On forward attractors for nonautonomous dynamical systems with application to the asymptotically autonomous Chafee-Infante equation

TL;DR

This paper establishes conditions under which pullback attractors also serve as forward attractors in nonautonomous systems, with specific application to the asymptotically autonomous Chafee-Infante equation, including differential inclusions.

Contribution

It provides the first general conditions linking pullback and forward attractors in nonautonomous systems, applied specifically to the Chafee-Infante equation.

Findings

Conditions for pullback attractors to be forward attractors

Application to asymptotically autonomous Chafee-Infante equation

Extensions to differential inclusions

Abstract

In this paper, for nonautonomous dynamical systems, we give first general conditions ensuring that a pullback attractor is a forward attractor as well in both the single and multivalued frameworks. In particular, we consider asymptotically autonomous systems. After that, and this is the main result of this paper, we apply these abstract theorems to the asymptotically autonomous Chafee-Infante equation. Finally, applications to ordinary and parabolic differential inclusions are given.