Strong stabilization of models incorporating the thermoelastic Reissner–Mindlin plate equations with second sound

Abstract

In this article we are concerned with the strong stabilization of models for the Reissner–Mindlin plate equations with second sound, that is, models that include thermal effects described according to Cattaneo's law of heat conduction instead of Fourier's law in classical thermoelasticity. Two models will be considered which are distinct with respect to the property of compactness or non-compactness of the resolvent of the generator of the underlying semigroup. In accordance with the compactness or non-compactness of the resolvent operator, a different criterion for strong stability is implemented to achieve the strong stabilization of each model. In the compact resolvent case we avail ourselves of a result given by Benchimol [C.D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optim. 16 (1978), pp. 373–379] and in the non-compact case we resort to a stability criterion Tomilov [Y. Tomilov, A resolvent approach to stability of operator semigroups, J. Operator Theory 46 (2001), pp. 63–98].

Keywords:

• Reissner–Mindlin plate equations
• Cattaneo
• second sound
• strong stability

AMS Subject Classifications::

• 35L70
• 93D15
• 35B40

Acknowledgements

The author is grateful to Reinhard Racke and Michael Pokojovy for stimulating discussions on this work and other problems of common interest during and after her visit to the University of Konstanz, Germany, July–August 2009. I also extend my sincere thanks to the referees for their constructive criticism and suggestions towards an improved version of this article.