# zbMATH — the first resource for mathematics

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 $$\Omega\subset {\mathbb R}^n$$ imposing Dirichlet boundary conditions. For the nonlinearity it is assumed that $$\liminf_{| u| \to\infty}f(u)/u>-\lambda_1$$, with $$\lambda_1$$ the first eigenvalue of $$-\Delta$$. In addition, the growth condition $$| f(u)| \leq C(1+| u| ^{n/(n-2)})$$ is supposed if $$n\geq 3$$, 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 $$V(u, u_t)=\int_\Omega\{(1/2)u_t^2+(1/2)| \nabla u| ^2+F(u)\}\,dx$$, 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.

##### MSC:
 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 35L70 Second-order nonlinear hyperbolic equations 37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations 35B41 Attractors
Full Text: