Evolution Equations and Control Theory (EECT)

Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III

Pages: 423 - 440, Volume 2, Issue 2, June 2013      doi:10.3934/eect.2013.2.423

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Belkacem Said-Houari - Division of Mathematical and Computer Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia (email)
Radouane Rahali - Mathematics Department, Annaba University, PO Box 12, Annaba, 23000, Algeria (email)

Abstract: In this paper, we investigate the decay property of a Timoshenko system in thermoelasticity of type III in the whole space where the heat conduction is given by the Green and Naghdi theory. Surprisingly, we show that the coupling of the Timoshenko system with the heat conduction of Green and Naghdi's theory slows down the decay of the solution. In fact we show that the $L^2$-norm of the solution decays like $(1+t)^{-1/8}$, while in the case of the coupling of the Timoshenko system with the Fourier or Cattaneo heat conduction, the decay rate is of the form $(1+t)^{-1/4}$ [25]. We point out that the decay rate of $(1+t)^{-1/8}$ has been obtained provided that the initial data are in $L^1( \mathbb{R})\cap H^s(\mathbb{R}), (s\geq 2)$. If the wave speeds of the first two equations are different, then the decay rate of the solution is of regularity-loss type, that is in this case the previous decay rate can be obtained only under an additional regularity assumption on the initial data. In addition, by restricting the initial data to be in $H^{s}\left( \mathbb{R}\right)\cap L^{1,\gamma }\left( \mathbb{R}\right) $ with $ \gamma \in \left[ 0,1\right] $, we can derive faster decay estimates with the decay rate improvement by a factor of $t^{-\gamma/4}$.

Keywords:  Decay rate, heat conduction, Timoshenko, thermoelasticity.
Mathematics Subject Classification:  35B35, 35L55, 74D05, 93D15, 93D20.

Received: September 2012;      Revised: December 2012;      Available Online: March 2013.