Asymptotic Behavior for a nonlocal diffusion equation on the half line

TL;DR

This paper analyzes the long-term behavior of solutions to a nonlocal diffusion equation on the half line, revealing asymptotic profiles in different spatial regions and deriving precise proportionality constants through a complex matching procedure.

Contribution

It provides a detailed asymptotic analysis of nonlocal diffusion on a half line, including the derivation of explicit constants and the handling of non-overlapping regions.

Findings

Outer region behavior matches a multiple of the dipole heat solution, decaying as O(t^{-1})

Inner region scaled by t^{3/2} converges to a stationary solution with linear growth at infinity

Matching procedure accounts for different decay rates across regions

Abstract

We study the large time behavior of solutions to a non-local diffusion equation, $u_t=J*u-u$ with $J$ smooth, radially symmetric and compactly supported, posed in $\mathbb{R}_+$ with zero Dirichlet boundary conditions. In sets of the form $x\ge \xi t^{1/2}$, $\xi>0$, the outer region, the asymptotic behavior is given by a multiple of the dipole solution for the local heat equation, and the solution is $O(t^{-1})$. The proportionality constant is determined from a conservation law, related to the asymptotic first momentum. On compact sets, the inner region, after scaling the solution by a factor $t^{3/2}$, it converges to a multiple of the unique stationary solution of the problem that behaves as $x$ at infinity. The precise proportionality factor is obtained through a matching procedure with the outer behavior. Since the outer and the inner region do not overlap, the matching is quite involved. It has to be done for the scaled function $t^{3/2}u(x,t)/x$, which takes into account that different scales lead to different decay rates.