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

We study the large time behavior of nonnegative solutions of the Cauchy problem $u_t=\int J(x-y)(u(y,t)-u(x,t))dy-u^p$, $u(x,0)=u_0(x)\in L^{\mathrm{\infty }}$, where $\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }{u}_0(x)\to A>0$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$. One of our main goals is the study of the critical case $p=1+2\mathrm{/}\alpha$ for $0<\alpha <N$, left open in previous articles, for which we prove that $t^{\alpha \mathrm{/}2}\mathrm{\mid }u(x,t)-U(x,t)\mathrm{\mid }\to 0$ where $U$ is the solution of the heat equation with absorption with initial datum $U(x,0)=C_{A,N}\mathrm{\mid }x{\mathrm{\mid }}^{-\alpha }$. 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 ($p=1+2\mathrm{/}N$) for bounded and integrable $u_0$.