About the structure of attractors for a nonlocal Chafee-Infante problem

TL;DR

This paper investigates the global attractor structure for a nonlocal reaction-diffusion equation, revealing bifurcation patterns, stability properties, and the composition of the attractor in terms of stationary points and heteroclinic connections.

Contribution

It extends the understanding of attractor structures to nonlocal reaction-diffusion equations with non-unique solutions, analyzing bifurcations and stability in this context.

Findings

The problem exhibits a bifurcation cascade similar to the classical Chafee-Infante equation.

The semiflow is shown to be dynamically gradient.

The attractor is composed of stationary points and heteroclinic connections.

Abstract

In this paper, we study the structure of the global attractor for the multivalued semiflow generated by a nonlocal reaction-diffusion equation in which we cannot guarantee uniqueness of the Cauchy problem. First, we analyse the existence and properties of stationary points, showing that the problem undergoes the same cascade of bifurcations as in the Chafee-Infante equation. Second, we study the stability of the fixed points and establish that the semiflow is dynamically gradient. We prove that the attractor consists of the stationary points and their heteroclinic connections and analyse some of the possible connections.