The singular limit dynamics of semilinear damped wave equations. (English) Zbl 0699.35177

Mora, Xavier; Solà-Morales, Joan

J. Differ. Equations 78, No. 2, 262-307 (1989).

The authors consider the following abstract problem: \[ \epsilon \ddot u+\dot u+Au=Fu,\quad u(0)=u_ 0,\quad \epsilon \dot u(0)=\epsilon v_ 0 \] in a Hilbert space E, where A is a linear operator while F is nonlinear. This problem includes, e.g., the case \(Au=-u_{xx}\), with some usual boundary conditions, and \(Fu=f(,u( ))\) (i.e. a semilinear damped wave equation). It is shown that there exist an integer n and an \({\bar \epsilon}>0\) such that, for every \(\epsilon\in [0,{\bar \epsilon})\), the global attractor of the corresponding dynamical system is contained in an invariant manifold of class \(C^ 1\) and dimension n and that for \(\epsilon\) \(\to 0\) both this manifold and the vector field on it converge in the \(C^ 1\) topology towards the ones corresponding to \(\epsilon =0\).

Reviewer: G.Moroşanu

MSC:

35L70	Second-order nonlinear hyperbolic equations
35K57	Reaction-diffusion equations

Keywords:

semilinear damped wave equation; global attractor; invariant manifold

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