Asymptotic behavior for a nonlocal diffusion equation in exterior domains: the critical two-dimensional case

Abstract

We study the long time behavior of bounded, integrable solutions to a nonlocal diffusion equation, ${\partial }_{t}u=J\ast u-u$, where $J$ is a smooth, radially symmetric kernel with support $B_d(0)\subset {\mathbb{R}}^2$. The problem is set in an exterior two-dimensional domain which excludes a hole $\mathcal{H}$, and with zero Dirichlet data on $\mathcal{H}$. In the far field scale, ${\xi }_1\le |x|t^{-1/2}\le {\xi }_2$ with ${\xi }_1,{\xi }_2>0$, the scaled function $\log tu(x,t)$ behaves as a multiple of the fundamental solution for the local heat equation with a certain diffusivity determined by $J$. The proportionality constant, which characterizes the first non-trivial term in the asymptotic behavior of the mass, is given by means of the asymptotic \lq logarithmic momentum' of the solution, ${\lim }_{t\to \infty }{\int }_{{\mathbb{R}}^2}u(x,t)\log |x|dx$. This asymptotic quantity can be easily computed in terms of the initial data. In the near field scale, $|x|\le t^{1/2}h(t)$ with ${\lim }_{t\to \infty }h(t)=0$, the scaled function $t(\log t{)}^{2}u(x,t)/\log |x|$ converges to a multiple of $\varphi (x)/\log |x|$, where $\varphi$ is the unique stationary solution of the problem that behaves as $\log |x|$ when $|x|\to \infty$. The proportionality constant is obtained through a matching procedure with the far field limit. Finally, in the very far field, $|x|\ge t^{1/2}g(t)$ with $g(t)\to \infty$, the solution is proved to be of order $o((t\log t{)}^{-1})$.

24 pages