Transmission Problems in (Thermo)Viscoelasticity with Kelvin-Voigt Damping: Nonexponential, Strong, and Polynomial Stability

We investigate transmission problems between a (thermo)viscoelastic system with Kelvin--Voigt damping, and a purely elastic system. It is shown that neither the elastic damping by Kelvin--Voigt mechanisms nor the dissipative effect of the temperature in one material can assure the exponential stability of the total system when it is coupled through transmission to a purely elastic system. The approach shows the lack of exponential stability using Weyl's theorem on perturbations of the essential spectrum. Instead, strong stability can be shown using the principle of unique continuation. To prove polynomial stability we provide an extended version of the characterizations in [A. Borichev and Y. Tomilov, Math. Ann., 347 (2009), pp. 455--478]. Observations on the lack of compacity of the inverse of the arising semigroup generators are included too. The results apply to thermoviscoelastic systems, to purely elastic systems as well as to the scalar case consisting of wave equations.

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