# 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 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.
 [1] Denis Mercier, Virginie Régnier. Decay rate of the Timoshenko system with one boundary damping. Evolution Equations and Control Theory, 2019, 8 (2) : 423-445. doi: 10.3934/eect.2019021 [2] Makram Hamouda, Ahmed Bchatnia, Mohamed Ali Ayadi. Numerical solutions for a Timoshenko-type system with thermoelasticity with second sound. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2975-2992. doi: 10.3934/dcdss.2021001 [3] Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control and Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014 [4] Sandra Carillo, Vanda Valente, Giorgio Vergara Caffarelli. Heat conduction with memory: A singular kernel problem. Evolution Equations and Control Theory, 2014, 3 (3) : 399-410. doi: 10.3934/eect.2014.3.399 [5] Antonio Magaña, Alain Miranville, Ramón Quintanilla. On the time decay in phase–lag thermoelasticity with two temperatures. Electronic Research Archive, 2019, 27: 7-19. doi: 10.3934/era.2019007 [6] Tamara Fastovska. Decay rates for Kirchhoff-Timoshenko transmission problems. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2645-2667. doi: 10.3934/cpaa.2013.12.2645 [7] Corrado Mascia. Stability analysis for linear heat conduction with memory kernels described by Gamma functions. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3569-3584. doi: 10.3934/dcds.2015.35.3569 [8] Xueke Pu, Boling Guo. Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction. Kinetic and Related Models, 2016, 9 (1) : 165-191. doi: 10.3934/krm.2016.9.165 [9] Micol Amar, Roberto Gianni. Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1739-1756. doi: 10.3934/dcdsb.2018078 [10] Claudio Giorgi, Diego Grandi, Vittorino Pata. On the Green-Naghdi Type III heat conduction model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2133-2143. doi: 10.3934/dcdsb.2014.19.2133 [11] Zhuangyi Liu, Ramón Quintanilla. Time decay in dual-phase-lag thermoelasticity: Critical case. Communications on Pure and Applied Analysis, 2018, 17 (1) : 177-190. doi: 10.3934/cpaa.2018011 [12] M. Carme Leseduarte, Ramon Quintanilla. Phragmén-Lindelöf alternative for an exact heat conduction equation with delay. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1221-1235. doi: 10.3934/cpaa.2013.12.1221 [13] Akram Ben Aissa. Well-posedness and direct internal stability of coupled non-degenrate Kirchhoff system via heat conduction. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 983-993. doi: 10.3934/dcdss.2021106 [14] Micol Amar, Daniele Andreucci, Paolo Bisegna, Roberto Gianni. Homogenization limit and asymptotic decay for electrical conduction in biological tissues in the high radiofrequency range. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1131-1160. doi: 10.3934/cpaa.2010.9.1131 [15] Mohammed Aassila. On energy decay rate for linear damped systems. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 851-864. doi: 10.3934/dcds.2002.8.851 [16] Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721 [17] Pavol Quittner. The decay of global solutions of a semilinear heat equation. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 307-318. doi: 10.3934/dcds.2008.21.307 [18] Xin Zhong. Singularity formation to the two-dimensional non-barotropic non-resistive magnetohydrodynamic equations with zero heat conduction in a bounded domain. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1083-1096. doi: 10.3934/dcdsb.2019209 [19] Zhiqiang Yang, Junzhi Cui, Qiang Ma. The second-order two-scale computation for integrated heat transfer problem with conduction, convection and radiation in periodic porous materials. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 827-848. doi: 10.3934/dcdsb.2014.19.827 [20] Jong-Shenq Guo, Bei Hu. Blowup rate estimates for the heat equation with a nonlinear gradient source term. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 927-937. doi: 10.3934/dcds.2008.20.927

2020 Impact Factor: 1.081