Attractors for strongly damped wave equations with critical nonlinearities

Abstract

In this paper we obtain global well-posedness results for the strongly damped wave equation utt +( −∆) θ ut =∆ u + f (u), for θ ∈ � 1 , 1 � ,i nH 1 0 (Ω) × L 2 (Ω) when Ω is a bounded smooth domain and the map f grows like |u| n+2 n−2 .I ff = 0, then this equation generates an analytic semigroup with generator −A(θ). Special attention is devoted to the case when θ =1 since in this case the generator −A(1) does not have compact resolvent, contrary to the case θ ∈ � 1 , 1 � . Under the dissipativeness condition lim sup |s|→∞ f (s) s ≤ 0 we prove the existence of compact global attractors for this problem. In the critical growth case we use Alekseev’s nonlinear variation of constants formula to obtain that the semigroup is asymptotically smooth.