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

For a process U(t,τ):Xτ→XtU(t,τ):Xτ→Xt 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 U(t,τ)U(t,τ) is T  -closed for some T>0T>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 Di Plinio et al. (2011) [13]. Finally, we exploit the new framework to study the longterm behavior of wave equations with time-dependent speed of propagation.