Vol. 207, No. 2, 2002

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Attractors for strongly damped wave equations with critical nonlinearities

Alexandre N. Carvalho and Jan W. Cholewa

Vol. 207 (2002), No. 2, 287–310

In this paper we obtain global well-posedness results for the strongly damped wave equation utt + (Δ)𝜃ut = Δu + f(u), for 𝜃 [1,1]
2, in H01(Ω) ×L2(Ω) when Ω is a bounded smooth domain and the map f grows like |u|nn+−22. If f = 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  )
2,1. Under the dissipativeness condition limsup|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.

Published: 1 December 2002
Alexandre N. Carvalho
Departamento de Matemática
Instituto de Ciências Matemáticas de São Carlos
Universidade de São Paulo - Campus de São Carlos
Caixa Postal 668
13.560-970 São Carlos SP, Brazil
Jan W. Cholewa
Institute of Mathematics
Silesian University
40-007 Katowice, Poland