Rauch, J. Arch. Rational Mech. Anal. (1976) 62: 77. doi:10.1007/BF00251857
The qualitative behavior of solutions of the mixed problem utt = Δu-a(x)ut in IR x Ω, u=0 on IR x ∂Ω, is studied in the case when a>0 and Ω⊂IRn is bounded. Roughly speaking, if a≧amin>0, then solutions decay at least as fast as exp t(ɛ −1/2amin), with the possible exception of a finite dimensional set of smooth solutions whose existence is associated with a phenomenon of overdamping. If amax is sufficiently small, depending on Ω, then no overdamping occurs.