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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 \(\Omega\subset {\mathbb R}^n\) 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.

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
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