TY - JOUR

T1 - Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III

AU - Rahali, Radouane

AU - Said-Houari, Belkacem

N1 - KAUST Repository Item: Exported on 2020-10-01
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.

PY - 2013/4/4

Y1 - 2013/4/4

N2 - 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).

AB - 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).

UR - http://hdl.handle.net/10754/575691

UR - http://aimsciences.org//article/doi/10.3934/eect.2013.2.423

UR - http://www.scopus.com/inward/record.url?scp=84922338311&partnerID=8YFLogxK

U2 - 10.3934/eect.2013.2.423

DO - 10.3934/eect.2013.2.423

M3 - Article

VL - 2

SP - 423

EP - 440

JO - Evolution Equations and Control Theory

JF - Evolution Equations and Control Theory

SN - 2163-2472

IS - 2

ER -