Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data: the subcritical case

In this paper we continue our study of the large time behavior of the bounded solution to the nonlocal diffusion equation with absorption \begin{align} \begin{cases} u_t = \mathcal{L} u-u^p\quad& \mbox{in}\quad \mathbb R^N\times(0,\infty),\\ u(x,0) = u_0(x)\quad& \mbox{in}\quad \mathbb R^N, \end{cases} \end{align} where $p>1$, $u_0\ge0$ and bounded and $$ \mathcal{L} u(x,t)=\int J(x-y)\left(u(y,t)-u(x,t)\right)\,dy $$ with $J\in C_0^{\infty}(\mathbb R^N)$, radially symmetric, $J\geq 0$ with $\int J=1$. Our assumption on the initial datum is that $0\le u_0\in L^\infty(\mathbb R^N)$ and $$ |x|^{\alpha}u_0(x)\to A>0\quad\mbox{as}\quad|x|\to\infty. $$ This problem was studied in the supercritical and critical cases $p\ge 1+2/\alpha$. %See also \cite{PR,TW2} for the case $u_0\in L^\infty(\mathbb R^N)\cap L^1(\mathbb R^N)$, $p\ge 1+2/N$. In the present paper we study the subcritical case $1<p<1+2/\alpha$. More generally, we consider bounded non-negative initial data such that \[ |x|^{\frac2{p-1}}u_0(x)\to\infty\quad\mbox{as}\quad |x|\to \infty \] and prove that \[t^{\frac1{p-1}} u(x,t)\to\Big(\frac1{p-1}\Big)^{\frac1{p-1}}\quad\mbox{as}\quad t\to\infty \] uniformly in $ |x|\le k\sqrt t$, for every $k>0$.