Large Time Behavior of a Nonlocal Diffusion Equation with Absorption and Bounded Initial Data

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 $u_0$.