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We study the large time behavior of nonnegative solutions of the Cauchyproblem $u_t=\int J(x-y)(u(y,t)-u(x,t))\,dy-u^p$, $u(x,0)=u_0(x)\in L^\infty$,where $|x|^{\alpha}u_0(x)\to A>0$ as $|x|\to\infty$. One of our main goals isthe study of the critical case $p=1+2/\alpha$ for $0<\alpha<N$, left open inprevious 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 initialdatum $U(x,0)=C_{A,N}|x|^{-\alpha}$. Our proof, involving sequences ofrescalings of the solution, allows us to establish also the large time behaviorof solutions having more general nonintegrable initial data $u_0$ in thesupercritical case and also in the critical case ($p=1+2/N$) for bounded andintegrable $u_0$.

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