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Global attractors for damped semilinear wave equations. (English) Zbl 1056.37084
The nonlinear damped wave equation $$ u_{tt}+\beta u_t-\Delta u+f(u)=0 $$ is considered on a bounded domain $\Omega\subset {\Bbb R}^n$ imposing Dirichlet boundary conditions. For the nonlinearity it is assumed that $\liminf_{\vert u\vert \to\infty}f(u)/u>-\lambda_1$, with $\lambda_1$ the first eigenvalue of $-\Delta$. In addition, the growth condition $\vert f(u)\vert \le C(1+\vert u\vert ^{n/(n-2)})$ is supposed if $n\ge 3$, whereas $f$ may grow exponentially for $n=2$. The main result of the paper asserts that the equation has a connected global attractor in $H_0^1(\Omega)\times L^2(\Omega)$, identifying $u$ with $(u, u_t)$. It is further shown that for each global orbit in the attractor the $\alpha$- resp. $\omega$-limit set is a connected subset of the critical points of the Lyapunov functional $V(u, u_t)=\int_\Omega\{(1/2)u_t^2+(1/2)\vert \nabla u\vert ^2+F(u)\}\,dx$, where $F'=f$. If the set of critical points is totally disconnected, then the solutions do not only approach the attractor as a set, but they converge to an individual critical point as $t\to\pm\infty$. The proofs rely on the application of suitable abstract results concerning the existence of attractors.

37L30Attractors and their dimensions, Lyapunov exponents
35L70Nonlinear second-order hyperbolic equations
37L05General theory, nonlinear semigroups, evolution equations
35B41Attractors (PDE)
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