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Global attractors for damped semilinear wave equations. (English) Zbl 1056.37084

The nonlinear damped wave equation

u tt +βu t -Δu+f(u)=0

is considered on a bounded domain Ω n imposing Dirichlet boundary conditions. For the nonlinearity it is assumed that lim inf |u| f(u)/u>-λ 1 , with λ 1 the first eigenvalue of -Δ. In addition, the growth condition |f(u)|C(1+|u| n/(n-2) ) is supposed if n3, 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 (Ω)×L 2 (Ω), identifying u with (u,u t ). It is further shown that for each global orbit in the attractor the α- resp.¬†ω-limit set is a connected subset of the critical points of the Lyapunov functional V(u,u t )= Ω {(1/2)u t 2 +(1/2)|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±. The proofs rely on the application of suitable abstract results concerning the existence of attractors.


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