Uniform stabilization of the higher-dimensional system of thermoelasticity with a nonlinear boundary feedback

Liu, Wei-Jiu; Zuazua, Enrique

doi:10.1090/qam/1828455

Authors: Wei-Jiu Liu and Enrique Zuazua
Journal: Quart. Appl. Math. 59 (2001), 269-314
MSC: Primary 74F05; Secondary 35B35, 35Q72, 74M05
DOI: https://doi.org/10.1090/qam/1828455
MathSciNet review: MR1828455
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Using multiplier techniques and Lyapunov methods, we derive explicit decay rates for the energy in the higher-dimensional system of thermoelasticity with a nonlinear velocity feedback on part of the boundary of a thermoelastic body, which is clamped along the rest of its boundary.

Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
Shmuel Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR 0178246
Fatiha Alabau and Vilmos Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems, SIAM J. Control Optim. 37 (1999), no. 2, 521–542. MR 1665070, DOI https://doi.org/10.1137/S0363012996313835
V. I. Arnol′d, Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein. MR 997295
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
Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR 0390843
E. Bisognin, V. Bisognin, G. Perla Menzala, and E. Zuazua, On exponential stability for von Kármán equations in the presence of thermal effects, Math. Methods Appl. Sci. 21 (1998), no. 5, 393–416. MR 1608076, DOI https://doi.org/10.1002/%28SICI%291099-1476%2819980325%2921%3A5%3C393%3A%3AAID-MMA958%3E3.3.CO%3B2-A
Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). Théorie et applications. [Theory and applications]. MR 697382
John A. Burns, Zhuangyi Liu, and Song Mu Zheng, On the energy decay of a linear thermoelastic bar, J. Math. Anal. Appl. 179 (1993), no. 2, 574–591. MR 1249839, DOI https://doi.org/10.1006/jmaa.1993.1370
Goong Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl. (9) 58 (1979), no. 3, 249–273. MR 544253
Francis Conrad and Bopeng Rao, Decay of solutions of the wave equation in a star-shaped domain with nonlinear boundary feedback, Asymptotic Anal. 7 (1993), no. 3, 159–177. MR 1226972
Constantine M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal. 29 (1968), 241–271. MR 233539, DOI https://doi.org/10.1007/BF00276727
Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 2, Springer-Verlag, Berlin, 1988. Functional and variational methods; With the collaboration of Michel Artola, Marc Authier, Philippe Bénilan, Michel Cessenat, Jean Michel Combes, Hélène Lanchon, Bertrand Mercier, Claude Wild and Claude Zuily; Translated from the French by Ian N. Sneddon. MR 969367
P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. MR 1173209
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
P. Grisvard, Contrôlabilité exacte des solutions de l’équation des ondes en présence de singularités, J. Math. Pures Appl. (9) 68 (1989), no. 2, 215–259 (French, with English summary). MR 1010769
Scott W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. Appl. 167 (1992), no. 2, 429–442. MR 1168599, DOI https://doi.org/10.1016/0022-247X%2892%2990217-2
Mary Ann Horn, Implications of sharp trace regularity results on boundary stabilization of the system of linear elasticity, J. Math. Anal. Appl. 223 (1998), no. 1, 126–150. MR 1627344, DOI https://doi.org/10.1006/jmaa.1998.5963
Mary Ann Horn, Sharp trace regularity for the solutions of the equations of dynamic elasticity, J. Math. Systems Estim. Control 8 (1998), no. 2, 11 pp.}, issn=1052-0600, review=\MR{1651449},.
Jong Uhn Kim, On the energy decay of a linear thermoelastic bar and plate, SIAM J. Math. Anal. 23 (1992), no. 4, 889–899. MR 1166563, DOI https://doi.org/10.1137/0523047
V. Komornik, Exact controllability and stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. The multiplier method. MR 1359765
Vilmos Komornik and Bopeng Rao, Boundary stabilization of compactly coupled wave equations, Asymptot. Anal. 14 (1997), no. 4, 339–359. MR 1461124
Vilmos Komornik and Enrike Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl. (9) 69 (1990), no. 1, 33–54. MR 1054123

A. D. Kovalenko, Thermoelasticity, Basic Theory and Applications, Wolters-Noordhoff Publishing, Gröningen, Netherlands, 1969

John Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations 50 (1983), no. 2, 163–182. MR 719445, DOI https://doi.org/10.1016/0022-0396%2883%2990073-6
John Lagnese, Boundary stabilization of linear elastodynamic systems, SIAM J. Control Optim. 21 (1983), no. 6, 968–984. MR 719524, DOI https://doi.org/10.1137/0321059
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, Uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary, Nonlinear Anal. 16 (1991), no. 1, 35–54. MR 1086824, DOI https://doi.org/10.1016/0362-546X%2891%2990129-O
J. E. Lagnese and G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback, J. Differential Equations 91 (1991), no. 2, 355–388. MR 1111180, DOI https://doi.org/10.1016/0022-0396%2891%2990145-Y
J. Lagnese and J.-L. Lions, Modelling analysis and control of thin plates, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 6, Masson, Paris, 1988. MR 953313
I. Lasiecka, Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary, J. Differential Equations 79 (1989), no. 2, 340–381. MR 1000694, DOI https://doi.org/10.1016/0022-0396%2889%2990107-1
I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations 6 (1993), no. 3, 507–533. MR 1202555
Gilles Lebeau and Enrique Zuazua, Sur la décroissance non uniforme de l’énergie dans le système de la thermoélasticité linéaire, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 4, 409–415 (French, with English and French summaries). MR 1440958, DOI https://doi.org/10.1016/S0764-4442%2897%2980077-8
G. Leugering, On boundary feedback stabilization of a viscoelastic membrane, Dynam. Stability Systems 4 (1989), no. 1, 71–79. MR 1023012, DOI https://doi.org/10.1080/02681118908806063

G. Leugering, On boundary feedback stabilization of a viscoelastic beam, Proc. Roy. Soc. Edinburgh, Sect. A 114, 57–69 (1990)

J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. II, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 182. MR 0350178
Weijiu Liu, Partial exact controllability and exponential stability in higher-dimensional linear thermoelasticity, ESAIM Control Optim. Calc. Var. 3 (1998), 23–48. MR 1610226, DOI https://doi.org/10.1051/cocv%3A1998113

W. J. Liu, Exact Controllability for Some Partial Differential Equations of Evolutional Type, Ph.D. Thesis, University of Wollongong, Australia, 1997

Weijiu Liu, The exponential stabilization of the higher-dimensional linear system of thermoviscoelasticity, J. Math. Pures Appl. (9) 77 (1998), no. 4, 355–386. MR 1623383, DOI https://doi.org/10.1016/S0021-7824%2898%2980103-7
Zhuangyi Liu and Song Mu Zheng, Exponential stability of the semigroup associated with a thermoelastic system, Quart. Appl. Math. 51 (1993), no. 3, 535–545. MR 1233528, DOI https://doi.org/10.1090/qam/1233528
Wei-Jiu Liu and Enrique Zuazua, Decay rates for dissipative wave equations, Ricerche Mat. 48 (1999), no. suppl., 61–75. Papers in memory of Ennio De Giorgi (Italian). MR 1765677

G. P. Menzala and E. Zuazua, Explicit exponential decay rates for solutions of von Kármán’s system of thermoelastic plates, C. R. Acad. Sci Paris Sér. I Math. 324, 49–54 (1997)

G. P. Menzala and E. Zuazua, Energy decay rates for the Von Kármán system of thermoelastic plates, Differential Integral Equations, to appear.

Kimiaki Narukawa, Boundary value control of thermoelastic systems, Hiroshima Math. J. 13 (1983), no. 2, 227–272. MR 707182
Luiz Augusto Fernandes de Oliveira, Exponential decay in thermoelasticity, Commun. Appl. Anal. 1 (1997), no. 1, 113–118. MR 1453945
Jaime E. Muñoz Rivera, Decomposition of the displacement vector field and decay rates in linear thermoelasticity, SIAM J. Math. Anal. 24 (1993), no. 2, 390–406. MR 1205533, DOI https://doi.org/10.1137/0524025
J. E. Muñoz Rivera, Asymptotic behaviour in $n$-dimensional thermoelasticity, Appl. Math. Lett. 10 (1997), no. 5, 47–53. MR 1471316, DOI https://doi.org/10.1016/S0893-9659%2897%2900082-7
Jaime E. Muñoz Rivera and Milton Lacerda Olivera, Stability in inhomogeneous and anisotropic thermoelasticity, Boll. Un. Mat. Ital. A (7) 11 (1997), no. 1, 115–127 (English, with Italian summary). MR 1438361
Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. McGraw-Hill Series in Higher Mathematics. MR 0344043
Han Kun Wang and Goong Chen, Asymptotic behavior of solutions of the one-dimensional wave equation with a nonlinear boundary stabilizer, SIAM J. Control Optim. 27 (1989), no. 4, 758–775. MR 1001918, DOI https://doi.org/10.1137/0327040
Enrike Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim. 28 (1990), no. 2, 466–477. MR 1040470, DOI https://doi.org/10.1137/0328025