References & Citations
Mathematics > Analysis of PDEs
Title: Large Time Behavior of a Nonlocal Diffusion Equation with Absorption and Bounded Initial Data
(Submitted on 5 Apr 2010 (v1), last revised 12 Apr 2010 (this version, v2))
Abstract: We study the large time behavior of nonnegative solutions of the Cauchy problem $u_t=\int J(x-y)(u(y,t)-u(x,t))\,dy-u^p$, $u(x,0)=u_0(x)\in L^\infty$, where $|x|^{\alpha}u_0(x)\to A>0$ as $|x|\to\infty$. One of our main goals is the study of the critical case $p=1+2/\alpha$ for $0<\alpha<N$, left open in previous articles, for which we prove that $t^{\alpha/2}|u(x,t)-U(x,t)|\to 0$ where $U$ is the solution of the heat equation with absorption with initial datum $U(x,0)=C_{A,N}|x|^{-\alpha}$. Our proof, involving sequences of rescalings of the solution, allows us to establish also the large time behavior of solutions having more general nonintegrable initial data $u_0$ in the supercritical case and also in the critical case ($p=1+2/N$) for bounded and integrable $u_0$.
Submission history
From: Joana Terra [view email][v1] Mon, 5 Apr 2010 21:03:23 GMT (20kb)
[v2] Mon, 12 Apr 2010 20:31:24 GMT (20kb)
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