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Stability in locally degenerate dual-phase-lag heat conduction. (English) Zbl 1455.35016
Summary: This work is devoted to the large time behaviour of a one-dimensional degenerate dual-phase-lag heat conduction system. Assume that those two delay parameters in the system are positive constants in a subregion of the spatial domain \([-1,1]\) and vanished in others. We obtain two stability results for this problem. First, we show that this system is exponentially stable under certain condition, which is consistent with the stability result for the system with delay parameters satisfying the same condition globally in the whole domain. Second, we show that the system is polynomially stable with decay rate \(t^{-2}\) for the critical case, which is faster than the corresponding one with globally positive delay parameters, the decay rate of which is \(t^{-1/2}\). The optimality of the decay rate \(t^{-2}\) is further verified by the asymptotic spectral analysis for the system operator. Finally, some numerical experiments are presented to support these stability results.
MSC:
35B40 Asymptotic behavior of solutions to PDEs
35K51 Initial-boundary value problems for second-order parabolic systems
93D20 Asymptotic stability in control theory
35B35 Stability in context of PDEs
Software:
Matlab
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[1] Lord, HW; Shulman, Y., A generalized dynamical theory of thermoelasticity, J Mech Phys Solids, 15, 299-309 (1967) · Zbl 0156.22702
[2] Green, AE; Lindsay, KA., Thermoelasticity, J Elasticity, 2, 1-7 (1972) · Zbl 0775.73063
[3] Green, AE; Naghdi, PM., On undamped heat waves in an elastic solid, J Therm Stress, 15, 253-264 (1992)
[4] Green, AE; Naghdi, PM., Thermoelasticity without energy dissipation, J Elasticity, 31, 189-208 (1993) · Zbl 0784.73009
[5] Green, AE; Naghdi, PM., A unified procedure for construction of theories of deformable media, I. Classical continuum physics, II. Generalized continua, III. Mixtures of interacting continua, Proc R Soc Lond A, 448, 335-388 (1995) · Zbl 0868.73013
[6] Hetnarski, RB; Ignaczak, J., Generalized thermoelasticity, J Therm Stress, 22, 451-470 (1999)
[7] Hetnarski, RB; Ignaczak, J., Nonclassical dynamical thermoelasticity, Int J Solids Struct, 37, 215-224 (1999) · Zbl 1075.74033
[8] Ignaczak, J, Ostoja-Starzewski, M.Thermoelasticity with finite wave speeds. Oxford: Oxford University Press; 2010. (Oxford Mathematical Monographs). · Zbl 1183.80001
[9] Straughan, B., Heat waves (2011), New York (NY): Springer-Verlag, New York (NY) · Zbl 1250.80001
[10] Gallego, FA; Munoz Rivera, JE., Decay rates of the solutions to the thermoelastic bresse system of types I and III, Electron J Differ Equ, 2017, 73, 1-26 (2017) · Zbl 1370.35046
[11] Tzou, DY., A unified approach for heat conduction from macro to micro-scales, ASME J Heat Transf, 117, 8-16 (1995)
[12] Quintanilla, R.; Racke, R., A note on stability in dual-phase-lag heat conduction, Int J Heat Mass Transfer, 49, 1209-1213 (2006) · Zbl 1189.80025
[13] Choudhuri, SKR., On a thermoelastic three-phase-lag model, J Therm Stress, 30, 231-238 (2007)
[14] Dreher, M.; Quintanilla, R.; Racke, R., Ill-posed problems in thermomechanics, Appl Math Lett, 22, 1374-1379 (2009) · Zbl 1173.80301
[15] Quintanilla, R., Exponential stability in the dual-phase-lag heat conduction theory, J Non-Equilib Thermodyn, 27, 217-227 (2002) · Zbl 1039.80002
[16] Liu, Z.; Quintanilla, R.; Wang, Y., On the phase-lag heat equation with spatial dependent lags, J Math Anal Appl, 455, 422-438 (2017) · Zbl 1378.35083
[17] Borgmeyer, K.; Quintanilla, R.; Racke, R., Phase-lag heat condition: decay rates for limit problems and well-posedness, J Evol Equ, 14, 863-884 (2014) · Zbl 1316.35033
[18] Liu, Z.; Quintanilla, R., Time decay in dual-phase-lag thermoelasticity critical case, Commun Pure Appl Anal, 17, 177-190 (2018) · Zbl 1375.35544
[19] Pazy, A.Semigroups of linear operators and applications to partial differential equations. New York (NY): Springer-Verlag; 1983. (Applied Mathematical Sciences; 44). · Zbl 0516.47023
[20] Adams, RA.Sobolev spaces. New York (NY): Academic Press; 1975. (Pure and Applied Mathematics; 65). · Zbl 0314.46030
[21] Kato, T., Perturbation theory for linear operators (1984), New York (NY): Springer-Verlag, New York (NY)
[22] Prüss, J., On the spectrum of \(####\)-semigroups, Trans Amer Math Soc, 284, 847-857 (1984) · Zbl 0572.47030
[23] Huang, FL., Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann Diff Eqs, 1, 43-56 (1985) · Zbl 0593.34048
[24] Gearhart, LM., Spectral theory for contraction semigroups on Hilbert space, Trans Amer Math Soc, 236, 385-394 (1978) · Zbl 0326.47038
[25] Borichev, A.; Tomilov, Y., Optimal polynomial decay of functions and operator semigroups, Mathematische Ann, 347, 455-478 (2010) · Zbl 1185.47044
[26] Liu, Z.; Zheng, S., Semigroups associated with dissipative systems (1999), Boca Raton (FL): Chapman&Hall/CRC, Boca Raton (FL) · Zbl 0924.73003
[27] Liu, Z.; Rao, B., Characterization of polynomial decay rate for the solution of linear evolution equation, Z Angew Math Phys, 56, 630-644 (2005) · Zbl 1100.47036
[28] Trefethen, LN., Spectral methods in Matlab (2000), Philadelphia (PA): SIAM, Philadelphia (PA)
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