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Communications on Pure & Applied Analysis

2013 , Volume 12 , Issue 2

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Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations
Françoise Demengel and O. Goubet
2013, 12(2): 621-645 doi: 10.3934/cpaa.2013.12.621 +[Abstract](193) +[PDF](480.6KB)
Abstract:
We prove here the existence of boundary blow up solutions for fully nonlinear equations in general domains, for a nonlinearity satisfying Keller-Osserman type condition. If moreover the nonlinearity is non decreasing, we prove uniqueness for boundary blow up solutions on balls for operators related to Pucci's operators.
The decay estimates of solutions for 1D compressible flows with density-dependent viscosity coefficients
Wuming Li, Xiaojun Liu and Quansen Jiu
2013, 12(2): 647-661 doi: 10.3934/cpaa.2013.12.647 +[Abstract](232) +[PDF](379.8KB)
Abstract:
We consider the Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients, we are mainly concerned with the decay estimates of the density and velocity as $t \rightarrow \infty$. Firstly, we obtain the decay estimates of $\rho-\bar{\rho}$ and u in $L^2(R)$ norm, then we obtain the decay estimate of $\rho-\bar{\rho}$ in $L^{\infty}(R)$ norm as $\bar{\rho}>0$. Secondly, we construct a functional and use the energy method to obtain the decay estimate of $\rho$ in $L^{\infty}(R)$ norm as $\bar{\rho}=0$.
Nonexistence of positive solution for an integral equation on a Half-Space $R_+^n$
Yanqin Fang and Jihui Zhang
2013, 12(2): 663-678 doi: 10.3934/cpaa.2013.12.663 +[Abstract](178) +[PDF](429.3KB)
Abstract:
Let $n, m$ be a positive integer and let $R_+^n$ be the $n$-dimensional upper half Euclidean space. In this paper, we study the following integral equation \begin{eqnarray} u(x)=\int_{R_+^n}G(x,y)u^pdy, \end{eqnarray} where \begin{eqnarray*} G(x,y)=\frac{C_n}{|x-y|^{n-2m}}\int_0^{\frac{4x_n y_n}{|x-y|^2}}\frac{z^{m-1}}{(z+1)^{n/2}}dz, \end{eqnarray*} $C_{n}$ is a positive constant, $0 <2m 1$. Using the method of moving planes in integral forms, we show that equation (1) has no positive solution.
Non degeneracy for solutions of singularly perturbed nonlinear elliptic problems on symmetric Riemannian manifolds
Marco Ghimenti and A. M. Micheletti
2013, 12(2): 679-693 doi: 10.3934/cpaa.2013.12.679 +[Abstract](184) +[PDF](403.3KB)
Abstract:
Given a symmetric Riemannian manifold $(M,g)$, we show some results of genericity for non degenerate sign changing solutions of singularly perturbed nonlinear elliptic problems with respect to the parameters: the positive number $\varepsilon$ and the symmetric metric $g$. Using these results we obtain a lower bound on the number of non degenerate solutions which change sign exactly once.
Quasilinear systems involving multiple critical exponents and potentials
Dongsheng Kang
2013, 12(2): 695-710 doi: 10.3934/cpaa.2013.12.695 +[Abstract](157) +[PDF](426.8KB)
Abstract:
In this paper, a quasilinear system of elliptic equations is investigated, which involves multiple critical Hardy--Sobolev exponents and Hardy--type terms. By variational methods and analytic technics, the existence of nontrivial solutions to the system is established.
Median values, 1-harmonic functions, and functions of least gradient
Matthew B. Rudd and Heather A. Van Dyke
2013, 12(2): 711-719 doi: 10.3934/cpaa.2013.12.711 +[Abstract](174) +[PDF](364.1KB)
Abstract:
Motivated by the mean value property of harmonic functions, we introduce the local and global median value properties for continuous functions of two variables. We show that the Dirichlet problem associated with the local median value property is either easy or impossible to solve, and we prove that continuous functions with this property are $1$-harmonic in the viscosity sense. We then close with the following conjecture: a continuous function having the global median value property and prescribed boundary values coincides with the function of least gradient having those same boundary values.
Multiple positive solutions for Kirchhoff type problems with singularity
Xing Liu and Yijing Sun
2013, 12(2): 721-733 doi: 10.3934/cpaa.2013.12.721 +[Abstract](340) +[PDF](356.6KB)
Abstract:
A class of Kirchhoff type problems containing both singular and superlinear terms is considered in a bounded domain in $R^3$: multiplicity results are obtained by variational methods.
Long-time dynamics of the parabolic $p$-Laplacian equation
Pelin G. Geredeli and Azer Khanmamedov
2013, 12(2): 735-754 doi: 10.3934/cpaa.2013.12.735 +[Abstract](269) +[PDF](397.2KB)
Abstract:
In this paper, we study the long-time behaviour of solutions of Cauchy problem for the parabolic $p$-Laplacian equation with variable coefficients. Under the mild conditions on the coefficient of the principal part and without upper growth restriction on the source function, we prove that this problem possesses a compact and invariant global attractor in $L^2(R^n)$.
The Riemann problem of conservation laws in magnetogasdynamics
Yanbo Hu and Wancheng Sheng
2013, 12(2): 755-769 doi: 10.3934/cpaa.2013.12.755 +[Abstract](327) +[PDF](415.3KB)
Abstract:
In this paper, we study the Riemann problem for a simplified model of one dimensional ideal gas in magnetogasdynamics. By using the characteristic analysis method, we prove the global existence of solutions to the Riemann problem constructively under the Lax entropy condition. The image of contact discontinuity in magnetogasdynamics is a curve with $u=Const.$ in the $(\tau,p,u)$ space. Its projection on the $(p,u)$ plane is a straight line that parallels to the $p$-axis. In contrast with the problem in gas dynamics, the result causes more complicated and difficult than that in gas dynamics.
Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part
Minbo Yang and Yanheng Ding
2013, 12(2): 771-783 doi: 10.3934/cpaa.2013.12.771 +[Abstract](247) +[PDF](401.8KB)
Abstract:
In the present paper we study the existence of solutions for a nonlocal Schrödinger equation \begin{eqnarray*} -\varepsilon^2\Delta u +V(x)u =(\int_{R^3} \frac{|u|^{p}}{|x-y|^{\mu}}dy)|u|^{p-2}u, \end{eqnarray*} where $0 < \mu < 3$ and $\frac{6-\mu}{3} < p < {6-\mu}$. Under suitable assumptions on the potential $V(x)$, if the parameter $\varepsilon$ is small enough, we prove the existence of solutions by using Mountain-Pass Theorem.
Multiplicity results for a class of elliptic problems with nonlinear boundary condition
Patrick Winkert
2013, 12(2): 785-802 doi: 10.3934/cpaa.2013.12.785 +[Abstract](194) +[PDF](436.0KB)
Abstract:
This paper provides multiplicity results for a class of nonlinear elliptic problems under a nonhomogeneous Neumann boundary condition. We prove the existence of three nontrivial solutions to these problems which depend on the Fučík spectrum of the negative $p$-Laplacian with a Robin boundary condition. Using variational and topological arguments combined with an equivalent norm on the Sobolev space $W^{1,p}$ it is obtained a smallest positive solution, a greatest negative solution, and a sign-changing solution.
Some results on two-dimensional Hénon equation with large exponent in nonlinearity
Chunyi Zhao
2013, 12(2): 803-813 doi: 10.3934/cpaa.2013.12.803 +[Abstract](224) +[PDF](359.6KB)
Abstract:
The Hénon equation on a bounded domain in $R^2$ with large exponent in the nonlinear term is studied in this paper. We investigate positive solution obtained by the variational method and give its asymptotic behavior as the nonlinear exponent gets large.
Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter
Salvatore A. Marano and Nikolaos S. Papageorgiou
2013, 12(2): 815-829 doi: 10.3934/cpaa.2013.12.815 +[Abstract](228) +[PDF](400.8KB)
Abstract:
A nonlinear elliptic equation with $p$-Laplacian, concave-convex reaction term depending on a parameter $\lambda>0$, and homogeneous boundary condition, is investigated. A bifurcation result, which describes the set of positive solutions as $\lambda$ varies, is obtained through variational methods combined with truncation and comparison techniques.
Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity
Soohyun Bae and Jaeyoung Byeon
2013, 12(2): 831-850 doi: 10.3934/cpaa.2013.12.831 +[Abstract](226) +[PDF](467.6KB)
Abstract:
We consider the singularly perturbed nonlinear elliptic problem \begin{eqnarray*} \varepsilon^2 \Delta v - V(x)v + f(v) =0, v > 0, \lim_{|x|\to \infty} v(x) = 0. \end{eqnarray*} Under almost optimal conditions for the potential $V$ and the nonlinearity $f$, we establish the existence of single-peak solutions whose peak points converge to local minimum points of $V$ as $\varepsilon \to 0$. Moreover, we exhibit a threshold on the condition of $V$ at infinity between existence and nonexistence of solutions.
Local existence of strong solutions to the three dimensional compressible MHD equations with partial viscosity
Yu-Zhu Wang and Yin-Xia Wang
2013, 12(2): 851-866 doi: 10.3934/cpaa.2013.12.851 +[Abstract](222) +[PDF](404.3KB)
Abstract:
In this paper we investigate three-dimensional compressible magnetohydrodynamic equations with partial viscosity. Local strong solutions to the compressible magnetohydrodynamic equations with large data are established.
Generalized Schrödinger-Poisson type systems
Antonio Azzollini, Pietro d’Avenia and Valeria Luisi
2013, 12(2): 867-879 doi: 10.3934/cpaa.2013.12.867 +[Abstract](169) +[PDF](406.3KB)
Abstract:
In this paper we study the boundary value problem \begin{eqnarray*} -\Delta u+ \varepsilon q\Phi f(u)=\eta|u|^{p-1}u \quad in \quad \Omega, \\ - \Delta \Phi=2 qF(u) \quad in \quad \Omega, \\ u=\Phi=0 \quad on \quad \partial \Omega, \end{eqnarray*} where $\Omega \subset R^3$ is a smooth bounded domain, $1 < p < 5$, $\varepsilon,\eta= \pm 1$, $q>0$, $f: R\to R$ is a continuous function and $F$ is the primitive of $f$ such that $F(0)=0.$ We provide existence and multiplicity results assuming on $f$ a subcritical growth condition. The critical case is also considered and existence and nonexistence results are proved.
A new class of $(H^k,1)$-rectifiable subsets of metric spaces
Roberta Ghezzi and Frédéric Jean
2013, 12(2): 881-898 doi: 10.3934/cpaa.2013.12.881 +[Abstract](158) +[PDF](456.0KB)
Abstract:
The main motivation of this paper arises from the study of Carnot--Carathéodory spaces, where the class of $1$-rectifiable sets does not contain smooth non-horizontal curves; therefore a new definition of rectifiable sets including non-horizontal curves is needed. This is why we introduce in any metric space a new class of curves, called continuously metric differentiable of degree $k$, which are Hölder but not Lipschitz continuous when $k>1$. Replacing Lipschitz curves by this kind of curves we define $(H^k,1)$-rectifiable sets and show a density result generalizing the corresponding one in Euclidean geometry. This theorem is a consequence of computations of Hausdorff measures along curves, for which we give an integral formula. In particular, we show that both spherical and usual Hausdorff measures along curves coincide with a class of dimensioned lengths and are related to an interpolation complexity, for which estimates have already been obtained in Carnot--Carathéodory spaces.
Local well-posedness of quasi-linear systems generalizing KdV
Timur Akhunov
2013, 12(2): 899-921 doi: 10.3934/cpaa.2013.12.899 +[Abstract](201) +[PDF](524.1KB)
Abstract:
In this article we prove local well-posedness of quasilinear dispersive systems of PDE generalizing KdV. These results adapt the ideas of Kenig-Ponce-Vega from the Quasi-Linear Schrödinger equations to the third order dispersive problems. The main ingredient of the proof is a local smoothing estimate for a general linear problem that allows us to proceed via the artificial viscosity method.
Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion
Yan Jia, Xingwei Zhang and Bo-Qing Dong
2013, 12(2): 923-937 doi: 10.3934/cpaa.2013.12.923 +[Abstract](212) +[PDF](426.3KB)
Abstract:
This article is concerned with the blow-up criterion for smooth solutions of three-dimensional Boussinesq equations with zero diffusion. It is shown that if the velocity field $u(x,t)$ satisfies \begin{eqnarray*} u\in L^p(0,T_1;B^r_{q,\infty}(R^3)),\quad \frac{2}{p}+\frac{3}{q}=1+r,\quad \frac{3}{1+r}< q \leq \infty, \quad -1 < r \leq 1, \end{eqnarray*} then the solution can be continually extended to the interval $(0,T)$ for some $T>T_1$.
Weak solutions for generalized large-scale semigeostrophic equations
Mahmut Çalik and Marcel Oliver
2013, 12(2): 939-955 doi: 10.3934/cpaa.2013.12.939 +[Abstract](218) +[PDF](443.6KB)
Abstract:
We prove existence, uniqueness and continuous dependence on initial data of global weak solutions to the generalized large-scale semigeostrophic equations with periodic boundary conditions. This family of Hamiltonian balance models for rapidly rotating shallow water includes the $L_1$ model derived by R. Salmon in 1985 and its 2006 generalization by the second author. The analysis is based on the vorticity formulation of the models supplemented by a nonlinear velocity-vorticity relation. The results are fundamentally due to the conservation of potential vorticity. While classical solutions are known to exist provided the initial potential vorticity is positive---a condition which is already implicit in the formal derivation of balance models, we can assert the existence of weak solutions only under the slightly stronger assumption that the potential vorticity is bounded below by $\sqrt{5}-2$ times the equilibrium potential vorticity. The reason is that the nonlinearities in the potential vorticity inversion are felt more strongly when working in weaker function spaces. Another manifestation of this effect is that point-vortex solutions are not supported by the model even in the special case when the potential vorticity inversion gains three derivatives in spaces of classical functions.
A general stability result in a memory-type Timoshenko system
Salim A. Messaoudi and Muhammad I. Mustafa
2013, 12(2): 957-972 doi: 10.3934/cpaa.2013.12.957 +[Abstract](163) +[PDF](376.7KB)
Abstract:
In this paper we consider the following Timoshenko system \begin{eqnarray*} \varphi _{t t}-(\varphi _{x}+\psi )_{x}=0,\quad (0,1)\times R^+\\ \psi _{t t}-\psi _{x x}+\varphi _{x}+\psi +\int_0^t g(t-\tau )\psi_{x x}(\tau )d\tau =0,\quad (0,1)\times R^{+} \end{eqnarray*} with Dirichlet boundary conditions where $g$ is a positive nonincreasing function satisfying \begin{eqnarray*} g'(t)\leq -H(g(t)) \end{eqnarray*} and $H$ is a function satisfying some regularity and convexity conditions. We establish a general stability result for this system.
Decay of solutions to fractal parabolic conservation laws with large initial data
Fengbai Li and Feng Rong
2013, 12(2): 973-984 doi: 10.3934/cpaa.2013.12.973 +[Abstract](198) +[PDF](376.1KB)
Abstract:
In this paper, we study the time-asymptotic behavior of solutions to the Cauchy problem for multi-dimensional parabolic conservation laws with fractional dissipation. For arbitrarily large initial data, we obtain the optimal decay rates in $L^2$ and homogeneous Sobolev spaces for solutions to the equation with the power of Laplacian $\frac{1}{2} < \alpha \le 1$ by using the time-frequency decomposition method and the energy method. The argument is based on a maximum principle.
Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations
Renjun Duan and Xiongfeng Yang
2013, 12(2): 985-1014 doi: 10.3934/cpaa.2013.12.985 +[Abstract](226) +[PDF](532.1KB)
Abstract:
In this paper, we are concerned with the initial boundary value problem on the two-fluid Navier-Stokes-Poisson system in the half-line $R_+$. We establish the global-in-time asymptotic stability of the rarefaction wave and the boundary layer both for the outflow problem under the smallness assumption on initial perturbation, where the strength of the rarefaction wave is not necessarily small while the strength of the boundary layer is additionally supposed to be small. Here, the large initial data with densities far from vacuum is also allowed in the case of the non-degenerate boundary layer. The results show that the large-time behavior of solutions coincides with the one for the single Navier-Stokes system in the absence of the electric field. The proof is based on the classical energy method. The main difficulty in the analysis comes from the slower time-decay rate of the system caused by the appearance of the electric field. To overcome it, we use the coupling property of the two-fluid equations to capture the dissipation of the electric field interacting with the nontrivial asymptotic profile.
Global attractors for strongly damped wave equations with subcritical-critical nonlinearities
Filippo Dell'Oro
2013, 12(2): 1015-1027 doi: 10.3934/cpaa.2013.12.1015 +[Abstract](227) +[PDF](450.0KB)
Abstract:
This paper is concerned with the nonlinear strongly damped wave equation \begin{eqnarray*} u_{t t}-\Delta u_t-\Delta u+f(u_t)+g(u)=h, \end{eqnarray*} with Dirichlet boundary conditions and a time-independent external force $h$. In the presence of nonlinearities $f$ and $g$ of subcritical and critical growth, respectively, the existence of a global attractor of optimal regularity is established.
The explicit nonlinear wave solutions of the generalized $b$-equation
Liu Rui
2013, 12(2): 1029-1047 doi: 10.3934/cpaa.2013.12.1029 +[Abstract](201) +[PDF](2248.4KB)
Abstract:
In this paper, we study the nonlinear wave solutions of the generalized $b$-equation involving two parameters $b$ and $k$. Let $c$ be constant wave speed, $c_5=$ $\frac{1}{2}(1+b-\sqrt{(1+b)(1+b-8k)})$, $c_6=\frac{1}{2}(1+b+\sqrt{(1+b)(1+b-8k)})$. We obtain the following results:

1. If $-\infty < k < \frac{1+b}{8}$ and $c\in (c_5, c_6)$, then there are three types of explicit nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic peakon wave solution and hyperbolic blow-up solution.

2. If $-\infty < k < \frac{1+b}{8}$ and $c=c_5$ or $c_6$, then there are two types of explicit nonlinear wave solutions, fractional peakon wave solution and fractional blow-up solution.

3. If $k=\frac{1+b}{8}$ and $c=\frac{b+1}{2}$, then there are two types of explicit nonlinear wave solutions, fractional peakon wave solution and fractional blow-up solution.

Not only is the existence of these solutions shown, but their concrete expressions are presented. We also reveal the relationships among these solutions. Besides, the correctness of these solutions is tested by using the software Mathematica.
Existence of a rotating wave pattern in a disk for a wave front interaction model
Jong-Shenq Guo, Hirokazu Ninomiya and Chin-Chin Wu
2013, 12(2): 1049-1063 doi: 10.3934/cpaa.2013.12.1049 +[Abstract](201) +[PDF](474.0KB)
Abstract:
We study the rotating wave patterns in an excitable medium in a disk. This wave pattern is rotating along the given disk boundary with a constant angular speed. To study this pattern we use the wave front interaction model proposed by Zykov in 2007. This model is derived from the FitzHugh-Nagumo equation and it can be described by two systems of ordinary differential equations for wave front and wave back respectively. Using a delicate shooting argument with the help of the comparison principle, we derive the existence and uniqueness of rotating wave patterns for any admissible angular speed with convex front in a given disk.
Travelling wave solutions of a free boundary problem for a two-species competitive model
Chueh-Hsin Chang and Chiun-Chuan Chen
2013, 12(2): 1065-1074 doi: 10.3934/cpaa.2013.12.1065 +[Abstract](331) +[PDF](391.7KB)
Abstract:
We study a di usive logistic system with a free boundary in ecology proposed by Mimura, Yamada and Yotsutani [10]. Motivated by the spreading-vanishing dichotomy obtained by Du and Lin [1], we suppose the spreading speed of the free boundary tends to a constant as time tends to in nity and consider the corresponding travelling wave problem. We establish the existence and uniqueness of a travelling wave solution for this free boundary problem.
Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution
Weiguo Zhang, Yan Zhao and Xiang Li
2013, 12(2): 1075-1090 doi: 10.3934/cpaa.2013.12.1075 +[Abstract](231) +[PDF](470.3KB)
Abstract:
In this paper, we apply the theory of planar dynamical systems to make a qualitative analysis to the traveling wave solutions of nonlinear Kakutani-Kawahara equation $u_t+uu_x+bu_{x x x}-a(u_t+uu_x)_x=0$ ($b>0, a\ge0$) and obtain the existent conditions of the bounded traveling wave solutions. In dispersion-dominant case, we find that the unique bounded traveling wave solution of this equation has not only oscillatory property but also damped property. Furthermore, according to the evolution of orbits in the global phase portraits, we present an approximate damped oscillatory solution for this equation by the undetermined coefficients method. Finally, by the idea of homogenization principles, we obtain an integral equation which reflects the relation between this approximate damped oscillatory solution and its exact solution, thereby having the error estimate. The error is an infinitesimal decreasing in exponential form. From the results in this paper, it can be seen that the damped oscillatory solution of Kakutani-Kawahara equation in dispersion-dominant case still remains some properties of solitary wave solution for KdV equation.
Limit cycles of non-autonomous scalar ODEs with two summands
José-Luis Bravo and Manuel Fernández
2013, 12(2): 1091-1102 doi: 10.3934/cpaa.2013.12.1091 +[Abstract](230) +[PDF](726.8KB)
Abstract:
We establish upper bounds for the number of limit cycles (isolated periodic solutions in the set of periodic solutions) of the two families of scalar ordinary differential equations $x'=(a(t) x +b(t)) f(x)$ and $x'=a(t) g(x) +b(t)f(x)$, where $f(x)$ and $g(x)$ are analytic funtions and $a(t)$, $b(t)$ are $T$--periodic continuous functions for which there exist $\alpha, \beta \in R$ such that $\alpha a(t)+\beta b(t)$ is not identically zero and does not change sign in $[0,T]$. As a consequence we obtain that generalized Abel equations $x'=a(t)x^n + b(t)x^m$, where $n> m \geq 1$ are natural numbers, have at most three limit cycles.
Blow-up for semilinear parabolic equations with critical Sobolev exponent
Li Ma
2013, 12(2): 1103-1110 doi: 10.3934/cpaa.2013.12.1103 +[Abstract](255) +[PDF](326.1KB)
Abstract:
In this paper, we study the global existence and blow-up results of semilinear parabolic equations with critical Sobolev exponent \begin{eqnarray*} u_t-\Delta u=|u|^{p-1}u, in \Omega\times (0,T) \end{eqnarray*} with the Dirichlet boundary condition $u=0$ on the boundary $\partial\Omega\times [0,T)$ and $u=\phi$ at $t=0$, where $\Omega\subset R^n$, $n\geq 3$, is a compact $C^1$ domain, $p=p_S=\frac{n+2}{n-2}$ is the critical Sobolev exponent, and $0 ≨ \phi \in C^1_0(\Omega)$ is a given smooth function. We show that there are two sets $\tilde{W}$ and $\tilde{Z}$ such that for $\phi\in\tilde{W}$, there is a global positive solution $u(t)\in \tilde{W}$ with $H^1$ omega limit $\{0\}$ and for $\phi\in \tilde{Z}$, the solution blows up at finite time.
Attractors in $H^2$ and $L^{2p-2}$ for reaction diffusion equations on unbounded domains
Ming Wang and Yanbin Tang
2013, 12(2): 1111-1121 doi: 10.3934/cpaa.2013.12.1111 +[Abstract](185) +[PDF](409.6KB)
Abstract:
We are concerned with a class of reaction diffusion equations with nonlinear terms of arbitrary growth on unbounded domains. The existence of an $L^2 - L^{2p-2} \cap H^2$ global attractor is proved. This improves the results in previous references, and the proof is shorter.
Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density
Sofía Nieto and Guillermo Reyes
2013, 12(2): 1123-1139 doi: 10.3934/cpaa.2013.12.1123 +[Abstract](185) +[PDF](449.7KB)
Abstract:
We study the long-time behavior of non-negative, finite-energy solutions to the initial value problem for the Porous Medium Equation with variable density, i.e. solutions of the problem \begin{eqnarray*} \rho (x) \partial_{t} u = \Delta u^{m}, \quad in \quad Q:= R^n \times R_+, \\ u(x,0)=u_{0}(x), \quad in\quad R^n, \end{eqnarray*} where $m>1$, $u_0\in L^1(R^n, \rho(x)dx)$ and $n\ge 3$. We assume that $\rho (x)\sim C|x|^{-2}$ as $|x|\to\infty$ in $R^n$. Such a decay rate turns out to be critical. We show that the limit behavior can be described in terms of a family of source-type solutions of the associated singular equation $|x|^{-2}u_t = \Delta u^{m}$. The latter have a self-similar structure and exhibit a logarithmic singularity at the origin.
Uniform $L^1$-stability of the relativistic Boltzmann equation near vacuum
Seung-Yeal Ha, Eunhee Jeong and Robert M. Strain
2013, 12(2): 1141-1161 doi: 10.3934/cpaa.2013.12.1141 +[Abstract](160) +[PDF](446.9KB)
Abstract:
We present the uniform $L^1$-stability estimate for the relativistic Boltzmann equation near vacuum. For this, we explicitly construct a relativistic counterpart of the nonlinear functional which is a linear combination of $L^1$-distance and a collision potential. This functional measures the $L^1$-distance between two continuous mild solutions. When the initial data is sufficiently small and decays exponentially fast, we show that the functional satisfies the uniform stability estimate leading to the uniform $L^1$-stability estimate with respect to initial data.

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