Strongly damped wave equations in $W^(1,p)_0 (\Omega) \times L^p(\Omega)$
Alexandre N. Carvalho - 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 (email) Abstract: 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 $\mu \in (0, 1)$.
Keywords: Analytic semigroup, well posedness, global attractor, regularity.
Received: September 2006; Revised: January 2007; Published: September 2007. |