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

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.