Regularity of solutions on the global attractor for a semilinear damped wave equation

We consider attractors Aη, η ∈ [0, 1], corresponding to a singularly perturbed damped wave equation utt +2ηA 1 2 ut + aut + Au = f (u) in H1 0 (Ω)×L2(Ω), where Ω is a bounded smooth domain in R3. For dissipative nonlinearity f ∈ C2(R,R) satisfying |f (s)| c(1 + |s|) with some c > 0, we prove that the family of attractors {Aη, η 0} is upper semicontinuous at η = 0 in H1+s(Ω) × Hs(Ω) for any s ∈ (0, 1). For dissipative f ∈ C3(R,R) satisfying lim|s|→∞ f (s) s = 0 we prove that the attractor A0 for the damped wave equation utt +aut +Au = f (u) (case η = 0) is bounded in H4(Ω)×H3(Ω) and thus is compact in the Hölder spaces C2+μ(Ω)×C1+μ(Ω) for every μ ∈ (0, 12 ). As a consequence of the uniform bounds we obtain that the family of attractors {Aη,η ∈ [0, 1]} is upper and lower semicontinuous in C2+μ(Ω)×C1+μ(Ω) for every μ ∈ (0, 12 ). © 2007 Elsevier Inc. All rights reserved.