Asymptotic Behavior for a Nonlocal Diffusion Equation in Domains with Holes

TL;DR

This paper investigates the long-term behavior of solutions to a nonlocal diffusion equation in domains with holes, revealing how solutions resemble heat kernels away from holes and harmonic functions near holes, with a unified approximation approach.

Contribution

It provides a detailed analysis of the asymptotic behavior of nonlocal diffusion solutions in perforated domains, including a matching procedure for inner and outer behaviors.

Findings

Solutions behave like heat kernels away from holes in dimensions three or more.

Near holes, solutions resemble L-harmonic functions that vanish outside the domain.

A global approximation unifies inner and outer solution behaviors.

Abstract

The paper deals with the asymptotic behavior of solutions to a non-local diffusion equation, $u_t=J*u-u:=Lu$, in an exterior domain, $\Omega$, which excludes one or several holes, and with zero Dirichlet data on $\mathbb{R}^N\setminus\Omega$. When the space dimension is three or more this behavior is given by a multiple of the fundamental solution of the heat equation away from the holes. On the other hand, if the solution is scaled according to its decay factor, close to the holes it behaves like a function that is $L$-harmonic, $Lu=0$, in the exterior domain and vanishes in its complement. The height of such a function at infinity is determined through a matching procedure with the multiple of the fundamental solution of the heat equation representing the outer behavior. The inner and the outer behavior can be presented in a unified way through a suitable global approximation.