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

Abstract

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 $p>1$, $u_0\ge 0$ 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, $J\ge 0$ with $\int J=1$. Our assumption on the initial datum is that $0\le u_0\in L^{\infty }({\mathbb{R}}^N)$ and $$ |x|^αu_0(x)\to A>0\quad\mbox{as}\quad|x|\to\infty. $$ 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 $k>0$.

14 pages