## Transmission problems in (thermo-)viscoelasticity with Kelvin-Voigt damping : non-exponential, strong and polynomial stability

2015
##### Authors
Munoz Rivera, Jaime E.
##### Series
Konstanzer Schriften in Mathematik; 336
##### Publication type
Working Paper/Technical Report
##### Abstract
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 by Borichev and Tomilov. Observations on the lack of compacity of the inverse of the arising semigroup generators are included too. The results apply to thermo-viscoelastic systems, to< purely elastic systems as well as to the scalar case consisting of wave equations.
510 Mathematics
##### Cite This
ISO 690MUNOZ RIVERA, Jaime E., Reinhard RACKE, 2015. Transmission problems in (thermo-)viscoelasticity with Kelvin-Voigt damping : non-exponential, strong and polynomial stability
BibTex
@techreport{MunozRivera2015Trans-30731,
year={2015},
series={Konstanzer Schriften in Mathematik},
title={Transmission problems in (thermo-)viscoelasticity with Kelvin-Voigt damping : non-exponential, strong and polynomial stability},
number={336},
author={Munoz Rivera, Jaime E. and Racke, Reinhard}
}

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<dcterms:abstract xml:lang="eng">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 by Borichev and Tomilov. Observations on the lack of compacity of the inverse of the arising semigroup generators are included too. The results apply to thermo-viscoelastic systems, to&lt; purely elastic systems as well as to the scalar case consisting of wave equations.</dcterms:abstract>
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Yes