Exponential-stability and super-stability of a thermoelastic system of type II with boundary damping

Lei  Wang; Zhong-Jie Han; Gen-Qi Xu

doi:10.3934/dcdsb.2015.20.2733

Article Contents

2015, Volume 20, Issue 8: 2733-2750. Doi: 10.3934/dcdsb.2015.20.2733

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Exponential-stability and super-stability of a thermoelastic system of type II with boundary damping

1.
Department of Mathematics, Tianjin University of Technology, Tianjin 300384
2.
Department of Mathematics, Tianjin University, Tianjin 300072

Received: September 30, 2014

Revised: November 30, 2014

Published: September 2015

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Abstract

In this paper, the stability of a one-dimensional thermoelastic system with boundary damping is considered. The theory of thermoelasticity under consideration is developed by Green and Naghdi, which is named as ``thermoelasticity of type II''. This system consists of two strongly coupled wave equations. By the frequency domain method, we prove that the energy of this system generally decays to zero exponentially. Furthermore, by showing the spectrum of the system is empty under certain condition and estimating the norm of the resolvent operator, we give a sufficient condition on the super-stability of this thermoelastic system. Under this condition, the solution to the system is identical to zero after finite time. Moreover, we also estimate the maximum existence time of the nonzero part of the solution. Finally, we give some numerical simulations.

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Mathematics Subject Classification: Primary: 93D20; Secondary: 93C20, 93D15.

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