A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor

In this paper we consider the strongly damped wave equation with time-dependent terms utt−Δu−γ(t)Δut+βε(t)ut=f(u),utt−Δu−γ(t)Δut+βε(t)ut=f(u), in a bounded domain Ω⊂RnΩ⊂Rn, under some restrictions on βε(t)βε(t), γ(t)γ(t) and growth restrictions on the nonlinear term ff. The function βε(t)βε(t) depends on a parameter εε, βε(t)⟶ε→00. We will prove, under suitable assumptions, local and global well-posedness (using the uniform sectorial operators theory), the existence and regularity of pullback attractors {Aε(t):t∈R}{Aε(t):t∈R}, uniform bounds for these pullback attractors, characterization of these pullback attractors and their upper and lower semicontinuity at ϵ=0ϵ=0.