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Decay and Global Existence for Nonlinear Wave Equations with Localized Dissipations in General Exterior Domains

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New Trends in the Theory of Hyperbolic Equations

Part of the book series: Operator Theory: Advances and Applications ((APDE,volume 159))

Abstract

In this article we consider the initial-boundary value problem for linear and nonlinear wave equations in an exterior domain Ω in RN with the homogeneous Dirichlet boundary condition. Under the effect of localized dissipation like a(x)ut we derive both of local and total energy decay estimates for the linear wave equation and apply these to the existence problem of global solutions of semilinear and quasilinear wave equations. We make no geometric condition on the shape of the boundary ∂Ω.

The dissipation a(x)ut is intended to be as weak as possible, and if the obstacle V = RN ∖ Ω is star-shaped our results based on local energy decay hold even if a(x) ≡ 0, while for the results concerning the total energy decay we need a(x) ≥ ɛ0 > 0 near ∞.

In the final section we consider the wave equation with a ‘half-linear’ dissipation σ(x, ut) which is like a(x)|ut|rut in a bounded area and which is linear like a(x)ut near ∞.

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  1. Graduate School of Mathematics, Kyushu University, Ropponmatsu, Fukuoka, 810-8560, Japan

    Mitsuhiro Nakao

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  1. Mitsuhiro Nakao
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  1. Fakultät für Mathematik und Informatik Institut für Angewandte Analysis, TU Bergakademie Freiberg, 09596, Freiberg, Germany

    Michael Reissig

  2. Institut für Mathematik, Universität Potsdam, 14415, Potsdam, Germany

    Bert-Wolfgang Schulze

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Nakao, M. (2005). Decay and Global Existence for Nonlinear Wave Equations with Localized Dissipations in General Exterior Domains. In: Reissig, M., Schulze, BW. (eds) New Trends in the Theory of Hyperbolic Equations. Operator Theory: Advances and Applications, vol 159. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7386-5_3

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