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 & 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. doi: 10.1080/00036811.2012.702341. Google Scholar

[2]

M. Aouadi, On uniform decay of a nonsimple thermoelastic bar with memory,, J. Math. Anal. Appl., 402 (2013), 745. doi: 10.1016/j.jmaa.2013.01.059. Google Scholar

[3]

M. Ciarletta and D. Ieşan, Non-classical Elastic Solids,, Pitman Research Notes in Mathematical Series, (1993). Google Scholar

[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. doi: 10.1016/j.ijengsci.2010.04.014. Google Scholar

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J. A. Gawinecki and J. Lazuka, Global solution on Cauchy problem in nonlinear non-simple thermoelastic materials,, Proc. Appl. Math. Mech., 6 (2006), 371. doi: 10.1002/pamm.200610167. Google Scholar

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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. Google Scholar

[7]

A. E. Green and R. S. Rivlin, Multipolar continuum mechanics,, Arch. Ration. Mech. Anal., 17 (1964), 113. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/S0898-1221(99)00303-X. Google Scholar

[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. doi: 10.1006/jmaa.1997.5544. Google Scholar

[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. doi: 10.1016/j.jfa.2005.02.006. Google Scholar

[12]

S. W. Hansen, Exponential energy decay in a linear thermoelastic rod,, J. Math. Anal. Appl., 167 (1992), 429. doi: 10.1016/0022-247X(92)90217-2. Google Scholar

[13]

S. W. Hansen and B.-Y. Zhang, Boundary control of a linear thermoelastic beam,, J. Math. Anal. Appl., 210 (1997), 182. doi: 10.1006/jmaa.1997.5437. Google Scholar

[14]

D. B. Henry, A. Perssinitto and O. Lopes, On the essential spectrum of a semigroup of thermoelasticity,, Nonlinear Anal. TMA, 21 (1993), 65. doi: 10.1016/0362-546X(93)90178-U. Google Scholar

[15]

D. Ieşan, Thermoelastic Models of Continua,, Kluwer Academic Publishers, (2004). doi: 10.1007/978-1-4020-2310-1. Google Scholar

[16]

H. Kolakowski and J. Lazuka, The Cauchy problem for the system of partial differential equations describing nonsimple thermoelasticity,, Appl. Math., 35 (2008), 97. doi: 10.4064/am35-1-6. Google Scholar

[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. doi: 10.1093/imamci/20.4.393. Google Scholar

[18]

Z. Y. Liu and S. M. Zheng, Exponential stability of semigroup associated with thermoelastic system,, Quart. Appl. Math., 51 (1993), 535. Google Scholar

[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. doi: 10.1137/S0363012991219006. Google Scholar

[20]

Yu. I. Lyubich and V. Q. Phóong, Asymptotic stability of linear differential equations in Banach spaces,, Studia Math., 88 (1988), 37. Google Scholar

[21]

V. Pata and R. Quintanilla, On the decay of solutions in nonsimple elastic solids with memory,, J. Math. Anal. Appl., 363 (2010), 19. doi: 10.1016/j.jmaa.2009.07.055. Google Scholar

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences 44, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[23]

R. Quintanilla, Thermoelasticity without energy dissipation of nonsimple materials,, Z. Angew. Math. Mech., 83 (2003), 172. doi: 10.1002/zamm.200310017. Google Scholar

[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., 136 (2006), 189. doi: 10.1017/S0308210500004510. Google Scholar

[25]

R. M. Young, An Introduction to Nonharmonic Fourier Series,, Academic Press, (1980). Google Scholar

show all references

References:
[1]

M. Aouadi, Stability aspects in a nonsimple thermoelastic diffusion problem,, Appl. Anal., 92 (2013), 1816. doi: 10.1080/00036811.2012.702341. Google Scholar

[2]

M. Aouadi, On uniform decay of a nonsimple thermoelastic bar with memory,, J. Math. Anal. Appl., 402 (2013), 745. doi: 10.1016/j.jmaa.2013.01.059. Google Scholar

[3]

M. Ciarletta and D. Ieşan, Non-classical Elastic Solids,, Pitman Research Notes in Mathematical Series, (1993). Google Scholar

[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. doi: 10.1016/j.ijengsci.2010.04.014. Google Scholar

[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. doi: 10.1002/pamm.200610167. Google Scholar

[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. Google Scholar

[7]

A. E. Green and R. S. Rivlin, Multipolar continuum mechanics,, Arch. Ration. Mech. Anal., 17 (1964), 113. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/S0898-1221(99)00303-X. Google Scholar

[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. doi: 10.1006/jmaa.1997.5544. Google Scholar

[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. doi: 10.1016/j.jfa.2005.02.006. Google Scholar

[12]

S. W. Hansen, Exponential energy decay in a linear thermoelastic rod,, J. Math. Anal. Appl., 167 (1992), 429. doi: 10.1016/0022-247X(92)90217-2. Google Scholar

[13]

S. W. Hansen and B.-Y. Zhang, Boundary control of a linear thermoelastic beam,, J. Math. Anal. Appl., 210 (1997), 182. doi: 10.1006/jmaa.1997.5437. Google Scholar

[14]

D. B. Henry, A. Perssinitto and O. Lopes, On the essential spectrum of a semigroup of thermoelasticity,, Nonlinear Anal. TMA, 21 (1993), 65. doi: 10.1016/0362-546X(93)90178-U. Google Scholar

[15]

D. Ieşan, Thermoelastic Models of Continua,, Kluwer Academic Publishers, (2004). doi: 10.1007/978-1-4020-2310-1. Google Scholar

[16]

H. Kolakowski and J. Lazuka, The Cauchy problem for the system of partial differential equations describing nonsimple thermoelasticity,, Appl. Math., 35 (2008), 97. doi: 10.4064/am35-1-6. Google Scholar

[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. doi: 10.1093/imamci/20.4.393. Google Scholar

[18]

Z. Y. Liu and S. M. Zheng, Exponential stability of semigroup associated with thermoelastic system,, Quart. Appl. Math., 51 (1993), 535. Google Scholar

[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. doi: 10.1137/S0363012991219006. Google Scholar

[20]

Yu. I. Lyubich and V. Q. Phóong, Asymptotic stability of linear differential equations in Banach spaces,, Studia Math., 88 (1988), 37. Google Scholar

[21]

V. Pata and R. Quintanilla, On the decay of solutions in nonsimple elastic solids with memory,, J. Math. Anal. Appl., 363 (2010), 19. doi: 10.1016/j.jmaa.2009.07.055. Google Scholar

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences 44, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[23]

R. Quintanilla, Thermoelasticity without energy dissipation of nonsimple materials,, Z. Angew. Math. Mech., 83 (2003), 172. doi: 10.1002/zamm.200310017. Google Scholar

[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., 136 (2006), 189. doi: 10.1017/S0308210500004510. Google Scholar

[25]

R. M. Young, An Introduction to Nonharmonic Fourier Series,, Academic Press, (1980). Google Scholar

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