Time decay in dual-phase-lag thermoelasticity: Critical case

Previous Article
Higher order eigenvalues for non-local Schrödinger operators
CPAA Home
This Issue
Next Article
Stability of standing waves for a nonlinear SchrÖdinger equation under an external magnetic field

January 2018, 17(1): 177-190. doi: 10.3934/cpaa.2018011

Time decay in dual-phase-lag thermoelasticity: Critical case

Zhuangyi Liu ^1, and Ramón Quintanilla ^2,,

1.	Department of Mathematics and Statistics, University of Minnesota, Duluth, MN 55812-2496, USA
2.	Department of Mathematics, UPC, Colom 11, 08222 Terrassa, Spain

* Corresponding authorRamón Quintanilla

Received March 2017 Revised June 2017 Published September 2017

Fund Project: The second author R. Q. is supported by the Projects "Análisis Matemático de las Ecuaciones en Derivada Parciales de la Termomecánica"(MTM2013-42004-P), "Análisis Matemático de Problemas de la Termomecánica"(MTM2016-74934-P), (AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness.

This note is devoted to the study of the time decay of the one-dimensional dual-phase-lag thermoelasticity. In this theory two delay parameters $τ_q$ and $τ_{θ}$ are proposed. It is known that the system is exponentially stable if $τ_q<2 τ_{θ}$ [22]. We here make two new contributions to this problem. First, we prove the polynomial stability in the case that $τ_q=2 τ_{θ}$ as well the optimality of this decay rate. Second, we prove that the exponential stability remains true even if the inequality only holds in a proper sub-interval of the spatial domain, when $τ_{θ}$ is spatially dependent.

Keywords: Phase-lag, thermoelasticity, polynomial stability, exponential stability.

Mathematics Subject Classification: Primary:35Q70, 35B40;Secondary:74F05.

Citation: 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

References:

[1]	K. Borgmeyer, R. Quintanilla and R. Racke, Phase-lag heat condition: Decay rates for limit problems and well-posedness, J. Evol. Equ., 14 (2014), 863-884. doi: 10.1007/s00028-014-0242-6.
[2]	A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Annal., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0.
[3]	D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature, Appl. Mech. Rev., 51 (1998), 705-729.
[4]	M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics, Appl. Math. Letters, 22 (2009), 1374-1379. doi: 10.1016/j.aml.2009.03.010.
[5]	J. N. Flavin and S. Rionero, Qualitative Estimates for Partial Differential Equations. An Introduction CRC Press Inc., Boca Raton, 1996.
[6]	A. E. Green and K. A. Lindsay, Thermoelasticity, J. Elasticity, 2 (1972), 1-7.
[7]	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.
[8]	A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208. doi: 10.1007/BF00044969.
[9]	A. E. Green and P. M. Naghdi, A unified procedure for contruction of theories of deformable media, Ⅰ. Classical continuum physics, Ⅱ. Generalized continua, Ⅲ. Mixtures of interacting continua, Proc. Royal Society London A, 448 (1995), 335-356,357-377,379-388. doi: 10.1098/rspa.1995.0022.
[10]	R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity, J. Thermal Stresses, 22 (1999), 451-470. doi: 10.1080/014957399280832.
[11]	R. B. Hetnarski and J. Ignaczak, Nonclassical dynamical thermoelasticity, International Journal of Solids and Structures, 37 (1999), 215-224. doi: 10.1016/S0020-7683(99)00089-X.
[12]	F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. of Diff. Eqs, 1 (1985), 43-56.
[13]	J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds Oxford Mathematical Monographs, Oxford, 2010.
[14]	K. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity, Z. Angew. Math. Phy., 53 (2002), 265-280. doi: 10.1007/s00033-002-8155-6.
[15]	Z. Liu, R. Quintanilla and Y. Wang, On the phase-lag heat equation with spatial dependent lags, Jour. Math. Anal. Appl., 455 (2017), 422-438. doi: 10.1016/j.jmaa.2017.05.050.
[16]	H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoealsticity, J. Mech. Phys. Solids, 15 (1967), 299-309.
[17]	Z. Liu and S. Zheng, Semigroup Associated with Dissipative System, Res. Notes Math. Vol 394, Chapman & Hall/CRC, Boca Raton, 1999.
[18]	A. Morro, L. E. Payne and B. Straughan, Decay, growth, continuous dependence and uniqueness of generalized heat conduction theories, Appl. Anal., 38 (1990), 231-243. doi: 10.1080/00036819008839964.
[19]	R. Quintanilla, Exponential stability in the dual-phase-lag heat conduction theory, Journal Non-Equilibrium Thermodynamics, 27 (2002), 217-227.
[20]	R. Quintanilla, A condition on the delay parameters in the one-dimensional dual-phase-lag thermoelastic theory, Journal Thermal Stresses, 26 (2003), 713-721.
[21]	R. Quintanilla and R. Racke, A note on stability in dual-phase-lag heat conduction, Int. J. Heat Mass Transfer, 49 (2006), 1209-1213.
[22]	R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag thermoelasticity, SIAM Journal Applied Mathematics, 66 (2006), 977-1001. doi: 10.1137/05062860X.
[23]	B. Straughan, Heat Waves Springer-Verlag. New York, 2011. doi: 10.1007/978-1-4614-0493-4.
[24]	D. Y. Tzou, A unified approach for heat conduction from macro to micro-scales, ASME J. Heat Transfer, 117 (1995), 8-16.

show all references

References:

[1]	K. Borgmeyer, R. Quintanilla and R. Racke, Phase-lag heat condition: Decay rates for limit problems and well-posedness, J. Evol. Equ., 14 (2014), 863-884. doi: 10.1007/s00028-014-0242-6.
[2]	A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Annal., 347 (2010), 455-478. doi: 10.1007/s00208-009-0439-0.
[3]	D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature, Appl. Mech. Rev., 51 (1998), 705-729.
[4]	M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics, Appl. Math. Letters, 22 (2009), 1374-1379. doi: 10.1016/j.aml.2009.03.010.
[5]	J. N. Flavin and S. Rionero, Qualitative Estimates for Partial Differential Equations. An Introduction CRC Press Inc., Boca Raton, 1996.
[6]	A. E. Green and K. A. Lindsay, Thermoelasticity, J. Elasticity, 2 (1972), 1-7.
[7]	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.
[8]	A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208. doi: 10.1007/BF00044969.
[9]	A. E. Green and P. M. Naghdi, A unified procedure for contruction of theories of deformable media, Ⅰ. Classical continuum physics, Ⅱ. Generalized continua, Ⅲ. Mixtures of interacting continua, Proc. Royal Society London A, 448 (1995), 335-356,357-377,379-388. doi: 10.1098/rspa.1995.0022.
[10]	R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity, J. Thermal Stresses, 22 (1999), 451-470. doi: 10.1080/014957399280832.
[11]	R. B. Hetnarski and J. Ignaczak, Nonclassical dynamical thermoelasticity, International Journal of Solids and Structures, 37 (1999), 215-224. doi: 10.1016/S0020-7683(99)00089-X.
[12]	F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. of Diff. Eqs, 1 (1985), 43-56.
[13]	J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds Oxford Mathematical Monographs, Oxford, 2010.
[14]	K. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity, Z. Angew. Math. Phy., 53 (2002), 265-280. doi: 10.1007/s00033-002-8155-6.
[15]	Z. Liu, R. Quintanilla and Y. Wang, On the phase-lag heat equation with spatial dependent lags, Jour. Math. Anal. Appl., 455 (2017), 422-438. doi: 10.1016/j.jmaa.2017.05.050.
[16]	H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoealsticity, J. Mech. Phys. Solids, 15 (1967), 299-309.
[17]	Z. Liu and S. Zheng, Semigroup Associated with Dissipative System, Res. Notes Math. Vol 394, Chapman & Hall/CRC, Boca Raton, 1999.
[18]	A. Morro, L. E. Payne and B. Straughan, Decay, growth, continuous dependence and uniqueness of generalized heat conduction theories, Appl. Anal., 38 (1990), 231-243. doi: 10.1080/00036819008839964.
[19]	R. Quintanilla, Exponential stability in the dual-phase-lag heat conduction theory, Journal Non-Equilibrium Thermodynamics, 27 (2002), 217-227.
[20]	R. Quintanilla, A condition on the delay parameters in the one-dimensional dual-phase-lag thermoelastic theory, Journal Thermal Stresses, 26 (2003), 713-721.
[21]	R. Quintanilla and R. Racke, A note on stability in dual-phase-lag heat conduction, Int. J. Heat Mass Transfer, 49 (2006), 1209-1213.
[22]	R. Quintanilla and R. Racke, Qualitative aspects in dual-phase-lag thermoelasticity, SIAM Journal Applied Mathematics, 66 (2006), 977-1001. doi: 10.1137/05062860X.
[23]	B. Straughan, Heat Waves Springer-Verlag. New York, 2011. doi: 10.1007/978-1-4614-0493-4.
[24]	D. Y. Tzou, A unified approach for heat conduction from macro to micro-scales, ASME J. Heat Transfer, 117 (1995), 8-16.

[1]	Marco Campo, Maria I. M. Copetti, José R. Fernández, Ramón Quintanilla. On existence and numerical approximation in phase-lag thermoelasticity with two temperatures. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2221-2245. doi: 10.3934/dcdsb.2021130
[2]	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
[3]	Ramon Quintanilla, Reinhard Racke. Stability in thermoelasticity of type III. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 383-400. doi: 10.3934/dcdsb.2003.3.383
[4]	Farah Abdallah, Denis Mercier, Serge Nicaise. Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems. Evolution Equations and Control Theory, 2013, 2 (1) : 1-33. doi: 10.3934/eect.2013.2.1
[5]	Alaa Hayek, Serge Nicaise, Zaynab Salloum, Ali Wehbe. Exponential and polynomial stability results for networks of elastic and thermo-elastic rods. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1183-1220. doi: 10.3934/dcdss.2021142
[6]	Margareth S. Alves, Rodrigo N. Monteiro. Stability of non-classical thermoelasticity mixture problems. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4879-4898. doi: 10.3934/cpaa.2020216
[7]	Ramon Quintanilla. Structural stability and continuous dependence of solutions of thermoelasticity of type III. Discrete and Continuous Dynamical Systems - B, 2001, 1 (4) : 463-470. doi: 10.3934/dcdsb.2001.1.463
[8]	Karim El Mufti, Rania Yahia. Polynomial stability in viscoelastic network of strings. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1421-1438. doi: 10.3934/dcdss.2022073
[9]	Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219
[10]	Shaoqiang Tang, Huijiang Zhao. Stability of Suliciu model for phase transitions. Communications on Pure and Applied Analysis, 2004, 3 (4) : 545-556. doi: 10.3934/cpaa.2004.3.545
[11]	Nanhee Kim. Uniqueness and Hölder type stability of continuation for the linear thermoelasticity system with residual stress. Evolution Equations and Control Theory, 2013, 2 (4) : 679-693. doi: 10.3934/eect.2013.2.679
[12]	Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Exponential stability of the solutions to the Boltzmann equation for the Benard problem. Kinetic and Related Models, 2012, 5 (4) : 673-695. doi: 10.3934/krm.2012.5.673
[13]	Sigurdur Freyr Hafstein. A constructive converse Lyapunov theorem on exponential stability. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 657-678. doi: 10.3934/dcds.2004.10.657
[14]	Hichem Kasri, Amar Heminna. Exponential stability of a coupled system with Wentzell conditions. Evolution Equations and Control Theory, 2016, 5 (2) : 235-250. doi: 10.3934/eect.2016003
[15]	Litan Yan, Wenyi Pei, Zhenzhong Zhang. Exponential stability of SDEs driven by fBm with Markovian switching. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6467-6483. doi: 10.3934/dcds.2019280
[16]	István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773
[17]	Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089
[18]	Guy Katriel. Stability of synchronized oscillations in networks of phase-oscillators. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 353-364. doi: 10.3934/dcdsb.2005.5.353
[19]	Cónall Kelly, Alexandra Rodkina. Constrained stability and instability of polynomial difference equations with state-dependent noise. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 913-933. doi: 10.3934/dcdsb.2009.11.913
[20]	Reza Kamyar, Matthew M. Peet. Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2383-2417. doi: 10.3934/dcdsb.2015.20.2383

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Zhuangyi Liu Ramón Quintanilla
on Google Scholar
Zhuangyi Liu Ramón Quintanilla

2020 Impact Factor: 1.916

Tools

Article outline

2020 Impact Factor: 1.916

Tools

Metrics

Other articles
by authors

on AIMS
Zhuangyi Liu Ramón Quintanilla
on Google Scholar
Zhuangyi Liu Ramón Quintanilla

[Back to Top]