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

Type
Article

Authors
Rahali, Radouane
Said-Houari, Belkacem

KAUST Department
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Online Publication Date
2013-04-04

Print Publication Date
2013

Date
2013-04-04

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 (R) boolean AND H-s (R); (s >= 2). If the wave speeds of the fi rst two equations are di ff erent, 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 (R) boolean AND L-1,L-gamma (R) with gamma is an element of [0; 1], we can derive faster decay estimates with the decay rate improvement by a factor of t(-gamma/4).

Citation
Said-Houari, B., & Rahali, R. (2013). Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III. Evolution Equations & Control Theory, 2(2), 423–440. doi:10.3934/eect.2013.2.423

Acknowledgements
The authors want to thank the referee for his useful remarks and for his careful reading of the proofs. The first author thanks KAUST for its support.

Publisher
American Institute of Mathematical Sciences (AIMS)

Journal
Evolution Equations & Control Theory

DOI
10.3934/eect.2013.2.423

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