Large-time behavior in a nonlocal heat equation with absorption. The absorption dominated case with fast decaying initial data

Abstract

We study the large-time behavior of nonnegative solutions to a nonlocal dispersal equation in ${\mathbb{R}}^N$ with an absorption term modeled by $-u^p$, with $1<p<1+\frac{2}{N}$. The initial datum $u_0$ is assumed to be bounded, and to satisfy $|x{|}^{\frac{2}{p-1}}{u}_0(x)\to A\ge 0$ as $|x|\to \infty$. Under these assumptions, we prove that the decay rate is that of the purely absorbing problem, while the limit profile is a very singular solution to a local diffusion problem with absorption if $A=0$, and a solution to this same local problem with initial datum $A|x{|}^{-\frac{2}{p-1}}$ if $A>0$.