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 equationutt+2ηA12ut+aut+Au=f(u) in H01(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 equationutt+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).