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

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 $1<p<1+\frac2N$. The initial datum $u_0$ is assumed to be bounded, and to satisfy $|x|^{\frac2{p-1}}u_0(x)\to A\ge0$ 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 $A|x|^{-\frac2{p-1}}$ if $A>0$.