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

Zhuangyi Liu; Ramón Quintanilla

doi:10.3934/cpaa.2018011

Article Contents

2018, Volume 17, Issue 1: 177-190. Doi: 10.3934/cpaa.2018011

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

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
* Corresponding authorRamón Quintanilla

Received: March 2017

Revised: June 2017

Published online: September 2017

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.

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

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:

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

Citation:

Full Text(HTML)

Related Papers

Cited by

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.