# American Institute of Mathematical Sciences

October  2015, 20(8): 2733-2750. doi: 10.3934/dcdsb.2015.20.2733

## Exponential-stability and super-stability of a thermoelastic system of type II with boundary damping

 1 Department of Mathematics, Tianjin University of Technology, Tianjin 300384, China 2 Department of Mathematics, Tianjin University, Tianjin 300072

Received  October 2014 Revised  December 2014 Published  August 2015

In this paper, the stability of a one-dimensional thermoelastic system with boundary damping is considered. The theory of thermoelasticity under consideration is developed by Green and Naghdi, which is named as thermoelasticity of type II''. This system consists of two strongly coupled wave equations. By the frequency domain method, we prove that the energy of this system generally decays to zero exponentially. Furthermore, by showing the spectrum of the system is empty under certain condition and estimating the norm of the resolvent operator, we give a sufficient condition on the super-stability of this thermoelastic system. Under this condition, the solution to the system is identical to zero after finite time. Moreover, we also estimate the maximum existence time of the nonzero part of the solution. Finally, we give some numerical simulations.
Citation: Lei Wang, Zhong-Jie Han, Gen-Qi Xu. Exponential-stability and super-stability of a thermoelastic system of type II with boundary damping. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2733-2750. doi: 10.3934/dcdsb.2015.20.2733
##### References:
 [1] M. Aouadi, Exponential stability in hyperbolic thermoelastic diffusion problem with second sound, International Journal of Differential Equations, 2011 (2011), Article ID 274843, 21pp. doi: 10.1155/2011/274843. [2] A. V. Balakrishnan, Smart structures and super-stability, in Computational Science for 21st Century, Wiley & Chichester, 1997, 660-669. [3] A. V. Balakrishnan, On superstable semigroup of operators, Dynamic Systems and Applications, 5 (1996), 371-383. [4] A. V. Balakrishnan, On superstability of semigroups, systems modelling and optimization, in Proceedings of the 18th IFIP Conference on System Modelling and Optimization (eds. M. P. Polis, et al.), CRC Research Notes in Mathematics, Chapman and Hall, 1999, 12-19. [5] A. V. Balakrishnan, Superstability of systems, Appl. Math. Comput., 164 (2005), 321-326. doi: 10.1016/j.amc.2004.06.052. [6] D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature, Appl. Mech. Rev., 51 (1998), 705-729. [7] D. S. Chandrasekharaiah, A note on the uniqueness of solution in the linear theory of thermoelasticity without energy dissipations, J. Elasticity, 43 (1996), 279-283. doi: 10.1080/01495739608946173. [8] D. S. Chandrasekharaiah, Complete solutions in the theory of thermoelasticity without energy dissipations, Mech. Res. Comm., 24 (1997), 625-630. doi: 10.1016/S0093-6413(97)00080-3. [9] C. M. Dafermos, On the existence and the asymptotic stability of solution to the equation of linear thermoelasticity, Arch. Rational Mech. Anal., 29 (1968), 241-271. doi: 10.1007/BF00276727. [10] A. Djebabla and N. Tatar, Exponential stabilization of the Timoshenko system by a thermo-viscoelastic damping, Journal of Dynamical and Control Systems, 16 (2010), 189-210. doi: 10.1007/s10883-010-9089-5. [11] L. H. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system, IMA Journal of Applied Mathematics, 75 (2010), 881-904. doi: 10.1093/imamat/hxq038. [12] H. D. Fernández Sare and J. E. Muñoz Rivera, Optimal rates of decay in 2-d thermoelasticity with second sound, J. Math. Phys., 53 (2012), Article ID 073509, 13pp. doi: 10.1063/1.4734239. [13] A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. R. Soc. Lond. Ser. A, 432 (1991), 171-194. doi: 10.1098/rspa.1991.0012. [14] A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, Journal of Elasticity, 31 (1993), 189-208. doi: 10.1007/BF00044969. [15] 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. [16] A. E. Green and P. M. Naghdi, A unified pocedure for contruction of theories of deformable media, I. Clasical continuum physics， Proc. R. Soc. Lond. Ser. A, 448 (1995), 335-356. doi: 10.1098/rspa.1995.0020. [17] L. M. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Transactions of the American Mathematical Society, 236 (1978), 385-394. doi: 10.1090/S0002-9947-1978-0461206-1. [18] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. [19] J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds, Oxford mathematical Monographs, Oxford University Press, New York, 2010. [20] S. Jiang and R. Racke, Evolution Equations in Thermoelasticity, Monographs and Surveys in Pure and Applied Mathematics, Vol. 112, Chapman and Hall/CRC, Boca Raton, 2000. [21] Z. Liu and R. Quintanilla, Energy decay rate of a mixed type II and type III thermoelastic system, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1433-1444. doi: 10.3934/dcdsb.2010.14.1433. [22] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, CRC Research Notes in Mathematics, Vol. 398, Chapman and Hall/CRC, Boca Raton, 1999. [23] B. Lazzari and R. Nibbi, On the exponential decay in thermoelasticity without energy dissipation and of type III in presence of an absorbing boundary, J. Math. Anal. Appl., 338 (2008), 317-329. doi: 10.1016/j.jmaa.2007.05.017. [24] M. C. Leseduarte, A. Magana and R. Quintanilla, On the time decay of solutions in porous-thermo-elasticity of type II, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 375-391. doi: 10.3934/dcdsb.2010.13.375. [25] I. Lasiecka and M. Wilke, Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system, Discrete Contin. Dyn. Sys., 33 (2013), 5189-5202. doi: 10.3934/dcds.2013.33.5189. [26] I. Lasiecka, S. Maad and A. Sasane, Existence and exponential decay of solutions to a quasilinear thermoelastic plate system, Nonlinear Differential Equations and Applications, 15 (2008), 689-715. doi: 10.1007/s00030-008-0011-8. [27] 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. [28] S. A. Messaoudi and A. Fareh, Energy decay in a Timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds, Arabian J. Math., 2 (2013), 199-207. doi: 10.1007/s40065-012-0061-y. [29] S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system with history in thermoelasticity of type III, Advanced in Differential Equations, 14 (2009), 375-400. [30] J. E. Muñoz Rivera, Energy decay rates in linear thermoelasticty, Funkcialaj Ekvacioj, 35 (1992), 19-30. [31] J. E. Muñoz Rivera, Asymptotic behaviour in inhomogeneous linear thermoelasticity, Applicable Analysis, 53 (1994), 55-65. doi: 10.1080/00036819408840243. [32] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1. [33] J. Prüss, On the spectrum of $C_0$-semigroups, Transactions of the American Mathematical Society, 284 (1984), 847-857. doi: 10.2307/1999112. [34] R. Quintanilla, Instability and non-existence in the nonlinear theory of thermoelasticity without energy dissipation, Continuum Mech. Thermodyn, 13 (2001), 121-129. doi: 10.1007/s001610100044. [35] R. Quintanilla and R. Racke, Stability in thermoelasticity of type III, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 383-400. doi: 10.3934/dcdsb.2003.3.383. [36] Y. F. Shang, D. Y. Liu and G. Q. Xu, Super-stability and the spectrum of one-dimensional wave equations on general feedback controlled networks, IMA Journal of Mathematical Control and Information, 31 (2014), 73-99. doi: 10.1093/imamci/dnt003. [37] L. N. Trefethen, Spectral Methods in Matlab, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598. [38] F. E. Udwadia, Boundary control, quiet boundaries, super-stability and super-instability, Appl. Math. Comput., 164 (2005), 327-349. doi: 10.1016/j.amc.2004.06.040. [39] F. E. Udwadia, On the longitudinal vibrations of a bar with viscous boundaries: Super-stability, superinstability, and loss of damping, Internat. J. Eng. Sci., 50 (2012), 79-100. doi: 10.1016/j.ijengsci.2011.09.001. [40] J. M. Wang and B. Z. Guo, On dynamic behavior of a hyperbolic system derived from a thermoelastic equation with memory type, Journal of the Franklin Institute-Engineering and Applied Mathematics, 344 (2007), 75-96. doi: 10.1016/j.jfranklin.2005.10.003. [41] G. Q. Xu, Stabilization of string system with linear boundary feedback, Nonlinear Anal.: Hybrid Syst., 1 (2007), 383-397. doi: 10.1016/j.nahs.2006.07.003. [42] X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III, Communications in Contemporary Mathematics, 5 (2003), 25-83. doi: 10.1142/S0219199703000896. [43] Y. X. Zhang and G. Q. Xu, Exponential and super stability of a wave network, Acta Appl. Math., 124 (2013), 19-41. doi: 10.1007/s10440-012-9768-1.

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##### References:
 [1] M. Aouadi, Exponential stability in hyperbolic thermoelastic diffusion problem with second sound, International Journal of Differential Equations, 2011 (2011), Article ID 274843, 21pp. doi: 10.1155/2011/274843. [2] A. V. Balakrishnan, Smart structures and super-stability, in Computational Science for 21st Century, Wiley & Chichester, 1997, 660-669. [3] A. V. Balakrishnan, On superstable semigroup of operators, Dynamic Systems and Applications, 5 (1996), 371-383. [4] A. V. Balakrishnan, On superstability of semigroups, systems modelling and optimization, in Proceedings of the 18th IFIP Conference on System Modelling and Optimization (eds. M. P. Polis, et al.), CRC Research Notes in Mathematics, Chapman and Hall, 1999, 12-19. [5] A. V. Balakrishnan, Superstability of systems, Appl. Math. Comput., 164 (2005), 321-326. doi: 10.1016/j.amc.2004.06.052. [6] D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature, Appl. Mech. Rev., 51 (1998), 705-729. [7] D. S. Chandrasekharaiah, A note on the uniqueness of solution in the linear theory of thermoelasticity without energy dissipations, J. Elasticity, 43 (1996), 279-283. doi: 10.1080/01495739608946173. [8] D. S. Chandrasekharaiah, Complete solutions in the theory of thermoelasticity without energy dissipations, Mech. Res. Comm., 24 (1997), 625-630. doi: 10.1016/S0093-6413(97)00080-3. [9] C. M. Dafermos, On the existence and the asymptotic stability of solution to the equation of linear thermoelasticity, Arch. Rational Mech. Anal., 29 (1968), 241-271. doi: 10.1007/BF00276727. [10] A. Djebabla and N. Tatar, Exponential stabilization of the Timoshenko system by a thermo-viscoelastic damping, Journal of Dynamical and Control Systems, 16 (2010), 189-210. doi: 10.1007/s10883-010-9089-5. [11] L. H. Fatori and J. E. Muñoz Rivera, Rates of decay to weak thermoelastic Bresse system, IMA Journal of Applied Mathematics, 75 (2010), 881-904. doi: 10.1093/imamat/hxq038. [12] H. D. Fernández Sare and J. E. Muñoz Rivera, Optimal rates of decay in 2-d thermoelasticity with second sound, J. Math. Phys., 53 (2012), Article ID 073509, 13pp. doi: 10.1063/1.4734239. [13] A. E. Green and P. M. Naghdi, A re-examination of the basic postulates of thermomechanics, Proc. R. Soc. Lond. Ser. A, 432 (1991), 171-194. doi: 10.1098/rspa.1991.0012. [14] A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, Journal of Elasticity, 31 (1993), 189-208. doi: 10.1007/BF00044969. [15] 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. [16] A. E. Green and P. M. Naghdi, A unified pocedure for contruction of theories of deformable media, I. Clasical continuum physics， Proc. R. Soc. Lond. Ser. A, 448 (1995), 335-356. doi: 10.1098/rspa.1995.0020. [17] L. M. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Transactions of the American Mathematical Society, 236 (1978), 385-394. doi: 10.1090/S0002-9947-1978-0461206-1. [18] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. [19] J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds, Oxford mathematical Monographs, Oxford University Press, New York, 2010. [20] S. Jiang and R. Racke, Evolution Equations in Thermoelasticity, Monographs and Surveys in Pure and Applied Mathematics, Vol. 112, Chapman and Hall/CRC, Boca Raton, 2000. [21] Z. Liu and R. Quintanilla, Energy decay rate of a mixed type II and type III thermoelastic system, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1433-1444. doi: 10.3934/dcdsb.2010.14.1433. [22] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, CRC Research Notes in Mathematics, Vol. 398, Chapman and Hall/CRC, Boca Raton, 1999. [23] B. Lazzari and R. Nibbi, On the exponential decay in thermoelasticity without energy dissipation and of type III in presence of an absorbing boundary, J. Math. Anal. Appl., 338 (2008), 317-329. doi: 10.1016/j.jmaa.2007.05.017. [24] M. C. Leseduarte, A. Magana and R. Quintanilla, On the time decay of solutions in porous-thermo-elasticity of type II, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 375-391. doi: 10.3934/dcdsb.2010.13.375. [25] I. Lasiecka and M. Wilke, Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system, Discrete Contin. Dyn. Sys., 33 (2013), 5189-5202. doi: 10.3934/dcds.2013.33.5189. [26] I. Lasiecka, S. Maad and A. Sasane, Existence and exponential decay of solutions to a quasilinear thermoelastic plate system, Nonlinear Differential Equations and Applications, 15 (2008), 689-715. doi: 10.1007/s00030-008-0011-8. [27] 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. [28] S. A. Messaoudi and A. Fareh, Energy decay in a Timoshenko-type system of thermoelasticity of type III with different wave-propagation speeds, Arabian J. Math., 2 (2013), 199-207. doi: 10.1007/s40065-012-0061-y. [29] S. A. Messaoudi and B. Said-Houari, Energy decay in a Timoshenko-type system with history in thermoelasticity of type III, Advanced in Differential Equations, 14 (2009), 375-400. [30] J. E. Muñoz Rivera, Energy decay rates in linear thermoelasticty, Funkcialaj Ekvacioj, 35 (1992), 19-30. [31] J. E. Muñoz Rivera, Asymptotic behaviour in inhomogeneous linear thermoelasticity, Applicable Analysis, 53 (1994), 55-65. doi: 10.1080/00036819408840243. [32] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1. [33] J. Prüss, On the spectrum of $C_0$-semigroups, Transactions of the American Mathematical Society, 284 (1984), 847-857. doi: 10.2307/1999112. [34] R. Quintanilla, Instability and non-existence in the nonlinear theory of thermoelasticity without energy dissipation, Continuum Mech. Thermodyn, 13 (2001), 121-129. doi: 10.1007/s001610100044. [35] R. Quintanilla and R. Racke, Stability in thermoelasticity of type III, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 383-400. doi: 10.3934/dcdsb.2003.3.383. [36] Y. F. Shang, D. Y. Liu and G. Q. Xu, Super-stability and the spectrum of one-dimensional wave equations on general feedback controlled networks, IMA Journal of Mathematical Control and Information, 31 (2014), 73-99. doi: 10.1093/imamci/dnt003. [37] L. N. Trefethen, Spectral Methods in Matlab, SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598. [38] F. E. Udwadia, Boundary control, quiet boundaries, super-stability and super-instability, Appl. Math. Comput., 164 (2005), 327-349. doi: 10.1016/j.amc.2004.06.040. [39] F. E. Udwadia, On the longitudinal vibrations of a bar with viscous boundaries: Super-stability, superinstability, and loss of damping, Internat. J. Eng. Sci., 50 (2012), 79-100. doi: 10.1016/j.ijengsci.2011.09.001. [40] J. M. Wang and B. Z. Guo, On dynamic behavior of a hyperbolic system derived from a thermoelastic equation with memory type, Journal of the Franklin Institute-Engineering and Applied Mathematics, 344 (2007), 75-96. doi: 10.1016/j.jfranklin.2005.10.003. [41] G. Q. Xu, Stabilization of string system with linear boundary feedback, Nonlinear Anal.: Hybrid Syst., 1 (2007), 383-397. doi: 10.1016/j.nahs.2006.07.003. [42] X. Zhang and E. Zuazua, Decay of solutions of the system of thermoelasticity of type III, Communications in Contemporary Mathematics, 5 (2003), 25-83. doi: 10.1142/S0219199703000896. [43] Y. X. Zhang and G. Q. Xu, Exponential and super stability of a wave network, Acta Appl. Math., 124 (2013), 19-41. doi: 10.1007/s10440-012-9768-1.
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