Attractors for processes on time-dependent spaces. Applications to wave equations

TL;DR

This paper introduces a new concept of attractors for processes on time-dependent spaces, proving invariance under weaker conditions and applying it to analyze the long-term behavior of wave equations with variable propagation speeds.

Contribution

It develops a minimality-based notion of time-dependent attractors and extends existing theories, enabling analysis under weaker assumptions and applying to wave equations with changing speeds.

Findings

Invariance of attractors under T-closed processes.

Generalization of attractor theory in time-dependent spaces.

Application to wave equations with time-varying propagation speed.

Abstract

For a process U(t,s) acting on a one-parameter family of normed spaces, we present a notion of time-dependent attractor based only on the minimality with respect to the pullback attraction property. Such an attractor is shown to be invariant whenever the process is T-closed for some T>0, a much weaker property than continuity (defined in the text). As a byproduct, we generalize the recent theory of attractors in time-dependent spaces developed in [10]. Finally, we exploit the new framework to study the longterm behavior of wave equations with time-dependent speed of propagation.