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Klein-Gordon type decay rates for wave equations with a time-dependent dissipation. (English) Zbl 1002.35076

Author’s abstract: This work is concerned with the proof of decay estimates for the solutions of the Cauchy problem for the wave equation with time-dependent dissipation \(u_{tt}-\lambda ^2(t)b^2(t)\triangle u+a\lambda (t)b(t)u_{t}=0.\) Here \(\lambda=\lambda (t)\) is an increasing smooth and positive function. The positive function \(b=b(t)\) is an oscillating one. The goal is to study under which assumptions for \(\lambda\) and \(b\) one can expect decay estimates generalizing the classical result of the article by A. Matsumura [Publ. Res. Inst. Math. Sci. Kyoto Univ. 12, 169-189 (1976; Zbl 0356.35008)] for the case \(\lambda (t)\equiv 1.\)

MSC:

35L15 Initial value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs

Citations:

Zbl 0356.35008
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