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

  • Mitsuhiro Nakao
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, 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 ∞.

Keywords

Nonlinear wave equations exterior domains decay global solutions 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Mitsuhiro Nakao
    • 1
  1. 1.Graduate School of MathematicsKyushu UniversityFukuokaJapan

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