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Decay of solutions of the wave equation with a local degenerate dissipation

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Abstract

We derive a precise decay estimate of the solutions to the initial-boundary value problem for the wave equation with a dissipation:u tt − Δu+a(x)u t =0 in Ω × [0, ∞) with the boundary conditionu/∂Ω, wherea(x) is a nonnegative function on\(\bar \Omega \) satisfying

$$a(x) > a.e. x \in \omega and\smallint _\omega \frac{1}{{a(x)^P }}dx< \infty for some 0< p< 1$$

for an open set\(\omega \subset \bar \Omega \) including a part of ϱΩ with a specific property. The result is applied to prove a global existence and decay of smooth solutions for a semilinear wave equation with such a weak dissipation.

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Correspondence to Mitsuhiro Nakao.

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Nakao, M. Decay of solutions of the wave equation with a local degenerate dissipation. Israel J. Math. 95, 25–42 (1996). https://doi.org/10.1007/BF02761033

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  • DOI: https://doi.org/10.1007/BF02761033

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