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Exponential decay for the semilinear wave equation with locally distributed damping. (English) Zbl 0716.35010

The paper deals with weak solutions u of the initial-boundary value problem for a multi-dimensional wave equation with nonlinearity f(u) and local damping \(a(x)u_ t\). The goal of the present paper is to give some sufficient conditions on f(u) and the domain where a(x) is nontrivial ensuring the uniform time-exponential decay of the energy E(t). There is a short survey of well-known results in this field. In essence the author extends the results of the linear case (i.e. \(f(u)=ku)\) to the semilinear wave equation with local damping term. The above mentioned estimates for the energy E(t) are proved for the two basic cases - f(u) is globally Lipschitz or superlinear. In conclusion some extensions of the stated problem for other types of differential equations and its links with that of the boundary stabilization are considered. References, 24 in number, fully cover the topic.
Reviewer: V.Chernyatin

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L70 Second-order nonlinear hyperbolic equations
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References:

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