Stabilization of Local Energy in an Exterior Domain for the Wave Equation with a Localized Dissipation

We derive an algebraic decay rate for the local energyEloc(t) of the solutions to the initial-boundary value problem of the wave equation with a localized dissipation in an exterior domainΩ. The dissipative terma(x) utis assumed to be effective only on a neighbourhood of a part of the boundaryΛ(x0)={x ∂Ω (x−x0)•ν(x)>0} for somex0, whereν(x) is unit outward normal atxwith respect toΩ. It should be noted that we make no geometrical condition on the boundary ∂ω. IfV≡RN/0 is star-shaped with respect tox0,Λ(x0) is empty and hence, we may assumea(x)≡0. Thus, our result is a natural extension of the well-known most classical result.