# American Institute of Mathematical Sciences

October  2016, 9(5): 1475-1492. doi: 10.3934/dcdss.2016059

## Asymptotic analysis of a nonsimple thermoelastic rod

 1 Ecole Nationale d'Ingénieurs de Bizerte, Université de Carthage, BP66, Campus Universitaire Menzel Abderrahman 7035 2 Faculté des Sciences de Bizerte, 7021 Zarzouna, Université de Carthage

Received  October 2014 Revised  July 2015 Published  October 2016

The asymptotic analysis of a one-dimensional nonsimple thermoelastic problem is considered in this paper. By a detailed spectral analysis, the asymptotic expressions for eigenvalues and eigenfunctions of the considered system are developed. It is shown that the eigenfunctions form a Riesz basis on the Hilbert space and the eigenvalues asymptotically fall on two branches. One branch is along the negative horizontal axis in the complex plane and the other branch is asymptotic to a vertical line that is parallel to the imaginary axis. This gives the spectrum-determined growth condition for the $C_0-$semigroup associated to the system, and consequently, the asymptotic and the exponential stability of the solutions are deduced. The approach developed in this paper confirms the already-existing results; furthermore, it can be extended to a larger field of applications such as coupled system of rod or beam with diffusion equation. The method will be illustrated by an example of thermoelastic beam equations with Dirichlet boundary conditions.
Citation: Moncef Aouadi, Taoufik Moulahi. Asymptotic analysis of a nonsimple thermoelastic rod. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1475-1492. doi: 10.3934/dcdss.2016059
##### References:
 [1] M. Aouadi, Stability aspects in a nonsimple thermoelastic diffusion problem, Appl. Anal., 92 (2013), 1816-1828. doi: 10.1080/00036811.2012.702341. [2] M. Aouadi, On uniform decay of a nonsimple thermoelastic bar with memory, J. Math. Anal. Appl., 402 (2013), 745-757. doi: 10.1016/j.jmaa.2013.01.059. [3] M. Ciarletta and D. Ieşan, Non-classical Elastic Solids, Pitman Research Notes in Mathematical Series, vol. 293, John Wiley & Sons, Inc., New York, 1993. [4] H. D. Fernàndez Sare, J. E. Munoz Rivera and R. Quintanilla, Decay of solutions in nonsimple thermoelastic bars, Int. J. Eng. Sci., 48 (2010), 1233-1241. doi: 10.1016/j.ijengsci.2010.04.014. [5] J. A. Gawinecki and J. Lazuka, Global solution on Cauchy problem in nonlinear non-simple thermoelastic materials, Proc. Appl. Math. Mech., 6 (2006), 371-372. doi: 10.1002/pamm.200610167. [6] M. Grasselli and M. Squassina, Exponential stability and singular limit for a linear thermoelastic plate with memory effects, Adv. Math. Sci. Appl., 16 (2006), 15-31. [7] A. E. Green and R. S. Rivlin, Multipolar continuum mechanics, Arch. Ration. Mech. Anal., 17 (1964), 113-147. [8] B. Z. Guo, Further result on a one-dimensional linear thermoelastic equation with Dirichlet-Dirichlet boundary condition, Aust. N. Z. Ind. Appl. Math., 43 (2002), 449-462. [9] B. Z. Guo and J. C. Chen, The first real eigenvalue of a one-dimensional linear thermoelastic system, J. Comput. Math. Appl., 38 (1999), 249-256. doi: 10.1016/S0898-1221(99)00303-X. [10] B. Z. Guo and S. P. Yung, Asymptotic behavior of the eigenfrequency of a one-dimensional linear thermoelastic system, J. Math. Anal. Appl., 213 (1997), 406-421. doi: 10.1006/jmaa.1997.5544. [11] B. Z. Guo and G. Q. Xu, Expansion of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition, J. Funct. Anal., 231 (2006), 245-268. doi: 10.1016/j.jfa.2005.02.006. [12] S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl., 167 (1992), 429-442. doi: 10.1016/0022-247X(92)90217-2. [13] S. W. Hansen and B.-Y. Zhang, Boundary control of a linear thermoelastic beam, J. Math. Anal. Appl., 210 (1997), 182-205. doi: 10.1006/jmaa.1997.5437. [14] D. B. Henry, A. Perssinitto and O. Lopes, On the essential spectrum of a semigroup of thermoelasticity, Nonlinear Anal. TMA, 21 (1993), 65-75. doi: 10.1016/0362-546X(93)90178-U. [15] D. Ieşan, Thermoelastic Models of Continua, Kluwer Academic Publishers, Dordrecht, 2004. doi: 10.1007/978-1-4020-2310-1. [16] H. Kolakowski and J. Lazuka, The Cauchy problem for the system of partial differential equations describing nonsimple thermoelasticity, Appl. Math., 35 (2008), 97-105. doi: 10.4064/am35-1-6. [17] H. Leiva, A necessary and sufficient algebraic condition for the controllability of a thermoelastic plate equation, IMA J. Math. Control and Information, 20 (2003), 393-410. doi: 10.1093/imamci/20.4.393. [18] Z. Y. Liu and S. M. Zheng, Exponential stability of semigroup associated with thermoelastic system, Quart. Appl. Math., 51 (1993), 535-545. [19] Z. Y. Liu and S. M. Zheng, Uniform exponential stability and approximation in control of a thermoelastic system, SIAM J. Control & Optim., 32 (1994), 1226-1246. doi: 10.1137/S0363012991219006. [20] Yu. I. Lyubich and V. Q. Phóong, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42. [21] V. Pata and R. Quintanilla, On the decay of solutions in nonsimple elastic solids with memory, J. Math. Anal. Appl., 363 (2010), 19-28. doi: 10.1016/j.jmaa.2009.07.055. [22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [23] R. Quintanilla, Thermoelasticity without energy dissipation of nonsimple materials, Z. Angew. Math. Mech., 83 (2003), 172-180. doi: 10.1002/zamm.200310017. [24] 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. Royal Soc. Edinb., Section A Mathematics, 136 (2006), 189-207. doi: 10.1017/S0308210500004510. [25] R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 1980.

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##### References:
 [1] M. Aouadi, Stability aspects in a nonsimple thermoelastic diffusion problem, Appl. Anal., 92 (2013), 1816-1828. doi: 10.1080/00036811.2012.702341. [2] M. Aouadi, On uniform decay of a nonsimple thermoelastic bar with memory, J. Math. Anal. Appl., 402 (2013), 745-757. doi: 10.1016/j.jmaa.2013.01.059. [3] M. Ciarletta and D. Ieşan, Non-classical Elastic Solids, Pitman Research Notes in Mathematical Series, vol. 293, John Wiley & Sons, Inc., New York, 1993. [4] H. D. Fernàndez Sare, J. E. Munoz Rivera and R. Quintanilla, Decay of solutions in nonsimple thermoelastic bars, Int. J. Eng. Sci., 48 (2010), 1233-1241. doi: 10.1016/j.ijengsci.2010.04.014. [5] J. A. Gawinecki and J. Lazuka, Global solution on Cauchy problem in nonlinear non-simple thermoelastic materials, Proc. Appl. Math. Mech., 6 (2006), 371-372. doi: 10.1002/pamm.200610167. [6] M. Grasselli and M. Squassina, Exponential stability and singular limit for a linear thermoelastic plate with memory effects, Adv. Math. Sci. Appl., 16 (2006), 15-31. [7] A. E. Green and R. S. Rivlin, Multipolar continuum mechanics, Arch. Ration. Mech. Anal., 17 (1964), 113-147. [8] B. Z. Guo, Further result on a one-dimensional linear thermoelastic equation with Dirichlet-Dirichlet boundary condition, Aust. N. Z. Ind. Appl. Math., 43 (2002), 449-462. [9] B. Z. Guo and J. C. Chen, The first real eigenvalue of a one-dimensional linear thermoelastic system, J. Comput. Math. Appl., 38 (1999), 249-256. doi: 10.1016/S0898-1221(99)00303-X. [10] B. Z. Guo and S. P. Yung, Asymptotic behavior of the eigenfrequency of a one-dimensional linear thermoelastic system, J. Math. Anal. Appl., 213 (1997), 406-421. doi: 10.1006/jmaa.1997.5544. [11] B. Z. Guo and G. Q. Xu, Expansion of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition, J. Funct. Anal., 231 (2006), 245-268. doi: 10.1016/j.jfa.2005.02.006. [12] S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl., 167 (1992), 429-442. doi: 10.1016/0022-247X(92)90217-2. [13] S. W. Hansen and B.-Y. Zhang, Boundary control of a linear thermoelastic beam, J. Math. Anal. Appl., 210 (1997), 182-205. doi: 10.1006/jmaa.1997.5437. [14] D. B. Henry, A. Perssinitto and O. Lopes, On the essential spectrum of a semigroup of thermoelasticity, Nonlinear Anal. TMA, 21 (1993), 65-75. doi: 10.1016/0362-546X(93)90178-U. [15] D. Ieşan, Thermoelastic Models of Continua, Kluwer Academic Publishers, Dordrecht, 2004. doi: 10.1007/978-1-4020-2310-1. [16] H. Kolakowski and J. Lazuka, The Cauchy problem for the system of partial differential equations describing nonsimple thermoelasticity, Appl. Math., 35 (2008), 97-105. doi: 10.4064/am35-1-6. [17] H. Leiva, A necessary and sufficient algebraic condition for the controllability of a thermoelastic plate equation, IMA J. Math. Control and Information, 20 (2003), 393-410. doi: 10.1093/imamci/20.4.393. [18] Z. Y. Liu and S. M. Zheng, Exponential stability of semigroup associated with thermoelastic system, Quart. Appl. Math., 51 (1993), 535-545. [19] Z. Y. Liu and S. M. Zheng, Uniform exponential stability and approximation in control of a thermoelastic system, SIAM J. Control & Optim., 32 (1994), 1226-1246. doi: 10.1137/S0363012991219006. [20] Yu. I. Lyubich and V. Q. Phóong, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42. [21] V. Pata and R. Quintanilla, On the decay of solutions in nonsimple elastic solids with memory, J. Math. Anal. Appl., 363 (2010), 19-28. doi: 10.1016/j.jmaa.2009.07.055. [22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [23] R. Quintanilla, Thermoelasticity without energy dissipation of nonsimple materials, Z. Angew. Math. Mech., 83 (2003), 172-180. doi: 10.1002/zamm.200310017. [24] 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. Royal Soc. Edinb., Section A Mathematics, 136 (2006), 189-207. doi: 10.1017/S0308210500004510. [25] R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 1980.
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