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 ${\partial }_{t}u=J\ast u-u$ 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 $o(t^{-1})$.

25 pages