# American Institute of Mathematical Sciences

2007, 2007(Special): 230-239. doi: 10.3934/proc.2007.2007.230

## Strongly damped wave equations in $W^(1,p)_0 (\Omega) \times L^p(\Omega)$

 1 Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil 2 Institute of Mathematics, Silesian University, 40-007 Katowice, Poland

Received  September 2006 Revised  January 2007 Published  September 2007

This paper is devoted to well posedness and regularity of the solutions of

$u_t_t + 2\nA^(1/2)u_t + A\u = \f(u)$

in $W^(1,p)_0 (\Omega) \times L^p(\Omega), p \in (1,\infty)$, where $\Omega \subset \mathbb {R}^N$ is a bounded smooth domain, $\n$ > 0 and -$\A$ is the Dirichlet Laplacian in $L^p(\Omega)$. We prove local well posedness result for nonlinearities $\f : \mathbb {R} \rightarrow \mathbb {R}$ satisfying $|f(s) - f(t)| \<= C|s - t|(1 + |s|^(p - 1) + |t|^(p - 1))$ with $\p < (N+p)/(N - p) (N > p)$, and show that the solutions are classical. If $f$ is dissipative and $p < (N+2)/(N - 2) (N \>= 3)$, we show that the associated semigroup has a global attractor $\cc{A}_(n,p)$ in $W^(1,p)_0(\Omega)\times L^p(\Omega)$, $p \in [2,\infty)$, which coincides with the attractor $\bb{A}_(n,2) =: \bb{A}_n$. We also obtain that $\bb{A}_n$ is compact in $C^(2+\mu)(bar(\Omega)) \times C^(1+\mu)(bar(\Omega))$ and attracts bounded subsets of $H^1_0(\Omega) \times L^2(\Omega)$ in $C^(2+\mu)(bar(\Omega)) \times C^(1+\mu)(bar(\Omega))$ for each $\mu \in (0, 1)$.

Citation: Alexandre N. Carvalho, Jan W. Cholewa. Strongly damped wave equations in $W^(1,p)_0 (\Omega) \times L^p(\Omega)$. Conference Publications, 2007, 2007 (Special) : 230-239. doi: 10.3934/proc.2007.2007.230
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