Stabilization for the Wave Equation on Exterior Domains

Abstract

The aim of this article is the study of stabilization for the wave equation on exterior domains, with Dirichlet boundary condition. More precisely, let O be a bounded, smooth domain of ℝⁿ (n odd); we consider the following wave equation on Ω = ^c Ō:

$$ \left( E \right)\left\{ \begin{gathered} \square u = \partial _t^2 u = 0 on \mathbb{R} \times \Omega \hfill \\ u\left( 0 \right) = f_1 ,\partial _t u\left( 0 \right) = f_2 on \Omega \hfill \\ u_{\partial \Omega \times \mathbb{R}} = 0 \hfill \\ \end{gathered} \right. $$

with the initial data f = (f ₁, f ₂) ∈ H(Ω) = H _D × L ², the completion of (C ^∞₀ (Ω))² for the energy norm.