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 and 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), 82-115. doi: 10.1016/S0022-0396(03)00185-2.

[2]

C. Cattaneo, Sulla conduzione del calore, Atti. Sem. Mat. Fis. Univ. Modena., 3 (1949), 83-101.

[3]

D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature, Appl. Mech. Rev., 51 (1998), 705-729. doi: 10.1115/1.3098984.

[4]

M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics, Appl. Math. Lett., 22 (2009), 1374-1379. doi: 10.1016/j.aml.2009.03.010.

[5]

A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Royal Society London. A,, 432 (1991), 171-194. doi: 10.1098/rspa.1991.0012.

[6]

A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253-264. doi: 10.1080/01495739208946136.

[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), 647-667. doi: 10.1142/S0218202508002802.

[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), 1001-1025. doi: 10.1142/S0218202508002930.

[9]

R. Ikehata, Diffusion phenomenon for linear dissipative wave equations in an exterior domain, J. Differential Equations, 186 (2002), 633-651. doi: 10.1016/S0022-0396(02)00008-6.

[10]

R. Ikehata, New decay estimates for linear damped wave equations and its application to nonlinear problem, Math. Meth. Appl. Sci., 27 (2004), 865-889. doi: 10.1002/mma.476.

[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), 414-420. doi: 10.1016/j.mechrescom.2008.04.001.

[12]

D. D. Joseph and L. Preziosi, Heat waves, Rev. Mod. Physics, 61 (1989), 41-73. doi: 10.1103/RevModPhys.61.41.

[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), 505-534. doi: 10.1002/mma.1049.

[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), 205-232. doi: 10.1002/mma.556.

[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), 298-307. doi: 10.1016/j.jmaa.2008.07.036.

[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), 375-400.

[17]

R. Quintanilla and R. Racke, Stability in thermoelasticity of type III, Discrete and Continuous Dynamical Systems B, 3 (2003), 383-400. doi: 10.3934/dcdsb.2003.3.383.

[18]

R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag thermoelasticity, SIAM J. Appl. Math., 66 (2006), 977-1001 (electronic). doi: 10.1137/05062860X.

[19]

R. Racke, "Lectures on Nonlinear Evolution Equations. Initial Value Problems," Aspects of Mathematics, E19, Friedrich Vieweg and Sohn, Braunschweig, 1992.

[20]

R. Racke, Thermoelasticity with second sound-exponential stability in linear and non-linear 1-d, Math. Methods. Appl. Sci., 25 (2002), 409-441. doi: 10.1002/mma.298.

[21]

R. Racke and Y. Wang, Nonlinear well-posedness and rates of decay in thermoelasticity with second sound, J. Hyperbolic Differ. Equ., 5 (2008), 25-43. doi: 10.1142/S021989160800143X.

[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), 1359-1381. doi: 10.1002/mma.619.

[23]

J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248-278. doi: 10.1016/S0022-247X(02)00436-5.

[24]

J. E. Muñoz Rivera and H. D. Fernández Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502. doi: 10.1016/j.jmaa.2007.07.012.

[25]

B. Said-Houari and A. Kasimov, Decay property of Timoshenko system in thermoelasticity, Math. Methods. Appl. Sci., 35 (2012), 314-333. doi: 10.1002/mma.1569.

[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), 221-251. doi: 10.1007/s00205-009-0220-2.

[27]

M. A. Tarabek, On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound, Quart. Appl. Math., 50 (1992), 727-742.

[28]

D. Y. Tzou, Thermal shock phenomena under high-rate response in solids, Annual Review of Heat Transfer, 4 (1992), 111-185.

[29]

D. Y. Tzou, A unified field approach for heat conduction from macro to micro-scales, J. Heat Transfer, 117 (1995), 8-16. doi: 10.1115/1.2822329.

[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), 189-207. doi: 10.1017/S0308210500004510.

[31]

X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III, Commun. Contemp. Math., 5 (2003), 25-83. doi: 10.1142/S0219199703000896.

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), 82-115. doi: 10.1016/S0022-0396(03)00185-2.

[2]

C. Cattaneo, Sulla conduzione del calore, Atti. Sem. Mat. Fis. Univ. Modena., 3 (1949), 83-101.

[3]

D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature, Appl. Mech. Rev., 51 (1998), 705-729. doi: 10.1115/1.3098984.

[4]

M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics, Appl. Math. Lett., 22 (2009), 1374-1379. doi: 10.1016/j.aml.2009.03.010.

[5]

A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. Royal Society London. A,, 432 (1991), 171-194. doi: 10.1098/rspa.1991.0012.

[6]

A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, J. Thermal Stresses, 15 (1992), 253-264. doi: 10.1080/01495739208946136.

[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), 647-667. doi: 10.1142/S0218202508002802.

[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), 1001-1025. doi: 10.1142/S0218202508002930.

[9]

R. Ikehata, Diffusion phenomenon for linear dissipative wave equations in an exterior domain, J. Differential Equations, 186 (2002), 633-651. doi: 10.1016/S0022-0396(02)00008-6.

[10]

R. Ikehata, New decay estimates for linear damped wave equations and its application to nonlinear problem, Math. Meth. Appl. Sci., 27 (2004), 865-889. doi: 10.1002/mma.476.

[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), 414-420. doi: 10.1016/j.mechrescom.2008.04.001.

[12]

D. D. Joseph and L. Preziosi, Heat waves, Rev. Mod. Physics, 61 (1989), 41-73. doi: 10.1103/RevModPhys.61.41.

[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), 505-534. doi: 10.1002/mma.1049.

[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), 205-232. doi: 10.1002/mma.556.

[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), 298-307. doi: 10.1016/j.jmaa.2008.07.036.

[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), 375-400.

[17]

R. Quintanilla and R. Racke, Stability in thermoelasticity of type III, Discrete and Continuous Dynamical Systems B, 3 (2003), 383-400. doi: 10.3934/dcdsb.2003.3.383.

[18]

R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag thermoelasticity, SIAM J. Appl. Math., 66 (2006), 977-1001 (electronic). doi: 10.1137/05062860X.

[19]

R. Racke, "Lectures on Nonlinear Evolution Equations. Initial Value Problems," Aspects of Mathematics, E19, Friedrich Vieweg and Sohn, Braunschweig, 1992.

[20]

R. Racke, Thermoelasticity with second sound-exponential stability in linear and non-linear 1-d, Math. Methods. Appl. Sci., 25 (2002), 409-441. doi: 10.1002/mma.298.

[21]

R. Racke and Y. Wang, Nonlinear well-posedness and rates of decay in thermoelasticity with second sound, J. Hyperbolic Differ. Equ., 5 (2008), 25-43. doi: 10.1142/S021989160800143X.

[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), 1359-1381. doi: 10.1002/mma.619.

[23]

J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems-global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248-278. doi: 10.1016/S0022-247X(02)00436-5.

[24]

J. E. Muñoz Rivera and H. D. Fernández Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502. doi: 10.1016/j.jmaa.2007.07.012.

[25]

B. Said-Houari and A. Kasimov, Decay property of Timoshenko system in thermoelasticity, Math. Methods. Appl. Sci., 35 (2012), 314-333. doi: 10.1002/mma.1569.

[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), 221-251. doi: 10.1007/s00205-009-0220-2.

[27]

M. A. Tarabek, On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound, Quart. Appl. Math., 50 (1992), 727-742.

[28]

D. Y. Tzou, Thermal shock phenomena under high-rate response in solids, Annual Review of Heat Transfer, 4 (1992), 111-185.

[29]

D. Y. Tzou, A unified field approach for heat conduction from macro to micro-scales, J. Heat Transfer, 117 (1995), 8-16. doi: 10.1115/1.2822329.

[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), 189-207. doi: 10.1017/S0308210500004510.

[31]

X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III, Commun. Contemp. Math., 5 (2003), 25-83. doi: 10.1142/S0219199703000896.

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