A Non-Autonomous Strongly Damped Wave Equation: Existence and Continuity of the Pullback Attractor

Abstract

In this paper we consider the strongly damped wave equation with time dependent terms utt − u − γ(t) ut + β"(t)ut = f(u), in a bounded domain ⊂ Rn, under some restrictions on β"(t), γ(t) and growth restrictions on the non-linear term f. The function β"(t) depends on a parameter ε, β"(t) "!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"(t) : t ∈ R}, uniform bounds for these pullback attractors, characterization of these pullback attractors and their upper and lower semicontinuity at ǫ = 0.