Stability in locally degenerate dual-phase-lag heat conduction

References

• Lord HW, Shulman Y. A generalized dynamical theory of thermoelasticity. J Mech Phys Solids. 1967;15:299–309. doi: 10.1016/0022-5096(67)90024-5
   Web of Science ®Google Scholar
• Green AE, Lindsay KA. Thermoelasticity. J Elasticity. 1972;2:1–7. doi: 10.1007/BF00045689
   Google Scholar
• Green AE, Naghdi PM. On undamped heat waves in an elastic solid. J Therm Stress. 1992;15:253–264. doi: 10.1080/01495739208946136
   Web of Science ®Google Scholar
• Green AE, Naghdi PM. Thermoelasticity without energy dissipation. J Elasticity. 1993;31:189–208. doi: 10.1007/BF00044969
   Web of Science ®Google Scholar
• 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. 1995;448:335–388. doi: 10.1098/rspa.1995.0020
   Web of Science ®Google Scholar
• Hetnarski RB, Ignaczak J. Generalized thermoelasticity. J Therm Stress. 1999;22:451–470. doi: 10.1080/014957399280832
   Web of Science ®Google Scholar
• Hetnarski RB, Ignaczak J. Nonclassical dynamical thermoelasticity. Int J Solids Struct. 1999;37:215–224. doi: 10.1016/S0020-7683(99)00089-X
   Web of Science ®Google Scholar
• Ignaczak J, Ostoja-Starzewski M. Thermoelasticity with finite wave speeds. Oxford: Oxford University Press; 2010. (Oxford Mathematical Monographs).
   Google Scholar
• Straughan B. Heat waves. New York (NY): Springer-Verlag; 2011.
   Google Scholar
• 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(2017:73);1–26.
   Google Scholar
• Tzou DY. A unified approach for heat conduction from macro to micro-scales. ASME J Heat Transf. 1995;117:8–16. doi: 10.1115/1.2822329
   Web of Science ®Google Scholar
• Quintanilla R, Racke R. A note on stability in dual-phase-lag heat conduction. Int J Heat Mass Transfer. 2006;49:1209–1213. doi: 10.1016/j.ijheatmasstransfer.2005.10.016
   Web of Science ®Google Scholar
• Choudhuri SKR. On a thermoelastic three-phase-lag model. J Therm Stress. 2007;30:231–238. doi: 10.1080/01495730601130919
   Web of Science ®Google Scholar
• Dreher M, Quintanilla R, Racke R. Ill-posed problems in thermomechanics. Appl Math Lett. 2009;22:1374–1379. doi: 10.1016/j.aml.2009.03.010
   Web of Science ®Google Scholar
• Quintanilla R. Exponential stability in the dual-phase-lag heat conduction theory. J Non-Equilib Thermodyn. 2002;27:217–227. doi: 10.1515/JNETDY.2002.012
   Web of Science ®Google Scholar
• Liu Z, Quintanilla R, Wang Y. On the phase-lag heat equation with spatial dependent lags. J Math Anal Appl. 2017;455:422–438. doi: 10.1016/j.jmaa.2017.05.050
   Web of Science ®Google Scholar
• Borgmeyer K, Quintanilla R, Racke R. Phase-lag heat condition: decay rates for limit problems and well-posedness. J Evol Equ. 2014;14:863–884. doi: 10.1007/s00028-014-0242-6
   Web of Science ®Google Scholar
• Liu Z, Quintanilla R. Time decay in dual-phase-lag thermoelasticity critical case. Commun Pure Appl Anal. 2018;17:177–190. doi: 10.3934/cpaa.2018011
   Web of Science ®Google Scholar
• Pazy A. Semigroups of linear operators and applications to partial differential equations. New York (NY): Springer-Verlag; 1983. (Applied Mathematical Sciences; 44).
   Google Scholar
• Adams RA. Sobolev spaces. New York (NY): Academic Press; 1975. (Pure and Applied Mathematics; 65).
   Google Scholar
• Kato T. Perturbation theory for linear operators. New York (NY): Springer-Verlag; 1984.
   Google Scholar
• Prüss J. On the spectrum of C0-semigroups. Trans Amer Math Soc. 1984;284:847–857. doi: 10.2307/1999112
   Web of Science ®Google Scholar
• Huang FL. Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces. Ann Diff Eqs. 1985;1:43–56.
   Google Scholar
• Gearhart LM. Spectral theory for contraction semigroups on Hilbert space. Trans Amer Math Soc. 1978;236:385–394. doi: 10.1090/S0002-9947-1978-0461206-1
   Web of Science ®Google Scholar
• Borichev A, Tomilov Y. Optimal polynomial decay of functions and operator semigroups. Mathematische Ann. 2010;347:455–478. doi: 10.1007/s00208-009-0439-0
   Web of Science ®Google Scholar
• Liu Z, Zheng S. Semigroups associated with dissipative systems. Boca Raton (FL): Chapman&Hall/CRC; 1999.
   Google Scholar
• Liu Z, Rao B. Characterization of polynomial decay rate for the solution of linear evolution equation. Z Angew Math Phys. 2005;56:630–644. doi: 10.1007/s00033-004-3073-4
   Web of Science ®Google Scholar
• Trefethen LN. Spectral methods in Matlab. Philadelphia (PA): SIAM; 2000.
   Google Scholar