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.

💡 Research Summary

This paper addresses a fundamental gap in the theory of nonautonomous dynamical systems: while pullback attractors are well‑understood and their existence follows from standard dissipativity and compactness assumptions, forward attractors—objects that describe the asymptotic behavior of solutions as time advances from a fixed initial moment—do not enjoy such general guarantees. The authors first develop abstract sufficient conditions under which a pullback attractor automatically becomes a forward attractor. The key idea is to compare two limit sets associated with the process: the forward ω‑limit set ω_f(B, t₀) of any bounded set B (the set of all possible limit points of trajectories starting from B as t → +∞) and the ω₀‑limit set ω₀(A) of the pullback attractor family A(t) (the set of points that stay arbitrarily close to A(t) for all sufficiently large times). If for every bounded B and every initial time t₀ one has ω_f(B, t₀) ⊂ ω₀(A), then the pullback attractor A(t) also attracts all bounded sets forward in time, i.e., it is a forward attractor. This result holds provided the process is forward asymptotically compact, meaning that any sequence of points taken from forward images of a bounded set at times t_n → +∞ possesses a convergent subsequence.

The paper then specializes to asymptotically autonomous systems, where the nonautonomous forcing converges to a time‑independent limit as t → +∞. In this setting, the authors show that if the pullback attractor A(t) converges (in the Hausdorff metric) to the global attractor A_∞ of the limiting autonomous semigroup, then ω₀(A) = A_∞ and the inclusion condition above is automatically satisfied. Consequently, the pullback attractor is a forward attractor.

These abstract theorems are applied to three concrete problems:

1. Asymptotically autonomous Chafee‑Infante equation
   ∂_t u – ∂_x² u = λ u – b(t) u³ on (τ,∞)×(0,π) with Dirichlet boundary conditions. The parameter λ>0 and the coefficient b(t) is uniformly continuous, bounded away from zero and infinity, and converges to a limit b∈