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

Research paper by **Ariel Salort, Joana Terra, Noemí Wolanski**

Indexed on: **11 Apr '14**Published on: **11 Apr '14**Published in: **Mathematics - Analysis of PDEs**

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#### 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 $p>1$, $u_0\ge0$ 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 $k>0$.