Asymptotic behavior of solutions for a critical heat equation with nonlocal reaction

TL;DR

This paper analyzes the long-term behavior of solutions to a critical nonlocal heat equation, establishing potential well structures, stability criteria, and uniform bounds for global solutions.

Contribution

It introduces the stable and unstable sets for the equation, proves a potential well structure, and studies the asymptotic behavior of solutions, including those not intersecting these sets.

Findings

Global solutions are bounded in L-infinity norm under natural conditions.

The potential well structure determines the stability and asymptotic behavior.

Solutions not intersecting stable or unstable sets exhibit specific asymptotic properties.

Abstract

In this paper, we consider the following nonlocal parabolic equation \begin{equation*} u_{t}-\Delta u=\left( \int_{\Omega}\frac{|u(y,t)|^{2^{\ast}_{\mu}}}{|x-y|^{\mu}}dy\right) |u|^{2^{\ast}_{\mu}-2}u,\ \text{in}\ \Omega\times(0,\infty), \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $0<\mu<N$ and $2^{\ast}_{\mu}=(2N-\mu)/(N-2)$ denotes the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. We first introduce the stable and unstable sets for the equation and prove that the problem has a potential well structure. Next, we investigate the global asymptotic behavior of the solutions. In particular, we study the behavior of the global solutions that intersect neither with the stable set nor the unstable set. Finally, we prove that global solutions have $L^{\infty}$-uniform bound under some natural conditions.