Asymptotic behavior for a one-dimensional nonlocal diffusion equation in exterior domains

Abstract

We study the long time behavior of solutions to the nonlocal diffusion equation $\partial_t u=J*u-u$ in an exterior one-dimensional domain, with zero Dirichlet data on the complement. In the far field scale, $\xi_1\le|x|t^{-1/2}\le\xi_2$, $\xi_1,\xi_2>0$, this behavior is given by a multiple of the dipole solution for the local heat equation with a diffusivity determined by $J$. However, the proportionality constant is not the same on $\mathbb{R}_+$ and $\mathbb{R}_-$: it is given by the asymptotic first momentum of the solution on the corresponding half line, which can be computed in terms of the initial data. In the near field scale, $|x|\le t^{1/2}h(t)$, $\lim_{t\to\infty}h(t)=0$, the solution scaled by a factor $t^{3/2}/(|x|+1)$ converges to a stationary solution of the problem that behaves as $b^\pm{x}$ as $x\to\pm\infty$. The constants $b^\pm$ are obtained through a matching procedure with the far field limit. In the very far field, $|x|{\ge}t^{1/2} g(t)$, $g(t)\to\infty$, the solution has order $o(t^{-1})$.