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On asymptotic stability of stationary solutions to nonlinear wave and Klein-Gordon equations. (English) Zbl 0863.35064

Authors’ abstract: We consider nonlinear wave and Klein-Gordon equations with general nonlinear terms, localized in space. Conditions are found which provide asymptotic stability of stationary solutions in local energy norms. These conditions are formulated in terms of spectral properties of the Schrödinger operator corresponding to the linearized problem. They are natural extensions to partial differential equations of the known Lyapunov condition. For the nonlinear wave equation in three-dimensional space we find asymptotic expansions, as \(t\to\infty\), of the solutions which are close enough to a stationary asymptotically stable solution.
Reviewer: R.Racke (Konstanz)

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35L30 Initial value problems for higher-order hyperbolic equations
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