New Trends in the Theory of Hyperbolic Equations pp 213-299 | Cite as

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

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## Abstract

In this article we consider the initial-boundary value problem for linear and nonlinear wave equations in an exterior domain Ω in **R**^{N} with the homogeneous Dirichlet boundary condition. Under the effect of localized dissipation like *a*(*x*)*u*_{t} 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*)*u*_{t} is intended to be as weak as possible, and if the obstacle *V* = **R**^{N} ∖ 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*) ≥ *ɛ*_{0} > 0 near ∞.

In the final section we consider the wave equation with a ‘half-linear’ dissipation *σ*(*x, u*_{t}) which is like *a*(*x*)|*u*_{t}|^{r}*u*_{t} in a bounded area and which is linear like *a*(*x*)*u*_{t} near ∞.

## Keywords

Nonlinear wave equations exterior domains decay global solutions## Preview

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