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Time decay in dual-phase-lag thermoelasticity: Critical case
1. | Department of Mathematics and Statistics, University of Minnesota, Duluth, MN 55812-2496, USA |
2. | Department of Mathematics, UPC, Colom 11, 08222 Terrassa, Spain |
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 $τ_q<2 τ_{θ}$ [
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.
|
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