Large Time Behavior of a Nonlocal Diffusion Equation with Absorption and Bounded Initial Data

TL;DR

This paper investigates the long-term behavior of solutions to a nonlocal diffusion equation with absorption, especially focusing on the critical case where the absorption term balances diffusion, and establishes convergence rates and asymptotic profiles.

Contribution

It provides a detailed analysis of the critical case for the nonlocal diffusion equation with absorption, including new convergence results and asymptotic descriptions for various initial data.

Findings

Proves convergence of solutions to heat equation with absorption in the critical case.

Establishes decay rates for solutions with nonintegrable initial data.

Analyzes large time behavior for bounded and integrable initial data in supercritical and critical cases.

Abstract

We study the large time behavior of nonnegative solutions of the Cauchy problem $u_t=\int J(x-y)(u(y,t)-u(x,t))\,dy-u^p$, $u(x,0)=u_0(x)\in L^\infty$, where $|x|^{\alpha}u_0(x)\to A>0$ as $|x|\to\infty$. One of our main goals is the study of the critical case $p=1+2/\alpha$ for $0<\alpha<N$, left open in previous articles, for which we prove that $t^{\alpha/2}|u(x,t)-U(x,t)|\to 0$ where $U$ is the solution of the heat equation with absorption with initial datum $U(x,0)=C_{A,N}|x|^{-\alpha}$. Our proof, involving sequences of rescalings of the solution, allows us to establish also the large time behavior of solutions having more general nonintegrable initial data $u_0$ in the supercritical case and also in the critical case ($p=1+2/N$) for bounded and integrable $u_0$.