On the time decay in phase–lag thermoelasticity with two temperatures

Antonio Magaña; Alain Miranville; Ramón Quintanilla

doi:10.3934/era.2019007

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2019, 27: 7-19. doi: 10.3934/era.2019007

On the time decay in phase–lag thermoelasticity with two temperatures

Antonio Magaña ^1, , Alain Miranville ^2,, and Ramón Quintanilla ^1,

1.	Departament de Matemàtiques, Universitat Politècnica de Catalunya, 08222 Terrassa, Barcelona, Spain
2.	Laboratoire de Mathématiques et Applications, Université de Poitiers, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France

^* Corresponding author: Alain Miranville

Received June 2019 Revised November 2019 Published September 2019

Fund Project: The first and the third authors are supported by the project "Análisis Matemático de Problemas de la Termomecánica" (MTM2016-74934-P), (AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness.

The aim of this paper is to study the time decay of the solutions for two models of the one-dimensional phase-lag thermoelasticity with two temperatures. The first one is obtained when the heat flux vector and the inductive temperature are approximated by a second-order and first-order Taylor polynomial, respectively. In this case, the solutions decay in a slow way. The second model that we consider is obtained taking first-order Taylor approximations for the inductive thermal displacement, the inductive temperature and the heat flux. The decay is, therefore, of exponential type.

Keywords: Phase-lag thermoelasticity, exponential stability, slow decay, spectral analysis.

Mathematics Subject Classification: Primary: 74F05, 74H40; Secondary: 74H20, 34B35, 35P20.

Citation: 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

References:

[1]	I. A. Abdallah, Dual phase lag heat conduction and thermoelastic properties of a semi-infinite medium induced by ultrashort pulsed laser, Progress in Physics, 3 (2009), 60-63. Google Scholar
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[4]	P. J. Chen and M. E. Gurtin, On a theory of heat involving two temperatures, J. Applied Mathematics and Physics (ZAMP), 19 (1968), 614-627. Google Scholar
[5]	P. J. Chen, M. E. Gurtin and W. O. Williams, A note on non-simple heat conduction, J. Applied Mathematics and Physics (ZAMP), 19 (1968), 969-970. doi: 10.1007/BF01602278. Google Scholar
[6]	P. J. Chen, M. E. Gurtin and W. O. Williams, On the thermodynamics of non-simple materials with two temperatures, J. Applied Mathematics and Physics (ZAMP), 20 (1969), 107-112. Google Scholar
[7]	S. K. R. Choudhuri, On a thermoelastic three-phase-lag model, J. Thermal Stresses, 30 (2007), 231-238. doi: 10.1080/01495730601130919. Google Scholar
[8]	M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics, Applied Mathematics Letters, 22 (2009), 1374-1379. doi: 10.1016/j.aml.2009.03.010. Google Scholar
[9]	M. A. Ezzat, A. S. El-Karamany and S. M. Ezzat, Two-temperature theory in magneto-thermoelasticity with fractional order dual-phase-lag heat transfer, Nuclear Engineering and Design, 252 (2012), 267-277. doi: 10.1016/j.nucengdes.2012.06.012. Google Scholar
[10]	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. Google Scholar
[11]	A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189–208. doi: 10.1007/BF00044969. Google Scholar
[12]	M. A. Hader, M. A. Al-Nimr and B. A. Abu Nabah, The Dual-Phase-Lag heat conduction model in thin slabs under a fluctuating volumetric thermal disturbance, Int. J. Thermophysics, 23 (2002), 1669-1680. Google Scholar
[13]	F. L. Huang, Strong asymptotic stability of linear dynamical systems in Banach spaces, J. Differential Equations, 104 (1993), 307-324. doi: 10.1006/jdeq.1993.1074. Google Scholar
[14]	R. Quintanilla and P. M. Jordan, A note on the two-temperature theory with dual-phase-lag decay: Some exact solutions, Mechanics Research Communications, 36 (2009), 796-803. doi: 10.1016/j.mechrescom.2009.05.002. Google Scholar
[15]	M. C. Leseduarte, R. Quintanilla and R. Racke, On (non-)exponential decay in generalized thermoelasticity with two temperatures, Applied Mathematics Letters, 70 (2017), 18-25. doi: 10.1016/j.aml.2017.02.020. Google Scholar
[16]	Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Research Notes in Mathematics, vol. 398, Chapman & Hall/CRC, Boca Raton, FL, 1999. Google Scholar
[17]	A. Magaña, A. Miranville and R. Quintanilla, On the stability in phase-lag heat conduction with two temperatures, J. of Evolution Equations, 18 (2018), 1697-1712. doi: 10.1007/s00028-018-0457-z. Google Scholar
[18]	J. E. Marsden and T. J. R. Hughes, Topics in the mathematical foundations of elasticity, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. II, 30-285, Res. Notes in Math., 27, Pitman, Boston, Mass.-London, 1978. Google Scholar
[19]	A. Miranville and R. Quintanilla, A phase-field model based on a three-phase-lag heat conduction, Applied Mathematics and Optimization, 63 (2011), 133-150. doi: 10.1007/s00245-010-9114-9. Google Scholar
[20]	S. Mukhopadhyay, R Prasad and R. Kumar, On the theory of Two-Temperature Thermoelaticity with Two Phase-Lags, J. Thermal Stresses, 34 (2011), 352-365. Google Scholar
[21]	M. A. Othman, W. M. Hasona and E. M. Abd-Elaziz, Effect of rotation on micropolar generalized thermoelasticity with two temperatures using a dual-phase-lag model, Canadian J. Physics, 92 (2014), 149-158. doi: 10.1139/cjp-2013-0398. Google Scholar
[22]	J. Prüss, On the spectrum of C₀-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112. Google Scholar
[23]	R. Quintanilla, Exponential stability in the dual-phase-lag heat conduction theory, J. Non-Equilibrium Thermodynamics, 27 (2002), 217-227. doi: 10.1515/JNETDY.2002.012. Google Scholar
[24]	R. Quintanilla, A well-posed problem for the Dual-Phase-Lag heat conduction, J. Thermal Stresses, 31 (2008), 260-269. Google Scholar
[25]	R. Quintanilla, A well-posed problem for the three-dual-phase-lag heat conduction, J. Thermal Stresses, 32 (2009), 1270-1278. doi: 10.1080/01495730903310599. Google Scholar
[26]	R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag thermoelasticity, SIAM J. Appl. Math., 66 (2006), 977-1001. doi: 10.1137/05062860X. Google Scholar
[27]	R. Quintanilla and R. Racke, A note on stability of dual-phase-lag heat conduction, Int. J. Heat Mass Transfer, 49 (2006), 1209-1213. doi: 10.1016/j.ijheatmasstransfer.2005.10.016. Google Scholar
[28]	R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag heat conduction, Proc. Royal Society London A, 463 (2007), 659-674. doi: 10.1098/rspa.2006.1784. Google Scholar
[29]	R. Quintanilla and R. Racke, A note on stability in three-phase-lag heat conduction, Int. J. Heat Mass Transfer, 51 (2008), 24-29. Google Scholar
[30]	R. Quintanilla and R. Racke, Spatial behavior in phase-lag heat conduction, Differential and Integral Equations, 28 (2015), 291-308. Google Scholar
[31]	S. A. Rukolaine, Unphysical effects of the dual-phase-lag model of heat conduction, Int. J. Heat and Mass Transfer, 78 (2014), 58-63. doi: 10.1016/j.ijheatmasstransfer.2014.06.066. Google Scholar
[32]	D. Y. Tzou, A unified approach for heat conduction from macro to micro-scales, ASME J. Heat Transfer, 117 (1995), 8-16. doi: 10.1115/1.2822329. Google Scholar
[33]	W. E. Warren and P. J. Chen, Wave propagation in two temperatures theory of thermoelaticity, Acta Mechanica, 16 (1973), 83-117. Google Scholar
[34]	Y. Zhang, Generalized dual-phase lag bioheat equations based on nonequilibrium heat transfer in living biological tissues, Int. J. of Heat and Mass Transfer, 52 (2009), 4829-4834. doi: 10.1016/j.ijheatmasstransfer.2009.06.007. Google Scholar

show all references

References:

[1]	I. A. Abdallah, Dual phase lag heat conduction and thermoelastic properties of a semi-infinite medium induced by ultrashort pulsed laser, Progress in Physics, 3 (2009), 60-63. Google Scholar
[2]	S. Banik and M. Kanoria, Effects of three-phase-lag on two temperatures generalized thermoelasticity for an infinite medium with a spherical cavity, Applied Mathematics and Mechanics, 33 (2012), 483-498. doi: 10.1007/s10483-012-1565-8. Google Scholar
[3]	K. Borgmeyer, R. Quintanilla and R. Racke, Phase-lag heat conduction: Decay rates for limit problems and well-posedness, J. Evolution Equations, 14 (2014), 863-884. doi: 10.1007/s00028-014-0242-6. Google Scholar
[4]	P. J. Chen and M. E. Gurtin, On a theory of heat involving two temperatures, J. Applied Mathematics and Physics (ZAMP), 19 (1968), 614-627. Google Scholar
[5]	P. J. Chen, M. E. Gurtin and W. O. Williams, A note on non-simple heat conduction, J. Applied Mathematics and Physics (ZAMP), 19 (1968), 969-970. doi: 10.1007/BF01602278. Google Scholar
[6]	P. J. Chen, M. E. Gurtin and W. O. Williams, On the thermodynamics of non-simple materials with two temperatures, J. Applied Mathematics and Physics (ZAMP), 20 (1969), 107-112. Google Scholar
[7]	S. K. R. Choudhuri, On a thermoelastic three-phase-lag model, J. Thermal Stresses, 30 (2007), 231-238. doi: 10.1080/01495730601130919. Google Scholar
[8]	M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics, Applied Mathematics Letters, 22 (2009), 1374-1379. doi: 10.1016/j.aml.2009.03.010. Google Scholar
[9]	M. A. Ezzat, A. S. El-Karamany and S. M. Ezzat, Two-temperature theory in magneto-thermoelasticity with fractional order dual-phase-lag heat transfer, Nuclear Engineering and Design, 252 (2012), 267-277. doi: 10.1016/j.nucengdes.2012.06.012. Google Scholar
[10]	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. Google Scholar
[11]	A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189–208. doi: 10.1007/BF00044969. Google Scholar
[12]	M. A. Hader, M. A. Al-Nimr and B. A. Abu Nabah, The Dual-Phase-Lag heat conduction model in thin slabs under a fluctuating volumetric thermal disturbance, Int. J. Thermophysics, 23 (2002), 1669-1680. Google Scholar
[13]	F. L. Huang, Strong asymptotic stability of linear dynamical systems in Banach spaces, J. Differential Equations, 104 (1993), 307-324. doi: 10.1006/jdeq.1993.1074. Google Scholar
[14]	R. Quintanilla and P. M. Jordan, A note on the two-temperature theory with dual-phase-lag decay: Some exact solutions, Mechanics Research Communications, 36 (2009), 796-803. doi: 10.1016/j.mechrescom.2009.05.002. Google Scholar
[15]	M. C. Leseduarte, R. Quintanilla and R. Racke, On (non-)exponential decay in generalized thermoelasticity with two temperatures, Applied Mathematics Letters, 70 (2017), 18-25. doi: 10.1016/j.aml.2017.02.020. Google Scholar
[16]	Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Research Notes in Mathematics, vol. 398, Chapman & Hall/CRC, Boca Raton, FL, 1999. Google Scholar
[17]	A. Magaña, A. Miranville and R. Quintanilla, On the stability in phase-lag heat conduction with two temperatures, J. of Evolution Equations, 18 (2018), 1697-1712. doi: 10.1007/s00028-018-0457-z. Google Scholar
[18]	J. E. Marsden and T. J. R. Hughes, Topics in the mathematical foundations of elasticity, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. II, 30-285, Res. Notes in Math., 27, Pitman, Boston, Mass.-London, 1978. Google Scholar
[19]	A. Miranville and R. Quintanilla, A phase-field model based on a three-phase-lag heat conduction, Applied Mathematics and Optimization, 63 (2011), 133-150. doi: 10.1007/s00245-010-9114-9. Google Scholar
[20]	S. Mukhopadhyay, R Prasad and R. Kumar, On the theory of Two-Temperature Thermoelaticity with Two Phase-Lags, J. Thermal Stresses, 34 (2011), 352-365. Google Scholar
[21]	M. A. Othman, W. M. Hasona and E. M. Abd-Elaziz, Effect of rotation on micropolar generalized thermoelasticity with two temperatures using a dual-phase-lag model, Canadian J. Physics, 92 (2014), 149-158. doi: 10.1139/cjp-2013-0398. Google Scholar
[22]	J. Prüss, On the spectrum of C₀-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112. Google Scholar
[23]	R. Quintanilla, Exponential stability in the dual-phase-lag heat conduction theory, J. Non-Equilibrium Thermodynamics, 27 (2002), 217-227. doi: 10.1515/JNETDY.2002.012. Google Scholar
[24]	R. Quintanilla, A well-posed problem for the Dual-Phase-Lag heat conduction, J. Thermal Stresses, 31 (2008), 260-269. Google Scholar
[25]	R. Quintanilla, A well-posed problem for the three-dual-phase-lag heat conduction, J. Thermal Stresses, 32 (2009), 1270-1278. doi: 10.1080/01495730903310599. Google Scholar
[26]	R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag thermoelasticity, SIAM J. Appl. Math., 66 (2006), 977-1001. doi: 10.1137/05062860X. Google Scholar
[27]	R. Quintanilla and R. Racke, A note on stability of dual-phase-lag heat conduction, Int. J. Heat Mass Transfer, 49 (2006), 1209-1213. doi: 10.1016/j.ijheatmasstransfer.2005.10.016. Google Scholar
[28]	R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag heat conduction, Proc. Royal Society London A, 463 (2007), 659-674. doi: 10.1098/rspa.2006.1784. Google Scholar
[29]	R. Quintanilla and R. Racke, A note on stability in three-phase-lag heat conduction, Int. J. Heat Mass Transfer, 51 (2008), 24-29. Google Scholar
[30]	R. Quintanilla and R. Racke, Spatial behavior in phase-lag heat conduction, Differential and Integral Equations, 28 (2015), 291-308. Google Scholar
[31]	S. A. Rukolaine, Unphysical effects of the dual-phase-lag model of heat conduction, Int. J. Heat and Mass Transfer, 78 (2014), 58-63. doi: 10.1016/j.ijheatmasstransfer.2014.06.066. Google Scholar
[32]	D. Y. Tzou, A unified approach for heat conduction from macro to micro-scales, ASME J. Heat Transfer, 117 (1995), 8-16. doi: 10.1115/1.2822329. Google Scholar
[33]	W. E. Warren and P. J. Chen, Wave propagation in two temperatures theory of thermoelaticity, Acta Mechanica, 16 (1973), 83-117. Google Scholar
[34]	Y. Zhang, Generalized dual-phase lag bioheat equations based on nonequilibrium heat transfer in living biological tissues, Int. J. of Heat and Mass Transfer, 52 (2009), 4829-4834. doi: 10.1016/j.ijheatmasstransfer.2009.06.007. Google Scholar

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On the time decay in phase–lag thermoelasticity with two temperatures

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