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

TL;DR

This paper analyzes the long-term behavior of solutions to a one-dimensional nonlocal diffusion equation in exterior domains, revealing different asymptotic regimes and their dependence on initial data and spatial scale.

Contribution

It provides a detailed asymptotic analysis of nonlocal diffusion in exterior domains, including far, near, and very far field behaviors, with explicit constants and matching procedures.

Findings

Far field behavior matches a scaled dipole solution with data-dependent constants.

Near field solutions converge to stationary solutions with linear behavior at infinity.

In very far field, solutions decay faster than any polynomial rate.

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 $o(t^{-1})$.