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.
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Dafermos, C., Contraction semigroups and trend to equalibrium in continuum mechanics, to appear.
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Partially supported by NSF grant NSF GP 34260.
This work was partially supported by the National Science Foundation under Grant No. GP 34260
Communicated by C. Dafermos
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Rauch, J. Qualitative behavior of dissipative wave equations on bounded domains. Arch. Rational Mech. Anal. 62, 77–85 (1976). https://doi.org/10.1007/BF00251857
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