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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 & 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.  Google Scholar

[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.  Google Scholar

[3]

D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature, Appl. Mech. Rev., 51 (1998), 705-729.   Google Scholar

[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.  Google Scholar

[5]

J. N. Flavin and S. Rionero, Qualitative Estimates for Partial Differential Equations. An Introduction CRC Press Inc., Boca Raton, 1996.  Google Scholar

[6]

A. E. Green and K. A. Lindsay, Thermoelasticity, J. Elasticity, 2 (1972), 1-7.   Google Scholar

[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.  Google Scholar

[8]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208.  doi: 10.1007/BF00044969.  Google Scholar

[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.  Google Scholar

[10]

R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity, J. Thermal Stresses, 22 (1999), 451-470.  doi: 10.1080/014957399280832.  Google Scholar

[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.  Google Scholar

[12]

F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. of Diff. Eqs, 1 (1985), 43-56.   Google Scholar

[13]

J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds Oxford Mathematical Monographs, Oxford, 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoealsticity, J. Mech. Phys. Solids, 15 (1967), 299-309.   Google Scholar

[17]

Z. Liu and S. Zheng, Semigroup Associated with Dissipative System, Res. Notes Math. Vol 394, Chapman & Hall/CRC, Boca Raton, 1999.  Google Scholar

[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.  Google Scholar

[19]

R. Quintanilla, Exponential stability in the dual-phase-lag heat conduction theory, Journal Non-Equilibrium Thermodynamics, 27 (2002), 217-227.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[23]

B. Straughan, Heat Waves Springer-Verlag. New York, 2011. doi: 10.1007/978-1-4614-0493-4.  Google Scholar

[24]

D. Y. Tzou, A unified approach for heat conduction from macro to micro-scales, ASME J. Heat Transfer, 117 (1995), 8-16.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

D. S. Chandrasekharaiah, Hyperbolic thermoelasticity: A review of recent literature, Appl. Mech. Rev., 51 (1998), 705-729.   Google Scholar

[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.  Google Scholar

[5]

J. N. Flavin and S. Rionero, Qualitative Estimates for Partial Differential Equations. An Introduction CRC Press Inc., Boca Raton, 1996.  Google Scholar

[6]

A. E. Green and K. A. Lindsay, Thermoelasticity, J. Elasticity, 2 (1972), 1-7.   Google Scholar

[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.  Google Scholar

[8]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy dissipation, J. Elasticity, 31 (1993), 189-208.  doi: 10.1007/BF00044969.  Google Scholar

[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.  Google Scholar

[10]

R. B. Hetnarski and J. Ignaczak, Generalized thermoelasticity, J. Thermal Stresses, 22 (1999), 451-470.  doi: 10.1080/014957399280832.  Google Scholar

[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.  Google Scholar

[12]

F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. of Diff. Eqs, 1 (1985), 43-56.   Google Scholar

[13]

J. Ignaczak and M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds Oxford Mathematical Monographs, Oxford, 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoealsticity, J. Mech. Phys. Solids, 15 (1967), 299-309.   Google Scholar

[17]

Z. Liu and S. Zheng, Semigroup Associated with Dissipative System, Res. Notes Math. Vol 394, Chapman & Hall/CRC, Boca Raton, 1999.  Google Scholar

[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.  Google Scholar

[19]

R. Quintanilla, Exponential stability in the dual-phase-lag heat conduction theory, Journal Non-Equilibrium Thermodynamics, 27 (2002), 217-227.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[23]

B. Straughan, Heat Waves Springer-Verlag. New York, 2011. doi: 10.1007/978-1-4614-0493-4.  Google Scholar

[24]

D. Y. Tzou, A unified approach for heat conduction from macro to micro-scales, ASME J. Heat Transfer, 117 (1995), 8-16.   Google Scholar

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