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

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.