of Theorem \\ref{Ex1T1} and the sequence of solutions  $h _{ n }$  obtained  by Theorem \\ref{Ex1T1}. Using  (\\ref{MAhn})  we know that $M_\\alpha (h_n(\\tau ))<\\infty$ for $\\tau >0$ and $n\\in \\mathbb{N}$. \n\nOur first step is to prove that (\\ref{AUXW}) holds also true for $\\varphi (x)=x^\\alpha $. Notice that $h_n$ solves now the equation (\\ref{AUXW}), with the operator $\\widetilde{\\mathscr Q}_3$ in the right-hand side, whose kernel is not compactly supported and the argument in the proof of (\\ref{MAhn}) must be slightly modified.\n\nIn order to use (\\ref{AUXW}) we consider a sequence $(\\varphi_k)\\subset C^1_b([0,\\infty))$ such that:\\begin{align}\n\\label{M7a5}\n&\\varphi_k\\to\\varphi\\quad\\text{as}\\quad k\\to\\infty\\\\\n\\label{M7a6}\n&\\varphi_k\\leq\\varphi_{k+1}\\leq \\varphi\\\\\n\\label{M7a7}\n&\\varphi'\\geq\\varphi_k'\\geq 0.\n\\end{align}\nLet us prove by the dominated convergence Theorem  that for all $\\tau\\geq 0$:\n\\begin{align}\n\\label{Q332}\n(i)\\quad&\\widetilde{\\mathscr{Q}}_3(\\varphi,h_n)\\in L^1(0,\\tau),\\\\\n\\label{Q333}\n(ii)\\quad&\\lim _{ k\\to \\infty }\\int_0^{\\tau}\\widetilde{\\mathscr{Q}}_3(\\varphi_k,h_n(\\sigma))d\\sigma=\\int_0^{\\tau}\\widetilde{\\mathscr{Q}}_3(\\varphi,h_n(\\sigma))d\\sigma.\n\\end{align}\nTo this end we first observe that, for $x\\ge y>0$:\n\\begin{align}\n\\label{MAJ1}\n\\lim _{ k\\to \\infty }\\Lambda(\\varphi_k)(x, y)=\\Lambda(\\varphi)\\qquad\\text{and}\\qquad \\lim _{ n\\to \\infty  }\\mathcal{L}(\\varphi_k)=\\mathcal{L}(\\varphi)(x)\n\\end{align}\nand, by the mean value Theorem:\n\\begin{align*}\n\\frac{\\Lambda(\\varphi_k)(x,y)}{\\sqrt{xy}}\\leq \\varphi_k'(\\xi_1)-\\varphi_k'(\\xi_2)\n\\end{align*}\nfor some $\\xi_1\\in (x,x+y)$ and $\\xi_2\\in(x-y,x)$. Using then (\\ref{M7a7}):\n\\begin{align}\n\\label{MAJ2}\n\\frac{|\\Lambda(\\varphi_k)(x,y)|}{\\sqrt{xy}}\\leq \\alpha\\big(2^{\\alpha-1}+1\\big)x^{\\alpha-1}\\qquad\\forall k\\in\\mathbb{N},\n\\end{align}\nand by (\\ref{M7a6}):\n\\begin{align*}\n\\frac{|\\mathcal{L}(\\varphi_k)(x)|}{\\sqrt{x}}\\leq \\left(\\frac{\\alpha+3}{\\alpha+1}\\right)x^{\\alpha+\\frac{1}{2}}\\qquad\\forall k\\in\\mathbb{N}.\n\\end{align*}\nSince by Theorem \\ref{Ex1T1}: $M_{\\alpha-1}(h_n(\\tau))<\\infty$ and $M_{\\alpha+1\/2}(h_n(\\tau))<\\infty$, for every fixed $n$  we may apply the Lebesgue's convergence Theorem \nto  the sequences $\\left\\{\\frac{\\Lambda(\\varphi_k)(x,y)}{\\sqrt{xy}}h_n(\\sigma , x)h_n(\\sigma , y)\\right\\} _{ k\\in \\mathbb{N} }$ and\n$\\left\\{\\frac{\\mathcal{L}(\\varphi_k)(x)}{\\sqrt{x}}h_n(\\sigma , x)\\right\\} _{ k\\in \\mathbb{N} }$ and obtain (\\ref{Q332}), (\\ref{Q333}).\n\nWe use now $\\varphi_k$ in (\\ref{AUXW}) and take the limit $k\\to\\infty$. We obtain from (\\ref{M7a5}), (\\ref{M7a6}), (\\ref{Q333}) and monotone convergence:\n\\begin{align}\nM_{\\alpha}(h_n(\\tau))=M_{\\alpha}(h _{ 0, n })+\\int_0^{\\tau}\\widetilde{\\mathscr{Q}}_3(\\varphi,h_n(\\sigma))d\\sigma\\qquad\\forall\\tau\\geq 0,\n\\end{align}\nand then, using (\\ref{Q332}):\n\\begin{align}\n\\label{EQMA}\n\\frac{d}{d\\tau}M_{\\alpha}(h_n(\\tau))=\\widetilde{\\mathscr{Q}}_3(\\varphi,h_n(\\tau))\\qquad a.e.\\;\\tau>0.\n\\end{align}\nIf we use  (\\ref{MAL}) and (\\ref{MAQ}) in the right hand side of (\\ref{EQMA}), we obtain\n\\begin{align*}\n\\frac{d}{d\\tau}M_{\\alpha}(h_n)\\leq 2^{\\alpha-2}\\alpha(\\alpha-1)E_n M_{\\alpha-2}(h_n)-\\left(\\frac{\\alpha-1}{\\alpha+1}\\right)M_{\\alpha+\\frac{1}{2}}(h_n),\n\\end{align*}\nwhere $E_n=M_1(h_{0,n})$.\nNow by H\\\"{o}lder's inequality:\n\\begin{align*}\n&M_{\\alpha-2}(h_n)\\leq E_n^{2\/(\\alpha-1)}M_{\\alpha}(h_n)^{(\\alpha-3)\/(\\alpha-1)}\\\\\n&M_{\\alpha}(h_n)\\leq E_n^{1\/(2\\alpha-1)}M_{\\alpha+\\frac{1}{2}}(h_n)^{2(\\alpha-1)\/(2\\alpha-1)}.\n\\end{align*}\nThen we obtain\n\\begin{align*}\n\\frac{d}{d\\tau}M_{\\alpha}(h_n)&\\leq 2^{\\alpha-2}\\alpha(\\alpha-1)E_n^{1+2\/(\\alpha-1)}M_{\\alpha}(h_n)^{(\\alpha-3)\/(\\alpha-1)}\\\\\n&-\\left(\\frac{\\alpha-1}{\\alpha+1}\\right)E_n^{-1\/(2(\\alpha-1))}M_{\\alpha}(h_n)^{(2\\alpha-1)\/(2(\\alpha-1))}.\\nonumber\n\\end{align*}\nSince $(\\alpha-3)\/(\\alpha-1)\\in [0,1)$ then\n\\begin{align*}\nM_{\\alpha}(h_n)^{(\\alpha-3)\/(\\alpha-1)}\\leq 1+M_{\\alpha}(h_n),\n\\end{align*}\nand :\n\\begin{align}\n\\label{MOM19}\n\\frac{d}{d\\tau}M_{\\alpha}(h_n)&\\leq 2^{\\alpha-2}\\alpha(\\alpha-1)E_n^{1+2\/(\\alpha-1)}\\big(1+M_{\\alpha}(h_n)\\big)\\\\\n&-\\left(\\frac{\\alpha-1}{\\alpha+1}\\right)E_n^{-1\/(2(\\alpha-1))}M_{\\alpha}(h_n)^{(2\\alpha-1)\/(2(\\alpha-1))},\\nonumber\n\\end{align}\nwhere $(2\\alpha-1)\/(2(\\alpha-1))>1$. If we define:\n\\begin{align*}\n&u(\\sigma)=M_{\\alpha}(h_n(\\tau)),\\quad\\sigma=C_1\\tau,\\quad q=2(\\alpha-1),\\\\\n&C_1=2^{\\alpha-2}\\alpha(\\alpha-1)E^{1+2\/(\\alpha-1)}\\\\\n&C_2=\\left(\\frac{\\alpha-1}{\\alpha+1}\\right)E^{-1\/(2(\\alpha-1))},\\quad C=\\frac {C_2} {C_1}.\n\\end{align*}\nWe deduce from (\\ref{MOM19}) that\n\\begin{align}\nu'\\leq1+u-Cu^{1+1\/q},\n\\end{align}\nand then by  Lemma 6.3 in \\cite{Bob}, for every $n\\in \\mathbb{N}$:\n\\begin{equation}\nM_{\\alpha}(h_n(\\tau))\\leq C(\\alpha,E_n)\\left(\\frac{1}{1-e^{-\\gamma(\\alpha,E_n)\\tau}}\\right)^{2(\\alpha-1)},\n\\end{equation}\nwhere the constants $C(\\alpha,E_n)$ and $\\gamma (\\alpha,E_n)$ are defined as in Theorem \\ref{S5T5R}. We may argue  now as in the proof of Theorem \\ref{Ex1T1}\nand  pass to the limit along a subsequence to obtain a limit $h\\in C\\big([0,\\infty), \\mathscr{M}_+ ([0,\\infty))\\big)$ satisfying (\\ref{lip loc h})--(\\ref{EE}) and (\\ref{S5Ealpha }). Using (\\ref{S5Ealpha }) and $h\\in C\\big([0,\\infty), \\mathscr{M}_+ ([0,\\infty))\\big)$ we deduce  as in the proof of Theorem \\ref{Ex1T1} that \n$h\\in C\\big((0,\\infty), \\mathscr{M}_+^\\alpha  ([0,\\infty))\\big)$ for all $\\alpha\\geq 1$.\n\\end{proof}\n\n\n\\begin{proof}[\\upshape\\bfseries{Proof of Corollary \\ref{S5C52R}}] \nWe first observe that  the map $\\tau\\mapsto M_{1\/2}(h(\\tau))$ is locally bounded. Indeed by H\\\"{o}lder's inequality, (\\ref{MMI}) and (\\ref{EE}):\n$$\nM_{1\/2}(h(\\tau))\\leq\\sqrt{M_1(h(\\tau))M_0(h(\\tau))}\\leq\\sqrt{E}\\bigg(\\frac{\\sqrt{E}}{2}\\tau+\\sqrt{N}\\bigg).\n$$\nThen by (\\ref{lip loc h}) it follows (\\ref{lip loc H}). Now for all $\\varphi\\in C^1_b([0,\\infty))$ and for a.e. $\\tau> 0$, we deduce from (\\ref{AUXW}): \n\\begin{align*}\n\\quad\\frac{d}{d\\tau}\\int_{[0,\\infty)}\\varphi(x)H(\\tau,x)dx&=\\widetilde{\\mathscr{Q}}_3(\\varphi,h(\\tau))-\\varphi(0)M_{1\/2}(h(\\tau))\\\\\n&=\\mathscr{Q}_3(\\varphi,h(\\tau)).\n\\end{align*}\nSince $H=h$ on $(0,\\infty)$ then $\\mathscr{Q}_3(\\varphi,H)\\equiv\\mathscr{Q}_3(\\varphi,h)$, and therefore (\\ref{AUXWH}) holds.\n\nNow for the initial data: $H(0)=h(0)=h_0$.\nThe conservation of mass (\\ref{MMH}) follows from (\\ref{AUXWH}) for $\\varphi=1$, since $\\Lambda(\\varphi)=0=\\mathcal{L}_0(\\varphi)$. The conservation of energy (\\ref{EEH}) follows directly from (\\ref{EE}) since $H=h$ on $(0,\\infty)$. \n\\end{proof}\n\n\\section{Properties of $h(\\tau , \\{0\\})$.}\n\\label{SectionC}\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\n\nIn all this Section we denote\n\\begin{equation}\n\\label{mZ}\nm(\\tau )=h(\\tau , \\{0\\}).\n\\end{equation}\nThe main result of this Section  is the following.\n\\begin{theorem}\n\\label{S1T4h}\nSuppose that $h\\in C([0, \\infty); \\mathscr{M}_+^1([0,\\infty))$ is a solution of (\\ref{S1E16ha}) with  $h(0)=h_0\\in\\mathscr{M}_+^1([0,\\infty))$, $N=M_0(h_0)>0$ and $E=M_1(h_0)>0$. \nThen $m$ is right continuous, a.e. differentiable and strictly increasing on $[0,\\infty)$.\n\\end{theorem}\nWe begin with the following properties of the function $m$ defined in (\\ref{mZ}).\n\\begin{lemma}\n\\label{S6L1'}\nThe function $m$ is nondecreasing, a.e. differentiable and right continuous on $[0,\\infty)$.\n\\end{lemma}\n\n\\begin{proof}\nGiven any $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST}, then for all $\\tau \\ge 0$\n\\begin{align}\n&m(\\tau)=\\lim _{ \\varepsilon \\to 0 }\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau,x)dx,\\label{S6L1E1}\n\\end{align}\nand by (\\ref{S1E16ha})-(\\ref{S1EB2})\n\\begin{align}\n\\frac{d}{d\\tau}\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau,x)dx=\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h(\\tau))\n-\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi_{\\varepsilon},h(\\tau)). \\label{S6E47}\n\\end{align}\nSince $\\varphi_{\\varepsilon}$ is convex, nonnegative and decreasing, it follows from Lemma \\ref{convex-positivity} \nthat $\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h)\\geq 0$ and \n$\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi_{\\varepsilon},h)\\leq 0$ for all $\\varepsilon>0$. \nThen by (\\ref{S6E47})\n\\begin{equation*}\n\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau_2,x)d x\\geq\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau_1,x)dx\\qquad\\forall\\tau_2\\geq\\tau_1\\geq 0.\n\\end{equation*}\nLetting $\\varepsilon\\to 0$ it follows from (\\ref{S6L1E1}) that $m$ is nondecreasing on $[0,\\infty)$ and, for any $\\tau\\geq 0$ and $\\delta>0$,\n\\begin{align}\n\\label{right continuity 1}\n\\liminf_{\\delta\\rightarrow 0^+}m(\\tau+\\delta)\\geq m(\\tau).\n\\end{align}\nUsing Lebesgue's Theorem (cf. \\cite{Roy}), $m$ is a.e. differentiable on $[0,\\infty)$. \nOn the other hand, if in (\\ref{S6E47}) the term $\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi_{\\varepsilon},h)$ is dropped,\n\\begin{align*}\n\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau+\\delta,x)&d x\\leq\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau,x)d x\n+\\int_{\\tau}^{\\tau+\\delta}\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h(\\sigma))d\\sigma.\n\\end{align*}\nUsing $\\mathds{1}_{\\{0\\}}\\leq\\varphi_{\\varepsilon}$ for all $\\varepsilon>0$, and  the bound \\eqref{lemma regularity 1}, we deduce\n\\begin{equation*}\nm(\\tau+\\delta)\\leq\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau,x)d x\n+\\frac {2\\delta} {\\varepsilon }(M_0(h(\\tau)))^2.\n\\end{equation*}\nIf we take now superior limits as $\\delta\\to 0^+$\nat $\\varepsilon >0$ fixed,\n\\begin{align*}\n\\limsup_{\\delta\\rightarrow 0^+}m(\\tau+\\delta)&\\leq\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau,x)d x\\qquad\\forall\\varepsilon>0.\n\\end{align*}\nWe may pass now to the limit  as $\\varepsilon \\to 0$ in the right hand side above\nand use (\\ref{S6L1E1}) to get,\n\\begin{align}\n\\label{right continuity 2}\n\\limsup_{\\delta\\rightarrow 0^+}m(\\tau+\\delta)\\leq m(\\tau).\n\\end{align}\nThe right continuity then follows from \\eqref{right continuity 1} and \\eqref{right continuity 2}.\n\\end{proof}\n\n\n\\begin{corollary}\n\\label{S6L1}\nThe map $\\tau\\mapsto H(\\tau,\\{0\\})$, defined for all $\\tau \\ge 0$, is right continuous on $[0,\\infty)$ and \n\\begin{align}\n\\label{JIM}\n\\limsup_{\\delta\\rightarrow 0^+}H(\\tau-\\delta,\\{0\\})\\leq H(\\tau,\\{0\\})\\qquad\\forall\\tau> 0.\n\\end{align}\n\\end{corollary}\n\n\\begin{proof}\nBy construction (cf.(\\ref{S1EdecompH})) it follows\n\\begin{align*}\nH(\\tau,\\{0\\})=m(\\tau)-\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma.\n\\end{align*}\nSince $M_{1\/2}(h)\\in L^1_{loc}(\\mathbb{R}_+)$ then $\\tau\\mapsto\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma$ is absolutely continuous, \nand since $m$ is right continuous by Lemma \\ref{S6L1'}, it follows that $\\tau\\mapsto H(\\tau,\\{0\\})$ is also right continuous. \nTo prove (\\ref{JIM}) we use the continuity of $\\tau\\mapsto\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma$ and the monotonicity of $h(\\tau,\\{0\\})$:\nfor all $\\tau>0$ and $\\delta\\in(0,\\tau)$,\n\\begin{align*}\n\\limsup_{\\delta\\to 0^+} H(\\tau-\\delta,\\{0\\})&=\\limsup_{\\delta\\to 0^+} m(\\tau-\\delta)-\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma\\\\\n&\\leq m(\\tau)-\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma=H(\\tau,\\{0\\}).\n\\end{align*}\n\\end{proof} \n\n\\begin{remark}\n\\label{S6Ej}\nWe do not know if the map $\\tau\\mapsto H(\\tau,\\{0\\})$ is continuous. By property \\eqref{JIM} however, $H(\\tau,\\{0\\})$ does not decrease through the possible discontinuities.\n\\end{remark}\n\nThe proof of Theorem \\ref{S1T4h} closely follows the proof of Proposition 1.21 in \\cite{AV1} (see also \\cite{KIER}, Ch. 3), where the authors proved the same result for the equation without the linear term $\\widetilde{\\mathscr Q}_3^{(1)}$. The main arguments in the proof are, on the one hand, the invariance of the problem (\\ref{S1E16ha}) with respect to time translation and under a suitable  scaling transformation. On the other hand, the fact that \n$\\Lambda(\\varphi)\\ge 0$ on $\\mathbb{R}_+^2$ for convex test functions $\\varphi $, and that the map $\\tau\\mapsto\\mathscr{Q}_3^{(2)}(\\varphi,h(\\tau))$ is locally integrable on $[0, T)$. When the linear term $\\widetilde{\\mathscr Q}_3^{(1)}$ is added, a slight modification of these argument still leads to the proof. \nSince by Lemma \\ref{convex-positivity}, for all nonnegative, convex decreasing test function $\\varphi \\in C_b^1([0, \\infty))$, we have\n$\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi,h)\\leq 0$, then solutions $h$ to (\\ref{S1E16ha}) are also super solutions (cf. Definition \\ref{SUPER}).\n\n\\begin{proposition}\n\\label{S5P1}\nLet $h$ be a super solution of (\\ref{S1E16ha}). Then for any $R>0$ and $\\theta\\in(0,1)$\n\\begin{align}\n\\int_{[0,R]}h(\\tau ,x)d x\\geq (1-\\theta)\\int_{[0,\\theta R]}h(\\tau_0 ,x)d x\\qquad\\forall \\tau\\geq\\tau_0\\geq 0.\\label{S5EP12}\n\\end{align}\n\\end{proposition}\n\n\\begin{proof}\nChose $\\varphi _R(x)=(1-x\/R)_+$ for $R>0$, and consider a sequence $(\\varphi_{R,n})_{n\\in\\mathbb{N}}\\subset C_b^1([0, \\infty))$ such that $\\varphi_{R,n}\\to\\varphi_R$,\n$\\varphi _{R,n}\\le\\varphi_{R}$ and $\\varphi_{R,n}(0)=1$ for all $n\\in\\mathbb{N}$. Since by convexity $\\mathscr{Q}_3^{(2)}(\\varphi_{R,n},h)\\geq 0$, then for all\n$\\tau$ and $\\tau_0$ with $\\tau\\geq\\tau_0\\geq 0$,\n\\begin{align*}\n\\int_{ [0, \\infty) } &\\varphi_{ R, n } (x)h(\\tau  , x)dx\\ge \\int_{ [0, \\infty) } \\varphi _{ R, n }(x)h(\\tau_0  , x)dx\\\\\n&\\ge  \\int_{ [0,\\theta R] } \\varphi _{ R, n }(x)h(\\tau_0  , x)dx\n\\ge \\varphi_{ R, n }(\\theta R) \\int_{ [0,\\theta R]}h(\\tau_0  , x)dx,\n\\end{align*}\nand (\\ref{S5EP12}) follows since, if we let $n\\to\\infty$,\n$$\n\\int_{[0,R]}h(\\tau ,x)d x\\geq \\int_{ [0,\\infty) }\\varphi_R(x)h(\\tau  , x)dx\n\\ge \\varphi _R(\\theta R) \\int_{ [0,\\theta R]}h(\\tau_0, x)dx.\n$$\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{S5L1}\nLet $h$ be a super solution of (\\ref{S1E16ha}). Let $R>0$ and consider a sequence $R:=a_0< a_1< a_2<...< a_n<...$ such that \n$|a_i-a_{i-1}|\\leq\\frac{R}{2}$ for all $i\\in\\{1, 2, 3,... \\}$. Then for all $\\tau \\geq\\tau_0 \\geq 0$ there holds\n\\begin{align}\n\\int_{[0,R]}h(\\tau,x)d x\\geq\\sum_{i=1}^\\infty \\frac{1}{2a_i}\\int_{\\tau _0}^{\\tau}\\bigg(\\int_{(a_{i-1},a_i]}h(\\sigma,x)d x\\bigg)^2d\\sigma. \\label{S5E97}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nWe chose $\\varphi _R$ and $\\varphi _{R, n }$ as in the proof of Proposition \\ref{S5P1} above. Since $h$ is a super solution of (\\ref{S1E16ha}), then\nfor all $n\\in\\mathbb{N}$,\n$$\n\\frac {d} {d\\tau }\\int  _{ [0, \\infty) }h(\\tau , x)\\varphi  _{ R, n }(x)dx\\ge \n{\\mathscr Q}_3^{(2)}(\\varphi  _{ R, n },h(\\tau)).\n$$\nWe have now:\n\\begin{eqnarray*}\n&&{\\mathscr Q}_3^{(2)}(\\varphi  _{ R, n },h(\\tau)) \\ge \\iint _{ (R, \\infty)^2 }h(\\tau, x)h(\\tau , y)\n\\frac {\\varphi  _{ R, n }(|x-y|)} {\\sqrt {x y}}dxdy\\\\\n&&\\ge \\sum_{ i=1 } ^\\infty\\frac { \\varphi  _{ R, n }(R\/2)} {a_i} \\iint _{(a _{ i-1 }, a_i]^2}h(\\tau, x)h(\\tau , y)dxdy\\\\\n&&=\\sum_{ i=1 } ^\\infty\\frac { \\varphi  _{ R, n }(R\/2)} {a_i} \\left(\\int _{(a _{ i-1 }, a_i]}h(\\tau, x)dx\\right)^2.\n\\end{eqnarray*}\nEstimate (\\ref{S5E97}) follows in the limit $n\\to\\infty$, since $\\varphi_{ R, n }(R\/2)\\to1\/2.$\n\\end{proof}\n\n\\begin{proposition}\n\\label{S5P2}\nLet $h$ be a super solution of (\\ref{S1E16ha}) with initial data $h_0\\in\\mathscr{M}_+^1([0,\\infty))$, and denote $N=M_0(h_0)$ and $E=M_1(h_0)$. Then for all $R>0$, \n$\\alpha\\in\\left(-\\frac{1}{2},\\infty\\right)$, and $\\tau_1$ and $\\tau_2$ with $0\\leq\\tau_1\\leq\\tau_2$:\n\\begin{align}\n\\label{MNEG6}\n\\int_{\\tau_1}^{\\tau_2}\\int_{(0,R]}x^{\\alpha}h(\\tau,x)dxd\\tau\n\\leq\\frac{2R^{\\frac{1}{2}+\\alpha}\\sqrt{\\tau_2-\\tau_1}}{1-\\left(\\frac{2}{3}\\right)^{\\frac{1}{2}+\\alpha}}\\left(\\frac{\\sqrt{E}}{2}\\tau_2+\\sqrt{N}\\right).\n\\end{align}\n\\end{proposition}\n\n\\begin{proof}\nSince h is a super solution of (\\ref{S1E16ha}), if we chose $\\varphi(x)=(1-x\/r)^2_+$ for $r>0$, then \n\\begin{align}\n\\label{MNEG4}\n\\int_{[0,\\infty)}\\varphi(x)h(\\tau_2,x)dx\\geq\\int_{\\tau_1}^{\\tau_2}\\mathscr{Q}_3^{(2)}(\\varphi,h(\\tau))d\\tau.\n\\end{align}\nSince \n$\\operatorname{supp}\\Lambda(\\varphi)=\\{(x,y)\\in[0,\\infty)^2:|x-y|\\leq r\\}$\nand\n$\\Lambda(\\varphi)(x,y)=\\varphi(|x-y|)$ for all $(x,y)\\in[r,\\infty)^2$,\nthen for all $\\tau\\geq 0$:\n\\begin{align*}\n\\mathscr{Q}_3^{(2)}(\\varphi,h(\\tau))&\\geq \\iint_{\\left(r,\\frac{3r}{2}\\right]^2}\\frac{\\varphi(|x-y|)}{\\sqrt{xy}}h(\\tau,x)h(\\tau,y)dxdy\\\\\n&\\geq\\frac{1}{4}\\left(\\int_{\\left(r,\\frac{3r}{2}\\right]}\\frac{h(\\tau,x)}{\\sqrt{x}}dx\\right)^2.\n\\end{align*}\nIf we use that $\\varphi\\leq1$ \nin the left hand side of (\\ref{MNEG4}), and  the estimate above in the right hand side, then\n\\begin{align*}\n\\int_{\\tau_1}^{\\tau_2}\\left(\\int_{\\left(r,\\frac{3r}{2}\\right]}\\frac{h(\\tau,x)}{\\sqrt{x}}dx\\right)^2d\\tau\\leq 4 M_0(h(\\tau_2)).\n\\end{align*} \nSince for any $\\alpha\\in(-1\/2,\\infty)$\n\\begin{align*}\n\\int_{\\left(r,\\frac{3r}{2}\\right]}\\frac{h(\\tau,x)}{\\sqrt{x}}dx\n\\geq \\left(\\frac{3r}{2}\\right)^{-\\alpha-\\frac{1}{2}}\\int_{\\left(r,\\frac{3r}{2}\\right]}x^{\\alpha}h(\\tau,x)dx,\n\\end{align*}\nwe then obtain\n\\begin{align}\n\\label{MNEG5}\n\\int_{\\tau_1}^{\\tau_2}\\left(\\int_{\\left(r,\\frac{3r}{2}\\right]}x^{\\alpha}h(\\tau,x)dx\\right)^2d\\tau\\leq 4M_0(h(\\tau_2))\\left(\\frac{3r}{2}\\right)^{1+2\\alpha}.\n\\end{align}\nFor any given $R>0$, using the decomposition \n$$(0,R]=\\bigcup_{k=0}^\\infty(a_{k+1},a_k],\\qquad a_k=\\left(\\frac{2}{3}\\right)^{k}\\!\\!\\!\\!R,$$\nand Cauchy-Schwarz inequality we obtain\n\\begin{align*}\n\\int_{\\tau_1}^{\\tau_2}\\int_{(0,R]}x^{\\alpha}h(\\tau,x)dxd\\tau\n\\leq\\sqrt{\\tau_2-\\tau_1}\\sum_{k=0}^{\\infty}\\left(\\int_{\\tau_1}^{\\tau_2}\\bigg(\\int_{(a_{k+1},a_k]}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!x^{\\alpha}h(\\tau,x)dx\\bigg)^2d\\tau\\right)^{\\frac{1}{2}}.\n\\end{align*}\nIf we chose $r=a_{k+1}$ so that $(a_{k+1},a_k]=(r,(3\/2)r]$ for every $k\\in\\mathbb{N}$, then by (\\ref{MNEG5}) we deduce\n\\begin{align*}\n\\int_{\\tau_1}^{\\tau_2}\\int_{(0,R]}x^{\\alpha}h(\\tau,x)dxd\\tau\n\\leq2\\sqrt{(\\tau_2-\\tau_1)M_0(h(\\tau_2))}\\sum_{k=0}^{\\infty}a_k^{\\frac{1}{2}+\\alpha}.\n\\end{align*}\nUsing the estimate (\\ref{MMI}) for $M_0(h(\\tau_2))$ and \n\\begin{align*}\n\\sum_{k=0}^{\\infty}a_k^{\\frac{1}{2}+\\alpha}\n=\\frac{R^{\\frac{1}{2}+\\alpha}}{1-\\left(\\frac{2}{3}\\right)^{\\frac{1}{2}+\\alpha}},\n\\end{align*}\nwe finally obtain (\\ref{MNEG6}).\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{lemma 2^n}\nLet $h$ be a super solution of (\\ref{S1E16ha}). Then for all $r>0$, $\\tau\\geq \\tau_0\\geq 0$ and $n\\in\\mathbb{N}$:\n\\begin{equation}\n\\label{e1}\n\\int_{[0,r]}h(\\tau,x)dx \\geq\\frac{1}{4^{n+1}r}\\int_{\\tau_0}^{\\tau}\\bigg(\\int_{(r,r2^n]}h(\\sigma,x)dx\\bigg)^2d\\sigma.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nConsider the decomposition\n$$\n\\left(r,2^nr\\right]=\\bigcup_{i=3}^{2^{n+1}}\\left(\\frac{r}{2}(i-1),\\frac{r}{2}i\\right].\n$$\nThen by Lemma \\ref{S5L1}, and Lemma 3.12 in \\cite{AV1}, we have\n\\begin{align*}\n\\int_{[0,r]}h(\\tau,x)dx&\\geq\\int_{\\tau_0}^{\\tau}\\sum_{i=3}^{2^{n+1}}\\frac{1}{ri}\\bigg(\\int_{\\left(\\frac{r}{2}(i-1),\\frac{r}{2}i\\right]}\nh(\\sigma,x)dx\\bigg)^2d\\sigma\\\\\n&\\geq\\int_{\\tau_0}^{\\tau}\\frac{1}{r}\\Bigg(\\sum_{i=3}^{2^{n+1}}i\\Bigg)^{-1}\\bigg(\\int_{(r,r2^n]}h(\\sigma,x)dx\\bigg)^2d\\sigma\\\\\n&\\geq\\frac{1}{(2^n-1)(2^{n+1}+3)r}\\int_{\\tau_0}^{\\tau}\\bigg(\\int_{(r,r2^n]}h(\\sigma,x)dx\\bigg)^2d\\sigma.\n\\end{align*}\nNotice that $(2^n-1)(2^{n+1}+3)\\leq 4^{n+1}$.\n\\end{proof}\n\nThe next Lemma takes into account the linear term $\\widetilde{\\mathscr Q}_3^{(1)}$.\n\\begin{lemma}\n\\label{S5L47} \nLet $h$ be a solution of (\\ref{S1E16ha}) with initial data $h_0\\in \\mathscr{M}_+^1([0,\\infty))$ satisfying\n\\begin{equation}\nm_0=\\int_{(0,\\infty)}h_0(x)dx>0. \\label{S5EL467}\n\\end{equation}\nThen, for any $\\tau _0\\ge 0$  there exist $R_1>0$, $C_1>0$  such that\n\\begin{equation}\n\\int_{[0,r]}h(\\tau ,x)d x\\geq C_1\\,r\\qquad\\forall r\\in[0,R_1],\\quad\\forall\\tau\\geq \\tau _0.\n\\end{equation}\n\\end{lemma} \n\n\\begin{proof}\nBy (\\ref{S5EL467}), there exist $0<a\\leq b<\\infty$ such that \n\\begin{align}\n\\label{mass of g0}\n\\int_{(a,b]}h_0(x)dx>\\frac{m_0}{2}.\n\\end{align}\nWe prove now\n\\begin{eqnarray}\n\\exists T'>0;\\,\\forall \\tau\\in [0, T'):\\,\\,\\,\\int_{\\left(\\frac{a}{2},2b\\right]}h(\\tau,x)dx\\geq\\frac{m_0}{4}. \\label{S5EL468}\n\\end{eqnarray}\nTo this end we use  (\\ref{S1E16ha}) with a test function\n $\\varphi\\in C^1_c([0,\\infty))$ such that $0\\leq\\varphi\\leq 1$, $\\varphi=1$ on $(a,b]$ and $\\varphi=0$ on $[0,\\infty)\\setminus\\left(\\frac{a}{2},2b\\right]$ and  (\\ref{mass of g0}) to obtain:\n\\begin{align}\n\\label{es1}\n\\int_{\\left(\\frac{a}{2},2b\\right]} h(\\tau,x)dx\n&\\geq\\frac{m_0}{2}+\\int_0^{\\tau}\\widetilde{\\mathscr{Q}}_3(\\varphi,h(\\sigma))d\\sigma.\n\\end{align}\nNow using (\\ref{lemma regularity 1}) and (\\ref{MMI}) we deduce\n$$\n\\left|\\mathscr{Q}_3^{(2)}(\\varphi,h(\\sigma))\\right|\\leq 2\\|\\varphi'\\|_{\\infty}\\bigg(\\frac{\\sqrt{M_1(h_0)}}{2}\\sigma+\\sqrt{M_0(h_0)}\\bigg)^4.\n$$\nUsing now $\\frac{|\\mathcal{L}(\\varphi)(x)|}{\\sqrt{x}}\\leq 3\\|\\varphi\\|_{\\infty}\\sqrt{x}$\nand $M_{1\/2}(h)\\leq \\sqrt{M_0(h)M_1(h)}$,\nwe have by the conservation of energy and the mass inequality\n$$\n\\left|\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi,h(\\sigma))\\right|\\leq 2\\|\\varphi\\|_{\\infty}\\sqrt{M_1(h_0)}\\left(\\frac{M_1(h_0)}{2}\\sigma+\\sqrt{M_0(h_0)}\\right).\n$$\nIt follows that  $ \\widetilde{\\mathscr{Q}}_3(\\varphi,h)\\in L^1_{loc}(\\mathbb{R}_+)$\nand  we deduce  (\\ref{S5EL468}) from (\\ref{es1}).\\\\\n\n\nBy Lemma \\ref{lemma 2^n} and (\\ref{S5EL468}), for any $r\\in\\left(0,\\frac{a}{2}\\right]$ and $n\\in\\mathbb{N}$ such that $r2^n\\in(2b,3b]$ we have\n\\begin{align}\n\\int_{[0,r]}h(\\tau,x)dx\n&\\geq\\int_0^\\tau\\frac{1}{4^{n+1}r}\\left(\\int_{\\left(\\frac{a}{2},2b\\right]}h(\\sigma,x)dx\\right)^2d\\sigma \\nonumber\\\\\n&\\geq\\frac{\\tau}{4^{n+1}r}\\left(\\frac{m_0}{4}\\right)^2  \\ge  \\frac{m_0^2}{4^3(3b)^2} \\tau\\,r \\qquad\\forall\\tau\\in[0,T']. \\label{1}\n\\end{align}\nwhere $\\left(\\frac{a}{2},2b\\right]\\subset (r,r2^n]$ has been used. \n\nFor any given   $\\tau _0\\ge 0$  define $\\tau'=\\min\\{\\tau_0,T'\\}$. Then by (\\ref{S5EP12}) in Proposition \\ref{S5P1} \nwith $\\theta=\\frac{1}{2}$ and $R=2r$, we deduce from (\\ref{1}):\n\\begin{equation}\n\\label{2}\n\\int_{[0,2r]}h(\\tau,x)dx \\geq\\frac{C\\tau'}{2}r \\qquad\\forall\\tau\\geq \\tau'.\n\\end{equation}\nand this proves the Lemma, where $R_1=a\/2$ and $C_1=C\\tau '\/4$.\n\\end{proof}\n\n\\begin{proposition}\n\\label{S4P47}\nLet $h$ and $h_0$ be as in Lemma \\ref{S5L47}. For all $L>0$ and  every $\\tau _1>0$  there exists $R_0=R_0(h,L,\\tau _1)>0$ such that\n\\begin{align}\n\\label{S4EP47}\n\\int_{[0,R_0]}h(\\tau ,x)dx\\geq LR_0\\qquad\\forall\\tau\\geq\\tau_1.\n\\end{align}\n\\end{proposition}\n\n\n\\begin{proof}\nBy Lemma \\ref{S5L47} for $\\tau _0=\\frac{\\tau_1}{2}$\n\\begin{align}\n\\label{by the lemma}\n\\exists C_1>0, \\, \\exists R_1>0;\\,\\,\\int_{[0,r]}h(\\tau,x)dx &\\geq C_1r,\\quad\\forall r\\in[0,R_1],\\,\\,\\,\\forall\\tau\\geq\\frac{\\tau_1}{2}.\n\\end{align}\nNow fix an integer $p\\geq 2$ such that $C_1p \\geq 8 L$. We divide the proof in two parts.  Assume first :\n\\begin{equation}\n\\label{assumption 1}\n\\exists r'\\in (0, R_1],\\; \\exists \\tau'\\in\\left[\\frac{\\tau_1}{2},\\tau_1\\right]:\\,\\,\\,\\int_{\\left[0,\\frac{r'}{p}\\right]}h(\\tau',x)dx\\geq \\frac{C_1r'}{2}.\n\\end{equation}\nIt follows from lemma \\ref{S5P1} with $\\theta=\\frac{1}{2}$ and $R=\\frac{2r'}{p}$ that\n$$\n\\int_{\\left[0,\\frac{2r'}{p}\\right]}h(\\tau,x)dx\\geq\\frac{C_1r'}{4}\\qquad\\forall \\tau\\geq\\tau',\n$$\nIf we take $R_0:=\\frac{2r'}{p}$, we have, by our choice of $p$,\n$$\n\\int_{[0,R_0]}h(\\tau,x)dx\\geq \\frac{C_1p}{8}R_0\\geq LR_0\\qquad\\forall \\tau\\geq\\tau',\n$$\nso  (\\ref{S4EP47})  holds.\n\nAssume now that \\eqref{assumption 1} does not hold,\nthen, by (\\ref{by the lemma}):\n\\begin{equation}\n\\label{assumption 2}\n\\int_{\\left(\\frac{r}{p},r\\right]}h(\\tau,x)dx\\geq\\frac{C_1r}{2}\\qquad\\forall r\\in(0,R_1],\\quad\\forall\\tau\\in\\left[\\frac{\\tau_1}{2},\\tau_1\\right].\n\\end{equation}\nTake now any $r\\in\\left(0,\\frac{R_1}{p}\\right]$, let $n\\in\\mathbb{N}$ be the largest integer such that \n$r p^n\\in\\left(\\frac{R_1}{p},R_1\\right]$, and consider now the following decomposition\n$$(r,rp^n]=\\bigcup_{i=p+1}^{p^{n+1}}\\left(\\frac{r}{p}(i-1),\\frac{r}{p}i\\right]\n=\\bigcup_{k=1}^n\\bigcup_{i=p^k+1}^{p^{k+1}}\\left(\\frac{r}{p}(i-1),\\frac{r}{p}i\\right].$$\nBy lemma \\ref{S5L1} on $(\\tau _1\/2, \\tau _1)$ with $a_i=ri\/p$, $i=p+1, \\cdots, p^{n+1}$:\n\\begin{align}\n\\label{ue1}\n\\int_{[0,r]}h(&\\tau_1,x)dx\n\\geq\\int_{\\frac{\\tau_1}{2}}^{\\tau_1}\\left[\\frac{p}{2r}\\sum_{i=p+1}^{p^{n+1}}\\frac{1}{i}\\left(\\int_{\\left(\\frac{r}{p}(i-1),\\frac{r}{p}i\\right]} h(\\sigma,x)dx\\right)^2\\right]d\\sigma\\nonumber\\\\\n&=\\int_{\\frac{\\tau_1}{2}}^{\\tau_1}\\left[\\frac{p}{2r}\\sum_{k=1}^n\\sum_{i=p^k+1}^{p^{k+1}}\\frac{1}{i}\\left(\\int_{\\left(\\frac{r}{p}(i-1),\\frac{r}{p}i\\right]} h(\\sigma,x)dx\\right)^2\\right]d\\sigma\\nonumber\\\\\n&\\geq\\int_{\\frac{\\tau_1}{2}}^{\\tau_1}\\left[\\frac{1}{2r}\\sum_{k=1}^n\\frac{1}{p^k}\\sum_{i=p^k+1}^{p^{k+1}}\\left(\\int_{\\left(\\frac{r}{p}(i-1),\\frac{r}{p}i\\right]} h(\\sigma,x)dx\\right)^2\\right]d\\sigma.\n\\end{align}\nWe use now  Lemma  3.12 in \\cite{AV1}\n\\begin{align*}\n&\\sum_{i=p^k+1}^{p^{k+1}}\\left(\\int_{\\left(\\frac{r}{p}(i-1),\\frac{r}{p}i\\right]} h(\\sigma,x)dx\\right)^2\n\\geq\\frac{1}{p^k(p-1)}\\times \\\\ \n&\\times \\left(\\sum_{i=p^k+1}^{p^{k+1}}\\int_{\\left(\\frac{r}{p}(i-1),\\frac{r}{p}i\\right]} h(\\sigma,x)dx\\right)^2\n\\geq\\frac{1}{p^{k+1}}\\left(\\int_{(rp^{k-1},rp^k]}h(\\sigma,x)dx\\right)^2\n\\end{align*}\nand deduce\n\\begin{align*}\n\\int_{[0,r]}h(\\tau_1,x)dx\\geq\\int_{\\frac{\\tau_1}{2}}^{\\tau_1}\\left[\\frac{1}{2r}\\sum_{k=1}^n\\frac{1}{p^{2k+1}}\\left(\\int_{(rp^{k-1},rp^k]}h(\\sigma,x)dx\\right)^2\\right]d\\sigma.\n\\end{align*}\nDue to the choice of the integer $n$, $r p^k\\in (0, R_1]$ for all $k=1, \\cdots, n$, and we can use (\\ref{assumption 2}) on each interval\n$(rp^{k-1},rp^k]$ to obtain:\n\\begin{align*}\n\\int_{[0,r]}h(\\tau_1,x)dx\n&\\geq\\int_{\\frac{\\tau_1}{2}}^{\\tau_1}\\left[ \\frac{1}{2r}\\sum_{k=1}^n\\frac{1}{p^{2k+1}}\\left(\\frac{C_1rp^k}{2}\\right)^2\\right]d\\sigma\n=\\frac{\\tau_1C_1^2n}{16p}\\;r.\n\\end{align*}\nIt then follows from lemma \\ref{S5P1} with $\\theta=\\frac{1}{2}$ and $R=2r$ that\n\\begin{align}\n\\label{estimate unbounded}\n\\int_{[0,2r]}h(\\tau,x)dx\\geq\\frac{\\tau_1C_1^2n}{32p}\\;r\\qquad\\forall\\tau\\geq\\tau_1.\n\\end{align}\nSince $r p^n\\in\\left(\\frac{R_1}{p},R_1\\right]$, then $n\\geq\\frac{\\log\\left(\\frac{R_1}{rp}\\right)}{\\log(p)}$, and\nwe chose $r>0$ small enough in order to have $r\\in (0, R_1\/p)$ and \n$$\\frac{\\tau_1C_1^2}{64p}\\frac{\\log\\left(\\frac{R_1}{rp}\\right)}{\\log p}\\geq L;$$\nand set $R_0:=2r$. The result then follows from (\\ref{estimate unbounded}).\n\\end{proof}\n\n\\begin{lemma}\n\\label{rescaled}\nLet $h$ be a solution of (\\ref{S1E16ha}) and, for any $\\kappa>0$ and $\\lambda>0$, consider the rescaled measure\n$h_{\\kappa,\\lambda}$ defined as:\n\\begin{equation}\n\\label{S5Escaled}\n\\int_{[0,\\infty)} \\!\\!\\!\\! h_{\\kappa,\\lambda}(\\tau,x)\\varphi(x)dx\n=\\kappa\\!\\int_{[0,\\infty)}\\!\\!\\!\\!h(\\kappa\\lambda\\tau, x)\\varphi\\left(\\frac{x}{\\lambda}\\right)dx,\\;\\forall \\varphi \\in C_b([0, \\infty)).\n\\end{equation}\nThen $h_{\\kappa,\\lambda}$ is a super solution of (\\ref{S1E16ha}).\n\\end{lemma}\n\n\\begin{proof}\nLet $\\varphi\\in C^1_b([0,\\infty))$ be nonnegative, convex and decreasing, $\\psi(x)=\\varphi(x\/\\lambda)$, and $\\eta=\\kappa\\lambda\\tau$.\nBy Lemma \\ref{convex-positivity}, \n$\\widetilde{\\mathscr{Q}}_3^{(1)}(\\psi,h)\\leq 0$, and by (\\ref{S1E16ha})\n\\begin{align*}\n\\frac{d}{d\\eta}&\\int_{[0,\\infty)}\\psi(x)h(\\eta, x)dx\\geq \\mathscr{Q}_3^{(2)}(\\psi,h(\\eta)).\n\\end{align*}\nSince $\\mathscr{Q}_3^{(2)}(\\psi,h(\\eta))=\\kappa^{-2}\\lambda^{-1}\\mathscr{Q}_3^{(2)}(\\varphi,h_{\\kappa,\\lambda}(\\tau))$, then\n\\begin{align*}\n\\frac{d}{d\\tau}\\int_{[0,\\infty)} \\varphi(x)h_{\\kappa,\\lambda}(\\tau,x)dx&=\\kappa^2\\lambda\\frac{d}{d\\eta}\\int_{[0,\\infty)}\\psi(x)h(\\eta, x)dx\n\\geq\\mathscr{Q}_3^{(2)}(\\varphi,h_{\\kappa,\\lambda}(\\tau)).\n\\end{align*}\n\\end{proof}\n\n\\begin{lemma}\n\\label{concentration lemma}\nLet $h$ be a super solution of (\\ref{S1E16ha}).\nSuppose that there exists $\\tau'>0$ such that\n\\begin{equation}\n\\label{S5HC}\n\\int_{[0,1]}h(\\tau,x)dx\\geq 1\\qquad\\forall \\tau\\geq\\tau'.\n\\end{equation}\nThen for any given $\\delta>0$ there exist  $\\tau _0$ such that\n\\begin{align}\n&\\tau '\\leq\\tau _0\\leq\\tau '+T_0(\\delta),\\qquad T_0(\\delta )=\\frac{64}{\\delta^3}\\left(1-\\frac{\\delta}{2}\\right) \\label{S5EX2}\\\\\n&\\hbox{and}\\,\\,\\,\\,\\int_{\\left[0,\\frac{\\delta}{4}\\right]}h(\\tau_0,x)dx\\geq 1-\\frac{\\delta}{2}. \\label{S5EX3}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\n The statement of the Lemma is  equivalent to show that the following set\n$$\nA:=\\left\\{\\tau\\in[\\tau', \\tau '+T_0( \\delta )]:\\int_ {\\left[0,\\frac{\\delta}{4}\\right]}h(\\tau,x)dx\\geq 1-\\frac{\\delta}{2}\\right\\}.\n$$\nis non empty, where $T_0(\\delta )$ is defined in (\\ref{S5EX2}). \nTo this end we first apply  Lemma \\ref{S5L1}  with  $a_0=\\frac{\\delta}{4}$, $a_i=\\frac{\\delta}{4}\\left(1+\\frac{i}{2}\\right)$ for $i\\in\\{1,...,n-1\\}$ and $a_n=1$. The number $n$ is chosen to be the largest integer such that $a_{n-1}<1$, which implies\n\\begin{equation}\n\\label{S5e0}\n\\frac{1}{n+1}>\\frac{\\delta}{8}.\n\\end{equation}\nThen, using ${a_i}^{-1}\\geq 1$ for all $i\\in\\{1,...,n\\}$:\n\\begin{align*}\n\\int_{\\left[0,\\frac{\\delta}{4}\\right]}h(\\tau,x)dx\n\\geq\\frac{1}{2}\\int_{\\tau'}^{\\tau} \\sum_{i=1}^n\\bigg(\\int_{(a_{i-1},a_i]}h(\\sigma,x)dx\\bigg)^2d\\sigma,\\,\\,\\,\\forall \\tau >\\tau '.\n\\end{align*}\nSince by Lemma 3.12 in \\cite{AV1} and (\\ref{S5e0}):\n\\begin{align*}\n\\sum_{i=1}^n\\bigg(\\int_{(a_{i-1},a_i]}h(\\sigma,x)dx\\bigg)^2\n\\geq\\frac{\\delta}{8}\\bigg(\\int_{\\left(\\frac{\\delta}{4},1\\right]}h(\\sigma,x)dx\\bigg)^2,\n\\end{align*}\nwe obtain, for all $\\tau >\\tau '$\n\\begin{align}\n\\label{estimate for contradiction}\n\\int_{\\left[0,\\frac{\\delta}{4}\\right]}h(\\tau,x)dx\n\\geq\\frac{\\delta}{16}\\int_{\\tau'}^{\\tau}\\bigg(\\int_{\\left(\\frac{\\delta}{4},1\\right]}h(\\sigma,x)dx\\bigg)^2d\\sigma.\n\\end{align}\nArguing by contradiction suppose that $A=\\emptyset$:\n\\begin{align*}\n\\int_{\\left(0,\\frac{\\delta}{4}\\right]}h(\\tau ,x)dx< 1-\\frac{\\delta}{2}\\qquad\\forall\\tau \\in[\\tau',\\tau '+T_0(\\delta )]\n\\end{align*}\nand by (\\ref{S5HC}):\n$$\n\\int_{\\left(\\frac{\\delta}{4},1\\right]}h(\\tau ,x)dx\\geq \\frac{\\delta}{2}\\qquad\\forall\\tau \\in[\\tau',\\tau '+T_0(\\delta )].\n$$\nIt  follows from (\\ref{estimate for contradiction}) that\n$\n1-\\frac{\\delta}{2}>\\frac{\\delta^3}{64}(\\tau-\\tau')$ for all $\\tau\\in[\\tau',\\tau '+T_0(\\delta )]$\nwhich is a contradiction for $\\tau=\\tau '+T_0(\\delta )$. \n\\end{proof}\n\n\n\\begin{proposition}\n\\label{S5LB}\nLet $h$ be a solution of  (\\ref{S1E16ha}). Suppose that there exist $m$, $R>0$ such that\n\\begin{equation}\n\\int_{[0,R]}h(\\tau,x)dx\\geq m\\qquad\\forall\\tau\\in[0,\\infty).\n\\end{equation}\nThen given any $\\alpha\\in(0,1)$ there exists $T_*=T_*(\\alpha)>0$ such that\n\\begin{align}\n\\label{S5ELB}\n\\int_{[0,r]}h(\\tau,x)dx\\geq \\frac{m}{(2R)^{\\alpha}}\\,r^{\\alpha}\\qquad\\forall r\\in[0,R],\\quad\\forall\\tau\\in\\bigg[\\frac{R T_*}{m},\\infty\\bigg).\n\\end{align}\n\\end{proposition}\n\\begin{proof}\nWe argue by induction and define first the scaled measure  $h_1=h _{ \\kappa_1, \\lambda _1 } $, defined as in (\\ref{S5Escaled}), that satisfies condition (\\ref{S5HC}) for $\\kappa_1=\\frac {1} {m},\\,\\,\\,\\lambda _1=R.$\nFrom Lemma \\ref{rescaled}, and Lemma \\ref{concentration lemma} with $\\tau '=0$, we deduce that for all $\\delta \\in (0, 1)$ there exists  $\\tau _1>0$ such that:\n\\begin{align*}\n&0\\leq\\tau _1\\leq T_0(\\delta),\\quad \\int_{\\left[0,\\frac{\\delta}{4}\\right]}h_1(\\tau_1,x)dx\\geq 1-\\frac{\\delta}{2}.\n\\end{align*}\nThen from Lemma \\ref{rescaled}, and Proposition \\ref{S5P1} with $\\theta=\\delta \/2$ and $R=1\/2$,\n\\begin{align}\n\\int_{\\left[0,\\frac{1}{2}\\right]}h_1(\\tau,x)dx\\geq \\left(1-\\frac{\\delta}{2}\\right)^2,\\,\\,\\,\\forall \\tau \\geq T_0(\\delta ),\\nonumber\\\\\n\\label{S5EIt1}\n\\int_{\\left[0,\\frac{R}{2}\\right]}h\\left(\\tau , x\\right)dx\\geq m\\left(1-\\delta \\right),\\,\\,\\,\\forall \\tau \\geq \\frac {R} {m}T_0(\\delta ).\n\\end{align}\nExactly as before we now define $h_2=h _{ \\kappa_2, \\lambda _2 } $ as in (\\ref{S5Escaled}), that satisfies condition (\\ref{S5HC}) for\n$\\kappa_2=\\frac {1} {m(1-\\delta )^2},\\,\\,\\,\\lambda _2=\\frac {R} {2},\\,\\,\\,\\tau '= 2(1-\\delta )T_0(\\delta ).$\nThe same argument gives then:\n\\begin{equation}\n\\label{S5EIt2}\n\\int_{\\left[0,\\frac{R}{4}\\right]}h\\left(\\tau , x\\right)dx\\geq m\\left(1-\\delta \\right)^2,\\quad\\forall \\tau \\geq \\frac {R T_0(\\delta )} {m}\\left(1+\\frac {1} {2(1-\\delta )} \\right).\n\\end{equation}\nWe deduce after $n$ iterations\n\\begin{align}\n\\label{S5EItn}\n\\int_{\\left[0,\\frac{R}{2^n}\\right]}h\\left(\\tau , x\\right)dx\\geq m\\left(1-\\delta \\right)^{n},\\quad\\forall \\tau \\geq \\frac {R T_0(\\delta )} {m}\n\\sum_{ k=0 }^{n-1}\\frac {1} {2^{k}(1-\\delta )^k}\n\\end{align}\nIf we chose $\\delta=1-2^{-\\alpha }$, for any $0<\\alpha <1$, we may  define\n\\begin{eqnarray}\n\\label{S5ET*}\nT_*=T_0(\\delta )\\sum_{ k=0 }^{\\infty}2^{-(1-\\alpha)  k}= \\frac {T_0(\\delta )} {1-2^{-(1-\\alpha) }}.\n\\end{eqnarray}\nSince for any $r\\in (0, R)$ there exists $n\\in \\mathbb{N}$ such that $r\\in \\left(\\frac{R}{2^n} ,\\frac{R}{2^{n-1}}\\right]$,\n\\begin{align*}\n\\int_{\\left[0, r \\right]}h\\left(\\tau , x\\right)dx\\geq m2^{-\\alpha n} \n,\\,\\,\\,\\forall \\tau > \\frac {R T_*} {m}\n\\end{align*}\nand using $2^{-n}>r\/2R$, (\\ref{S5ELB}) follows.\n\\end{proof}\n\n\n\\begin{proposition}\n\\label{S5PLB}\nLet $h$ be a solution of  (\\ref{S1E16ha}).  Then, for all $\\tau_0>0$ and for any  $\\alpha\\in(0,1)$ there exists \n$R_*=R_*(h,\\tau_0,\\alpha)>0$ such that\n\\begin{align}\n\\int_{[0,r]}h(\\tau,x)\\,\\mathrm{d} x\\geq C\\,r^{\\alpha}\\qquad\\forall r\\in[0,R_*]\\quad\\forall \\tau\\in[\\tau_0,\\infty),\n\\end{align} \nwhere $C=\\frac{T_*(\\alpha)}{\\tau_0}(2R_*)^{1-\\alpha}$, and $T_*(\\alpha)$ is given by Proposition \\ref{S5LB}. \n\\end{proposition}\n\\begin{proof}\nBy Proposition \\ref{S4P47} with $L>0$ and for $\\tau _1=\\tau _0\/2 $, there exists $R_0(h,L,\\tau _1)>0$ such that\n\\begin{align*}\n\\int_{[0,R_0]}h(\\tau ,x)dx\\geq LR_0\\qquad\\forall\\tau\\geq\\frac{\\tau_0}{2}.\n\\end{align*}\nThen by Proposition \\ref{S5LB}, with $m=LR_0$ and $R=R_0$, we obtain that for any given $\\alpha\\in(0,1)$ there exists $T_*=T_*(\\alpha)>0$ such that\n\\begin{align*}\n\\int_{[0,r]} h(\\tau,x)\\,\\mathrm{d} x\\geq \\frac{LR_0}{(2R_0)^{\\alpha}}\\,r^{\\alpha}\\qquad\\forall r\\in[0,R_0],\\quad\\forall \\tau\\in\\bigg[\\frac{\\tau_0}{2}+\\frac{ T_*}{L},\\infty\\bigg).\n\\end{align*}\nIf we chose $L=2T_*\/\\tau_0$, then the Proposition follows with $R_*=R_0$.\n\\end{proof}\n\n\\begin{proof}[\\upshape{\\bfseries{Proof of Theorem \\ref{S1T4h}}}]\nBy Lemma \\ref{S6L1'} the map $\\tau\\mapsto h(\\tau,\\{0\\})$ is right continuous, nondecreasing and a.e. differentiable on $[0,\\infty)$. It remains to prove that it is actually strictly increasing.\nWe  first suppose that  $h_0$ is such that \n\\begin{eqnarray}\n\\label{S5E54}\n\\int_{\\{0\\}}h_0(x)dx=0,\\qquad \\int_{(0,\\infty)}h_0(x)dx>0,\n\\end{eqnarray}\nand prove \n\\begin{equation}\n\\label{S1ET4h}\nh(\\tau,\\{0\\})>0\\qquad\\forall \\tau>0.\n\\end{equation}\nArguing by contradiction, if we suppose that there exists $\\tau_0>0$ such that $h(\\tau_0,\\{0\\})=0$, by monotonicity \n$h(\\tau,\\{0\\})=0$ for all $\\tau\\in[0,\\tau_0]$. In particular\n\\begin{align}\n\\int_{\\frac{\\tau_0}{2}}^{\\tau_0}\\int_{[0,r]}h(\\sigma,x)dxd\\sigma=\\int_{\\frac{\\tau_0}{2}}^{\\tau_0}\\int_{(0,r]}h(\\sigma,x) dxd\\sigma\n\\end{align}\nfor all $r>0$. Now using  Proposition \\ref{S5P2} with $\\alpha=0$, and Proposition \\ref{S5PLB},  we deduce that, for any $\\alpha\\in(0,1\/2)$, there exists $R_*=R_*(h, \\tau _0\/2, \\alpha )$ such that\n\\begin{align*}\n&C_2\\,r^{\\alpha}\\leq \\int_{\\frac{\\tau_0}{2}}^{\\tau_0}\\int_{(0,r]}h(\\sigma,x)dxd\\sigma \\leq C_1\\sqrt{r},\n\\qquad\\forall r\\in[0,R_*];\\\\\n&C_1=8\\, \\sqrt{\\frac{\\tau_0}{2}} \\left( \\frac {\\sqrt{M_1(h_0)}} {2}\\tau_0 +\\sqrt{M_0(h_0)}\\right),\\quad C_2=\\frac{T_*(\\alpha)}{2}(2R_*)^{1-\\alpha},\n\\end{align*}\nand that leads to a contradiction for $r$ small enough.\n\nConsider now a general initial data $h_0$ such that $\\int_{\\{0\\}}h_0(x)dx>0$. Let $h$ be a solution of (\\ref{S1E16ha}) with initial data $h_0$ and define\n$$\n\\tilde h(\\tau )=h(\\tau )-h_0(\\{0\\})\\delta _0.\n$$\nThen, on the one hand, the initial data of $\\tilde h$  satisfies $\\tilde h(0, \\{0\\})=0$. On the other hand we claim that $\\tilde h$ is still a solution of   (\\ref{S1E16ha}). Notice indeed that $\\tilde h_\\tau \\equiv h_\\tau $ and, moreover,\n$\\widetilde{\\mathscr{Q}}_3(\\varphi,h(\\tau))=\\widetilde{\\mathscr{Q}}_3(\\varphi,\\tilde h(\\tau))$. Using the previous case\n$$\n\\int  _{ \\{0\\}}\\tilde h(\\tau, x)dx>0,\\quad\\forall \\tau >0,\n$$\nand then\n$$\n\\int  _{ \\{0\\}} h(\\tau, x)dx>\\int  _{ \\{0\\}} h_0(x)dx,\\quad\\forall \\tau >0.\n$$\nThe Theorem follows using now the time translation invariance of the equation.\n\\end{proof}\n\n\nThe last result of this section describes the relation between the Lebesgue-Stieltjes measure associated to the (right continuous and strictly increasing) function\n$m(\\tau)=h(\\tau,\\{0\\})$, and the equation for $h$ (\\ref{S1E16ha}).\n\n\\begin{proposition}\n\\label{Stieltjes1}\nLet $h$ be a solution of (\\ref{S1E16ha}) for a initial data $h_0\\in\\mathscr{M}_+^1([0,\\infty))$ with $N=M_0(h_0)>0$ and $E=M_1(h_0)>0$.\nIf we denote $m(\\tau)=h(\\tau,\\{0\\})$ and $\\lambda$ is the Lebesgue-Stieltjes measure associated to $m$, then for all $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST} and for all $\\tau_1$ and $\\tau_2$ with $0\\leq\\tau_1<\\tau_2$:\n\\begin{align}\n\\label{mm1}\n&m(\\tau_2)-m(\\tau_1)=\\lambda((\\tau_1,\\tau_2]),\\\\\n\\label{Stieltjes3}\n&\\lambda((\\tau_1,\\tau_2])=\\lim_{\\varepsilon\\to 0}\\int_{\\tau_1}^{\\tau_2}\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h(\\tau))d\\tau,\n\\end{align}\n\\begin{flalign}\n\\label{Stieltjes0}\n&\\text{and}&&0<\\lambda((\\tau_1,\\tau_2]))<\\infty.&&\n\\end{flalign} \nFurthermore, for all $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST}\n\\begin{align}\n\\label{Stieltjes7}\n\\lim_{\\varepsilon\\to 0}\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h)\\in\\mathscr{D}'(0,\\infty),\n\\end{align}\nand if we denote $m'$ the  derivative in the sense of Distributions of $m$, then \n\\begin{align}\n\\label{Stieltjes2}\nm'=\\lambda=\\lim_{\\varepsilon\\to 0}\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h)\\quad\\text{in}\\quad\\mathscr{D}'(0,\\infty).\n\\end{align}\n\\end{proposition}\n\n\\begin{proof}\nBy Lemma \\ref{S6L1'}, $m$ is right continuous and nondecreasing on $[0,\\infty)$. Then it has a Lebesgue-Stieltjes measure associated to it, $\\lambda$, that satisfies (\\ref{mm1}) (c.f.  \\cite{Fo} Ch.1).\n\nOn the other hand, since $h$ is a solution of (\\ref{S1E16ha}), using $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST} and taking the limit $\\varepsilon\\to 0$, it follows from (\\ref{limQ31b}) in Lemma \\ref{convergence lemma} that for all $\\tau_1$ and $\\tau_2$ with $0\\leq\\tau_1<\\tau_2$:\n\\begin{align}\n\\label{ZE3}\nm(\\tau_2)-m(\\tau_1)=\\lim_{\\varepsilon\\to 0}\\int_{\\tau_1}^{\\tau_2}\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h(\\sigma))d\\sigma,\n\\end{align}\nand then (\\ref{Stieltjes3}) follows from (\\ref{mm1}). Moreover, since by Theorem \\ref{S1T4h} $m$ is strictly increasing, then (\\ref{Stieltjes0}) holds.\n\nNotice that the limit in (\\ref{ZE3}) is independent of the choice of the test function $\\varphi_{\\varepsilon}$. Indeed, if $\\psi_{\\varepsilon}$ is another test function as in Remark \\ref{TEST}, since for all $\\tau\\geq 0$\n\\begin{align*}\n\\lim_{\\varepsilon\\to 0}\\int_{[0,\\infty)}\\psi_{\\varepsilon}(x)h(\\tau,x)dx=m(\\tau)=\\lim_{\\varepsilon\\to 0}\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau,x)dx,\n\\end{align*}\nit follows from (\\ref{ZE3}) that for all $0\\leq \\tau_1\\leq\\tau_2$\n\\begin{align*}\n\\lim_{\\varepsilon\\to 0}\\int_{\\tau_1}^{\\tau_2}\\mathscr{Q}_3^{(2)}(\\psi_{\\varepsilon},h(\\sigma))d\\sigma=\n\\lim_{\\varepsilon\\to 0}\\int_{\\tau_1}^{\\tau_2}\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h(\\sigma))d\\sigma.\n\\end{align*}\n\nNow, for all $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST}, consider the absolutely continuous function \n\\begin{align*}\n\\theta_{\\varepsilon}(\\tau)=\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)h(\\tau,x)dx.\n\\end{align*}\nThen the equation in (\\ref{AUXW}) reads $\\theta_{\\varepsilon}'(\\tau)=\\widetilde{\\mathscr{Q}}_3(\\varphi_{\\varepsilon},h(\\tau))\n$.\nUsing integration by parts we deduce that for all $\\varepsilon>0$:\n\\begin{align*}\n-\\int_0^{\\infty}\\phi'(\\tau)\\theta_{\\varepsilon}(\\tau)d\\tau=\\int_0^{\\infty}\\phi(\\tau)\\widetilde{\\mathscr{Q}}_3(\\varphi_{\\varepsilon},h(\\tau))d\\tau\n\\quad\\forall \\phi\\in C^{\\infty}_c(0,\\infty).\n\\end{align*}\nTaking the limit $\\varepsilon\\to 0$ it then follows from Lemma \\ref{convergence lemma} that\n\\begin{align*}\n-\\int_0^{\\infty}\\phi'(\\tau)m(\\tau)d\\tau=\\lim_{\\varepsilon\\to 0}\\int_0^{\\infty}\\phi(\\tau)\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h(\\tau))d\\tau,\n\\end{align*}\nhence, $m'=\\lim_{\\varepsilon\\to 0}\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},h)$. On the other hand, by Fubini's theorem\n\\begin{align*}\n\\int_0^{\\infty}\\phi(\\tau)d\\lambda(\\tau)=\\int_0^{\\infty}\\int_0^{\\tau}\\phi'(\\sigma)d\\sigma d\\lambda(\\tau)\n=-\\int_0^{\\infty}\\phi'(\\sigma)m(\\sigma)d\\sigma\n\\end{align*}\nfor all $\\phi\\in C^{\\infty}_c(0,\\infty)$ (cf. \\cite{WR3}, Example 6.14), thus $m'=\\lambda$.\n\\end{proof}\n\n\\section{Existence of solutions $G$, proof of  Theorem \\ref{S1T1}.}\n\\label{sectionG}\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\n\nGiven a initial data $G_0\\in \\mathscr M_+^1$ as in Theorem \\ref{S1T1}, let $h\\in C\\big([0,\\infty), \\mathscr{M}_+([0,\\infty))\\big)$ satisfy (\\ref{lip loc h})--(\\ref{EE}),  (\\ref{S5Ealpha }), given by (\\ref{S5C52R}) and $H$ defined by (\\ref{DEFH}) and satisfying  (\\ref{lip loc H})--(\\ref{EEH}), (\\ref{S5EalphaR }) by Corollary \\ref{S5C52R}.  It is natural, in view of the change of variables  (\\ref{S1E45b})  to define now, \n\\begin{equation}\n\\label{S6EG1}\nG(t) =H(\\tau ),\\quad\\tau =\\int _0^tG(s, \\{0\\})ds.\n\\end{equation}\nNotice nevertheless that since $G(s, \\{0\\})$ is still unknown,  (\\ref{S6EG1}) does not define $G(t)$ actually.  What we know is rather, given $\\tau >0$, what would be the value of $t$ such that \n\\begin{equation}\n\\label{S6EtCh}\nt =\\int _0^\\tau \\frac {d\\sigma  } {H(\\sigma  , \\{0\\})},\n\\end{equation}\nsince we expect to have $G(s,\\{0\\})=H(\\sigma , \\{0\\})$ for $s$ and $\\sigma $ such that\n\\begin{eqnarray*}\n\\sigma =\\int _0^sG(r, \\{0\\})dr,\\quad\\hbox{or}\\quad s=\\int _0^\\sigma \\frac {d\\rho } {H(\\rho , \\{0\\})}.\n\\end{eqnarray*}\nIf $G$ is going to be defined in that way it is then necessary first to check that the range of values taken by the variable $t$ in (\\ref{S6EtCh}) is all of $[0, \\infty)$. By definition (\\ref{DEFH}),\n\\begin{equation}\n\\label{{DEFH}2}\nH(\\tau,\\{0\\})=h(\\tau, \\{0\\})-\\int_0^\\tau M_{1\/2}(h(\\sigma))d\\sigma.\n\\end{equation}\nSince both terms in the right hand side are nonnegative, $H(\\tau, \\{0\\})$ has no a priori  definite sign. We must then consider that question in some detail.\nOur first step is to prove the following\n\n\\begin{lemma}\n\\label{S6C1}\nIf $G_0(\\{0\\})>0$, then\n\\begin{align}\n\\label{TAU1}\n&\\tau_*=\\inf\\{\\tau>0:H(\\tau,\\{0\\})=0\\}>0,\\\\\n\\label{TAU2}\n&H(\\tau_*,\\{0\\})=0,\\\\\n\\label{TAU3}\n&H(\\tau,\\{0\\})>0\\qquad\\forall\\tau\\in[0,\\tau_*).\n\\end{align}\n\\end{lemma}\n\\begin{proof}\n$H(0)=G_0$ by (\\ref{S5IDh}), and then, using $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST}, we deduce $H(0,\\{0\\})=G_0(\\{0\\})$, which is strictly positive by hypothesis. Then (\\ref{TAU1}) follows from the right continuity of $H(\\tau,\\{0\\})$ (cf. Corollary \\ref{S6L1}).\n\nIn order to prove (\\ref{TAU2}) we use a minimizing sequence $(\\tau _n)_{ n\\in \\mathbb{N} }$, i.e., $\\tau _n\\ge \\tau _*$, $H(\\tau _n,\\{0\\})=0$ for every $n\\in \\mathbb{N}$,  and $\\tau _n\\to \\tau _*$ as $n\\to \\infty$. Then from the right continuity (\\ref{TAU2}) holds.\n\nLet us prove now (\\ref{TAU3}). If $H(\\tau_0 , \\{0\\})<0$ for some $\\tau_0 \\in (0, \\tau_*)$, then $\\tau _0$ must be a left discontinuity point of $H(\\tau, \\{0\\})$ and\n\\begin{equation*}\n\\limsup_{\\delta\\rightarrow 0^+}H(\\tau_0-\\delta,\\{0\\})> H(\\tau_0,\\{0\\}),\n\\end{equation*}\nand this would contradict (\\ref{JIM}). That proves (\\ref{TAU3}).\n\\end{proof}\n\nIt follows from Lemma \\ref{S6C1} that the function:\n\\begin{equation}\nt =\\xi (\\tau )=\\int _0^\\tau \\frac {d\\sigma  } {H(\\sigma  , \\{0\\})}\n\\end{equation}\nintroduced  in (\\ref{S6EtCh}) is well defined, monotone nondecreasing  and continuous on  the interval $ [0, \\tau _*)$. We  then define,\n\\begin{align}\n\\label{S6DG2}\n\\forall t\\in [0, \\xi (\\tau_*)):\\quad G(t)=H(\\xi ^{-1}(t)).\n\\end{align}\nBy (\\ref{S6DG2}) and (\\ref{{DEFH}2}), if $G(t)=G(t,\\{0\\})\\delta _0+g(t)$ and $H(\\tau)=H(\\tau , \\{0\\})\\delta _0+\\tilde h(\\tau )$, then\n\\begin{align}\n&G(t,\\{0\\})=H(\\tau , \\{0\\}), \\label{S6DG2bis}\\\\\n&\\tilde h(\\tau )=h(\\tau )-h(\\tau , \\{0\\})\\delta_0, \\label{S6DG20}\\\\\n&g(t)=\\tilde h(\\tau ).\\label{ght}\n\\end{align}\n\n\\begin{remark}\nFormula (\\ref{S6DG2}) defines the function $G$ at time $t\\in (0, \\xi (\\tau _*))$ from the knowledge of the function $H(\\tau )$ for $\\tau >0$ such that $\\tau =\\xi ^{-1}(t)$. Moreover,\n\\begin{equation}\n\\forall t\\in (0, \\xi (\\tau _*)): \\quad \\xi ^{-1}(t)=\\int _0^t G(s, \\{0\\})ds.\n\\end{equation}\n\\end{remark}\nWe have now,\n\n\\begin{proposition}\n\\label{S6P3G1}\nThe function  $G$ defined by (\\ref{S6DG2}) is  such that\n\\begin{eqnarray}\nG\\in C\\big([0, \\xi (\\tau _* )),\\mathscr{M}_+^1([0,\\infty))\\big),\\,\\,\\,G(0)=G_0\n\\end{eqnarray} \nand satisfies (\\ref{S1ED3S}), (\\ref{S1E16}), (\\ref{S1E210}) and (\\ref{S1E220}) on the time interval $[0, \\xi (\\tau_* ))$.\n\\end{proposition}\n\n\\begin{proof}\nWe first prove that $G(t)$ is a positive measure for all $t\\in[0,\\xi(\\tau_*))$. By (\\ref{TAU3}) and (\\ref{S6DG2bis}) we have $G(t,\\{0\\})>0$ for all $t\\in[0,\\xi(\\tau_*))$. Then, since $h(\\tau)$ is a positive measure for all $\\tau\\in[0,\\infty)$, we deduce from (\\ref{ght}) and (\\ref{S6DG20}) that $g(t)$ is a positive measure for all $t\\in[0,\\xi(\\tau_*)).$ Hence $G(t)=G(t,\\{0\\})\\delta_0+g(t)$ is also positive. \n\n\nAll the properties of $G(t)$ at $t\\in [0, \\xi (\\tau_*))$ fixed follow from the corresponding property of $H(\\tau )$ with $t=\\xi(\\tau )$. The only property where $t$ is not fixed are (\\ref{S1ED2S}) and (\\ref{S1ED3S}). Since\n\\begin{equation*}\n\\left|\\frac {\\partial G(t)} {\\partial t}\\right|= \\left|\\frac {\\partial \\tau } {\\partial t}\\frac {\\partial H(\\tau )} {\\partial \\tau }\\right|\n\\le |H(\\tau , \\{0\\})|\\left|\\frac {\\partial H(\\tau )} {\\partial \\tau }\\right|\n\\end{equation*}\nBy definition,\n\\begin{eqnarray*}\n|H(\\tau , \\{0\\})|\\le |h(\\tau, \\{0\\})|+\\int_0^\\tau M_{1\/2}(h(\\sigma))d\\sigma.\n\\end{eqnarray*}\nSince $h\\in C([0, \\infty), \\mathscr M_+^1)$ it follows using also (\\ref{MMI}),  (\\ref{EE}) and H\\\"older inequality that \n$H(\\tau , \\{0\\})\\in L^\\infty _{ loc }([0, \\infty))$.Then, by  (\\ref{lip loc h}), $G(t)$ is locally Lipschitz on $[0, \\xi (\\tau _*))$ and satisfies (\\ref{S1ED3S}).\nSince $H$ satisfies (\\ref{AUXW}) the change of variables ensures that $G$ satisfies (\\ref{S1E16}).\n\\end{proof}\nWe prove now the following property of the function $G$ defined in (\\ref{S6DG2}).\n\\begin{proposition}\n\\label{origin G}\nLet  $G$ be the function defined in  (\\ref{S6DG2}) for $t\\in (0, \\xi (\\tau _*))$. Then the map $t\\mapsto G(t,\\{0\\})$ is right continuous and differentiable for almost every   $ t\\in [0,\\xi (\\tau _*))$ and, for all $t_0\\in (0, \\xi (\\tau _*))$ \n\\begin{align}\n\\label{CMI}\nG(t,\\{0\\})\\geq G(t_0,\\{0\\})e^{-\\int_{t_0}^tM_{1\/2}(g(s))d s}\\quad\\forall t\\in (t_0, \\xi (\\tau _*)).\n\\end{align} \nIn particular, if $G(0, \\{0\\})>0$, then $G(t,\\{0\\})> 0$ for all $t\\in (0, \\xi (\\tau _*))$.  \n\\end{proposition}\n\\begin{proof}\nUsing (\\ref{S1E16}) and (\\ref{S1EB1}) with $\\varphi _{\\varepsilon}$ as in Remark \\ref{TEST}, we have\n\\begin{align}\n\\frac{d}{dt}\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(t,x)d x+G(t,\\{0\\})M_{1\/2}&(G(t))\n=G(t,\\{0\\}) \\widetilde{\\mathscr Q}_3(\\varphi _\\varepsilon,G(t))\n.\\label{S6E78}\n\\end{align}\nWe use now that for all $\\varepsilon >0$:\n\\begin{align}\n\\label{ZAS}\nG(t,\\{0\\})\\leq \\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(t,x)d x,\n\\end{align}\nand we deduce from (\\ref{S6E78}), using  $J(t)=\\exp\\left({\\int_0^t M_{1\/2}(G(s))d s}\\right)$,\n\\begin{align}\n\\label{S6E79}\n&\\frac{d}{d t}\\bigg(J(t)\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(t,x)d x\\bigg) \\geq G(t,\\{0\\})J(t)  \\widetilde{\\mathscr Q}_3(\\varphi _\\varepsilon,G(t)).\n\\end{align}\nBy Lemma \\ref{convex-positivity} the right hand side of (\\ref{S6E79}) is nonnegative, and we deduce\n\\begin{equation*}\n\\label{S6E80}\nJ(t)\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(t,x)d x \\geq J(t_0)\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(t_0,x)d x\n\\end{equation*}\nfor all $t\\in (t_0, \\xi (\\tau _*))$ and all $\\varepsilon >0$.  If we pass now to the limit as $\\varepsilon \\to 0$:\n\\begin{equation}\n\\label{S6E80}\nJ(t)G(t,\\{0\\}) \\geq J(t_0)G(t_0,\\{0\\}),\n\\end{equation}\nand this proves the estimate (\\ref{CMI}). It  also follows from Lebesgue's Theorem that $J(t)G(t,\\{0\\})$ is differentiable for almost every $t\\in (0, \\xi (\\tau _*))$ (cf. \\cite{Roy},  Theorem 2). On the other hand, since $J(t)$ is a.e differentiable and $J(t)>0$ for all $t\\in [0, \\xi (\\tau _*))$, we deduce that $G(t,\\{0\\})$ is also differentiable for almost every \n$t\\in [0, \\xi (\\tau _*))$.\n\nWe prove now the  right continuity  of $G(t, \\{0\\})$. It follows from  (\\ref{S6E80}),\n\\begin{equation*}\nJ(t+\\delta)G(t+\\delta,\\{0\\}) \\geq J(t)G(t,\\{0\\}),\\quad\\forall \\delta>0\\,\\,\\forall t>0.\n\\end{equation*}\nIf we take inferior limits and use that $J$ is continuous and strictly positive we obtain,\n\\begin{equation}\n\\label{S6E578}\n\\liminf _{ \\delta\\to 0 }G(t+\\delta,\\{0\\}) \\geq G(t,\\{0\\}),\\quad\\forall t>0.\n\\end{equation}\n\nSince $\\mathcal L_0(\\varphi_{\\varepsilon})\\ge 0$ by convexity (cf. Lemma \\ref{convex-positivity}), we deduce \n\\begin{align*}\n\\frac{d}{d t}&\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(t,x)d x\n\\leq G(t,\\{0\\})\\iint_{(0,\\infty)^2}\\frac{\\Lambda(\\varphi_{\\varepsilon})(x,y)}{\\sqrt{x y}}G(t,x)G(t,y)d xd y,\n\\end{align*}\nand the argument  follows now as in the proof of the right continuity of $H$. \nFrom the inequality (\\ref{ZAS}), the bound \\eqref{lemma regularity 1} and the conservation of mass, we deduce for all  $t\\in [0, \\xi (\\tau _*))$ fixed  and  $\\delta\\in[0, \\xi (\\tau _*)-t)$,\n$$\nG(t+\\delta,\\{0\\})\\leq\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(t,x)d x\n+\\frac {2N^2\\delta} {\\varepsilon }\\int_0^t G(s,\\{0\\})ds.\n$$\nIf we take superior limits as $\\delta\\to 0$, and then let $\\varepsilon \\to 0$ we obtain, using  (\\ref{S6L1E1}) with $G$ instead of $H$:\n$$\n\\limsup _{ \\delta\\to 0 }G(t+\\delta,\\{0\\})\\leq G(t, \\{0\\}).\n$$\nand this combined with (\\ref{S6E578}) proves that  $G(t, \\{0\\})$ is right continuous on $[0, \\xi (\\tau _*))$.\n\\end{proof}\n\n\nIn the next Lemma we prove that the function $G$ defined by (\\ref{S6DG2}) is actually well defined for all $t>0$.\n\\begin{lemma}\n\\label{S6LG}\n\\begin{equation}\n\\lim _{ \\tau \\to \\tau _*^- }\\xi (\\tau )=\\infty.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nSince the function $\\xi (\\tau )$ is monotone  nondecreasing and continuous on $[0, \\tau _*)$, its limit as $\\tau \\to \\tau _*^-$ exists in $\\overline{\\mathbb{R}}_+$. Let us call it $\\ell$ and  suppose  $\\ell\\in \\mathbb{R}_+$. Now, from (\\ref{CMI}) and the fact that $G$ satisfies :\n$0\\le M _{ 1\/2 }(G(s))\\le \\sqrt {N E}$,\nwe deduce \n\\begin{equation}\n\\label{S6C56E1}\n\\limsup _{t\\to \\ell^-  }G(t, \\{0\\})\\ge e^{-\\sqrt{NE} \\ell}G(0, \\{0\\})>0,\n\\end{equation}\nand by (\\ref{JIM})\n$$\nH(\\tau_*, \\{0\\})\\ge \\limsup _{\\tau \\to \\tau _*^-  }H(\\tau, \\{0\\})=\\limsup _{ t\\to \\ell^- }G(t, \\{0\\})>0,\n$$\nand this contradicts (\\ref{TAU2}). This proves that $\\ell=\\infty$.\n\\end{proof}\n\n\\begin{proof}[\\upshape\\bfseries{Proof of Theorem \\ref{S1T1}}] \n By Lemma \\ref{S6LG} the function $G$ is defined for all $t>0$. As we have seen in the proof of Lemma \\ref{S6LG}, $G(t)\\in \\mathscr M_+([0, \\infty))$ for all $t>0$. It then follows from Proposition \\ref{S6P3G1} that $G$ satisfies now all the conditions (\\ref{S1ED2S})--(\\ref{S1E16}) and (\\ref{S1T1E0})--(\\ref{S1E220}).\nProperty (\\ref{S1E23}) follows from the corresponding estimate (\\ref{MAh}) for $h$. Similarly, property (\\ref{S5Ealphahh}) follows from the property (\\ref{S5Ealpha }) of $h$.\nWe prove now the point (iv). Suppose then $\\alpha \\in (1, 3]$ and condition (\\ref{PRO112}). For $\\varphi(x)=x^{\\alpha}$ we have,\n$$\n\\mathscr{Q}_3^{(1)}(\\varphi,G(t))=\\left(\\frac{\\alpha-1}{\\alpha+1}\\right) M_{\\alpha+\\frac{1}{2}}(G(t)).\n$$\nOn the other hand, for $0\\le y\\le x$,\n$$\n\\Lambda(\\varphi)(x,y)=x^{\\alpha}\\left(\\big(1+z\\big)^{\\alpha}+\\big(1-z\\big)^{\\alpha}-2\\right),\\quad z=\\frac{y}{x}\\in[0,1],\n$$\nIf  $\\alpha\\in(1,2]$, for all $x\\ge y >0$,\n$$\n\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}\\leq (2^{\\alpha}-2) x^{\\alpha-\\frac{3}{2}}y^{\\frac{1}{2}}\\le  (2^{\\alpha}-2) (xy)^{\\frac{\\alpha-1}{2}}.\n$$\nWe deduce\n$$\n\\mathscr{Q}_3^{(2)}(\\varphi,G(t))\\leq (2^{\\alpha}-2) \\Big( M_{\\frac{\\alpha-1}{2}}(G(t))\\Big)^2.\n$$\nand obtain\n\\begin{align*}\n\\frac{d}{dt}M_{\\alpha}(G(t))\\leq G(t,\\{0\\})\\left[C_{1,1}\\Big(M_{\\frac{\\alpha-1}{2}}(G(t))\\Big)^2-C_2M_{\\alpha+\\frac{1}{2}}(G(t))\\right],\n\\end{align*}\nwhere $C_{1,1}=2^{\\alpha}-2$ and $C_2=(\\alpha-1)\/(\\alpha+1).$\nUsing H\\\"{o}lder's inequality\n\\begin{align}\n\\label{CT98}\n\\frac{d}{dt}M_{\\alpha}(G(t))\\leq G(t,\\{0\\})\\Big[C_{1,1}N^{3-\\alpha}E^{\\alpha-1}\n-C_2E^{(2\\alpha+1)\/2}N^{(1-2\\alpha)\/2}\\Big].\n\\end{align}\nBy (\\ref{PRO112}), the right hand side of (\\ref{CT98}) is negative, and then $M_{\\alpha}(G(t))$ is decreasing on $(0,\\infty)$.\n\nFor $\\alpha\\in[2,3]$ we use the estimate (\\ref{MAQ}) with $C_{1,2}=\\alpha(\\alpha-1)$ instead of $C_{\\alpha}$. Then we proceed as in the previous case to obtain\n\\begin{equation}\n\\label{CT99}\n\\frac{d}{dt}M_{\\alpha}(G(t))\\leq G(t,\\{0\\})\\Big[C _{ 1,2 }N^{3-\\alpha}E^{\\alpha-1}\n-C_2E^{(2\\alpha+1)\/2}N^{(1-2\\alpha)\/2}\\Big].\n\\end{equation}\nAs before, (\\ref{PRO112}) implies that the right hand side of (\\ref{CT99}) is negative, and then $M_{\\alpha}(G(t))$ is decreasing.\n\\end{proof}\n\n\n\\begin{proof}[\\upshape\\bfseries{Proof of Theorem \\ref{S1Treg}}] \nBy construction\n\\begin{align*}\nG(t)=H(\\tau)=h(\\tau)-\\left(\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma\\right)\\delta_0,\n\\end{align*}\nwhere $\\tau$ and $t$ are related by\n\\begin{align}\n\\label{CHANGE}\nt=\\xi(\\tau)=\\int_0^{\\tau}\\frac{d\\sigma}{H(\\sigma,\\{0\\})};\\qquad \\tau=\\xi^{-1}(t)=\\int_0^t G(s,\\{0\\})ds.\n\\end{align}\nTherefore $G(t,x)=h(\\tau,x)$ for $x\\in(0,\\infty)$, and \n\\begin{align*}\n\\int_0^{T}G(t,\\{0\\})\\int_{(0,\\infty)}x^{\\alpha}G(t,x)dxdt\n=\\int_0^{\\xi^{-1}(T)}\\int_{(0,\\infty)}x^{\\alpha}h(\\tau,x)dxd\\tau.\n\\end{align*} \nThe result  then follows from Proposition \\ref{S5P2}.\n\\end{proof}\n\n\\begin{remark}\nOne could try  to directly solve the  system (\\ref{PR}), (\\ref{PR19}), written in $(g, n)$ variables. First, to obtain a sequence of solutions $(g_k, n_k)$ of an approximated system where the factor $x^{-1\/2}$ is modified by truncation and regularization, and then pass to the limit. However, the limit obtained in that way, say $(g, n)$ is not a solution of (\\ref{PR}), (\\ref{PR19}). The reason is that all the solutions $g_k$ of the approximated system will be functions with a bounded moment of order $-1\/2$. Then, the right hand side of the equation (\\ref{S1E9}) is equal to $M _{ 1\/2 }(g_k)$ and by passage to the limit the equation for $n$ will be $n'(t)=-n(t)M _{ 1\/2 }(g(t))$, and  the total mass will not be conserved. \n\\end{remark}\n\n\n\\section{Proofs of Theorems \\ref{THn01}, \\ref{MU1} and \\ref{EQUIV}.}\n\\label{SectionK}\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\n\nWe first prove Theorem \\ref{THn01}. \n\n\\begin{proof}[\\upshape\\bfseries{Proof of Theorem \\ref{THn01} }]\nWe already know by Proposition \\ref{origin G} and Lemma \\ref{S6LG} that $n$ is right continuous and a.e. differentiable on $[0,\\infty)$.\nThen, by construction\n\\begin{align*}\nG(t)=H(\\tau)=h(\\tau)-\\left(\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma\\right)\\delta_0,\n\\end{align*}\nwhere $\\tau\\in[0,\\tau^*)$ and $t\\in[0,\\infty)$ are related by (\\ref{CHANGE}).\nHence\n\\begin{align}\n\\label{ZE2}\nn(t)=m(\\tau)-\\int_0^{\\tau}M_{1\/2}(h(\\sigma))d\\sigma=m(\\tau)-\\int_0^t n(s)M_{1\/2}(g(s))ds.\n\\end{align}\nSince $n(0)=m(0)$, it then follows from Proposition \\ref{Stieltjes1} that for all $t>0$:\n\\begin{align}\n\\label{ZEaa}\nn(t)-n(0)+\\int_0^t n(s)M_{1\/2}(g(s))ds=\\lambda((0,\\tau]),\n\\end{align}\nand using (\\ref{CHANGE})\n\\begin{align}\n\\label{ZEbb}\n\\lambda((0,\\tau])=\\lim_{\\varepsilon\\to 0}\\int_0^t n(s)\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},g(s))ds.\n\\end{align}\nIf we denote $\\mu=\\xi_{\\#}\\lambda$ (c.f. \\cite{AMB}, Ch. 5), i.e., the push-forward  of $\\lambda$ through the function $\\xi:[0,\\tau^*)\\to[0,\\infty)$ in (\\ref{CHANGE}), then from the definition of $\\mu$ we obtain\n\\begin{equation}\n\\label{Mul}\n\\mu((0,t])=\\lambda((0,\\tau])\\qquad\\forall t>0.\n\\end{equation}\nThen (\\ref{ZE02}) and  (\\ref{ZE01}) follows from (\\ref{ZEaa}), (\\ref{ZEbb}) and (\\ref{Mul}). Moreover, (\\ref{ZE00}) follows from (\\ref{Stieltjes0}) in Proposition \\ref{Stieltjes1}. \n\\end{proof}\n\nThe following properties of $n(t)$ follows by   the same arguments used in the proofs of properties (\\ref{Stieltjes7}) and (\\ref{Stieltjes2}) of Proposition \\ref{Stieltjes1} \n\n\\begin{proposition} \nLet $G$, $g$, and $n(t)$ be as in Theorem \\ref{THn01}.  Then, for all $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST}, the following limit exists \nin $\\mathscr{D}'(0,\\infty)$:\n\\begin{flalign}\n\\label{ZE04}\n&&&\\lim_{\\varepsilon\\to 0} n\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},g)=T(G),&&\\\\\n\\text{and}&&& \n\\label{ZE05}\nn'+nM_{1\/2}(g)=T(G)\\quad\\text{in}\\quad\\mathscr{D}'(0,\\infty).&&\n\\end{flalign}\n\\end{proposition}\n\n\\begin{proof}\nConsider, for all  $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST}, the absolutely continuous functions \n\\begin{align}\n\\eta_{\\varepsilon}(t)=\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(t,x)dx.\n\\end{align}\nThen equation (\\ref{S1E16}) becomes $\n\\eta_{\\varepsilon}'=n\\mathscr{Q}_3(\\varphi_{\\varepsilon},g).$\nUsing integration by parts,\n\\begin{align*}\n-\\int_0^{\\infty}\\phi'(t)\\eta_{\\varepsilon}(t)dt=\\int_0^{\\infty}\\phi(t)n(t)\\mathscr{Q}_3(\\varphi_{\\varepsilon},g(t))dt\n\\quad\\forall \\phi\\in C^{\\infty}_c(0,\\infty).\n\\end{align*}\nTaking the limit $\\varepsilon\\to 0$ we deduce, using Lemma \\ref{convergence lemma}, that\n\\begin{align*}\n-\\int_0^{\\infty}\\phi'(t)n(t)dt=&\\lim_{\\varepsilon\\to 0}\\int_0^{\\infty}\\phi(t)n(t)\\mathscr{Q}_3^{(2)}(\\varphi_{\\varepsilon},g(t))dt\\\\\n&-\\int_0^{\\infty}\\phi(t)n(t)M_{1\/2}(g(t))dt,\n\\end{align*}\nand then (\\ref{ZE04}), (\\ref{ZE05}) follows. \n\\end{proof}\n\n\\begin{remark}\nIf we take distributional derivatives in both sides of  (\\ref{ZE02}) we obtain:\n$$\nn'+n M _{ 1\/2 }(g)=\\mu\\quad\\text{in}\\quad\\mathscr{D}'(0,\\infty),$$\nand by (\\ref{ZE05}), $\\mu =T(G)$.\n\\end{remark}\n\n\\begin{proof}[\\upshape\\bfseries{Proof of Theorem  \\ref{MU1}}]\nThe statement of the Theorem follows from (\\ref{Stieltjes0}) in Proposition \\ref{Stieltjes1} and (\\ref{Mul}).\n\\end{proof}\n\n\\begin{proof}[\\upshape\\bfseries{Proof of Theorem \\ref{EQUIV}}]\nProof of part (i).\nBy Theorem \\ref{THn01}, $n$ is given by (\\ref{ZE02}) and (\\ref{ZE01}). On the other hand, since $G$ satisfies (\\ref{S1E16}), and for all\n$\\varphi\\in C^1_b([0,\\infty))$ such that $\\varphi(0)=0$:\n\\begin{align}\n\\label{G=g}\n\\int_{[0,\\infty)}\\varphi(x)G(t,x)dx=\\int_{[0,\\infty)}\\varphi(x)g(t,x)dx,\n\\end{align}\nthen $g$ satisfies (\\ref{2E765}). In order to prove part (ii) we first show the existence of the limit in (\\ref{ZE01}). To this end we write $\\varphi_\\varepsilon  = (1-\\psi _\\varepsilon )$, where  $\\psi _\\varepsilon$ is as in Remark \\ref{TEST}. Then $\\varphi _\\varepsilon (0)=0$, and by (\\ref{2E765}) and (\\ref{S1EB1}),\nusing that $\\mathscr{Q}_3(1-\\psi_{\\varepsilon},g)=\\mathscr{Q}_3(1,g)-\\mathscr{Q}_3(\\psi_{\\varepsilon},g)$, and $\\mathscr{Q}_3(1,g)=0$, we deduce\n\\begin{align}\n\\label{Bht}\n\\int _0^tn(s)\\widetilde{\\mathscr Q}_3(\\psi _\\varepsilon,g(s))ds&=\\int  _{ (0, \\infty) }\\varphi _\\varepsilon (x)\\left(g(0, x)-g(t, x)\\right) dx\\nonumber \\\\\n&+\\int _0^tn(s)M _{ 1\/2 }(g(s))ds.\n\\end{align}\nThe existence of the limit in (\\ref{ZE01}) follows and, if we pass to the limit,\n\\begin{align}\n\\lim _{ \\varepsilon \\to 0 }\\int _0^tn(s) \\mathscr Q_3^{(2)}(\\psi _\\varepsilon,g(s))ds&=\\int  _{ (0, \\infty) }(g(0, x)-g(t, x))dx\\nonumber\\\\\n&+\\int _0^tn(s)M _{ 1\/2 }(g(s))ds. \\label{Bhtw}\n\\end{align}\nWe now check that, if $n$ satisfies the equation (\\ref{ene1}) then  $G$ satisfies equation (\\ref{S1E16}) for a.e. $t>0$ and for every $\\varphi \\in C_b^{1}([0, \\infty))$. \nIf $\\varphi (0)=0$ this follows from (\\ref{2E765}) and (\\ref{G=g}).\n\n\nFor $\\varphi (0)\\not= 0$ we may assume without loss of generality that $\\varphi (0)=1$, and write $\\varphi =(\\varphi -\\psi _\\varepsilon )+\\psi _\\varepsilon $, where \n$\\psi _\\varepsilon$ is as in Remark \\ref{TEST}. \nSince  $(\\varphi -\\psi _\\varepsilon )(0)=0$, using (\\ref{2E765}) and  (\\ref{S1ED3S})\n\\begin{align}\n\\label{S5P10E1'}\n\\int_{[0,\\infty)}(\\varphi-\\psi_{\\varepsilon})(x)g(t,x)dx&=\\int_{[0,\\infty)}(\\varphi-\\psi_{\\varepsilon})(x)g(0,x)dx\\nonumber\\\\\n&+\\int_0^t n(s)\\widetilde{\\mathscr{Q}}_3((\\varphi-\\psi_{\\varepsilon}),g(s))ds.\n\\end{align}\nIn order to pas to the limit as $\\varepsilon\\to 0$, we first use\n$\\widetilde {\\mathscr{Q}}_3((\\varphi -\\psi _\\varepsilon ),g)=\\widetilde {\\mathscr{Q}}_3(\\varphi,g)-\\widetilde{\\mathscr{Q}}_3(\\psi_{\\varepsilon},g).$\nThen, since for all $t\\geq 0$\n\\begin{align}\n\\label{gzero}\n\\lim_{\\varepsilon\\to 0}\\int_{[0,\\infty)}\\psi_{\\varepsilon}(x)g(t,x)dx=0,\n\\end{align}\nand $n$ satisfies (\\ref{ene1}), we deduce from (\\ref{S5P10E1'}) and Lemma \\ref{convergence lemma}:\n\\begin{align*}\n\\int_{[0,\\infty)}\\varphi(x)g(t,x)dx=&\\int_{[0,\\infty)}\\varphi(x)g(0,x)dx+\\int_0^t n(s)\\widetilde{\\mathscr{Q}}_3(\\varphi,g(s))ds\\\\\n&+n(0)-n(t)-\\int_0^t n(s)M_{1\/2}(g(s))ds.\n\\end{align*}\nSince $\\widetilde {\\mathscr{Q}}_3(\\varphi,G)-M_{1\/2}(g)=\\mathscr Q _3(\\varphi,G)$,\nit follows that $G$ satisfies \n\\begin{align*}\n\\int_{[0,\\infty)}\\varphi(x)G(t,x)dx=\\int_{[0,\\infty)}\\varphi(x)G(0,x)dx+\\int_0^t n(s)\\mathscr{Q}_3(\\varphi,g(s))ds,\n\\end{align*}\nthus (\\ref{S1E16}) holds for a.e. $t>0$.\n\nIn order to check that $G$ satisfies (\\ref{S1ED2S}) we first use (\\ref{S1E16})  with $\\varphi =1\\in C_b^1([0, \\infty))$. For that choice of $\\varphi $ we have $\\Lambda(\\varphi)=\\mathcal L_0(\\varphi )\\equiv 0$\nand then:\n$$\n\\int  _{ [0, \\infty) }G(t, x)dx=\\int  _{ [0, \\infty) }G_0(x)dx.\n$$\nBecause:\n$$\n\\int  _{ [0, \\infty) }x\\,G(t, x)dx=\\int  _{ [0, \\infty) }x\\,g(t, x)dx,\n$$\n$G$ satisfies (\\ref{S1ED2S}) since by hypothesis so does $g$.\n\\end{proof}\n\n\\begin{remark} If $G$ is a weak radial solution of (\\ref{PA}), (\\ref{PB}), we know by  Theorem \\ref{EQUIV} that  $g$ satisfies (\\ref{2E765}). It is straightforward to check that it also satisfies,\n\\begin{equation*}\n\\frac{d}{dt}\\int_{(0,\\infty)}\\varphi(x)g(t,x)dx=n(t)\\widetilde{\\mathscr{Q}} _3(\\varphi,g(t))-\\varphi (0)\\frac{d}{dt}\\mu((0,t]), \\label{LE02'}\n\\end{equation*}\nwhere $\\mu$ is as in Theorem \\ref{THn01}, and $\\widetilde{\\mathscr{Q}}_3$ is defined in (\\ref{S1EB2})--(\\ref{S1E21R}).\n\\end{remark}\n\n\\begin{proof}[\\upshape\\bfseries{Proof of Corollary \\ref{S1C31}}]  If we prove that $n$ satisfies  (\\ref{ene1}), the conclusion of the Corollary will follow from part (ii) of Theorem \\ref{EQUIV}. \nBy the hypothesis and part (ii) of Theorem \\ref{EQUIV}, the limit in (\\ref{ZE01}) exists, and (\\ref{Bhtw}) holds, that we write:\n\\begin{align*}\n\\lim _{ \\varepsilon \\to 0 }\\int _0^tn(s) \\mathscr Q_3^{(2)}(\\psi _\\varepsilon,g(s))ds-\\int _0^tn(s)M _{1\/2}(g(s))ds=\\\\\n=\\int  _{ [0, \\infty) }(G(0, x)-G(t, x))dx+n(t)-n(0).\n\\end{align*}\nUsing  the conservation of mass (\\ref{MBC}) it follows that $n$ satisfies equation (\\ref{ene1}).\n\\end{proof}\n\n\\begin{proposition}\n\\label{LM-1\/2}\nLet $G\\in\\mathscr{M}_+([0,\\infty))$. If $G$ has no atoms on $(0,\\infty)$ and $\\int_{(0,\\infty)}\\frac{G(x)}{\\sqrt{x}}dx<\\infty$, \nthen, for all $\\varphi _\\varepsilon $ as in Remarrk \\ref{TEST},\n\\begin{equation*}\n\\mathscr{T}(G)=\\lim_{\\varepsilon\\to 0}  \\mathscr Q_3^{(2)}(\\varphi_{\\varepsilon},G)=0.\n\\end{equation*}\n\\end{proposition}\n\n\\begin{proof}\nBy definition \n$$\n\\mathscr{T}(G)=\\lim_{\\varepsilon\\to 0}\\iint_{(0,\\infty)^2}\\frac{\\Lambda(\\varphi_{\\varepsilon})(x,y)}{\\sqrt{xy}}G(x)G(y)dxdy,\n$$\nSince $\\Lambda(\\varphi_{\\varepsilon})\\leq 1$ for all $\\varepsilon>0$ and \n\\begin{align*}\n\\lim _{ \\varepsilon \\to 0 }\\Lambda(\\varphi_{\\varepsilon})(x,y)=\\mathds{1}_{\\{x=y>0\\}}(x,y)\\qquad\\forall (x,y)\\in(0,\\infty)^2,\n\\end{align*}\nand $\\int_{(0,\\infty)}\\frac{G(x)}{\\sqrt{x}}dx<\\infty$, then by dominated convergence\n$$\n\\mathscr{T}(G)=\\iint_{\\{x=y>0\\}}\\frac{G(x)G(y)}{\\sqrt{xy}}dxdy.\n$$\nSince $G$ has no atoms on $(0,\\infty)$, i.e., $G(\\{x\\})=0$ for all $x>0$, by Fubini's theorem\n\\begin{align*}\n\\iint_{\\{x=y>0\\}}\\frac{G(x)G(y)}{\\sqrt{xy}}dxdy\n&=\\int_{(0,\\infty)}\\frac{G(x)}{x}G(\\{x\\})dx=0.\n\\end{align*}\n\\end{proof}\n\n\\begin{remark}\n\\label{HH}\nFrom Proposition \\ref{LM-1\/2}, if  $M_{-1\/2}(g)<\\infty$ and $g$ has no atoms, then $\\mu((0,t])=0$ for all $t>0$. \nIf  $g\\in L^1(0, \\infty)$ and $x=0$  is a Lebesgue point of $g$ then $\\mathscr{T}(g)=0$\n(cf. \\cite{Nouri1}) and again $\\mu((0,t])=0$ for all $t>0$. \nIf $g(x)=x^{-1\/2}$, then $\\mathscr{T}(g)=\\pi^2\/6$,\n(cf. \\cite{Lu3}), and a similar result holds if \n$\\lim _{ x\\to 0 }\\sqrt x g(x)=C>0$ (cf. \\cite{Spohn}). In that case,  $\\mu((0,t])=\\pi ^2\/6\\int _0^t n(s)ds$.\n\\end{remark}\n\n\\section{Proof of Theorem \\ref{S1T5}}\n\\label{SectionD}\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\n\n\\begin{proof}\nBy (\\ref{CT98})  and (\\ref{CT99}), we deduce that for all $t>t_0>0$:\n\\begin{align}\n&\\int _{ t_0 }^t G(s,\\{0\\})ds\\leq \\big(M_{\\alpha}(G(t_0))-M_{\\alpha}(G(t))\\big)C(N,E,\\alpha) \\nonumber \\\\\n\\label{PRO116}\n&C(N,E,\\alpha)=\\left[\\left(\\frac{\\alpha-1}{\\alpha+1}\\right)E^{(2\\alpha+1)\/2}N^{(1-2\\alpha)\/2}-C_{1}N^{3-\\alpha}E^{\\alpha-1}\\right]^{-1},\n\\end{align}\nwhere $C_{1}=2^{\\alpha}-2$ for $\\alpha\\in(1,2]$ and $C_1=\\alpha(\\alpha-1)$ for $\\alpha\\in[2,3]$.\nSince by part (i), $0\\leq M_{\\alpha}(G(t_0))-M{\\alpha}(G(t))\\leq M_{\\alpha}(G(t_0))$ for every $t>t_0$, we immediately deduce (\\ref{S1ET5B}).\n\n\nWe prove now (\\ref{S1ET5C}). \nSince, as we have seen in (\\ref{S6E80}), the function $n(t)J(t)$ is monotone nondecreasing, from where, for all $t>0$ and $s\\in (0, t)$:\n\\begin{equation*}\nn(t)\\ge e^{-\\int _s^tM _{ 1\/2 }(g(r))dr}n(s).\n\\end{equation*}\nAs we have $M _{ 1\/2 }(g(r))\\le \\sqrt{NE}$ for all $r\\ge 0$,\n\\begin{equation}\n\\label{S80E20}\nn(t)\\ge e^{-\\sqrt {NE}(t-s)}n(s).\n\\end{equation}\nBy (\\ref{S1ET5B}) we already have a sequence of times $\\theta_k$ such that $\\theta_k \\to \\infty $ and  $n(\\theta_k)\\to 0$ as $k\\to \\infty$.\nSuppose that there exists, for some $\\rho >0$, an increasing sequence of times $(s_k) _{ k\\in \\mathbb{N} }$ such that $s_k\\to\\infty$ as $k\\to\\infty$ and :\n\\begin{align*}\n\\forall k,\\;n(s_k)\\ge \\rho\\quad\\hbox{and}\\quad s _{ k+1 }-s_k>\\frac {\\log 2} {\\sqrt {NE}}.\n\\end{align*} \nThen, if we denote $t_k=s_k+\\frac {\\log 2} {\\sqrt {NE}}$,\nwe deduce from (\\ref{S80E20}) that for all $t\\in (s_k, t_k)$:\n\\begin{eqnarray*}\nn(t)\\ge e^{-\\sqrt {NE}(t-s_k)}n(s_k)\\ge e^{-\\sqrt {NE}(t_k-s_k)}\\rho =\\frac {\\rho } {2}.\n\\end{eqnarray*}\nThis would imply\n\\begin{equation*}\n\\int _0^\\infty n(t)dt\\ge \\sum_{ k=0 } ^\\infty \\int  _{ s_k }^{t_k}n(t)dt=\\infty,\n\\end{equation*}\nand this contradiction proves (\\ref{S1ET5C}).\n\\end{proof}\n\n\n\n\n\\section{Appendix}\n\\label{Appendix}\n\\setcounter{equation}{0}\n\\setcounter{theorem}{0}\nWe have gathered in this Section several results that are important and useful, but not directly related to the main results. For the sake of clarity, we present them in two different Sub Sections. In the first one, we find results that are used all along the manuscript, perhaps several times. In the second, we present results that are needed in Section \\ref{model}. \n\n\\subsection{A1}\n\n\\begin{lemma}[Convex-positivity]\n\\label{convex-positivity}\nLet $\\varphi\\in C([0,\\infty))$. If $\\varphi$ is convex then $\\Lambda(\\varphi)(x,y)\\geq 0$ for all $(x,y)\\in[0,\\infty)^2$ and $\\mathcal{L}_0(\\varphi)(x)\\geq 0$ for all $x\\in[0,\\infty)$. If $\\varphi$ is nonnegative and nonincreasing, then $\\mathcal{L}(\\varphi)(x)\\leq 0$ for all $x\\in[0,\\infty)$.\n\\end{lemma}\n\n\\begin{proof}\nSince $\\Lambda(\\varphi)(x,y)$ is symmetric we may reduce the proof to the case $0\\leq y\\leq x$. Putting\n$x=\\frac{x+y}{2}+\\frac{x-y}{2},$\nthen by the very definition of convexity\n$$\\varphi(x)\\leq\\frac{\\varphi(x+y)}{2}+\\frac{\\varphi(x-y)}{2},$$\ntherefore $\\Lambda(\\varphi)(x,y)\\geq 0$. \n\n\nThe positivity of $\\mathcal{L}_0(\\varphi)$ is equivalent to prove \n\\begin{align}\n\\label{positivity weak L}\n\\frac{1}{x}\\int_0^x\\varphi(y)d y\\leq\\frac{\\varphi(0)+\\varphi(x)}{2}\\qquad\\forall x\\in[0,\\infty).\n\\end{align}\nSince for any $0\\leq y\\leq x$ we may trivially write $y=\\left(1-\\frac{y}{x}\\right)\\, 0+\\frac{y}{x}\\, x$, then by convexity \n$\\varphi(y)\\leq \\left(1-\\frac{y}{x}\\right)\\varphi(0)+\\frac{y}{x}\\varphi(x)$, which implies\n\\eqref{positivity weak L}.\\\\\nIf $\\varphi$ is nonnegative and nonincreasing, then  $\\mathcal{L}(\\varphi)(x)\\leq -x\\,\\varphi(x)\\leq 0$ for all\\ $x\\in[0,\\infty).$\n\\end{proof}\n\n\\begin{remark}\n\\label{concave-negativity}\nBy linearity and Lemma \\ref{convex-positivity}, it follows that for all $\\varphi\\in C([0,\\infty))$ concave,\n$\\Lambda(\\varphi)(x,y)\\leq 0$ for all $(x,y)\\in[0,\\infty)^2$ and $\\mathcal{L}_0(\\varphi)(x)\\leq 0$ for all $x\\in[0,\\infty)$.\n\\end{remark}\n\n\\begin{lemma}\n\\label{lemma regularity}\nConsider the operators $\\Lambda(\\cdot)$, $\\mathcal{L}_0(\\cdot)$ and $\\mathcal{L}(\\cdot)$ given in (\\ref{S1E154}), (\\ref{S1E155}) and (\\ref{S1E21R}) respectively. Then\n\\begin{enumerate}[(i)]\n\\item\nIf $\\varphi\\in\\emph{Lip}([0,\\infty))$ with Lipschitz constant $L$, then  \n\\begin{align}\n\\label{lemma regularity 1}\n\\frac{|\\Lambda(\\varphi)(x,y)|}{\\sqrt{xy}}\\leq 2L\\qquad\\forall (x,y)\\in[0,\\infty)^2.\n\\end{align}\n\\item\nIf $\\varphi\\in C^1([0,\\infty))$, then the map\n$(x,y)\\mapsto\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}$ belongs to $C([0,\\infty)^2)$ and \n\\begin{align}\n\\label{lemma regularity 2}\n\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}=0\\qquad\\forall (x,y)\\in\\partial[0,\\infty)^2.\n\\end{align}\n\\item \nIf $\\varphi\\in C([0,\\infty))$ then the maps $x\\mapsto\\frac{\\mathcal{L}_0(\\varphi)(x)}{\\sqrt{x}}$ and  \n $x\\mapsto\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}$ belong to $C([0,\\infty))$ and \n$\\frac{\\mathcal{L}_0(\\varphi)(x)}{\\sqrt{x}}= \\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}=0$ at $x=0$. If in addition \n$\\varphi$ is bounded, then\n\\begin{align}\n\\frac{|\\mathcal{L}_0(\\varphi)(x)|}{\\sqrt{x}}\\leq 4\\|\\varphi\\|_{\\infty}\\sqrt{x}\\qquad\\forall x\\in[0,\\infty), \\label{lemma regularity 4}\\\\\n\\frac{|\\mathcal{L}(\\varphi)(x)|}{\\sqrt{x}}\\leq 3\\|\\varphi\\|_{\\infty}\\sqrt{x}\\qquad\\forall x\\in[0,\\infty). \\label{lemma regularity 4B}\n\\end{align}\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n(i) By the symmetry of $\\Lambda(\\varphi)$ we can assume that $0\\leq y\\leq x$, and directly from the Lipschitz continuity\n\\begin{align*}\n|\\Lambda(\\varphi)(x,y)|&\\leq |\\varphi(x+y)-\\varphi(x)|+|\\varphi(x-y)-\\varphi(x)|\\leq 2L\\,y,\n\\end{align*}\nwhich implies \\eqref{lemma regularity 1}.\\\\\n(ii) The only possible problem for the continuity is on the boundary of $[0,\\infty)^2$.\nAgain by the symmetry of $\\Lambda(\\varphi)$ we can assume $0\\leq y\\leq x$. Then by the mean value theorem\n$\\Lambda(\\varphi)(x,y)=y\\,(\\varphi'(\\xi_1)-\\varphi'(\\xi_2))$ for some $\\xi_1\\in(x,x+y)$ and $\\xi_2\\in(x-y,x)$. \nHence\n\\begin{align*}\n\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}\\leq \\varphi'(\\xi_1)-\\varphi'(\\xi_2),\n\\end{align*}\nand the continuity of $\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}$ on $[0,\\infty)^2$ and \\eqref{lemma regularity 2} follow from the continuity of $\\varphi'$.\\\\\n(iii) The continuity of $\\frac{\\mathcal{L}_0(\\varphi)(x)}{\\sqrt{x}}$  and $\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}$ are clear for $x>0$.  Using that $\\frac{1}{x}\\int_0^x \\varphi(y)d y\\rightarrow \\varphi(0)$ as $x\\rightarrow 0$ by Lebesgue differentiation Theorem, it follows the continuity at $x=0$ and that $\\frac{\\mathcal{L}_0(\\varphi)(x)}{\\sqrt{x}}=\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}=0$ for $x=0$. The bounds (\\ref{lemma regularity 4}) and (\\ref{lemma regularity 4B})   are straightforward  for $\\varphi\\in C_b([0,\\infty))$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{regularised operators converge uniformly}\nConsider the operators $\\Lambda(\\cdot)$ and $\\mathcal{L}_0(\\cdot)$ given in (\\ref{S1E154}) and (\\ref{S1E155}), and a sequence \n$(\\phi_n)_{n\\in\\mathbb{N}}\\subset C_c([0,\\infty))$ as in Cutoff \\ref{cut-off}.\n\\begin{enumerate}[(i)]\n\\item\nIf $\\varphi\\in C^1([0,\\infty))$ then $\\Lambda(\\varphi)(x,y)\\phi_n(x)\\phi_n(y)\\xrightarrow[n\\rightarrow\\infty]{}\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}$ uniformly on the compact sets of $[0,\\infty)^2$.\n\\item\nIf $\\varphi\\in C([0,\\infty))$ then $\\mathcal{L}(\\varphi)(x)\\phi_n(x)\\xrightarrow[n\\rightarrow\\infty]{}\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}$\nuniformly on the compact sets of $[0,\\infty)$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n(i) The pointwise convergence on $[0,\\infty)^2$ is trivial since $\\phi_n(x)\\rightarrow x^{-1\/2}$ as $n\\rightarrow\\infty$. Then, let $\\varepsilon>0$ and $R>0$. For $n\\geq R$ there holds $\\phi_n(x)=x^{-1\/2}$ for all \n$x\\in[1\/n,R]$, so we only need to show the uniform convergence on the regions $(x,y)\\in[0,R]\\times[0,1\/n]$ and $(x,y)\\in[0,1\/n]\\times[0,R]$. \nBy the symmetry of $\\Lambda(\\varphi)$, we may study only one region. \n\nUsing that $\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}$ is continuous (hence uniformly continuous on compacts) and vanishes when  \n$(x,y)\\in\\partial[0,\\infty)^2$\n(c.f. Lemma \\ref{lemma regularity}), there holds for all $(x,y)\\in[0,R]\\times [0,1\/n]$ that, for $n$ large enough,\n\\begin{align*}\n\\left|\\frac{\\Lambda(\\varphi)(x,y)}{\\sqrt{xy}}-\\Lambda(\\varphi)(x,y)\\phi_n(x)\\phi_n(y)\\right|\\leq\\frac{|\\Lambda(\\varphi)(x,y)|}{\\sqrt{xy}}\\leq \\varepsilon\n\\end{align*}\n\n(ii) Let $\\varepsilon>0$ and $R>0$. Since for $n\\geq R$ there holds $\\phi_n(x)=x^{-1\/2}$ for all $x\\in[1\/n,R]$, we only need to prove the uniform convergence on the region $[0,1\/n]$. Using that $\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}$ is continuous (hence uniformly continuous on compacts) and vanishes when $x\\rightarrow 0$ (cf. Lemma \\ref{lemma regularity}), we have\n\\begin{align*}\n\\left|\\frac{\\mathcal{L}(\\varphi)(x)}{\\sqrt{x}}-\\mathcal{L}(\\varphi)(x)\\phi_n(x)\\right|\\leq \\frac{|\\mathcal{L}(\\varphi)(x)|}{\\sqrt{x}}\\leq\\varepsilon\n\\qquad\\forall x\\in[0,1\/n]\n\\end{align*}\nfor $n$ large enough.\n\\end{proof}\n\n\n\n\n\nThe following Lemma is about the approximation of a measure by functions, keeping the mass and the energy constants. It is taken from \\cite{Lu1} with  minor modifications.\n\\begin{lemma}\n\\label{APROXDATA}\nLet $h_0\\in \\mathscr{M}_+^{\\alpha}([0,\\infty))$ for some $\\alpha\\geq 1$.\nThen there exists a sequence of functions $(f_n) _{n\\in\\mathbb{N}}\\subset C([0,\\infty))\\cap L^1\\big(\\mathbb{R}_+,(1+x^{\\alpha})dx\\big)$ with $f_n>0$ such that\n\\begin{align}\n\\label{GROWTHA}\n&\\forall \\varphi \\in C([0,\\infty)):\\quad\\sup_{x\\geq 0}\\frac{|\\varphi(x)|}{1+x^{\\alpha}}<\\infty,\\\\\n\\label{APROXDATA1}\n&\\lim _{ n\\to \\infty  }\\int_0^{\\infty}\\varphi(x)f_n(x)dx=\\int_{[0,\\infty)}\\varphi(x)h_0(x)dx.\n\\end{align}\nMoreover, if $M_1(h_0)>0$, then for all $n\\in\\mathbb{N}$:\n\\begin{align}\n\\label{APROXDATA2}\nM_0(f_n)=M_0(h_0)\\qquad\\text{and}\\qquad M_1(f_n)=M_1(h_0).\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nFor $a>0$ and $b>0$, let\n\\begin{align*}\nJ_{a,b}(x)=a e^{-bx^2},\\qquad(x\\geq 0)\n\\end{align*}\nand let, for $n\\in\\mathbb{N}$,\n\\begin{align*}\nf_n(x)=e^n\\int_0^{\\frac{x}{1-e^{-n}}}J_{a,b}\\left(e^n\\big(x-y(1-e^{-n})\\big)\\right)h(y)dy,\\qquad(x\\geq0).\n\\end{align*}\nSince $J_{a,b}$ is bounded and $M_0(h_0)<\\infty$, then $f_n$ is well defined. The continuity and the strict positivity of $J_{a,b}$, together with $M_0(h)>0$, implies that $f_n$ is continuous and $f_n>0$.\nNow for any $\\varphi\\in C([0,\\infty))$ satisfying (\\ref{GROWTHA}), using Fubini and the change of variables $z=e^n\\big(x-y(1-e^{-n})\\big)$:\n\\begin{align}\n\\label{APROXDATA3}\n&\\int_0^{\\infty}\\varphi(x)f_n(x)dx=\\int_{[0,\\infty)}I_n(\\varphi)(y)h_0(y)dy,\\\\\n\\label{APROXDATA33}\n&I_n(\\varphi)(y)=\\int_0^{\\infty}\\varphi\\big(ze^{-n}+y(1-e^{-n})\\big)J_{a,b}(z)dz.\n\\end{align}\nSince by (\\ref{GROWTHA}):\n\\begin{align*}\n\\big|\\varphi\\big(ze^{-n}+y(1-e^{-n})\\big)\\big|&\\leq C\\Big(1+\\big(ze^{-n}+y(1-e^{-n})\\big)^{\\alpha}\\Big)\\\\\n&\\leq C2^{\\alpha}\\big(1+y^{\\alpha}+z^{\\alpha}\\big)\\\\\n&\\leq C2^{\\alpha}\\big(1+y^{\\alpha}\\big) \\big(1+z^{\\alpha}\\big),\n\\end{align*}\nand $h_0\\in\\mathscr{M}_+^{\\alpha}([0,\\infty))$, we deduce from (\\ref{APROXDATA3})-(\\ref{APROXDATA33}) that \n$f_n\\in L^1\\big(\\mathbb{R}_+,(1+x^{\\alpha})\\big)$. \nWe also deduce using dominated convergence that\n\\begin{align}\n&\\lim _{ n\\to \\infty }I_n(\\varphi)(y)=\\varphi(y),\\,\\,\\forall y\\geq 0,\\nonumber \\\\\n\\label{APROXDATA4}\n&\\lim _{ n\\to \\infty  }\\int_{[0,\\infty)}I_n(\\varphi)(y)h_0(y)dy=\\bigg(\\int_0^{\\infty} J_{a,b}(z)dz\\bigg)\\!\\!\\int_{[0,\\infty)}\\varphi(y)h_0(y)dy.\n\\end{align}\nWe now fix $a>0$ so that $\\int_0^{\\infty}J_{a,b}(z)dz=1$. Namely, $a=2\\sqrt{b\/\\pi}$. Then (\\ref{APROXDATA1}) follows from (\\ref{APROXDATA3}) and (\\ref{APROXDATA4}). \n\nIf we chose $\\varphi=1$ in (\\ref{APROXDATA3})-(\\ref{APROXDATA33}), the first part of (\\ref{APROXDATA2}) follows.\nIf we chose now $\\varphi(y)=y$ then\n\\begin{align*}\nM_1(f_n)=M_1(h)+e^{-n}\\left(\\frac{M_0(h_0)}{\\sqrt{b\\pi}}-M_1(h_0)\\right).\n\\end{align*}\nWe now fix $b=\\pi^{-1}(M_0(h_0)\/M_1(h_0))^2$ to obtain the second part of (\\ref{APROXDATA2}).\n\\end{proof}\n\n\\begin{corollary}\n\\label{APD1}\nLet $h_0\\in\\mathscr{M}^{\\alpha} _+([0,\\infty))$ for some $\\alpha \\ge 1$. Then, there exists a sequence of nonnegative functions \n$(h_{0,n})_{n\\in\\mathbb{N}}\\subset C_c([0,\\infty))$ such that\n\\begin{align}\n\\label{APD56}\n\\limsup_{n\\to\\infty}M_{\\alpha}(h_{0,n})\\leq M_{\\alpha}(h_0),\n\\end{align}\nand for all $\\varphi\\in C_b([0,\\infty))$\n\\begin{align}\n\\label{APD3}\n&\\lim _{ n\\to \\infty }\\int_0^{\\infty}\\varphi(x)h_{0,n}(x)dx=\\int_{[0,\\infty)}\\varphi(x)h_0(x)dx.\n\\end{align}\n\n\\end{corollary}\n\n\\begin{proof}\nWe consider the sequence $(f_n)$ given by Lemma \\ref{APROXDATA} and a smooth cutoff $\\zeta_n\\in C([0,\\infty))$ such that $0\\leq\\zeta_n\\leq 1$, $\\zeta_n(x)=1$ for $x\\in[0,n]$ and $\\zeta_n(x)=0$ for $x\\geq n+1$. Then we define for all $n\\in\\mathbb{N}$:\n\\begin{align}\n\\label{APD33}\nh_{0,n}(x)=f_n(x)\\zeta_n(x)\n\\end{align}\nIt then follows that $h_{0,n}$ is continuous, nonnegative and with compact support. \nThe property (\\ref{APD56}) follows directly from (\\ref{APROXDATA1}) in Lemma \\ref{APROXDATA} since $h_{0,n}\\leq f_n$.\nNow let $\\varphi\\in C_b([0,\\infty))$. Since $f_n$ satisfies (\\ref{APROXDATA1}), in order to prove (\\ref{APD3}) it is sufficient to prove\n\\begin{align}\n\\label{APD4}\n\\lim _{ n\\to \\infty }\\bigg|\\int_0^{\\infty}\\varphi(x)h_{0,n}(x)dx-\\int_0^{\\infty}\\varphi(x)f_n(x)(x)dx\\bigg|=0,\n\\end{align}\nand (\\ref{APD4}) follows from\n\\begin{align*}\n\\lim_{n\\to\\infty}\\int_n^{\\infty}\\varphi(x)f_n(x)dx\\leq\\lim_{n\\to\\infty} \\frac{\\|\\varphi\\|_{\\infty}M_1(f_n)}{n}=0.\n\\end{align*}\n\\end{proof}\n\n\\begin{definition}\n\\label{definition operators strong}\nLet $h$, $\\phi_n$ and $\\varphi$ be real-valued functions with domain $\\mathbb{R}_+$. Then, let\n\\begin{align}\n\\widetilde{\\mathscr{Q}}_{3,n}(\\varphi,h)=\\mathscr{Q}_{3,n}^{(2)}(\\varphi,h)-\\widetilde{\\mathscr{Q}}_{3,n}^{(1)}(\\varphi,h), \\label{Aq3tilden}\n\\end{align}\nwhere\n\\begin{align}\n\\label{Q3n2}\n&\\mathscr{Q}_{3,n}^{(2)}(\\varphi,h)=\\int_0^{\\infty}\\!\\!\\!\\int_0^{\\infty} \\Lambda(\\varphi) (x, y)\\phi_n(x)\\phi_n(y)h(x)h(y)dxdy,\\\\\n\\label{Q3n1w}\n&\\widetilde{\\mathscr{Q}}_{3,n}^{(1)}(\\varphi,h)=\\int_0^{\\infty}\\mathcal {L}(\\varphi )(x)\\phi_n(x)h(x)dx,\n\\end{align}\nand let, for $x\\in\\mathbb{R}_+$:\n\\begin{align}\n\\label{A1E32}\nJ_{3,n}(h)(x)&=K_n(h)(x)+L_n(h)(x)-h(x)A_n(h)(x),\n\\end{align} \nwhere\n\\begin{align}\nK_n(h)(x)&=\\int_0^x h(x-y)h(y)\\phi_n(x-y)\\phi_n(y)d y \\nonumber\\\\\n&+2\\int_x^{\\infty} h(y)h(y-x)\\phi_n(y)\\phi_n(y-x)d y,\\label{A1E33}\\\\\nL_n(h)(x)&=2\\int_x^{\\infty}h(y)\\phi_n(y)d y,\\label{A1E34}\\\\\nA_n(h)(x)&=\\phi_n(x)\\Big(x+4\\int_0^x h(y)\\phi_n(y)d y\\Big).\\label{A1E35}\n\\end{align}\n\\end{definition}\n\n\\begin{lemma}\n\\label{convergence lemma}\nLet $G\\in\\mathscr{M}_+([0,\\infty))$, $\\varphi_{\\varepsilon}$ as in Remark \\ref{TEST}, and $\\phi_n$ as in Cutoff \\ref{cut-off}. Then\n\\begin{align}\n&G(\\{0\\})=\\lim_{\\varepsilon\\to 0}\\int_{[0,\\infty)}\\varphi_{\\varepsilon}(x)G(x)dx,\\label{convergence 1}\\\\\n&\\lim_{\\varepsilon\\to 0}\\widetilde{\\mathscr{Q}}_{3,n}^{(1)}(\\varphi_{\\varepsilon},G)=0\\qquad\\forall n\\in\\mathbb{N}.\\label{convergence 3}\n\\end{align}\nIf in addition $G$ has no singular part in $(0,\\infty)$, then\n\\begin{align}\n\\lim_{\\varepsilon\\to 0}\\mathscr{Q}_{3,n}^{(2)}(\\varphi_{\\varepsilon},G)=0\\qquad\\forall n\\in\\mathbb{N}.\\label{convergence 4}\n\\end{align}\nFurthermore, if $G\\in\\mathscr{M}_+^{1\/2}([0,\\infty))$, then\n\\begin{align}\n\\label{limQ31a}\n&\\lim_{\\varepsilon\\to 0}\\mathscr{Q}_3^{(1)}(\\varphi_{\\varepsilon},G)=M_{1\/2}(G),\\\\\n\\label{limQ31b}\n&\\lim_{\\varepsilon\\to 0}\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi_{\\varepsilon},G)=0,\n\\end{align}\nwhere $\\mathscr{Q}_3^{(1)}$ and $\\widetilde{\\mathscr{Q}}_3^{(1)}$ are defined in (\\ref{S1E1Q31}) and (\\ref{S1E20R}) respectively.\n\\end{lemma}\n\n\\begin{proof}\nThe proof only uses dominated convergence. Since $\\varphi_{\\varepsilon}\\leq 1$ for all $\\varepsilon>0$, and $M_0(G)<\\infty$, and \n$\\varphi_{\\varepsilon}\\to\\mathds{1}_{\\{0\\}}$ as $\\varepsilon\\to 0$, then (\\ref{convergence 1}) holds.\nThen, since for all $x\\in[0,\\infty)$ it follows from dominated convergence that\n\\begin{align}\n\\label{LIN1}\n\\lim_{\\varepsilon\\to 0}\\mathcal{L}_0(\\varphi_{\\varepsilon})(x)=x\n\\qquad\\text{and}\\qquad\n\\lim_{\\varepsilon\\to 0}\\mathcal{L}(\\varphi_{\\varepsilon})(x)=0,\n\\end{align}\nand $\\phi_n$ is compactly supported, then (\\ref{convergence 3}) follows. \nAlso, since for all $(x,y)\\in[0,\\infty)^2$, $\\Lambda(\\varphi_{\\varepsilon})(x,y)\\leq 1$ for all $\\varepsilon>0$, and \n\\begin{align*}\n\\lim_{\\varepsilon\\to 0}\\Lambda(\\varphi_{\\varepsilon})(x,y)=\\mathds{1}_{\\{x=y>0\\}}(x,y),\n\\end{align*}\nthen\n\\begin{align*}\n\\lim_{\\varepsilon\\to 0}\\mathscr{Q}_{3,n}^{(2)}(\\varphi_{\\varepsilon},G)=\\iint_{\\{x=y>0\\}}\\phi_n(x)\\phi_n(y)G(x)G(y)d xd y,\n\\end{align*}\nUsing that $G$ has no singular part on $(0,\\infty)$, (\\ref{convergence 4}) follows.\n\nLastly, since \n\\begin{align}\n\\widetilde{\\mathscr{Q}}_3^{(1)}(\\varphi_{\\varepsilon},G)\\leq \\mathscr{Q}_3^{(1)}(\\varphi_{\\varepsilon},G)=\\int_{(0,\\infty)}\\frac{\\mathcal{L}_0(\\varphi_{\\varepsilon})(x)}{\\sqrt{x}}G(x)dx,\n\\end{align}\nand by (\\ref{lemma regularity 4})\n\\begin{align*}\n\\int_{(0,\\infty)}\\frac{|\\mathcal{L}_0(\\varphi_{\\varepsilon})(x)|}{\\sqrt{x}}G(x)dx\\leq 4M_{1\/2}(G)\\qquad\\forall\\varepsilon>0.\n\\end{align*}\nthen (\\ref{limQ31a}) and (\\ref{limQ31b}) follows from (\\ref{LIN1}) and dominated convergence.\n\\end{proof}\n\n\n\\begin{lemma}\n\\label{well defined operators}\nConsider $n\\in\\mathbb{N}$, $\\phi_n\\in C_c([0,\\infty))$ nonnegative and  $\\rho\\in L^1_{loc}(\\mathbb{R}_+)$ nonnegative. Then for every nonnegative functions \n$h$, $h_1$ and $h_2$ in $L^{\\infty}(\\mathbb{R}_+)$, the functions $K_n(h)$, $L_n(h)$, $A_n(h)$ and $hA_n(h)$ are also nonnegative, belong to \n$L^{\\infty}(\\mathbb{R}_+)\\cap L^1_{\\rho}(\\mathbb{R}_+)$, and there exists a positive constant $C(n,\\rho)$ such that:\n\\begin{align}\n&\\|K_n(h_1)-K_n(h_2)\\|_{L^{\\infty}\\cap L^1_{\\rho}}\\leq C(n,\\rho)\\|h_1\\|_{\\infty}\\|h_1-h_2\\|_{\\infty}\\label{SAE100}\\\\\n&\\|L_n(h)\\|_{L^{\\infty}\\cap L^1_{\\rho}}\\leq C(n,\\rho)\\|h\\|_{\\infty}\\label{bound L}\\\\\n&\\|A_n(h)\\|_{L^{\\infty}\\cap L^1_{\\rho}}\\leq C(n,\\rho)\\big(1+\\|h\\|_{\\infty}\\big)\\label{SaE120}\\\\\n&\\|A_n(h_1)-A_n(h_2)\\|_{L^{\\infty}\\cap L^1_{\\rho}}\\leq C(n,\\rho)\\|h_1-h_2\\|_{\\infty}.\\label{SaE121}\n\\end{align}\n\\begin{flalign}\n\\text{Moreover}&&J_{3,n}(h)\\in L^{\\infty}(\\mathbb{R}_+)\\cap L^1_{\\rho}(\\mathbb{R}_+).&&\n\\end{flalign}\n\\end{lemma}\n\n\\begin{proof}\nThe positivity of the operators is clear from their definitions. Notice that since $\\phi_n$ is bounded and compactly supported on $\\mathbb{R}_+$ and $\\rho\\in L^1_{loc}(\\mathbb{R}_+)$, there exist two positive constants $C(n)$ and $C(n,\\rho)$ such that\n\\begin{align*}\n&\\sup_{x\\geq 0}\\int_0^{\\infty} \\phi_n(|x-y|)\\phi_n(y)d y\\leq C(n),\\\\\n&\\int_0^{\\infty}\\!\\!\\!\\int_0^{\\infty}\\rho(x)\\phi_n(|x-y|)\\phi_n(y)d yd x\\leq C(n,\\rho).\n\\end{align*}\n\n\\noindent1. Estimates for $K_n$. For all $x\\geq 0$:\n\\begin{align*}\nK_n(h)(x)&\\leq 3\\|h\\|_{\\infty}^2\\int_0^{\\infty} \\phi_n(|x-y|)\\phi_n(y)d y\\leq 3\\|h\\|_{\\infty}^2C(n),\n\\end{align*}\nand\n\\begin{align*}\n\\|K_n(h)\\|_{L^1_{\\rho}}\n&\\leq 3\\|h\\|_{\\infty}^2\\int_0^{\\infty}\\!\\!\\! \\int_0^{\\infty}  \\rho(x)\\phi_n(|x-y|)\\phi_n(y)d yd x\\leq 3\\|h\\|_{\\infty}^2 C(n,\\rho).\n\\end{align*}\nThen for all $x\\geq 0$:\n\\begin{align}\n\\label{KNs}\n&\\big|K_n(h_1)(x)-K_n(h_2)(x)\\big|\\\\\n&\\leq3\\int_0^{\\infty}\\phi_n(|x-y|)\\phi_n(y)\\big|h_1(|x-y|)h_1(y)-h_2(|x-y|)h_2(y)\\big|d y.\\nonumber\n\\end{align}\nWithout loss of generality we assume that $\\|h_1\\|_{\\infty}\\geq \\|h_2\\|_{\\infty}$. Using \n\\begin{align*}\n&\\big|h_1(|x-y|)h_1(y)-h_2(|x-y|)h_2(y)\\big|\\leq 2\\|h_1\\|_{\\infty}\\|h_1-h_2\\|_{\\infty}\n\\end{align*}\nin (\\ref{KNs}) then (\\ref{SAE100}) follows.\\\\\n2. Estimates for $L_n$. Since $\\phi_n$ is bounded and compactly supported and $\\rho\\in L^1_{loc}(\\mathbb{R}_+)$, there exist two positive constants $C(n)$ and \n$C(n,\\rho)$ such that\n\\begin{align*}\n\\int_0^{\\infty}\\phi_n(x)d x\\leq C(n)\\quad\\text{and}\\quad\n\\int_0^{\\infty}\\rho(x)\\int_x^{\\infty}\\phi_n(y) dy dx\\leq C(n,\\rho)\n\\end{align*}\nand (\\ref{bound L}) follows.\\\\\n3. Estimates for $A_n$.\nThe estimate  (\\ref{SaE120}) follows from\n\\begin{align*}\n\\|A_n(h)\\|_{\\infty}&\\leq \\|x\\,\\phi_n(x)\\|_{\\infty}+4\\|\\phi_n\\|_{\\infty}^2\\|h\\|_{\\infty}|\\operatorname{supp}(\\phi_n)|\\leq C(n)(1+\\|h\\|_{\\infty}),\n\\end{align*}\nand\n\\begin{align*}\n\\|A_n(h)\\|_{L^1_{\\rho}}&\\leq \\int_0^{\\infty}\\rho(x)\\,x\\,\\phi_n(x)d x+4\\,\\|h\\|_{\\infty}\\int_0^{\\infty}\\rho(x)\\,\\phi_n(x)\\int_0^x \\phi_n(y)d yd x\\\\\n&\\leq C(n,\\rho)(1+\\|h\\|_{\\infty}).\n\\end{align*}\nFor all $x\\geq 0$,\n\\begin{align*}\n|A_n(h_1)(x)-A_n(h_2)(x)|&\\leq 4\\|h_1-h_2\\|_{\\infty}\\phi_n(x)\\int_0^x\\phi_n(y)d y\\\\\n&\\leq C(n) \\|h_1-h_2\\|_{\\infty}.\n\\end{align*}\nWe also have, \n\\begin{align*}\n\\|A_n(h_1)-A_n(h_2)\\|_{L^1_{\\rho}}&\\leq 4\\|h_1-h_2\\|_{\\infty}\\int_0^{\\infty}\\rho(x)\\phi_n(x)\\int_0^x\\phi_n(y)d yd x\\\\\n&\\leq C(n,\\rho)\\,\\|h_1-h_2\\|_{\\infty},\n\\end{align*}\nand then, (\\ref{SaE121}) follows.\\\\\n4. Since $h\\in L^{\\infty}(\\mathbb{R}_+)$ and $A_n(h)\\in L^{\\infty}(\\mathbb{R}_+)\\cap L^1_{\\rho}(\\mathbb{R}_+)$, then $hA_n(h)\\in L^{\\infty}(\\mathbb{R}_+)\\cap L^1_{\\rho}(\\mathbb{R}_+)$.\\\\\n5. It also follows from points 1 to 4 that $J_{3,n}(h)$ has the desired regularity.\n\\end{proof}\n\n\\subsection{A2}\n\n\\begin{lemma}\n\\label{representation of Deltavarphi}\nLet $\\varphi\\in C ^{1.1}([0,\\infty))$. Then, for all $(x_1,x_2,x_3)\\in[0,\\infty)^3$ such that $x_1+x_2\\geq x_3$:\n\\begin{align*}\n&\\Delta\\varphi(x_1,x_2,x_3)\n=(x_1-x_3)(x_2-x_3)\\times  \\nonumber \\\\\n&\\qquad \\times \\int_0^1\\int_0^1\\varphi''\\big(x_3+t(x_1-x_3)+s(x_2-x_3)\\big)dsdt.\n\\end{align*}\nMoreover, if $\\varphi\\in C^{1.1}_b([0,\\infty))$, then for all $(x_1,x_2,x_3)\\in [0,\\infty)^3$\n\\begin{align}\n|\\Delta\\varphi(x_1,x_2,x_3)| \\leq\\min\\left\\{A, B, C, D\\right\\}. \\label{S2E2}\n\\end{align}\n\\begin{flalign}\n\\text{where} &&A&=4\\|\\varphi\\|_{\\infty},\\quad B=2\\|\\varphi'\\|_{\\infty}|x_1-x_3|,\\quad C=2\\|\\varphi'\\|_{\\infty}|x_2-x_3|,\\nonumber\\\\\n&&D&=\\|\\varphi''\\|_{\\infty}|x_1-x_3||x_2-x_3|.\\nonumber \n\\end{flalign}\n\\end{lemma}\n\n\\begin{proof}\nLet $(x_1,x_2,x_3)\\in[0,\\infty)^3$ be such that $x_1+x_2\\geq x_3$. By the fundamental Theorem of calculus\n\\begin{align*}\n\\Delta &\\varphi(x_1,x_2,x_3)=\\big[\\varphi(x_4)-\\varphi(x_2)\\big]-\\big[\\varphi(x_1)-\\varphi(x_3)\\big]\\\\\n&  =\\int_0^1\\frac{d}{dt}\\varphi\\big(x_2+t(x_1-x_3)\\big)dt-\\int_0^1\\frac{d}{dt}\\varphi\\big(x_3+t(x_1-x_3)\\big)dt\\\\\n&=(x_1-x_3)\\int_0^1\\big[\\varphi'\\big(x_2+t(x_1-x_3)\\big)-\\varphi'\\big(x_3+t(x_1-x_3)\\big)\\big]dt\\\\\n&=(x_1-x_3)\\int_0^1\\int_0^1\\frac{d}{ds}\\varphi'\\big(x_3+t(x_1-x_3)+s(x_2-x_3)\\big)dsdt\\\\\n&=(x_1-x_3)(x_2-x_3)\\int_0^1\\int_0^1\\varphi''\\big(x_3+t(x_1-x_3)+s(x_2-x_3)\\big)dsdt.\n\\end{align*}\nAssume now that $\\varphi\\in C^{1.1}_b([0,\\infty))$. Using the first, the third, and the fifth line above, estimate (\\ref{S2E2}) follows. \n\\end{proof}\n\nWe now consider the function $w$ given in (\\ref{S1E6'}) and define\n\\begin{align}\n\\label{S2E3}\nW(x_1,x_2,x_3)=\\left\\{\n\\begin{array}{ll}\n\\frac{w(x_1,x_2,x_3)}{\\sqrt{x_1x_2x_3}}&\\!\\!\\text{if}\\quad(x_1,x_2,x_3)\\in (0,\\infty)^3\\\\\n\\frac{1}{\\sqrt{x_1x_2}}&\\!\\!\\!\\!\\!\\!\\text{if}\\quad x_3=0,\\quad (x_1,x_2)\\in(0,\\infty)^2\\\\\n\\frac{1}{\\sqrt{x_ix_3}}&\\!\\!\\!\\!\\!\\!\\text{if}\\; x_j=0,\\, x_i>x_3>0;\\,\\{i, j\\}=\\{1,2\\} \\\\\n0&\\!\\!\\!\\text{otherwise}.\n\\end{array}\n\\right.\n\\end{align}\n\nWe then have:\n\\begin{lemma}\n\\label{S2L1}\nConsider the function $\\Phi_{\\varphi}=W\\Delta\\varphi$, where $\\Delta\\varphi$  and $W$  are defined in (\\ref{S2E1}) and (\\ref{S2E3}) respectively.\n\\begin{enumerate}[(i)]\n\\item\nIf $\\varphi\\in C^{1.1}([0,\\infty))$ then $\\Phi_{\\varphi}\\in C([0,\\infty)^3)$.\n\\item\nIf $\\varphi\\in C^{1.1}_b([0,\\infty))$ then $\\Phi_{\\varphi}\\in C_0([0,\\infty)^3)$. In particular $\\Phi_{\\varphi}$ is uniformly continuous on $[0,\\infty)^3$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\\textbf{Proof of (i).} By definition $\\Phi _\\varphi\\in C ((0, \\infty)^3)$. Therefore it only remains to study the behaviour of $\\Phi _\\varphi$ in a neighborhood of the boundary $\\partial [0, \\infty)^3$ of $[0, \\infty)^3$. First we show that $\\Phi_{\\varphi}$ is continuous on $\\partial [0, \\infty)^3$. \\\\\nThanks to the symmetry of $\\Phi_{\\varphi}$ in the $x_1$, $x_2$ variables, we just need to prove:\\\\\n(i)for all $(x_1,x_2)\\in (0,\\infty)^2$, \n\\begin{align}\n\\label{boundary1}\n\\Phi_{\\varphi}(x_1,x_2,0)=\\frac{\\Delta\\varphi(x_1,x_2,0)}{\\sqrt{x_1x_2}}\\longrightarrow 0\n\\end{align}\nwhenever $x_1\\rightarrow 0$ or $x_2\\rightarrow 0$ or $(x_1,x_2)\\rightarrow (0,0)$, and\\\\\n(ii) for all $x_1>x_3>0$,\n\\begin{align}\n\\label{boundary2}\n\\Phi_{\\varphi}(x_1,0,x_3)=\\frac{\\Delta\\varphi(x_1,0,x_3)}{\\sqrt{x_1x_3}}\\longrightarrow 0\n\\end{align}\nwhenever $x_1\\rightarrow x_3$ or $x_3\\rightarrow 0$ or $(x_1,x_3)\\rightarrow (0,0)$.\n\nBy (\\ref{S2E2}) $|\\Delta\\varphi(x_1,x_2,0)|\\leq \\|\\varphi''\\|_{\\infty}x_1x_2$ for all $(x_1,x_2)\\in (0,\\infty)^2$, which implies (\\ref{boundary1}). Also $|\\Delta\\varphi(x_1,0,x_3)|\\leq \\|\\varphi''\\|_{\\infty}x_3(x_1-x_3)$ for all $x_1>x_3>0$. Hence\n\\begin{align*}\n\\frac{|\\Delta\\varphi(x_1,0,x_3)|}{\\sqrt{x_1x_3}}\\leq  \\|\\varphi''\\|_{\\infty}\\sqrt{\\frac{x_3}{x_1}}(x_1-x_3)\\leq \\|\\varphi''\\|_{\\infty}(x_1-x_3),\n\\end{align*}\nwhich implies (\\ref{boundary2}).\n\nThen we prove that for any $x\\in\\partial[0,\\infty)^3$ and for any \n$(x_n)_{n\\in\\mathbb{N}}\\subset (0,\\infty)^3$ such that $x_n\\rightarrow x$, then $\\Phi_{\\varphi}(x_n)\\rightarrow\\Phi_{\\varphi}(x)$ as $n\\to\\infty$. Let us denote\n$$\\Omega=\\{(x_1,x_2,x_3)\\in(0,\\infty)^3:x_1+x_2\\leq x_3\\}.$$ Since $x_4$ is defined as $x_4=(x_1+x_2-x_3)_+$, then for all $(x_1,x_2,x_3)\\in(0,\\infty)^3$,\n$$(x_1,x_2,x_3)\\in\\Omega\\quad\\text{if and only if}\\quad x_4=0.$$\nIt might happen that the sequence $(x_n)_{n\\in\\mathbb{N}}$ ``jumps'' from $\\Omega$ to $\\Omega^c$. If in every neighbourhood of $x$ the sequence has points in both regions, then we may consider two subsequences, each one contained in one region only. For the sequel, the main estimate is the following:\nif we denote $x_n=(x_1^n,x_2^n,x_3^n)$ and $w(x_n)=\\min\\left\\{\\sqrt{x_1^n},\\sqrt{x_2^n},\\sqrt{x_3^n},\\sqrt{x_4^n}\\right\\}$, then by (\\ref{S2E2}) \n\\begin{align}\n\\label{main estimate to prove continuity}\n|\\Phi_{\\varphi}(x_n)|\\leq\\|\\varphi''\\|_{\\infty}\n\\frac{w(x_n)}{\\sqrt{x_1^nx_2^nx_3^n}}\n\\big|x_1^n-x_3^n\\big|\\big|x_2^n-x_3^n\\big|.\n\\end{align}\nWe study case by case depending on where $x$ lies.\n\nCase $x=(0,0,0)$. If $(x_n)\\subset\\Omega$ then $x_4^n=0$, \n$w(x_n)=\\sqrt{x_4^n}=0$ and thus $\\Phi_{\\varphi}(x_n)=0=\\Phi_{\\varphi}(x)$.\\\\\nIf $\\{x_n\\}\\subset\\Omega^c$ then $x_4^n>0$ and we study case by case depending on the relative order of $x_1^n$, $x_2^n$, and $x_3^n$. \nSince $\\Phi_{\\varphi}$\nis symmetric in the $x_1$, $x_2$ variables, we may assume without loss of generality that $x_1^n\\leq x_2^n$.  \nNote by \\eqref{main estimate to prove continuity} that we also may assume $x_3^n\\neq x_1^n$, $x_3^n\\neq x_2^n$; otherwise the result follows directly.\n\nIf $x_1^n\\leq x_2^n<x_3^n$,\nthen $w(x_n)=\\sqrt{x_4^n}$ and by \\eqref{main estimate to prove continuity}\n\\begin{align*}\n|\\Phi_{\\varphi}(x_n)|&\\leq\\|\\varphi''\\|_{\\infty}\n\\frac{\\sqrt{x_4^n}}{\\sqrt{x_1^nx_2^nx_3^n}}\\big(x_3^n-x_1^n\\big)\\big(x_3^n-x_2^n\\big)\\\\\n&\\leq\\|\\varphi''\\|_{\\infty}\\left(\\frac{\\sqrt{x_4^n}\\big(x_3^n\\big)^{3\/2}}{\\sqrt{x_1^nx_2^n}}+\\frac{\\sqrt{x_4^n x_1^n x_2^n}}{\\sqrt{x_3^n}}\\right)\\\\\n&\\leq\\|\\varphi''\\|_{\\infty}\\left( \\frac{\\big(x_3^n\\big)^{3\/2}}{\\sqrt{x_2^n}}+\\sqrt{x_1^n x_2^n}\\right).\n\\end{align*}\nSince $x_n\\to x=0$, then $\\sqrt{x_1^n x_2^n}\\to 0$. Moreover, since $x_n\\in\\Omega^c$ and $x_1^n\\leq x_2^n$, then \n$x_3^n<2 x_2^n$, and so \n$$ \\frac{\\big(x_3^n\\big)^{3\/2}}{\\sqrt{x_2^n}}\\leq 2^{3\/2} x_2^n\\longrightarrow 0\\qquad\\text{as}\\quad n\\rightarrow\\infty.$$\n\nIf $x_1^n<x_3^n< x_2^n$, then $w(x_n)=\\sqrt{x_1^n}$ and by \\eqref{main estimate to prove continuity}\n\\begin{align*}\n|\\Phi_{\\varphi}(x_n)|&\\leq\\|\\varphi''\\|_{\\infty}\\frac{(x_2^n-x_3^n)(x_3^n -x_1^n)}{\\sqrt{x_2^n x_3^n}}\\\\\n&\\leq \\|\\varphi''\\|_{\\infty}\\left(\\sqrt{x_2^n x_3^n}+\\frac{x_1^n\\sqrt{x_3^n}}{\\sqrt{x_2^n}}\\right)\\\\\n&\\leq \\|\\varphi''\\|_{\\infty}\\left(\\sqrt{x_2^n x_3^n}+\\sqrt{x_1^n x_3^n}\\right)\\longrightarrow 0\\qquad\\text{as}\\quad n\\rightarrow\\infty.\n\\end{align*}\n\nLastly, if $x_3^n<x_1^n\\leq x_2^n$, then $w(x_n)=\\sqrt{x_3^n}$ and by \\eqref{main estimate to prove continuity}\n\\begin{align*}\n|\\Phi_{\\varphi}(x_n)|&\\leq\\|\\varphi''\\|_{\\infty}\\frac{(x_1^n-x_3^n)(x_2^n-x_3^n)}{\\sqrt{x_1^n x_2^n}}\\\\\n&\\leq \\|\\varphi''\\|_{\\infty}\\left(\\sqrt{x_1^n x_2^n}+\\frac{\\big(x_3^n\\big)^2}{\\sqrt{x_1^n x_2^n}}\\right)\\\\\n&\\leq 2\\|\\varphi''\\|_{\\infty}\\left(\\sqrt{x_1^n x_2^n}+x_1\\right)\\longrightarrow 0\\qquad\\text{as}\\quad n\\rightarrow\\infty.\n\\end{align*}\nHence, in the three cases above $\\Phi_{\\varphi}(x_n)\\rightarrow 0=\\Phi_{\\varphi}(x).$\n\nCase $x=(x_1,0,0)$ with $x_1>0$. \nThen $w(x_n)=\n\\min\\left\\{\\sqrt{x_2^n},\\sqrt{x_3^n}\\right\\}$ for $n$ large enough.\nOn the other hand\n\\begin{align*}\n\\big|x_2^n-x_3^n\\big|&=\\big(\\sqrt{x_2^n}+\\sqrt{x_3^n}\\big)\\big|\\sqrt{x_2^n}-\\sqrt{x_3^n}\\big|\\\\\n&\\leq 2\\max\\left\\{\\sqrt{x_2^n},\\sqrt{x_3^n}\\right\\}\\big|\\sqrt{x_2^n}-\\sqrt{x_3^n}\\big|.\n\\end{align*}\nSince $\\min\\left\\{\\sqrt{x_2^n},\\sqrt{x_3^n}\\right\\}\\max\\left\\{\\sqrt{x_2^n},\\sqrt{x_3^n}\\right\\}=\\sqrt{x_2^nx_3^n}$, then by \\eqref{main estimate to prove continuity} \n\\begin{align*}\n|\\Phi_{\\varphi}(x_n)|\\leq2\\|\\varphi''\\|_{\\infty}\\frac{\\big|x_1^n-x_3^n\\big|}{\\sqrt{x_1^n}}\\big|\\sqrt{x_2^n}-\\sqrt{x_3^n}\\big|\n\\end{align*}\nfor $n$ large enough. It then follows $\\Phi_{\\varphi}(x_n)\\rightarrow 0=\\Phi_{\\varphi}(x)$ as $n\\to\\infty$.\n\nThe case $x=(0,x_2,0)$ with $x_2>0$ is analogous to the previous one thanks to the symmetry of $\\Phi_{\\varphi}$ in the $x_1$, $x_2$ variables.\n\nCase $x=(0,0,x_3)$ with $x_3>0$. Then $x_n\\in\\Omega$ for $n$ large enough, $x_4^n=0$ and\n$w(x_n)=\\sqrt{x_4^n}=0$. Thus\n$\\Phi_{\\varphi}(x_n)=0=\\Phi_{\\varphi}(x)$ for $n$ large enough.\n\nCase $x=(0,x_2,x_3)$ with $x_2>0$ and $x_3>0$. If $x_2>x_3$ then\n$w(x_n)=\\sqrt{x_1^n}$ for $n$ large enough and \n\\begin{align*}\n|\\Phi_{\\varphi}(x_n)-\\Phi_{\\varphi}(x)|=\\left|\\frac{1}{\\sqrt{x_2^nx_3^n}}\\Delta\\varphi(x_1^n,x_2^n,x_3^n)\n-\\frac{1}{\\sqrt{x_2x_3}}\\Delta\\varphi(0,x_2,x_3)\\right|,\n\\end{align*}\nwhich clearly goes to zero as $n\\rightarrow\\infty$. If $x_2<x_3$ then $x_4^n=0$ for $n$ large enough and \n$w(x_n)=\\sqrt{x_4^n}=0$, thus $\\Phi_{\\varphi}(x_n)=0=\\Phi_{\\varphi}(x)$.\nIf $x_2=x_3$ and $(x_n)\\subset\\Omega$ for $n$ large enough, then $x_4^n=0$, thus\n$\\Phi_{\\varphi}(x_n)=0=\\Phi_{\\varphi}(x)$.\\\\ \nIf $x_2=x_3$ and $(x_n)\\subset\\Omega^c$ for $n$ large enough, then\n$w(x_n)\n=\\min\\left\\{\\sqrt{x_1^n},\\sqrt{x_4^n}\\right\\}$, and by \\eqref{main estimate to prove continuity} \n\\begin{align*}\n|\\Phi_{\\varphi}(x_n)|\\leq\\|\\varphi''\\|_{\\infty}\n\\frac{\\min\\left\\{\\sqrt{x_1^n},\\sqrt{x_4^n}\\right\\}}{\\sqrt{x_1^nx_2^nx_3^n}}\n\\big|x_1^n-x_3^n\\big|\\big|x_2^n-x_3^n\\big|.\n\\end{align*}\nOn the one hand \n\\begin{align*}\n\\big|x_1^n-x_3^n\\big|\n\\leq 2\\max\\left\\{\\sqrt{x_1^n},\\sqrt{x_3^n}\\right\\}\\big|\\sqrt{x_1^n}-\\sqrt{x_3^n}\\big|.\n\\end{align*}\nOn the other hand \n$\\min\\left\\{\\sqrt{x_1^n},\\sqrt{x_4^n}\\right\\}\\leq\\min\\left\\{\\sqrt{x_1^n},\\sqrt{x_3^n}\\right\\}$ for $n$ large enough. \nSince $\\min\\left\\{\\sqrt{x_1^n},\\sqrt{x_3^n}\\right\\}\\max\\left\\{\\sqrt{x_1^n},\\sqrt{x_3^n}\\right\\}=\\sqrt{x_1^n x_3^n}$, then\n\\begin{align*}\n|\\Phi_{\\varphi}(x_n)|\\leq2\\|\\varphi''\\|_{\\infty}\\frac{\\big|x_2^n-x_3^n\\big|}{\\sqrt{x_2^n}}\\big|\\sqrt{x_1^n}-\\sqrt{x_3^n}\\big|,\n\\end{align*}\nwhich goes to zero as $n\\rightarrow\\infty$ since $x_2=x_3$. Thus $\\Phi_{\\varphi}(x_n)\\rightarrow 0=\\Phi_{\\varphi}(x)$.\n\nThe case $x=(x_1,0,x_3)$ with $x_1>0$ and $x_3>0$ is analogous to the previous one thanks to the symmetry of $\\Phi_{\\varphi}$ in the $x_1$, $x_2$ variables.\n\nCase $x=(x_1,x_2,0)$ with $(x_1,x_2)\\in (0,\\infty)^2$. Then $w(x_n)=\\sqrt{x_3^n}$ for $n$ large enough and\n\\begin{align*}\n|\\Phi_{\\varphi}(x_n)-\\Phi_{\\varphi}(x)|=\\left|\\frac{1}{\\sqrt{x_1^nx_2^n}}\\Delta\\varphi(x_1^n,x_2^n,x_3^n)\n-\\frac{1}{\\sqrt{x_1x_2}}\\Delta\\varphi(x_1,x_2,0)\\right|,\n\\end{align*}\nwhich clearly goes to zero as $n\\rightarrow\\infty$.\n\n\n\\textbf{Proof of (ii).}\nBy part (i) $\\Phi_{\\varphi}\\in C([0,\\infty)^3)$. Let us show now that for any given $\\varepsilon>0$ there exists $R(\\varepsilon)>0$ such that \n$|\\Phi_{\\varphi}(x)|\\leq\\varepsilon$ for all $x\\in[0,\\infty)^3\\setminus[0,R(\\varepsilon)]^3$. \n\nGiven $R>0$ and $\\alpha>0$, let $(x_1,x_2,x_3)\\in[0,\\infty)^3\\setminus[0,R]^3$ and denote\n$x_i=\\min\\{x_1,x_2,x_3\\}$, $x_k=\\max\\{x_1,x_2,x_3\\}$ and $x_j$ neither $x_i$ nor $x_k$.\nNotice that $x_k>R$ and the function $W$ defined in (\\ref{S2E3}) satisfies $W(x_1,x_2,x_3)\\leq\\frac{1}{\\sqrt{x_jx_k}}$.\nIf $x_i>\\alpha$ or $x_j>\\alpha$ then by (\\ref{S2E2}) \n\\begin{align*}\n|\\Phi_{\\varphi}(x_1,x_2,x_3)|\\leq\\frac{|\\Delta\\varphi(x_1,x_2,x_3)|}{\\sqrt{x_j x_k}}\\leq\\frac{4\\|\\varphi\\|_{\\infty}}{\\sqrt{\\alpha R}}\\leq\\varepsilon,\n\\end{align*}\nprovided $R\\geq\\frac{16\\|\\varphi\\|^2_{\\infty}}{\\alpha\\varepsilon^2}.$\nIf $x_i\\leq\\alpha$ and $x_j\\leq\\alpha$ we study case by case depending on the relative position of $x_1$, $x_2$, $x_3$. Since $\\Phi_{\\varphi}$ is symmetric in variables $x_1$ and $x_2$, we may assume without loss of generality that $x_2\\leq x_1$.\nIf $x_k=x_1$, using (\\ref{S2E2})\n\\begin{align*}\n|\\Phi_{\\varphi}(x_1,x_2,x_3)|&\\leq\\frac{2\\|\\varphi'\\|_{\\infty}(x_j-x_i)}{\\sqrt{x_1 x_j}}\n\\leq\\frac{2\\|\\varphi'\\|_{\\infty}\\sqrt{x_j}}{\\sqrt{x_1}}\n\\leq\\frac{2\\|\\varphi'\\|_{\\infty}\\sqrt{\\alpha}}{\\sqrt{R}}\\leq\\varepsilon,\n\\end{align*}\nprovided $R\\geq\\frac{4\\|\\varphi'\\|^2_{\\infty}\\alpha}{\\varepsilon^2}.$\nIf $x_k=x_3$ and $x\\in\\Omega$ then $x_4=0$ and $\\Phi_{\\varphi}(x)=0$.\nIf $x_k=x_3$ and $x\\in\\Omega^c$, then\n$x_1\\geq R\/2$ and\n\\begin{align*}\n|\\Phi_{\\varphi}(x_1,x_2,x_3)|&\\leq\\frac{4\\|\\varphi\\|_{\\infty}}{\\sqrt{x_1x_3}}\\leq\\frac{4\\sqrt{2}\\|\\varphi\\|_{\\infty}}{R}\\leq\\varepsilon,\n\\end{align*}\nprovided $R\\geq \\frac{4\\sqrt{2}\\|\\varphi\\|_{\\infty}}{\\varepsilon}$.\n\nFinally, if we chose $R\\geq\\max\\left\\{\\frac{16\\|\\varphi\\|^2_{\\infty}}{\\alpha\\varepsilon^2},\\frac{4\\|\\varphi'\\|^2_{\\infty}\\alpha}{\\varepsilon^2},\n\\frac{4\\sqrt{2}\\|\\varphi\\|_{\\infty}}{\\varepsilon}\\right\\}$ \nthen $\\Phi_{\\varphi}\\in C_0([0,\\infty)^3)$ and in particular, $\\Phi_{\\varphi}$ is uniformly continuous in $[0,\\infty)^3$.\n\\end{proof}\n\n\\noindent\n\\textbf{Acknowledgments.}\nThe research of the first author is supported by the Basque Government through the BERC 2014-2017 program, by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323, and by MTM2014-52347-C2-1-R of DGES. The research of the second author is supported by grants MTM2014-52347-C2-1-R of DGES and  IT641-13 of the Basque Government. The authors acknowledge the valuable remarks and helpful comments received from A. H. M. Kierkels and Pr. J. J. L. Vel\\'azquez, as well as the hospitality of  IAM at the University of Bonn.\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAll existing experimental data in particle physics are in good agreement\nwith the Standard Model predictions. However, the problems exist which\ncould not be resolved within the SM and it is obviously not a complete\nor final theory. It is unquestionable that the SM should be the low-energy\nlimit of some higher symmetry. The question is what could be this symmetry.\nAnd the main question is what is the mass scale of this symmetry restoration.\nA gloomy prospect is the restoration of this higher symmetry at once on\na very high mass scale, the so-called 'gauge desert'. A concept of a\nconsecutive symmetry restoration is much more attractive. It looks\nnatural in this case to suppose a correspondence of the hierarchy\nof symmetries and the hierarchy of the mass scales of their restoration.\nNow we are on the first step of some stairway of symmetries\nand we try to guess what could be the next one.\nIf we consider some well--known higher symmetries from this point of view,\ntwo questions are pertinent. First, isn't the supersymmetry as the\nsymmetry of bosons and fermions, higher than the symmetry within the\nfermion sector, namely, the quark--lepton symmetry\\cite{3}, or the\nsymmetry within the boson sector, namely, the left--right\nsymmetry\\cite{4}? Second, wouldn't the supersymmetry restoration be\nconnected with a higher mass scale than the others?\n\nWe should like to analyse a possibility when the quark-lepton symmetry\nis the next step beyond the SM. We take a minimal symmetry of the\nPati-Salam type with the lepton number as the fourth color\\cite{3},\n$SU(4)_V \\otimes SU(2)_L \\otimes G_R$. The fermions are combined into the\nfundamental representations of the $SU(4)_V$ group, the neutrinos with\nthe {\\it up} quarks and the charged leptons with the {\\it down} quarks.\nSome attractive features of this symmetry should be pointed out.\n\ni) The renormalizability of the SM demands some quark-lepton symmetry,\nnamely, the fermions have to be combined into generations for the\ncancellation of the triangle anomalies.\n\nii) The proton decay is absent.\n\niii) A natural explanation for the quark fractional hypercharge takes\nplace.\n\n\\noindent Really, the 15-th generator of the $SU(4)$ group can be written\nin the form  $\\qquad \\qquad T_{15} = \\sqrt{3\/8} \\, diag \\, ( 1\/3 \\, , \\,\n1\/3 \\, , \\, 1\/3 \\, , \\, -1 )$.\nIt is traceless and the values of the left hypercharge $Y_L$ appear to\nbe placed on the diagonal. Let us call it the vector hypercharge,\n$Y_L = Y_V$.\n\niv) Let us suppose that $G_R \\, = \\, U(1)_R$. If we take the\nwell--known values\nof the SM hypercharge of the left and right, and $up$ and $down$ quarks and\nleptons, then from the equation $Y_{SM} \\, = \\, Y_V \\, + \\, Y_R$ the values\nof the right\nhypercharge $Y_R$ occur to be equal $\\pm 1$ for the $up$ and $down$ fermions,\nboth quarks and leptons.\nIt is tempting to interpret this fact as the evidence for the\nright hypercharge to be actually the doubled third component of the right\nisospin. Hence the $G_R$ group is possibly $SU(2)_R$.\n\nThe most exotic object of the Pati--Salam type symmetry is the charged and\ncolored gauge $X$ boson named leptoquark.\nIts mass $M_X$ should be the scale of reducing\nof $SU(4)_V$ to $SU(3)_c$. The bounds on the vector leptoquark\nmass\\cite{5} were obtained from the data on the $\\pi \\rightarrow e \\nu$\ndecay and from the upper limit on $K^0_L \\rightarrow \\mu e$ decay.\nIn fact, these estimations\nwere not comprehensive because the phenomenon of a mixing in the lepton-quark\ncurrents was not considered there. It can be shown that such a mixing\ninevitably occurs in the theory.\n\n\\section{Three types of fermion mixing}\n\nThree fermion generations are\ncombined into the \\{4,2\\} representations of the semi-simple group\n$SU(4)_V \\otimes SU(2)_L$ of the type\n\n\\begin{equation}\n\\left ( \\begin{array}{c} u^c \\\\ \\nu \\end{array} \\,\n\\begin{array}{c} d^c \\\\ \\ell \\end{array}\n\\right )_i , \\qquad (i=1,2,3) , \\label{eq:d}\n\\end{equation}\n\n\\medskip\n\n\\noindent where $c$ is the color index.\nThe mixing in the quark interaction with the $W$ bosons being depicted by\nthe Cabibbo-Kobayashi-Maskawa matrix is sure to exist in Nature.\nIf one starts from the diagonal $d, \\nu, \\ell$ states and the $u$\nstates mixed by the CKM matrix than at the one--loop level the $d$\nstates are mixed due to the conversion $d \\to W + u (c,t) \\to d'$ and\nthen the $\\ell$ states are mixed also, $\\ell \\to X + d (s,b) \\to \\ell'$, etc.\nConsequently, it is necessary for the renormalizability of the model\nto include all kinds of mixing at the tree-level.\nIn the general case, none of the $\\, u,d,\\nu,\\ell \\,$ components is the mass\neigenstate.\nDue to the identity of the three representations~(\\ref{eq:d})\nthey always could be regrouped so that one of the components was\ndiagonalized with respect to mass. If we diagonalize the charged\nlepton mass matrix, then the representations~(\\ref{eq:d}) can be rewritten\nto the form where the $\\nu, u, $ and $d$ states are not the mass\neigenstates and are included\ninto the same representations as the charged leptons $\\, \\ell$,\n$\\nu_\\ell  =  {\\cal K}_{\\ell i} \\nu_i , \\; u_{\\ell}  =\n{\\cal U}_{\\ell p} u_p , \\; d_{\\ell}  =  {\\cal D}_{\\ell n} d_n$ .\nHere  $\\nu_i, u_p, $ and $d_n \\, (i, p, n = 1,2,3)$  are the mass\neigenstates, and ${\\cal K}_{\\ell i} , \\, {\\cal U}_{\\ell p}$, and\n${\\cal D}_{\\ell n}$ are the unitary mixing matrices.\nThe standard Cabibbo-Kobayashi-Maskawa\nmatrix is seen to be $V \\, = \\, {\\cal U}^+ \\cal D$.\nThis is as far as we know about $\\cal U$ and $\\cal D$ matrices.\n$\\cal K$ is the mixing matrix in the lepton sector.\n\n\\section{Bounds from the low--energy experiments}\n\nSubsequent to the spontaneous $SU(4)_V$ symmetry breaking up to $SU(3)_c$\non the $M_X$ scale six massive vector bosons are separated from the 15-plet\nof the gauge fields to generate three charged and colored leptoquarks.\nTheir interaction with the fermions has the form\n\n\\begin{equation}\n{\\cal L}_X \\, = \\, \\frac{g_S(M_X)}{\\sqrt 2} \\big [\n{\\cal D}_{\\ell n}\n\\big ( \\bar \\ell \\gamma_{\\alpha} d^c_n \\big ) +\n\\big ( {\\cal K^+ \\; \\cal U} \\big )_{i p}\n\\big ( \\bar{\\nu_i} \\gamma_{\\alpha} u^c_p \\big ) \\big ] X^c_{\\alpha} +\nh.c.\n\\label{eq:Lx}\n\\end{equation}\n\n\\noindent The constant\n\\, $g_S(M_X)$ \\, can be expressed in terms of the strong coupling constant\n\\, $\\alpha_S$ \\, at the leptoquark mass scale\n$M_X, \\quad g_S^2(M_X)\/4 \\pi = \\alpha_S(M_X)$.\n\nIf the momentum transferred is \\, $q \\ll M_X$, \\, then the Lagrangian\n{}~(\\ref{eq:Lx}) in the second order leads to the effective four-fermion\nvector-vector interaction of quarks and leptons. By using the Fiertz\ntransformation, the scalar,\npseudoscalar, vector and axial-vector terms may be separated in the\neffective Lagrangian.\nThe QCD correction amounts to the appearance of the magnifying factor\n$Q(\\mu)$ at the scalar and pseudoscalar terms,\n$Q(\\mu) \\, = \\, ( {\\alpha_S(\\mu)}\/{\\alpha_S(M_X)})^{4\/\\bar b}$.\nHere $\\alpha_S(\\mu)$ is the effective strong coupling constant\nat the hadron mass scale $\\mu \\sim 1~GeV$,\n$\\; \\bar b \\, = \\, 11 \\, - \\, \\frac {2}{3} \\bar n_f, \\; \\bar n_f $\nis the averaged number of the quark\nflavors at the scales $\\mu^2 \\le q^2 \\le M_X^2$. If the condition\n$M_X^2 \\gg m_t^2$ is valid, then we have $\\, \\bar n_f \\, \\simeq \\, 6$,\nand $\\bar b \\, \\simeq \\, 7$.\n\nAs the analysis shows, the tightest restrictions on the leptoquark mass\n$M_X$ and the mixing matrix $\\cal D$ elements\ncan be obtained from the experimental data on rare $\\pi$ and $K$ decays and\n$\\mu^- \\rightarrow e^-$ conversion in nuclei.\nIn the description of the interactions of $\\pi$ and $K$ mesons it is\nsufficient to take the scalar and pseudoscalar terms only. As we\nshall see later, these terms acquire, in addition to the QCD corrections,\nan extra enhancement at the amplitude by the small quark current masses.\n\nOne can easily see that the leptoquark contribution to the $\\pi\n\\rightarrow e \\nu$ decay is not suppressed by the electron mass\nin contrast to the $W-$contribution.\nTaking into account the interference of the leptoquark and $W-$\nexchange amplitudes we get the following expression for the ratio\n\n\\begin{equation}\nR \\, = \\,\n\\frac{\\Gamma (\\pi \\rightarrow e \\nu)}\n{\\Gamma (\\pi \\rightarrow \\mu \\nu)} \\, = \\,\nR_{SM} \\left [ 1 \\, - \\,\n\\frac{2 \\sqrt 2 \\pi \\, \\alpha_S(M_X) \\, m^2_{\\pi} \\, Q(\\mu)}\n{G_F M^2_X m_e (m_u(\\mu) + m_d(\\mu) )} \\; Re \\left ( \\frac{{\\cal D}_{e d}\n{\\cal U}^*_{e u}}{V_{u d}} \\right ) \\right ],\n\\label{eq:R}\n\\end{equation}\n\n\\noindent where $R_{SM} = (1.2345 \\pm 0.0010) \\cdot 10^{-4}$ is\nthe value of the ratio\nin the Standard Model\\cite{6}, $m_{u,d}(\\mu)$ are the running current\nmasses. To the $\\mu \\simeq 1~GeV$ scale there correspond the well-known\nvalues $m_u \\simeq 4~MeV, m_d \\simeq 7~MeV$ and $m_s \\simeq 150~MeV$.\nUsing the experimental data\\cite{7},  $R^{exp} = (1.2310 \\pm 0.0037)\n\\cdot 10^{-4}$, we get\nthe following lower bound on the leptoquark mass\n\n\\begin{equation}\nM_X > (210~TeV) \\cdot |Re  ( {{\\cal D}_{e d}{\\cal U}^*_{e u}} \/\n{V_{u d}} ) |^{1\/2}.\n\\label{eq:X1}\n\\end{equation}\n\n{}From the data on the $K \\to e \\nu$ decay we obtain similarly\n\n\\begin{equation}\nM_X > (55~TeV) \\cdot |Re ({{\\cal D}_{e s}{\\cal U}^*_{e u}}\n\/ {V_{u s}} ) |^{1\/2}.\n\\label{eq:55T}\n\\end{equation}\n\nOne can establish the following limits from the data\\cite{8}\non the rare decays $K^0_L \\rightarrow \\mu e$ and\n$K^0_L \\rightarrow e^+ e^-$\n\n\\begin{equation}\nM_X > (1200~TeV) \\cdot |{\\cal D}_{e d}\n{\\cal D}^*_{\\mu s} \\; + \\;\n{\\cal D}_{e s}\n{\\cal D}^*_{\\mu d} |^{1\/2}, \\qquad\n\\label{eq:X2}\n\\end{equation}\n\n\\begin{equation}\nM_X > (1400~TeV) \\cdot |Re \\big ( {\\cal D}_{e d}\n{\\cal D}^*_{e s} \\big )|^{1\/2}.\n\\label{eq:Kee}\n\\end{equation}\n\nThe situation with another rare $K$ decay,\n$K^0_L \\rightarrow \\mu^+ \\mu^-$, is rather intriguing.\nThe recent measurements of the branching ratio at BNL\\cite{9}\nlowered its value closely to the unitary limit $Br_{abs} = 6.8 \\cdot 10^{-9}$,\nand thus the decay amplitude has no real part. However, it was\nshown\\cite{10} that the real part could not be small in the\nSM with a heavy top quark. Isn't it a signal for a new physics, e.g.\nleptoquark? In this regard the discontinuance of the experiment KEK E137\nwhere the $K^0_L \\rightarrow \\mu^+ \\mu^-$ decay rate was also measured,\nis regrettable.\n\nA low-energy process under an intensive experimental searches, where the\nleptoquark could manifest itself is the $\\mu e$ conversion in nuclei.\nWe estimate the branching ratio of the conversion in titanium and\nestablish the bound on the model parameters\non the base of the experimental data\\cite{11}\n\n\\begin{equation}\nM_X > (680~TeV) \\cdot |{\\cal D}_{e d}\n{\\cal D}^*_{\\mu d} |^{1\/2}.\n\\label{eq:Xmue}\n\\end{equation}\n\nThe above restrictions on the model parameters contain the elements of the\nunknown unitary mixing matrices $\\cal D$ and $\\cal U$, which are connected\nby the condition ${\\cal U}^+ {\\cal D} = V$ only.\nThus the possibility is not excluded, in principle, that the bounds obtained\ndid not restrict $\\, M_X \\,$ at all, e.g. if the elements ${\\cal D}_{e d}$\nand ${\\cal D}_{\\mu d}$ were rather small. It would correspond to the\nconnection of the $\\tau$ lepton largely with the $d$ quark in the ${\\cal D}$\nmatrix, and the electron and the muon with the $s$ and $b$ quarks.\nIn general, it is not contradictory to anything even if it appears to be\nunusual. In this case a leptoquark could give a more noticeable\ncontribution to the flavor-changing decays of the $\\tau$ lepton and\n$B$ mesons. However, an accuracy of these data is relatively poor.\n{}From the experimental limits\\cite{12} on the decays\n$\\tau^- \\to \\mu^- K^0$,\n$\\tau^- \\to e^- K^0$, and\n$B^+ \\to K^+ \\mu^+ e^-$,\n$B^+ \\to K^+ \\mu^- e^+$,\nwhich are possible via the leptoquark exchange without suppression by the\nelements ${\\cal D}_{e d}$ and ${\\cal D}_{\\mu d}$, we obtain\n\n\\begin{equation}\nM_X > (1~TeV) \\cdot |{\\cal D}_{\\mu s}\n{\\cal D}^*_{\\tau d} |^{1\/2},\n\\qquad\nM_X > (1~TeV) \\cdot |{\\cal D}_{e s}\n{\\cal D}^*_{\\tau d} |^{1\/2},\n\\label{eq:tau}\n\\end{equation}\n\n\\begin{equation}\nM_X > (2.4~TeV) \\cdot |{\\cal D}_{e s}\n{\\cal D}^*_{\\mu b} |^{1\/2},\n\\qquad\nM_X > (2.4~TeV) \\cdot |{\\cal D}_{\\mu s}\n{\\cal D}^*_{e b} |^{1\/2}.\n\\label{eq:BK}\n\\end{equation}\n\nIn the recent paper\\cite{13} the limits on the Pati--Salam leptoquark\nwere also considered in the specific cases when every charged\nlepton is connected with only one quark in the currents. For the most part\nthe results of ref.\\cite{13} agree with ours\\cite{1}.\n\n\\section {Mixing--independent bound}\n\nWe could find only one occasion when the mixing-independent lower bound\non the leptoquark mass arises, namely, from the decay\n$\\pi^0 \\rightarrow \\nu \\bar \\nu$. The best laboratory limit\\cite{14}\n on this decay is\n$\\; Br(\\pi^0 \\rightarrow \\nu \\bar \\nu ) < 8.3 \\cdot 10^{-7}$.\nIn the papers\\cite{15} the almost coinciding cosmological and\nastrophysical estimations of the width of this decay were found:\n$Br(\\pi^0 \\rightarrow \\nu \\bar \\nu ) < 3 \\cdot 10^{-13}$ .\nWithin the Standard Model this value is proportional to $m^2_{\\nu}$.\nThe process is also possible through the leptoquark mediation, without the\nsuppression by the smallness of neutrino mass.\nOn summation over all neutrino species the decay probability\nis mixing-independent. As a result the bound on the leptoquark mass occurs\n$M_X > 18~TeV$. However, in the recent paper\\cite{16} a criticism\nhas been expressed on both the cosmological and astrophysical limits.\nTherefore, only the laboratory limit\\cite{14} is reliable to establish\nthe bound $M_X > 440~GeV$.\n\n\\section {Rare muon decays}\n\nThe lepton--number violating decays $\\mu \\rightarrow e \\gamma$,\n$\\mu \\rightarrow e \\gamma \\gamma$, $\\mu \\rightarrow e e \\bar e$\nare under the intensive experimental searches. Let us point out,\nhowever, that these decay modes are strongly suppressed in the SM with\nlepton mixing due to\nthe well--known GIM cancellation\\cite{17} by the factor\n$(m_{\\nu}\/m_W)^4 \\; \\sim \\; 10^{-39} \\cdot\n(m_{\\nu}\/20 \\,eV)^4$.\n\nThese processes arise in the model with vector leptoquark at the loop\nlevel via the virtual $d, s, b$ quarks\\cite{2}.\nAs the analyses of the radiative muon decays show, the two--photon decay\ndominates the one--photon decay\n\n\\begin{equation}\n\\frac{\\Gamma(\\mu\\to e\\gamma\\gamma)}{\\Gamma(\\mu\\to e\\gamma)} \\; \\sim \\;\n\\frac{\\alpha}{\\pi} \\, \\left( \\frac{M_X}{m_b} \\right) ^{4} \\; \\gg \\; 1.\n\\label{eq:GIM1}\n\\end{equation}\n\n\\noindent The magnitude of the $\\mu \\rightarrow e \\gamma \\gamma$ decay width\ncould be estimated using the bound~(\\ref{eq:Xmue}):\n\n\\begin{equation}\nBr(\\mu \\rightarrow e \\gamma \\gamma) \\; < \\; 1.0 \\cdot 10^{-18}.\n\\label{eq:BrSD}\n\\end{equation}\n\n\\noindent A similar analysis of the $\\mu \\to e e \\bar e$ decay\nwithin the above restrictions on the model parameters provides\n\n\\begin{equation}\nBr(\\mu \\rightarrow e e \\bar e)  \\; < \\; 1.0 \\cdot 10^{-15}.\n\\label{eq:Br3e}\n\\end{equation}\n\nAlthough the predicted values of the branches of the $\\mu \\to e \\gamma\n\\gamma$ and $\\mu \\rightarrow e e \\bar e$ decays\nare essentially less then the existing\nexperimental limits\\cite{18}\n$Br(\\mu \\rightarrow e \\gamma \\gamma)_{exp} < 7.2 \\cdot 10^{-11}$,\n$Br(\\mu \\rightarrow e e \\bar e)_{exp} < 1.0 \\cdot 10^{-12}$,\nthey are not as small as the predictions of the SM with lepton mixing,\nand a hope for their observation in the future still remains.\n\n\\section {Conclusions}\n\n$\\bullet$ The bounds on the Pati--Salam leptoquark mass were reexamined\nwith taking account of the mixing in the quark--lepton currents.\n\n\\noindent $\\bullet$\nSome semileptonic meson decays strongly suppressed within the SM\ncould be induced by vector leptoquark. Their further experimental\ninvestigations are very important.\n\n\\noindent $\\bullet$\nThe only mixing independent bound on the vector leptoquark mass arises\nfrom the limits on the invisible $\\pi^0 \\rightarrow \\nu \\bar \\nu$ decay.\nIt is $M_X > 440~GeV$ from the laboratory limit and\n$M_X > 18~TeV$ from the cosmological limit, but the last has to be verified.\n\n\\noindent $\\bullet$\nThe specific hierarchy of the rare muon decay probabilities via vector\nleptoquark takes place and the branching ratios are not as small as the\npredictions of the SM with lepton mixing.\n\n\\bigskip\n\n\\noindent {\\bf Acknowledgements.}\nWe are grateful to L.B.~Okun, J.~Ritchie, V.A.~Rubakov, A.D.~Smirnov,\nK.A.~Ter-Martirosian and S.~Willenbrock for helpful discussions.\n\nThe research described in this publication was made possible in part by\nGrant No. RO3000 from the International Science Foundation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction.}\n\\label{sec01}\n\n\nDynamics of elastic lattices is of exceptional importance in a wide range of applications in problems of structural mechanics. \nThe pioneering work of N.F. Morozov and his co-authors has addressed scale effects and dynamic  trapping for cracks propagating in transient regimes through structured media, as well as formation of nanocracks in crystalline solids \\cite{MPU1, MP, MPU2, Morozov_Petrov,Morozov_Petrov1}. An important concept of the dynamic fracture toughness at a stage of the crack initiation has been fully investigated, and an ``incubation time'' principle has been introduced by N.F. Morozov and his co-authors to describe time-dependent fracture in structured materials for the general types of loading. \n\n\n\n\n\nA significant impact has been made by the investigations of N.F. Morozov {\\em et al.}\\ \\cite{MT1,MT2,MT3,MT4} on the dynamics and stability of elastic rods subjected to transient longitudinal loads.   A new insight has been presented by Morozov and Tovstik, who had addressed the influence of longitudinal loads on the dynamics of elastic rods and elastic waves. This also includes important features such as parametric resonances and the dynamic loss of stability for the loads, whose magnitude is less than the classical Eulerian load. These important studies also extend to the non-linear regimes to analyse the growth of the post-critical deformations. \n\nAlthough most of micro-structured solids with defects are not periodic, they may possess structured interfaces or large subdomains, which include locally periodic patterns. Dynamic response of such elastic systems may be very unusual and sometimes counter-intuitive. The theory of Floquet-Bloch waves has been successfully used to explain the dynamic anisotropy, dispersion in multi-structured solids and localisation.       In particular, Slepyan \\cite{Slepyan_2002} and Slepyan and Ryvkin \\cite{Slepyan_Ryvkin_2010} have studied Floquet-Bloch waves in conjunction with the problems of dynamic fracture.\nStructured interfaces, asymptotic analysis and the transmission problems for solids, containing structured interfaces, have been analysed by  Bigoni and Movchan \\cite{Bigoni_2002}. For dynamic formulations in structured media where the long-wave approximations are not valid, the  high-frequency homogenisation is required, as explained in \\cite{Craster_2014}, and this may also identify the anisotropy and localisation of waveforms in micro-structured solids. The so-called  band gap Green's functions, together with the dynamic defect modes, were  analysed in \\cite{Movchan_Slepyan}.\n\n\n\n\nIn the previous papers by Piccolroaz and Movchan \\cite{PM_2014} and Piccolroaz {\\em et al.} \\cite{PMC2017a}, Floquet-Bloch elastic waves were considered for elastic networks of elastic beams, with an additional rotational inertia. The authors also drew a connection between dynamic models of couple-stress elastic materials and structured Rayleigh beams. In particular, the Rayleigh beam model includes effects of the rotational inertia, that is excluded from the theory of the classical Euler-Bernoulli beams. It was also demonstrated that the rotational inertia is apparently significant for the high-frequency dynamic response, especially in problems involving the Dirac cone steering and analysis of the dispersion degeneracies.   \n\n\n\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=120mm]{honeycomb_lattice_interface_R.pdf}\n\\caption{\\footnotesize A composite honeycomb lattice of flexural beams, which contains a structured interface consisting of the Rayleigh beams with an additional rotational inertia.}\n\\label{honeycomb_lattice_interface}\n\\end{figure}\n\n\n\nFig.~\\ref{honeycomb_lattice_interface} shows a honeycomb flexural lattice containing a structured interface, where the classical Euler-Bernoulli beams are used to construct the ambient network, whereas the structured interface is built of the Rayleigh beams with an appropriately designed rotational inertia. \nWhen a point source is applied in the proximity of such a structured interface, it is demonstrated that the dynamic response of such a multi-scale solid may show localisation, negative refraction as well as dynamic anisotropy and interfacial edge waves.\n\nThe structure of the paper is as follows. The main definitions and the geometry of the Rayleigh beam network are given in Section \n\\ref{sec02}. The elementary cell and the Bloch-Floquet junction conditions are discussed in Section \\ref{BFjc}. \nSection \\ref{sec03} is devoted to the dispersion of waves in the networks of the Euler-Bernoulli and of the Rayleigh beams. The dynamic anisotropy and the slowness contours corresponding to the dispersion diagrams are discussed in Section \\ref{sec04}. The study of the forced vibrations and of the dynamic response of structured interfaces is presented in Section \\ref{forced}. Finally, Section \\ref{sec05} includes concluding remarks and discussion.  \n\n\n\n\\section{The Rayleigh beam lattice.}\n\\label{sec02}\n\n\nWe begin by considering a doubly periodic honeycomb lattice consisting of the Rayleigh beams, which possess a rotational inertia. The geometry of the periodic structure is shown in Fig.~\\ref{fig01}, which includes a periodic honeycomb lattice consisting of Rayleigh beams. The elementary cell, of a parallelogram shape is also shown, together with the basis vectors of the lattice, $\\mbox{\\boldmath $v$}_1=(\\sqrt{3}\/2\\ h, 3\/2\\ h)$ and $\\mbox{\\boldmath $v$}_2=(-\\sqrt{3}\/2\\ h, 3\/2\\ h)$ where $h$ is the length of the beams.\n\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=120mm]{Reticolo_R.pdf}\n\\caption{\\footnotesize  A periodic honeycomb lattice consisting of Rayleigh beams. The elementary cell, of a parallelogram shape is shown, together with the basis vectors of the lattice.}\n\\label{fig01}\n\\end{figure}\n\n\n\nWithin a one-dimensional ligament of the lattice, consider a time harmonic flexural wave, with the out-of-plane displacement \n$U(x, t) = u(x) \\exp(i \\omega t),$ with $x$ being a local spatial coordinate and $t$ denoting time. \n\nIn the Rayleigh beam, the amplitude function $u(x)$ satisfies the governing equation\n\\begin{equation}\n\\label{eq:gov}\nEI u''''(x) - (P - \\rho I \\omega^2) u'' + (\\beta - \\rho A \\omega^2) u = 0\n\\end{equation}\nHere the following notations are in use: $E$ is the Young modulus, $\\beta$ is the stiffness of a Winkler type elastic foundation, $P$ the prestress, $\\rho$ the mass density, $A$ the area of the cross-section, and $I$ the area moment of inertia of the cross-section.\n\nLet $M$ and $V$ denote respectively the internal bending moment and the shear force, as follows: \n\\begin{equation}\nM(x) = -EI u''(x), \\quad V(x) = -EI u'''(x) + (P - \\rho I \\omega^2) u'(x)\n\\end{equation}\nIf the Rayleigh beam is of infinite extent and prestress and elastic foundation are absent, then the solution  of (\\ref{eq:gov}) can be written as\n\\begin{equation}\nu(x) = \\sum_{q=1}^{4} C_q e^{i \\kappa_q x} \\label{u_repr}\n\\end{equation}\nwhere \n\\begin{equation}\n\\kappa_{1,2,3,4} = \\pm \\frac{1}{r} \\sqrt{\\alpha\\frac{R\\omega^2}{2} \\pm \\sqrt{\\alpha\\frac{R^2\\omega^4}{4} + R\\omega^2}}\n\\label{kappa}\n\\end{equation}\nThe following notations are used here:\n\\begin{equation}\nr = \\sqrt{\\frac{I}{A}}, \\quad\nR = \\frac{\\rho r^2}{E} = \\frac{\\rho I}{EA}\n\\end{equation}\nHere $\\alpha=1$ for the Rayleigh beams, and $\\alpha=0$ for the Euler-Bernoulli beams. \n\n\n\n\n\\section{The Floquet-Bloch and junction conditions within the elementary cell.}\n\\label{BFjc}\n\n\nFor the flexural lattice ligaments, shown in Fig.~\\ref{fig01}, it is convenient to introduce local coordinates $x_j$ for each ligament and to refer to the junction points E and F, where three ligaments are connected.  \n\nFor the structure of five beams, shown in Fig.~\\ref{fig01}(c), the local displacement amplitudes $u_q(x_q)$, $q= 1,\\ldots,5,$  (there is no summation with respect to the repeated index $q$) are\n\\begin{equation}\nu_q (x_q) = \\sum_{p=1}^4 C_{pq} \\exp(i \\kappa_{pq} x_q), ~ q=1,\\ldots, 5 \\label{disp_nodal}\n\\end{equation}\nand the quantities $\\kappa_{pq}$ are defined by (\\ref{kappa}) for each flexural ligament of the elementary cell. \nHence, \nthere are 20 constants, $C_{pq}$, $p=1,\\cdots,4$, $q=1,\\cdots,5$, which can be found by solving the system of equations derived from the 8 Floquet-Bloch conditions at the boundary of the unit cell (points A, B, C, D), and from the 12 junction conditions at the points E and F) .\n\nIn particular, for the points $A-B$ of the unit cell, the quasi-periodic conditions at $x_2 = h\/2$ and $x_4 = h\/2$ hold\n\\begin{equation}\nu_2\\left(\\frac{h}{2}\\right) = u_4\\left(\\frac{h}{2}\\right) e^{ih(\\sqrt{3}\/2k_x+3\/2k_y)} \\label{eq6}\n\\end{equation}\n\\begin{equation}\nu_2'\\left(\\frac{h}{2}\\right) = -u_4'\\left(\\frac{h}{2}\\right) e^{ih(\\sqrt{3}\/2k_x+3\/2k_y)}\n\\end{equation}\n\\begin{equation}\n- EI u_2''\\left(\\frac{h}{2}\\right) = - EI u_4''\\left(\\frac{h}{2}\\right) e^{ih(\\sqrt{3}\/2k_x+3\/2k_y)}\n\\end{equation}\n\\begin{equation}\n-EI u_2'''\\left(\\frac{h}{2}\\right) - \\rho I \\omega^2 u_2'\\left(\\frac{h}{2}\\right) =\n- \\left[ -EI u_4'''\\left(\\frac{h}{2}\\right) - \\rho I \\omega^2 u_4'\\left(\\frac{h}{2}\\right) \\right] e^{ih(\\sqrt{3}\/2k_x+3\/2k_y)}\n\\end{equation}\nwhich prescribe the Floquet-Bloch shift across the unit cell for flexural displacement, rotation, internal moment and internal shear force. \n\nAnalogous quasi-periodic boundary conditions apply to the points $C-D$ of the unit cell,\n\\begin{equation}\nu_3\\left(\\frac{h}{2}\\right) = u_5\\left(\\frac{h}{2}\\right) e^{ih(-\\sqrt{3}\/2k_x+3\/2k_y)}\n\\end{equation}\n\\begin{equation}\nu_3'\\left(\\frac{h}{2}\\right) = -u_5'\\left(\\frac{h}{2}\\right) e^{ih(-\\sqrt{3}\/2k_x+3\/2k_y)}\n\\end{equation}\n\\begin{equation}\n- EI u_3''\\left(\\frac{h}{2}\\right) = - EI u_5''\\left(\\frac{h}{2}\\right) e^{ih(-\\sqrt{3}\/2k_x+3\/2k_y)}\n\\end{equation}\n\\begin{equation}\n-EI u_3'''\\left(\\frac{h}{2}\\right) - \\rho I \\omega^2 u_3'\\left(\\frac{h}{2}\\right) =\n- \\left[ -EI u_5'''\\left(\\frac{h}{2}\\right)- \\rho I \\omega^2 u_5'\\left(\\frac{h}{2}\\right) \\right] e^{ih(-\\sqrt{3}\/2k_x+3\/2k_y)}\n\\end{equation}\nThe junction conditions at the node $E$ are as follows.\nContinuity of displacement implies\n\\begin{equation}\nu_2(0) = u_1\\left(\\frac{h}{2}\\right)\n\\end{equation}\n\\begin{equation}\nu_3(0) = u_1\\left(\\frac{h}{2}\\right)\n\\end{equation}\nand the continuity of rotations yields\n\\begin{equation}\nu_1'\\left(\\frac{h}{2}\\right) = u_2'(0)+u_3'(0)\n\\end{equation}\nequations of motion for the node $E$ become\n\\begin{multline}\n\\left[ -EI u_1'''\\left(\\frac{h}{2}\\right)- \\rho I \\omega^2 u_1'\\left(\\frac{h}{2}\\right) \\right]=\n\\left[ -EI u_2'''(0) - \\rho I \\omega^2 u_2'(0) \\right] +\n\\left[ -EI u_3'''(0) - \\rho I \\omega^2 u_3'(0) \\right]\n\\end{multline}\n\\begin{equation}\nEI u_1''\\left(\\frac{h}{2}\\right)=\\frac{EI}{2} u_2''(0)+\\frac{EI}{2} u_3''(0)\n\\end{equation}\n\\begin{equation}\nEI u_2''(0)=EI u_3''(0)\n\\end{equation}\n\n\\noindent\nThe junction conditions at the node $F$ include: continuity of displacement\n\\begin{equation}\nu_4(0) = u_1\\left(-\\frac{h}{2}\\right)\n\\end{equation}\n\\begin{equation}\nu_5(0) = u_1\\left(-\\frac{h}{2}\\right)\n\\end{equation}\ncontinuity of rotations\n\\begin{equation}\nu_1'\\left(-\\frac{h}{2}\\right)+u_4'(0)+u_5'(0)=0\n\\end{equation}\nand the equations of motion for the node $F$\n\\begin{multline}\n\\left[ -EI u_1'''\\left(-\\frac{h}{2}\\right)- \\rho I \\omega^2 u_1'\\left(-\\frac{h}{2}\\right) \\right]=\n\\left[ -EI u_4'''(0) - \\rho I \\omega^2 u_4'(0) \\right] +\n\\left[ -EI u_5'''(0) - \\rho I \\omega^2 u_5'(0) \\right]\n\\end{multline}\n\\begin{equation}\nEI u_1''\\left(-\\frac{h}{2}\\right)=\\frac{EI}{2} u_4''(0)+\\frac{EI}{2} u_5''(0)\n\\end{equation}\n\\begin{equation}\nEI u_4''(0)=EI u_5''(0) \\label{eq25}\n\\end{equation}\n\n\n\n\n\n\n\\noindent\nEquations (\\ref{eq6})--(\\ref{eq25}) comprise a homogeneous linear algebraic system for the 20 unknown constants, and the requirement of the vanishing of the determinant of the matrix of this algebraic system yields the dispersion equation for the Floquet-Bloch waves in the honeycomb periodic lattice of the Rayleigh beams. \n\n\\section{Lower-dimensional model, dispersion properties.}\n\\label{sec03}\n\nIn the earlier paper \\cite{PMC2017a}, which addressed the networks of flexural beams for square and rectangular lattices, it has been noted that the \nanalysis of Floquet-Bloch waves contributes to the studies of the dynamic response of structured solids.\nHere we analyse the dispersion properties of flexural waves in honeycomb systems with rotational inertia, and also make a comparison between periodic networks of different geometries. \n\n\n\\subsection{The algebraic system.}\n\nSubstitution of the representation\n\\eq{disp_nodal} into the 20 relations \n(\\ref{eq6})--(\\ref{eq25}) \nleads to a system of linear algebraic equations with respect to the variables  $C_{pq}$. Introducing a vector $$\\mbox{\\boldmath $C$} = \\Big(C_{11}, C_{12}, C_{13}, C_{14}, \\ldots, C_{51}, C_{52}, C_{53}, C_{54}\\Big)$$ the matrix form of the equations is \n\\begin{equation}\n\\mbox{\\boldmath $\\mathcal{A}$}(\\omega, \\mbox{\\boldmath $k$}) \\mbox{\\boldmath $C$}^T =0\n\\label{alg_syst}\n\\end{equation}\nwhere $\\mbox{\\boldmath $\\mathcal{A}$}(\\omega, \\mbox{\\boldmath $k$})$ is a $20 \\times 20$ matrix  function, whose arguments are the radian frequency $\\omega$ and the Bloch vector $\\mbox{\\boldmath $k$}=(k_x,k_y)$. The dispersion equation has the form\n\\begin{equation}\n\\det \\mbox{\\boldmath $\\mathcal{A}$}(\\omega, \\mbox{\\boldmath $k$}) = 0\n\\label{disp_eq}\n\\end{equation}\nThe above equation defines implicitly the dispersion surfaces, and for any given $\\mbox{\\boldmath $k$}$ from the Brillouin zone of the reciprocal lattice we identify a countable set of values of the spectral parameter $\\omega$.\nThe dispersion equation has been solved and the results concerning the dispersion surfaces and, in particular, the Dirac cones and saddle points are discussed below.\n\n\\subsection{Dirac cones and standing waves.}\n\nHoneycomb lattice of the Rayleigh beams has many attractive features, due to its high-order symmetry and possession of a dynamic response linked to dynamic anisotropy and to standing waves.\n\nIn particular, Figs.~\\ref{fig3dh}, \\ref{fig3d} shows the surfaces, which include a conical point, which corresponds to a finite non-zero frequency. The conical surfaces, adjacent to that point, are referred to as {\\em Dirac cones}. It is also noted that a standing wave occurs, represented by a surface of zero slope traversing through the Dirac cone vertex.\n\n\n\n\n\n\nComparing the cases of the honeycomb networks of the Rayleigh beams, with an additional rotational inertia, and the classical case of the Euler-Bernoulli beams, we observe a clear distinction between the dispersion diagrams.\nThe first three dispersion surfaces for the Floquet-Bloch waves in the lattice, consisting of the Rayleigh beams, are placed at much lower frequencies compared to those for the Euler-Bernoulli beams,  as demonstrated in Fig.~\\ref{fig3dh}. \n\nWe also provide a comparison between the cases of different lattice geometries: the honeycomb lattice and the square lattice of flexural beams are discussed here. The results, concerning the dispersion of the Floquet-Bloch waves in the square flexural lattice were presented in the earlier work \\cite{PMC2017a}. \nIn particular, for the square lattice of elastic beams (both cases of Euler-Bernoulli and Rayleigh beams) the smoothness of dispersion surfaces and their degeneracies were investigated.\nThe effects of rotational inertia, attributed to the Rayleigh beams, deserve special attention, as illustrated in the text below.\n\n\n\n\n\n\nFor the honeycomb lattice, where the normalised physical and geometrical parameters are chosen as $E=1$, $\\rho=1$, $A=1$, $I=1$, $h=1$, $P=0$, $\\beta=0$, the dispersion equation, which relates the radian frequency $\\omega$ and the wave vector $(k_x, k_y)$,  takes the form\n\\begin{multline}\n\\sin ({\\Omega_1 h}) \\sinh ({\\Omega_2 h}) \\left[4 \\cos \\left(\\frac{\\sqrt{3}}{2}{k_x h}-\\frac{3}{2} {k_y h}\\right) \\right.\n+4 \\cos \\left(\\frac{\\sqrt{3}}{2}{k_x h}+\\frac{3}{2} {k_y h}\\right) \\\\ \\left.  +4 \\cos \\left(\\sqrt{3} {k_x h}\\right)-9 \\cos (2 {\\Omega_1 h})-3\\right]\n\\left[4 \\cos \\left(\\frac{\\sqrt{3}}{2}{k_x h}-\\frac{3}{2} {k_y h}\\right) \\right. \\\\ \\left. +4 \\cos \\left(\\frac{\\sqrt{3}}{2}{k_x h}+\\frac{3}{2} {k_y h}\\right)+4 \\cos \\left(\\sqrt{3} {k_x h}\\right)-9 \\cos (2 {\\Omega_2 h})-3\\right]=0 \\label{disp_honey}\n\\end{multline}\nwhere\n\\begin{equation}\n\\Omega_1 = \\frac{1}{r} \\sqrt{\\sqrt{\\alpha\\frac{R^2\\omega^4}{4} + R\\omega^2}+\\alpha\\frac{R\\omega^2}{2}},\n\\quad\n\\Omega_2 = \\frac{1}{r} \\sqrt{\\sqrt{\\alpha\\frac{R^2\\omega^4}{4} + R\\omega^2}-\\alpha\\frac{R\\omega^2}{2}}\n\\end{equation}\nin which $\\alpha=1$ for Rayleigh beams and $\\alpha=0$ for Euler-Bernoulli beams.\n\n\n\n\n\n\n\nThe high degree of symmetry of the honeycomb structure leads to a nicely factorised form of the dispersion equation, as above. In particular, two of the factors in the left-hand side are ${\\mbox{\\boldmath $k$}}$-independent, and there exists a countable set of standing waves characterised by the flat dispersion surfaces at $\\Omega_1(\\omega) h = \\pi k$ for any integer $k$.\nOne of such  flat surfaces is shown in Fig.~\\ref{fig3dh}, together with the two dispersion surfaces that are not smooth and have conical points. This figure includes several parts: (a) the Euler-Bernoulli beam structure, (b) the Rayleigh beam structure. Neither prestress nor elastic foundations are present in this computation. The physical and geometrical parameters are chosen as follows: $E=1$, $\\rho=1$, $A=1$, $I=1$, $h=1$. The effect of the rotational inertia, inherent to the Rayleigh beams, is significant,   as the first several dispersion surfaces occur at much lower frequencies compared to the corresponding surfaces for the Euler-Bernoulli's beams as in part (a).\n\n\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=48mm]{fig3dVEBh.pdf}\n\\includegraphics[width=48mm]{fig3dVRAh.pdf}\n\\caption\n{\\footnotesize Dispersion surfaces for Floquet-Bloch waves in a periodic honeycomb lattice: (a)  the Euler-Bernoulli beam structure, (b) the Rayleigh beam structure. Neither prestress nor elastic foundations are present in this computation. The physical and geometrical parameters are chosen as follows: \n$E=1$, $\\rho=1$, $A=1$, $I=1$, $h=1$.\nThe effect of the rotational inertia, inherent to the Rayleigh beams, is significant,   as the first several dispersion surfaces occur at much lower frequencies compared to the corresponding surfaces for the Euler-Bernoulli's beams as in part (a).\n}\n\\label{fig3dh}\n\\end{figure}\n\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=48mm]{fig3dVEB.pdf}\n\\includegraphics[width=48mm]{fig3dVRA.pdf}\n\\caption{\\footnotesize The case of a square lattice. Dispersion surfaces are presented for \nthe Euler-Bernoulli beam structure (a), and for the \nnetwork of Rayleigh beams  (b).\nThe parameter values, used in this combination, are $E=1$, $\\rho=1$, $A=1$, $I=1$, ${h}=1$.\n} \n\\label{fig3d}\n\\end{figure}\n\n\nAlthough a square lattice of flexural beams possesses interesting dispersion properties for the Floquet-Bloch waves, as shown in Fig.~\\ref{fig3d}, there are significant differences compared to the case of the honeycomb flexural lattice. Dispersion surfaces are presented for the Euler-Bernoulli beam structure in part (a), and for the network of Rayleigh beams in part (b). The parameter values, used in this combination, are $E=1$, $\\rho=1$, $A=1$, $I=1$, ${h}=1$.\nIn particular, the dispersion equation cannot be factorized to a similar form as in (\\ref{disp_honey}), and the flat dispersion surfaces characterising standing waves are absent. Instead, the emphasis is made on the dynamic anisotropy, and special dispersion features of the Floquet-Bloch waves in the neighbourhoods of the vertices of the cones.  \n Following \\cite{McPhedran_2015} we refer to these conical surfaces as ``Dirac cones'', and the radian frequency $\\omega$ corresponding to the vertex of the cone is identified as a special resonant frequency, corresponding to a multiple root of the dispersion equation.  \n\n\nIn particular, if the rotational inertia, the pre-stress and the stiffness of the elastic foundation are equal to zero, i.e.\\ the Rayleigh beam becomes the classical Euler-Bernoulli beam, the dispersion equation \\eq{disp_eq} for Floquet-Bloch waves in the square lattice takes the form\n\\begin{multline}\n\\sin \\left(\\Omega {h}\\right)\n\\left[\\cos (k_x {h}) + \\cos (k_y {h}) - 2 \\cos \\left(\\Omega {h}\\right)\\right]\n\\left[\\cos (k_x {h}) - \\cosh \\left(\\Omega {h}\\right)\\right]\n\\left[\\cos (k_y {h}) - \\cosh \\left(\\Omega {h}\\right)\\right] \\\\\n-\\sinh \\left(\\Omega {h}\\right)\n\\left[\\cos (k_x {h}) + \\cos (k_y {h}) - 2 \\cosh \\left(\\Omega {h}\\right)\\right]\n\\left[\\cos (k_x {h}) - \\cos \\left(\\Omega {h}\\right)\\right]\n\\left[\\cos (k_y {h}) - \\cos \\left(\\Omega {h}\\right)\\right] = 0\n\\end{multline}\nwhere ${h}$ is the length of the ligaments in the square lattice, $\\Omega = \\sqrt{\\omega} \\sqrt[4]{\\frac{\\rho A}{EI}}$, and the corresponding dispersion diagram is shown in Fig.~\\ref{fig3d}a.\n\n\\subsection{Resonant modes for elementary ligaments.}    \nIt turns out that the frequencies, corresponding to the Dirac cone vertices, and some of the corresponding vibration modes (i.e. standing waves) are identified as the natural frequencies and the eigenmodes of a simply supported beam, respectively. It also explains why when the size of beam ligaments and their physical properties are the same, the Dirac cone vertices for the honeycomb and for the square lattice occur at the same frequencies: \n\n\\begin{equation}\n\\omega_n = \\frac{n^2 \\pi^2}{{h}^2} \\sqrt{\\frac{EI}{\\rho A}} \\qquad \\text{for the Euler-Bernoulli beam} \\label{swEB}\n\\end{equation}\n\n\\begin{equation}\n\\omega_n = \\frac{n^2 \\pi^2}{{h}} \\sqrt{\\frac{EI}{\\rho (A{h}^2+n^2\\pi^2I)}} \\qquad \\text{for the Rayleigh beam} \\label{swR}\n\\end{equation}\n\n\n\nThe honeycomb lattice, even in the low-frequency regime, shows a very different dynamic response compared to the square lattice. It follows from the diagrams of Fig.~\\ref{fig3dh} and Fig.~\\ref{fig3d}, which present the dispersion surfaces for the Floquet-Bloch waves, that the honeycomb lattice is locally isotropic in the neighbourhood of the Dirac cone frequencies as well as at low frequencies. On the contrary, the orthotropy of the square lattice is apparent, and in particular, it leads to an unusual directional preference in the vicinity of the Dirac cones frequencies. \n\n\nAlso the effect of the rotational inertia, which is present in the Rayleigh beam structure, becomes important, and consequently the dispersion surfaces representing structures consisting of the Rayleigh beams and structures consisting of the Euler-Bernoulli beams, become different, as seen in Figs.~\\ref{fig3dh},~\\ref{fig3d}.\nFloquet-Bloch waves in a Rayleigh beam structure would show zero group velocity at much lower frequencies compared to the similar structure of the Euler-Bernoulli beams. At higher frequencies the Rayleigh beams show much richer behaviour, especially when it is concerned with the degeneracies and formation of the Dirac cones. \n\n\n\n\nFigs.~\\ref{figband2h}, \\ref{figband2} complement the  dispersion surfaces diagrams by the cross-sectional plots along the boundaries of the irreducible  Brillouin zones in the reciprocal lattices constructed for the honeycomb and the square networks of flexural beams.\nIn particular, Fig.~\\ref{figband2h} includes the cross-sectional dispersion diagrams along the boundary of the irreducible Brillouin zone, for Floquet-Bloch waves in the honeycomb lattice comprised of the Euler-Bernoulli beams in part (a) and the Rayleigh beams in part (b). \n{The inset on the right shows the contour $\\Gamma M K \\Gamma$ within the first Brillouin zone in the elementary cell of the reciprocal lattice; the dotted square corresponds to the computational window chosen to draw the dispersion surfaces in Fig.~\\ref{fig3dh}.}\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=150mm]{disp_diagrams.pdf}\n\\caption{\n\\footnotesize The cross-sectional dispersion diagrams along the boundary of the irreducible Brillouin zone, for Floquet-Bloch waves in the honeycomb lattice comprised of the Euler-Bernoulli beams (a) and the Rayleigh beams (b).\n{The inset on the right shows the contour $\\Gamma M K \\Gamma$ within the first Brillouin zone in the elementary cell of the reciprocal lattice; the dotted square corresponds to the computational window chosen to draw the dispersion surfaces in Fig.~\\ref{fig3dh}.}\n}\n\\label{figband2h}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=150mm]{figband2.pdf}\n\\caption{\n\\footnotesize The cross-sectional dispersion diagrams along the boundary of the irreducible Brillouin zone, for Floquet-Bloch waves in the square networks of the Euler-Bernoulli beams (a) and the Rayleigh beams (b).\n{The inset on the right shows the contour $\\Gamma X M \\Gamma$ within the first Brillouin zone in the elementary cell of the reciprocal lattice; the dotted square corresponds to the computational window chosen to draw the dispersion surfaces in Fig.~\\ref{fig3d}.}\n}\n\\label{figband2}\n\\end{figure}\n\n\nSimilar diagrams for square networks are shown in Fig.~\\ref{figband2} where the case of the Euler-Bernoulli beams is shown in part (a) and the Rayleigh beams case is displayed in part (b).\n{The inset on the right shows the contour $\\Gamma X M \\Gamma$ within the first Brillouin zone in the elementary cell of the reciprocal lattice; the dotted square corresponds to the computational window chosen to draw the dispersion surfaces in Fig.~\\ref{fig3d}.\n\n\n\n\n\n\nSpecial attention is given to the Dirac cones, represented by the intersecting dispersion curves. In particular, for the first intersection the case of the square lattice shows a triple root of the dispersion equation, with the standing wave being represented by the flat band crossing through the Dirac cone vertex. On the contrary, the Floquet-Bloch waves in the honeycomb network flexural waves also posses the Dirac cone mode, but for the first intersection depicted in Fig.~\\ref{figband2h} the standing wave is absent. We note that the standing wave for honeycomb network of flexural beams occurs at the second intersection, and this mode is represented by the flat band at the frequency defined by formulae \\eq{swEB},  \\eq{swR}.\nAlso, we remark that the additional rotational inertia attributed to the Rayleigh beams leads to the dispersion curves being compressed towards the horizontal axis compared to those on the diagrams presented for the Euler-Bernoulli beams. This feature is clearly visible from the comparison of the parts (a) and (b) on the dispersion diagrams shown in Figs.~\\ref{figband2h} and \\ref{figband2}.\n\n\n\n\n\n\n\n\n\n\\section{Floquet-Bloch waves in a honeycomb network. Saddle points and slowness contours.}\n\\label{sec04}\n\n\nThe slowness contours (often referred to as isofrequency contours) are useful to identify stationary points on the dispersion surface as well as the dynamic anisotropy of the structured medium. %\nA comprehensive  exposition of such an approach for flexural waves in periodic square lattices is discussed in \\cite{PMC2017a}.\n\n\n\n\n\n\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=120mm]{fig1EB.pdf} \\\\[6mm]\n\\includegraphics[width=120mm]{fig1RA.pdf}\n\\caption{\n\\footnotesize\nFirst dispersion surface and the corresponding isofrequencies contours for the Euler-Bernoulli beam honeycomb lattice (parts (a) and (b)) and for the honeycomb lattice of the Rayleigh beams (parts (c) and (d)). The Dirac cone is shown in both configurations. The slowness contours around the origin bound non-convex domains in the diagram (d), for the Rayleigh beams, in contrast with the diagram (b), corresponding to the Euler-Bernoulli beams. \n}\n\\label{slowness_contours}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=120mm]{fig2EB.pdf} \\\\[6mm]\n\\includegraphics[width=120mm]{fig2RA.pdf}\n\\caption{\\footnotesize\nThe second dispersion surface, including the Dirac cone, presented for the Floquet-Bloch waves in the case of the Euler-Bernoulli beam square lattice (parts (a) and (b)) and of the Rayleigh beam square lattice (parts (c) and (d)). Preferential directions along the coordinate axes are clearly identified.\n}\n\\label{slowness_contours2}\n\\end{figure}\n\n\n\nFor Floquet-Bloch flexural waves in the honeycomb lattice, the dispersion surfaces and the corresponding slowness contours, are presented in Figs.~\\ref{slowness_contours}, \\ref{slowness_contours2} for the cases of the Rayleigh beams and the Euler-Bernoulli beams. \nIn particular, the diagrams in Fig.~\\ref{slowness_contours} represent the first dispersion surface (so-called ``acoustic band'') for the networks of Euler-Bernoulli beams and of the Rayleigh beams, whereas the diagrams in  Fig.~\\ref{slowness_contours2} correspond to the second dispersion surface (the ``optical band'') for the networks of Euler-Bernoulli beams and of the Rayleigh beams. The cases of the Euler-Bernoulli beam honeycomb lattice correspond to parts (a) and (b) and the computations for the honeycomb lattice of the Rayleigh beams are presented in parts (c) and (d). \n\nThese dispersion surfaces show the presence of the conical points (vertices of the Dirac cones) as well as the saddle points. Locally isotropic pattern is observed in the neighbourhood of the Dirac cones vertices, whereas a strong dynamic anisotropy is featured at frequencies corresponding to the saddle points.\n\nThe additional rotational inertia, attributed to the case of the Rayleigh beams, leads to lower frequencies of the Dirac cone vertices and the saddle points compared to similar networks of the Euler-Bernoulli beams.\n\nIn the following section, we give an illustration of the dynamic response of the honeycomb flexural lattice. In particular, we consider the case of a structured interface built of the Rayleigh beams embedded into the ambient geometrically identical lattice comprised of the Euler-Bernoulli beams, as illustrated in Fig.~\\ref{honeycomb_lattice_interface}.\n\n\n\n\n\n\n\n\n\n\\section{Forced vibrations of a honeycomb flexural network.}\n\\label{forced}\n\n\n\n\n\n\n\n\n\nHere we present the results of the finite element  simulations for the Euler-Bernoulli and for the Rayleigh beams programmed in COMSOL Multiphysics for a honeycomb network subjected to a time-harmonic transverse point force or several point forces applied at the lattice junctions\nThe forces are applied in the direction perpendicular to the $(x, y)-$plane.\n\nTo simulate a dynamic response of an infinite lattice with a finite-size computational window and to avoid wave reflection at the boundaries, Rayleigh and Euler-Bernoulli beams with damping were also programmed and introduced at the boundary of the computational domain. This was achieved by replacing the Young modulus $E$ by a complex value, $E(1+i\\eta)$. These beams were used to build a damping layer around the perimeter of the finite-size lattice, and the viscous parameter $\\eta$ was chosen so that to suppress the wave reflection.\n\n\\subsection{A homogeneous flexural network of Rayleigh beams.}\n\nIn Fig.~\\ref{fig_homo_RA} we consider a uniform network where six identical time-harmonic point forces are applied at the junctions of the hexagonal cell.\n\n\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=160mm]{fig_homo_RA_R.pdf}\n\\caption{\n\\footnotesize\nThe field patterns in the network of the Rayleigh beams, where  forced vibrations are generated by six time-harmonic identical forces applied at the junction points of the hexagonal cell.\n(a) The radian frequency $\\omega = 2 \\pi f = 0.314$ is sufficiently low, and the dynamic response appears to be isotropic. (b) The radian frequency $\\omega = 0.942$ is in the neighbourhood of the first saddle point frequency, where strong dynamic anisotropy is observed. (c) The radian frequency $\\omega = 1.319$ is in the neighbourhood of the Dirac cone vertex, and the waveform is localised. (d) Another group of saddle points occurs at the radian frequency of  $\\omega = 1.696$, where a strong dynamic anisotropy is observed. Here and in the following figures, inserts represent slowness contour diagrams for Floquet-Bloch waves at the given frequencies.\n}\n\\label{fig_homo_RA}\n\\end{figure}\n\n\nSeveral field patterns are observed for different values of the input frequency.\nNamely, when the frequency is sufficiently low, as in part (a) of Fig.~\\ref{fig_homo_RA}, the radial wave pattern appears to be isotropic, as expected in the low-frequency regime for the honeycomb lattice.\nHowever, the increase of the input frequency leads to radical, but predictable  changes in the frequency response of the network of the Rayleigh beams.  \nIn the cases (b) and (d) of Fig.~\\ref{fig_homo_RA}, the frequency values are close to those of the saddle points on the dispersion surfaces, and hence strong dynamic anisotropy is observed. On the contrary, strong localisation shown in the case (c) corresponds to the vertex of the Dirac cone. \nThe frequency values are chosen as follows. (a) The radian frequency $\\omega = 2 \\pi f = 0.314$ is sufficiently low, and the dynamic response appears to be isotropic. (b) The radian frequency $\\omega = 0.942$ is in the neighbourhood of the first saddle point frequency, where strong dynamic anisotropy is observed. (c) The radian frequency $\\omega = 1.319$ is in the neighbourhood of the Dirac cone vertex, and the waveform is localised. (d) Another group of saddle points occurs at the radian frequency of  $\\omega = 1.696$, where a strong dynamic anisotropy is observed. Here and in the following figures, inserts represent slowness contour diagrams for Floquet-Bloch waves at the given frequencies.\n\n\n\\subsection{Structured interface possessing rotational inertia.}\n\nHere we consider a geometrically homogeneous lattice of flexural beams, but we assume that within a layer of finite thickness the classical Euler-Bernoulli beams are replaced by the Rayleigh beams, which possess a rotational inertia, as described by the governing equation  \\eq{eq:gov}.\nA point force excitation is applied at a junction, outside the structured layer, as depicted in Fig.~\\ref{honeycomb_lattice_interface}. \nIn different frequency regimes, the dynamic response of the structured layer is investigated here.\n\nFirst, in Fig.~\\ref{fig_inter1} we present two cases, which include the low frequency excitation (part (a)) and an excitation at a higher frequency (part (b)), where a negative refraction is observed. \n\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=150mm]{fig_inter1_R.pdf}\n\\caption{\n\\footnotesize\nA dynamic response of the structured layer, built of the Rayleigh beams. The ambient lattice, surrounding the structured interface, consists of the classical Euler-Bernoulli beams.      The field patterns are presented for the two cases, where a point force is applied at the junction separated from the structured interface by the distance equal to the double diameter of the elementary cell of the honeycomb lattice (``distance 2''), as shown in Fig.~\\ref{honeycomb_lattice_interface}.\n(a) A sufficiently low radian frequency $\\omega = 2 \\pi f = 0.314$ corresponds to the isotropic response, where the action of the interface layer is negligibly small. (b) The normalised radian frequency $\\omega = 4.082$ corresponds to the case of Floquet-Bloch waves, which occur near saddle points on the dispersion diagrams for both the Euler-Bernoulli and the Rayleigh beams.  In this case, the rotational inertia of the interface layer becomes important, as it leads to the negative refraction and consequently focussing of the elastic flexural wave across the interface.\n}\n\\label{fig_inter1}\n\\end{figure}\n\n\n\nThis is understandable, as in the low frequency regime the Euler-Bernoulli and the Rayleigh beams appear to be very similar in terms of their dynamic response to an external load, and hence the structured interface is ``invisible''.\nOn the other hand, as the frequency increases, the corresponding dispersion diagrams in Fig.~\\ref{fig3dh} for the Floquet-Bloch waves  show the presence of the saddle points as well the Dirac cones. Consequently, this leads to an interesting dynamic response of the structured interface:\nat the normalised radian frequency of $\\omega = 4.082$, we observe saddle points on the dispersion diagrams for both the Euler-Bernoulli and for the Rayleigh beam networks. Such a regime implies a strong dynamic anisotropy, and consequently a negative refraction and focussing by the structured interface are observed in this simulation.\nThe field patterns are presented for the two cases, where a point force is applied at the junction separated from the structured interface by the distance equal to the double diameter of the elementary cell of the honeycomb lattice (``distance 2''), as shown in Fig.~\\ref{honeycomb_lattice_interface}.\n(a) A sufficiently low radian frequency $\\omega = 2 \\pi f = 0.314$ corresponds to the isotropic response, where the action of the interface layer is negligibly small. (b) The normalised radian frequency $\\omega = 4.082$ corresponds to the case of Floquet-Bloch waves, which occur near saddle points on the dispersion diagrams for both the Euler-Bernoulli and the Rayleigh beams.  In this case, the rotational inertia of the interface layer becomes important, as it leads to the negative refraction and consequently focussing of the elastic flexural wave across the interface.\n\n\n\n\n\n\nFurthermore, another Fig.~\\ref{fig_inter2} displays the field plots and shows examples of strong dynamic anisotropy as well localisation within the structured interface consisting of the Rayleigh beams. The time-harmonic point force is placed at the distance 1 from the interface, as illustrated in Fig.~\\ref{honeycomb_lattice_interface}. \nIn particular, the part (a) of  Fig.~\\ref{fig_inter2} shows the localisation within the structured interface, parts (c) and (d) show the edge-wave modes, and the field pattern of the part (b) shows the wave blockage by the structured interface and a strong dynamic anisotropy exhibited by the ambient lattice of the Euler-Bernoulli beams.      \nThe field patterns are presented for four cases, where a point force is applied at the junction separated from the structured interface by the distance equal to a diameter of the elementary cell of the honeycomb lattice (``distance 1''), as shown in Fig.~\\ref{honeycomb_lattice_interface}.\n(a) The normalised radian frequency $\\omega = 2 \\pi f = 1.005$ corresponds to a frequency below the first saddle point for the ambient lattice, where the dynamic response is almost isotropic, but the same frequency corresponds to the saddle point of the Rayleigh beam network used in the structured interface; hence the strong anisotropy is shown within the structured interface. (b) The normalised radian frequency $\\omega = 1.508$  corresponds to the saddle point on the dispersion diagram for the Floquet-Bloch waves in the ambient lattice, and the same frequency corresponds to the neighbourhood of the Dirac cone vertex for the network of the Rayleigh beams. (c) The normalised radian frequency $\\omega = 2.513$ corresponds to the neighbourhoods of the Dirac cone vertices for both Euler-Bernoulli and the Rayleigh beams on the first dispersion surface and the second dispersion surface, respectively  (d)  The forced excitation at the normalised radian frequency $\\omega = 9.111$ gives a waveguide vibration mode, and it is shown that this regime is close to the Dirac cones of different orders for the Euler-Bernoulli and the Rayleigh beam networks, respectively.\n\n\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=150mm]{fig_inter2_R.pdf}\n\\caption{\n\\footnotesize\nA dynamic response of the structured layer, built of the Rayleigh beams. The ambient lattice, surrounding the structured interface, consists of the classical Euler-Bernoulli beams.      The field patterns are presented for four cases, where a point force is applied at the junction separated from the structured interface by the distance equal to a diameter of the elementary cell of the honeycomb lattice (``distance 1''), as shown in Fig.~\\ref{honeycomb_lattice_interface}.\n(a) The normalised radian frequency $\\omega = 2 \\pi f = 1.005$ corresponds to a frequency below the first saddle point for the ambient lattice, where the dynamic response is almost isotropic, but the same frequency corresponds to the saddle point of the Rayleigh beam network used in the structured interface; hence the strong anisotropy is shown within the structured interface. (b) The normalised radian frequency $\\omega = 1.508$  corresponds to the saddle point on the dispersion diagram for the Floquet-Bloch waves in the ambient lattice, and the same frequency corresponds to the neighbourhood of the Dirac cone vertex for the network of the Rayleigh beams. (c) The normalised radian frequency $\\omega = 2.513$ corresponds to the neighbourhoods of the Dirac cone vertices for both Euler-Bernoulli and the Rayleigh beams on the first dispersion surface and the second dispersion surface, respectively  (d)  The forced excitation at the normalised radian frequency $\\omega = 9.111$ gives a waveguide vibration mode, and it is shown that this regime is close to the Dirac cones of different orders for the Euler-Bernoulli and the Rayleigh beam networks, respectively.\n}\n\\label{fig_inter2}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion.}\n\\label{sec05}\n\nThe modelling of the beam networks which possess rotational inertia has revealed novel and counter-intuitive properties related to the dynamic response of the multi-scale solids.\nSpecifically, we have focused our attention on the regimes, which correspond to the saddle points or neighbourhoods of the Dirac cones on the dispersion diagrams, constructed for the Floquet-Bloch waves in the periodic networks of the Rayleigh or the Euler-Bernoulli beams.\nThe waveforms, and in particular standing waves also depend on the geometry of the periodic network, and we have given the comparative outline for the cases of square and honeycomb lattices, with the emphasis on the dynamic anisotropy and localisation.\nClosed form asymptotic estimates for frequencies corresponding to the vertices of the Dirac cones and of the standing waves provide an additional useful tool, which is essential in problems of optimal design of phononic filters and polarisers of elastic waves.\nFinally, the dynamic response  of the structured interfaces with an additional rotational inertia has been investigated, with the emphasis on edge waves, negative refraction and wave trapping.\nThis work naturally extends to the area of metamaterials design, and to control of flexural waves by multi-scale elastic networks. \n\n\n\\vspace{6mm}\n{\\bf Acknowledgements}. AP would like to acknowledge financial support from the\nEuropean Union's Seventh Framework Programme FP7\/2007-2013\/ under REA grant\nagreement number PCIG13-GA-2013-618375-MeMic. \nMajor part of the work was carried out while AM was visiting the University of Trento in 2016 with the support from the\nEuropean Union's Grant ERC-2013-ADG-340561-INSTABILITIES, which is gratefully acknowledged.\nAM also acknowledges the support from the UK EPSRC Program Grant  EP\/L024926\/1.\nLC acknowledges financial support from the University of Trento, within the research project 2014 entitled ``3D printed metallic foams for biomedical applications: understanding and improving their mechanical behavior''.\n\n\\bibliographystyle{jabbrv_unsrt}\n\\bibliography\n{%\nroaz1}\n\n\n\n\n\n\\end{document} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}