Asymptotic analysis of two thermoelastic plates with dissipative histories. (English) Zbl 1532.35053
Summary: In this paper we study the time decay of the solutions for the problems determined by two plates where the dissipation mechanisms are given by the history of the material. To be precise we consider the thermo-viscolestic plate with heat conduction of the Green-Naghdi type II and the thermoelastic plate when the heat equation is described by the history-dependent Moore-Gibson-Thompson equation. In both cases we prove the well-posedness of the problems by means of semigroup theory. In the first case we also prove that the solutions decay in an exponential way by means of Prüss characterizations of exponential stable semigroups. In the second case we prove that the solutions decay in a polynomial way with optimal rates of decay, which is proved by Tomilov-Borichev characterizations of polynomial stable semigroups.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35L57 | Initial-boundary value problems for higher-order hyperbolic systems |
| 35R09 | Integro-partial differential equations |
| 74F05 | Thermal effects in solid mechanics |
| 74K20 | Plates |
References:
| [1] | Borichev, A.; Tomilov, Y., Optimal polynomial decay of functions and operator semigroup. Math. Ann., 455-478 (2010) · Zbl 1185.47044 |
| [2] | Conti, M.; Pata, V.; Pellicer, M.; Quintanilla, R., On the analyticity of the MGT-viscoelastic plate with heat conduction. J. Differ. Equ., 7862-7880 (2020) · Zbl 1442.35073 |
| [3] | Conti, M.; Pata, V.; Pellicer, M.; Quintanilla, R., A new approach to MGT thermoviscoelasticity. Discrete Contin. Dyn. Syst., 4645-4666 (2021) · Zbl 1479.35855 |
| [4] | Conti, M.; Pata, V.; Quintanilla, R., Thermoelasticity of Moore-Gibson-Thompson type with history dependence in the temperature. Asymptot. Anal., 1-21 (2020) · Zbl 1458.35405 |
| [5] | Coti, M.; Dell’Oro, F.; Pata, V., Energy decay of type III linear thermoelastic plates with memory. J. Math. Anal. Appl., 357-366 (2013) · Zbl 1364.35357 |
| [6] | Dafermos, C. M., Asymptotic stability in viscoleasticity. Arch. Ration. Mech. Anal., 297-308 (1970) · Zbl 0214.24503 |
| [7] | Fernández Sare, H. D.; Muñoz Rivera, J., Stability from Timoshenko systems with past history. J. Math. Anal. Appl., 482-502 (2008) · Zbl 1132.45008 |
| [8] | Fernández Sare, H. D.; Quintanilla, R., Porous-elastic plates: Fourier versus type III. Appl. Math. Optim., 1055-1085 (2021) · Zbl 1477.35264 |
| [9] | Fernández Sare, H. D.; Quintanilla, R., Moore Gibson Thompson thermoelastic plates: comparisons. J. Evol. Equ., 70 (2023) · Zbl 1527.35417 |
| [10] | Green, A. E.; Naghdi, P. M., Thermoelasticity without energy dissipation. J. Elast., 189-208 (1993) · Zbl 0784.73009 |
| [11] | Gurtin, M. E., Time-reversal and symmetry in the thermodynamics of materials with memory. Arch. Ration. Mech. Anal., 387-399 (1972) · Zbl 0249.73003 |
| [12] | Huang, F. L., Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ., 1, 43-56 (1985) · Zbl 0593.34048 |
| [13] | Liu, Z.; Quintanilla, R., Analyticity of solutions in type III thermoelastic plates. IMA J. Appl. Math., 356-365 (2010) · Zbl 1375.74027 |
| [14] | Liu, Z.; Renardy, M., A note on the equations of a thermoelastic plates. Appl. Math. Lett., 1-6 (1995) · Zbl 0826.35048 |
| [15] | Liu, Z.; Quintanilla, R.; Wang, Y., On the regularity and stability of three-phase-lag thermoelastic plates. Appl. Anal., 5376-5385 (2022) · Zbl 1497.35460 |
| [16] | Liu, Z.; Zheng, S., Semigroups Associated with Dissipative Systems. \(π\) Research Notes Math. (1999), Chapman&Hall/CRC: Chapman&Hall/CRC Boca Raton · Zbl 0924.73003 |
| [17] | Muñoz Rivera, J. E.; Ochoa, E.; Quintanilla, R., Time decay of viscoelastic plates with type II heat conduction. J. Math. Anal. Appl. (2023) · Zbl 1536.74047 |
| [18] | Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0516.47023 |
| [19] | Prüss, J., On the spectrum of \(C_0\)-semigroups. Trans. Am. Math. Soc., 2, 847-857 (1984) · Zbl 0572.47030 |
| [20] | Quintanilla, R.; Racke, R.; Ueda, Y., Decay for thermoelastic Green-Lindsay plates in bounded and unbounded domains. Commun. Pure Appl. Anal., 167-191 (2023) · Zbl 1512.35577 |
| [21] | Racke, R.; Ueda, Y., The Cauchy problem for thermoelastic plates with two temperatures. Z. Anal. Anwend., 103-129 (2020) · Zbl 1445.35047 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
