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 u t t − Δ u − γ ( t ) Δ u t + β ε ( t ) u t = f ( u ) , in a bounded domain Ω ⊂ R n , under some restrictions on β ε ( t ) , γ ( t ) and growth restrictions on the nonlinear term f . The function β ε ( t ) depends on a parameter ε , β ε ( t ) ⟶ ε → 0 0 . We will prove, under suitable assumptions, local and global well-posedness (using the uniform sectorial operators theory), the existence and regularity of pullback attractors { A ε ( t ) : t ∈ R } , uniform bounds for these pullback attractors, characterization of these pullback attractors and their upper and lower semicontinuity at ϵ = 0 .