# American Institute of Mathematical Sciences

June  2013, 2(2): 423-440. doi: 10.3934/eect.2013.2.423

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

 1 Division of Mathematical and Computer Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900 2 Mathematics Department, Annaba University, PO Box 12, Annaba, 23000, Algeria

Received  September 2012 Revised  December 2012 Published  March 2013

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 $t^{-\gamma/4}$.
Citation: Belkacem Said-Houari, Radouane Rahali. Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III. Evolution Equations & Control Theory, 2013, 2 (2) : 423-440. doi: 10.3934/eect.2013.2.423
##### References:
 [1] F. Ammar-Khodja, A. Benabdallah, J. E. Muñoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type,, J. Differential Equations, 194 (2003), 82.  doi: 10.1016/S0022-0396(03)00185-2.  Google Scholar [2] C. Cattaneo, Sulla conduzione del calore,, Atti. Sem. Mat. Fis. Univ. Modena., 3 (1949), 83.   Google Scholar [3] D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature,, Appl. Mech. Rev., 51 (1998), 705.  doi: 10.1115/1.3098984.  Google Scholar [4] M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics,, Appl. Math. Lett., 22 (2009), 1374.  doi: 10.1016/j.aml.2009.03.010.  Google Scholar [5] A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics,, Proc. Royal Society London. A, 432 (1991), 171.  doi: 10.1098/rspa.1991.0012.  Google Scholar [6] A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid,, J. Thermal Stresses, 15 (1992), 253.  doi: 10.1080/01495739208946136.  Google Scholar [7] K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system,, Math. Mod. Meth. Appl. Sci., 18 (2008), 647.  doi: 10.1142/S0218202508002802.  Google Scholar [8] K. Ide and S. Kawashima, Decay property of regularity-loss type and nonlinear effects for dissipative Timoshenko system,, Math. Mod. Meth. Appl. Sci., 18 (2008), 1001.  doi: 10.1142/S0218202508002930.  Google Scholar [9] R. Ikehata, Diffusion phenomenon for linear dissipative wave equations in an exterior domain,, J. Differential Equations, 186 (2002), 633.  doi: 10.1016/S0022-0396(02)00008-6.  Google Scholar [10] R. Ikehata, New decay estimates for linear damped wave equations and its application to nonlinear problem,, Math. Meth. Appl. Sci., 27 (2004), 865.  doi: 10.1002/mma.476.  Google Scholar [11] P. M Jordan, W. Dai and R. E Mickens, A note on the delayed heat equation: Instability with respect to initial data,, Mechanics Research Communications, 35 (2008), 414.  doi: 10.1016/j.mechrescom.2008.04.001.  Google Scholar [12] D. D. Joseph and L. Preziosi, Heat waves,, Rev. Mod. Physics, 61 (1989), 41.  doi: 10.1103/RevModPhys.61.41.  Google Scholar [13] S. A. Messaoudi, M. Pokojovy and B. Said-Houari, Nonlinear damped Timoshenko systems with second sound - global existence and exponential stability,, Math. Meth. Appl. Sci., 32 (2009), 505.  doi: 10.1002/mma.1049.  Google Scholar [14] S. A. Messaoudi and B. Said-Houari, Exponential stability in one-dimensional non-linear thermoelasticity with second sound,, Math. Methods Appl. Sci., 28 (2005), 205.  doi: 10.1002/mma.556.  Google Scholar [15] S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system of thermoelasticity of type III,, J. Math. Anal. Appl., 348 (2008), 298.  doi: 10.1016/j.jmaa.2008.07.036.  Google Scholar [16] S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system with history in thermoelasticity of type III,, Advances in Differential Equations, 14 (2009), 375.   Google Scholar [17] R. Quintanilla and R. Racke, Stability in thermoelasticity of type III,, Discrete and Continuous Dynamical Systems B, 3 (2003), 383.  doi: 10.3934/dcdsb.2003.3.383.  Google Scholar [18] R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag thermoelasticity,, SIAM J. Appl. Math., 66 (2006), 977.  doi: 10.1137/05062860X.  Google Scholar [19] R. Racke, "Lectures on Nonlinear Evolution Equations. Initial Value Problems,", Aspects of Mathematics, E19 (1992).   Google Scholar [20] R. Racke, Thermoelasticity with second sound-exponential stability in linear and non-linear 1-d,, Math. Methods. Appl. Sci., 25 (2002), 409.  doi: 10.1002/mma.298.  Google Scholar [21] R. Racke and Y. Wang, Nonlinear well-posedness and rates of decay in thermoelasticity with second sound,, J. Hyperbolic Differ. Equ., 5 (2008), 25.  doi: 10.1142/S021989160800143X.  Google Scholar [22] M. Reissig and G. Y. Wang, Cauchy problems for linear thermoelastic systems of type III in one space variable,, Math. Methods Appl. Sci., 28 (2005), 1359.  doi: 10.1002/mma.619.  Google Scholar [23] J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability,, J. Math. Anal. Appl., 276 (2002), 248.  doi: 10.1016/S0022-247X(02)00436-5.  Google Scholar [24] J. E. Muñoz Rivera and H. D. Fernández Sare, Stability of Timoshenko systems with past history,, J. Math. Anal. Appl., 339 (2008), 482.  doi: 10.1016/j.jmaa.2007.07.012.  Google Scholar [25] B. Said-Houari and A. Kasimov, Decay property of Timoshenko system in thermoelasticity,, Math. Methods. Appl. Sci., 35 (2012), 314.  doi: 10.1002/mma.1569.  Google Scholar [26] H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law,, Arch. Rational Mech. Anal., 194 (2009), 221.  doi: 10.1007/s00205-009-0220-2.  Google Scholar [27] M. A. Tarabek, On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound,, Quart. Appl. Math., 50 (1992), 727.   Google Scholar [28] D. Y. Tzou, Thermal shock phenomena under high-rate response in solids,, Annual Review of Heat Transfer, 4 (1992), 111.   Google Scholar [29] D. Y. Tzou, A unified field approach for heat conduction from macro to micro-scales,, J. Heat Transfer, 117 (1995), 8.  doi: 10.1115/1.2822329.  Google Scholar [30] Y.-G. Wang and L. Yang, $L^p$-$L^q$ decay estimates for Cauchy problems of linear thermoelastic systems with second sound in three dimensions,, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 189.  doi: 10.1017/S0308210500004510.  Google Scholar [31] X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III,, Commun. Contemp. Math., 5 (2003), 25.  doi: 10.1142/S0219199703000896.  Google Scholar

show all references

##### References:
 [1] F. Ammar-Khodja, A. Benabdallah, J. E. Muñoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type,, J. Differential Equations, 194 (2003), 82.  doi: 10.1016/S0022-0396(03)00185-2.  Google Scholar [2] C. Cattaneo, Sulla conduzione del calore,, Atti. Sem. Mat. Fis. Univ. Modena., 3 (1949), 83.   Google Scholar [3] D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature,, Appl. Mech. Rev., 51 (1998), 705.  doi: 10.1115/1.3098984.  Google Scholar [4] M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics,, Appl. Math. Lett., 22 (2009), 1374.  doi: 10.1016/j.aml.2009.03.010.  Google Scholar [5] A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics,, Proc. Royal Society London. A, 432 (1991), 171.  doi: 10.1098/rspa.1991.0012.  Google Scholar [6] A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid,, J. Thermal Stresses, 15 (1992), 253.  doi: 10.1080/01495739208946136.  Google Scholar [7] K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system,, Math. Mod. Meth. Appl. Sci., 18 (2008), 647.  doi: 10.1142/S0218202508002802.  Google Scholar [8] K. Ide and S. Kawashima, Decay property of regularity-loss type and nonlinear effects for dissipative Timoshenko system,, Math. Mod. Meth. Appl. Sci., 18 (2008), 1001.  doi: 10.1142/S0218202508002930.  Google Scholar [9] R. Ikehata, Diffusion phenomenon for linear dissipative wave equations in an exterior domain,, J. Differential Equations, 186 (2002), 633.  doi: 10.1016/S0022-0396(02)00008-6.  Google Scholar [10] R. Ikehata, New decay estimates for linear damped wave equations and its application to nonlinear problem,, Math. Meth. Appl. Sci., 27 (2004), 865.  doi: 10.1002/mma.476.  Google Scholar [11] P. M Jordan, W. Dai and R. E Mickens, A note on the delayed heat equation: Instability with respect to initial data,, Mechanics Research Communications, 35 (2008), 414.  doi: 10.1016/j.mechrescom.2008.04.001.  Google Scholar [12] D. D. Joseph and L. Preziosi, Heat waves,, Rev. Mod. Physics, 61 (1989), 41.  doi: 10.1103/RevModPhys.61.41.  Google Scholar [13] S. A. Messaoudi, M. Pokojovy and B. Said-Houari, Nonlinear damped Timoshenko systems with second sound - global existence and exponential stability,, Math. Meth. Appl. Sci., 32 (2009), 505.  doi: 10.1002/mma.1049.  Google Scholar [14] S. A. Messaoudi and B. Said-Houari, Exponential stability in one-dimensional non-linear thermoelasticity with second sound,, Math. Methods Appl. Sci., 28 (2005), 205.  doi: 10.1002/mma.556.  Google Scholar [15] S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system of thermoelasticity of type III,, J. Math. Anal. Appl., 348 (2008), 298.  doi: 10.1016/j.jmaa.2008.07.036.  Google Scholar [16] S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system with history in thermoelasticity of type III,, Advances in Differential Equations, 14 (2009), 375.   Google Scholar [17] R. Quintanilla and R. Racke, Stability in thermoelasticity of type III,, Discrete and Continuous Dynamical Systems B, 3 (2003), 383.  doi: 10.3934/dcdsb.2003.3.383.  Google Scholar [18] R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag thermoelasticity,, SIAM J. Appl. Math., 66 (2006), 977.  doi: 10.1137/05062860X.  Google Scholar [19] R. Racke, "Lectures on Nonlinear Evolution Equations. Initial Value Problems,", Aspects of Mathematics, E19 (1992).   Google Scholar [20] R. Racke, Thermoelasticity with second sound-exponential stability in linear and non-linear 1-d,, Math. Methods. Appl. Sci., 25 (2002), 409.  doi: 10.1002/mma.298.  Google Scholar [21] R. Racke and Y. Wang, Nonlinear well-posedness and rates of decay in thermoelasticity with second sound,, J. Hyperbolic Differ. Equ., 5 (2008), 25.  doi: 10.1142/S021989160800143X.  Google Scholar [22] M. Reissig and G. Y. Wang, Cauchy problems for linear thermoelastic systems of type III in one space variable,, Math. Methods Appl. Sci., 28 (2005), 1359.  doi: 10.1002/mma.619.  Google Scholar [23] J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability,, J. Math. Anal. Appl., 276 (2002), 248.  doi: 10.1016/S0022-247X(02)00436-5.  Google Scholar [24] J. E. Muñoz Rivera and H. D. Fernández Sare, Stability of Timoshenko systems with past history,, J. Math. Anal. Appl., 339 (2008), 482.  doi: 10.1016/j.jmaa.2007.07.012.  Google Scholar [25] B. Said-Houari and A. Kasimov, Decay property of Timoshenko system in thermoelasticity,, Math. Methods. Appl. Sci., 35 (2012), 314.  doi: 10.1002/mma.1569.  Google Scholar [26] H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law,, Arch. Rational Mech. Anal., 194 (2009), 221.  doi: 10.1007/s00205-009-0220-2.  Google Scholar [27] M. A. Tarabek, On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound,, Quart. Appl. Math., 50 (1992), 727.   Google Scholar [28] D. Y. Tzou, Thermal shock phenomena under high-rate response in solids,, Annual Review of Heat Transfer, 4 (1992), 111.   Google Scholar [29] D. Y. Tzou, A unified field approach for heat conduction from macro to micro-scales,, J. Heat Transfer, 117 (1995), 8.  doi: 10.1115/1.2822329.  Google Scholar [30] Y.-G. Wang and L. Yang, $L^p$-$L^q$ decay estimates for Cauchy problems of linear thermoelastic systems with second sound in three dimensions,, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 189.  doi: 10.1017/S0308210500004510.  Google Scholar [31] X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III,, Commun. Contemp. Math., 5 (2003), 25.  doi: 10.1142/S0219199703000896.  Google Scholar
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