Qualitative behavior of dissipative wave equations on bounded domains

Abstract

The qualitative behavior of solutions of the mixed problem u tt = Δu-a(x)u t in IR x Ω, u=0 on IR x ∂Ω, is studied in the case when a>0 and Ω⊂IR n is bounded. Roughly speaking, if a≧a min >0, then solutions decay at least as fast as exp t(ɛ −1/2a min ), with the possible exception of a finite dimensional set of smooth solutions whose existence is associated with a phenomenon of overdamping. If a max is sufficiently small, depending on Ω, then no overdamping occurs.