, Volume 62, Issue 1, pp 77-85

Qualitative behavior of dissipative wave equations on bounded domains

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Abstract

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.

Communicated by C. Dafermos
Partially supported by NSF grant NSF GP 34260.
This work was partially supported by the National Science Foundation under Grant No. GP 34260