A transmission problem for thermoelastic plates

Muñoz Rivera, Jaime E.; Portillo Oquendo, Higidio

doi:10.1090/qam/2054600

Authors: Jaime E. Muñoz Rivera and Higidio Portillo Oquendo
Journal: Quart. Appl. Math. 62 (2004), 273-293
MSC: Primary 74F05; Secondary 35B35, 35B40, 35Q72, 74H40, 74K20
DOI: https://doi.org/10.1090/qam/2054600
MathSciNet review: MR2054600
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Abstract: In this paper we study a transmission problem for thermoelastic plates. We prove that the problem is well-posed in the sense that there exists only one solution which is as regular as the initial data. Moreover, we prove that the local thermal effect is strong enough to produce uniform rate of decay of the solution. More precisely, there exist positive constants $C$ and $\gamma$ such that the total energy $E\left ( t \right )$ satisfies \[ E\left ( t \right ) \le CE\left ( 0 \right ){e^{ - \gamma t}}\].

Mohammed Aassila, Exact boundary controllability of the plate equation, Differential Integral Equations 13 (2000), no. 10-12, 1413–1428. MR 1787074
George Avalos and Irena Lasiecka, Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal. 29 (1998), no. 1, 155–182. MR 1617180, DOI https://doi.org/10.1137/S0036141096300823
Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
John E. Lagnese, Boundary stabilization of thin plates, SIAM Studies in Applied Mathematics, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1061153
John E. Lagnese, Boundary controllability in problems of transmission for a class of second order hyperbolic systems, ESAIM Control Optim. Calc. Var. 2 (1997), 343–357. MR 1487483, DOI https://doi.org/10.1051/cocv%3A1997112
J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 2, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 9, Masson, Paris, 1988 (French). Perturbations. [Perturbations]. MR 963060

J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Collection Études Mathématiques, Dunod Paris 1969.

Weijiu Liu and Graham Williams, The exponential stability of the problem of transmission of the wave equation, Bull. Austral. Math. Soc. 57 (1998), no. 2, 305–327. MR 1617324, DOI https://doi.org/10.1017/S0004972700031683
Weijiu Liu and Graham H. Williams, Exact controllability for problems of transmission of the plate equation with lower-order terms, Quart. Appl. Math. 58 (2000), no. 1, 37–68. MR 1738557, DOI https://doi.org/10.1090/qam/1738557

J. E. Muñoz Rivera and Ma To Fu, Exponential stability for a transmission problem, To appear.

Jaime E. Muñoz Rivera and Higidio Portillo Oquendo, The transmission problem of viscoelastic waves, Acta Appl. Math. 62 (2000), no. 1, 1–21. MR 1778015, DOI https://doi.org/10.1023/A%3A1006449032100
Jaime E. Muñoz Rivera and Higidio Portillo Oquendo, The transmission problem for thermoelastic beams, J. Thermal Stresses 24 (2001), no. 12, 1137–1158. MR 1866412, DOI https://doi.org/10.1080/014957301753251665

J. E. Muñoz Rivera and H. Portillo Oquendo, The transmission problem of viscoelastic beams, To appear in Advances in Mathematical Science and Applications.

K. Oh, A theorical and experimental study of modal interactions in metallic and laminated composite plates, Ph.D. thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, 1994.

Martin Schechter, A generalization of the problem of transmission, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 14 (1960), 207–236. MR 131063