Asymptotic Behavior for a Nonlocal Diffusion Equation with Absorption and Nonintegrable Initial Data. the Supercritical Case

TL;DR

This paper investigates the long-term behavior of solutions to a nonlocal diffusion equation with absorption and nonintegrable initial data, showing that in the supercritical case, solutions resemble those of the heat equation asymptotically.

Contribution

It establishes the asymptotic equivalence of solutions to a nonlocal diffusion-absorption equation and the heat equation in the supercritical regime with nonintegrable initial data.

Findings

Solutions behave like heat equation solutions for large time

Asymptotic behavior is characterized in the supercritical case p>1+2/α

Initial data with negative power decay influences long-term dynamics

Abstract

In this paper we study the asymptotic behavior as time goes to infinity of the solution to a nonlocal diffusion equation with absorption modeled by a powerlike reaction $-u^p$, $p>1$ and set in $\R^N$. We consider a bounded, nonnegative initial datum $u_0$ that behaves like a negative power at infinity. That is, $|x|^\alpha u_0(x)\to A>0$ as $|x|\to\infty$ with $0<\alpha\le N$. We prove that, in the supercritical case $p>1+2/\alpha$, the solution behaves asymptotically as that of the heat equation --with diffusivity $\a$ related to the nonlocal operator-- with the same initial datum.