Asymptotic behavior for a nonlocal diffusion problem with Neumann boundary conditions and a reaction term

Abstract

In this paper, we consider the following initial value problem $u_t(x,t)=\int_{\Omega}J(x-y)(u(y,t)-u(x,t))dy-\gamma u^{p}(x,t)& \mbox{in}& \overline{\Omega}\times(0,\infty)$,
$u(x,0)=u_{0}(x)>0& \mbox{in}& \overline{\Omega}$,            where $\gamma\ in \{-1,1\}$ is a parameter, $\Omega$ is a bounded domain in      $\mathbb{R}^{N}$ with smooth boundary $\partial\Omega$, $p>1$, $J$:      $\mathbb{R}^N\longrightarrow\mathbb{R}$ is a kernel which is nonnegative,      measurable, symmetric, bounded and $\int_{\mathbb{R}^N}J(z)dz=1, $ the      initial datum $u_0 \ in C^0(\overline{\Omega})$, $u_0(x)>0$ in      $\overline{\Omega}$. We show that, if $\gamma=1$, then the solution $u$      of the above problem tends to zero as $t\rightarrow\infty$ uniformly in      $x\in\overline{\Omega}$, and a description of its asymptotic behavior is      given. We also prove that, if $\gamma=-1$, then the solution $u$ blows up      in a finite time, and its blow-up time goes to that of the solution of a      certain ODE as the $L^{\infty}$ norm of the initial datum goes to      infinity.