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Global existence of smooth solutions in one-dimensional nonlinear thermoelasticity

Published online by Cambridge University Press:  14 November 2011

Song Jiang
Song Jiang
Affiliation:
Institut für Angewandte Mathematik der Universität Bonn, Wegelerstrasse 10, D-5300 Bonn 1, B.R.D., and Department of Mathematics, Xi'an Jiaotong University, Xi'an, Shaanxi Province, P.R.China
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Synopsis

We consider the initial boundary value problem for the equations of one-dimensional nonlinear thermoelasticity in ℝ+; and prove a global existence-uniqueness theorem for small smooth data. The asymptotic behaviour is simultaneously obtained.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 115 , Issue 3-4 , 1990 , pp. 257 - 274
Copyright
Copyright © Royal Society of Edinburgh 1990

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References

1Adams, R. A.. Sobolev Spaces (New York: Academic Press, (1975).Google Scholar
2Carlson, D. E.. Linear Thermoelasticity. Handbuch der Physik Via, pp. 297–345. (Berlin: Springer, 1972).Google Scholar
3Coleman, B. D.. Thermodynamics of materials with memory. Arch. Rational Mech. Anal. 17 (1964), 1–46.CrossRefGoogle Scholar
4Dafermos., C. M.. Contemporary Issues in the Dynamic Behaviour of Continuous Media (Providence, R.I.: Lefschetz Center for Dynamical Systems Report, Brown University, 1985).Google Scholar
5Dafermos, C. M. and Hsiao, L.. Development of singularities in solutions of the equations of nonlinear thermoelasticity. Q. Appl. Math. 44 (1986), 463–474.CrossRefGoogle Scholar
6Jiang, S.. Global existence and asymptotic behavior of smooth solutions in one-dimensional nonlinear thermoelasticity. Bonner Math. Schrif. 192 (1989).Google Scholar
7John, F.. Blow-up for quasi-linear wave equations in three space dimensions. Comm. Pure Appl. Math. 34 (1981), 29–51.CrossRefGoogle Scholar
8Kawashima, S.. Systems of a hyperbolic-parabolic composite type, with applications to the equations of Magnethohydrodynamics (Thesis, Kyoto university, 1983).Google Scholar
9Klainerman, S. and Majda, A.. Formation of singularities for wave equations including the nonlinear vibrating string. Comm. Pure Appl. Math. 33 (1980), 241–263.CrossRefGoogle Scholar
10Klainerman, S. and Ponce, G.. Global, small amplitude solutions to nonlinear evolution equations. Comm. Pure Appl. Math. 36 (1983), 133–141.CrossRefGoogle Scholar
11Leis, R.. Initial Boundary Value Problems in Mathematical Physics (New York: John Wiley, 1986).CrossRefGoogle Scholar
12Maccamy, R. C. and Mizel, V. J.. Existence and non-existence in the large of solutions of quasilinear wave equations. Arch. Rational Mech. Anal. 25 (1967), 299–320.CrossRefGoogle Scholar
13Parkus, H.. Thermoelasticity, Chapter 5 (New York: Springer, 1976).CrossRefGoogle Scholar
14Racke, R.. Initial boundary value problems in one-dimensional non-linear thermoelasticity. Math. Meth. Appl. Sci. 10 (1988), 517–529.CrossRefGoogle Scholar
15Racke, R.. On the Cauchy problem in nonlinear 3-d-thermoelasticity. To appear in: Math. Z. (1990).CrossRefGoogle Scholar
16Racke, R.. Blow-up in nonlinear three-dimensional thermoelasticity. Math. Meth. Appl. Sci. 12 (1990), 267–273.CrossRefGoogle Scholar
17Shatah, J.. Global existence of small solutions to nonlinear evolution equations. Differential Equations 46 (1982), 409–425.CrossRefGoogle Scholar
18Shen, W. X. and Zheng, S. M.. Global smooth solutions to the system of one-dimensional thermoelasticity with dissipation boundary conditions. Chinese Ann. Math. Ser. B 7(3) (1986), 303–317.Google Scholar
19Slemrod, M.. Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear thermoelasticity. Arch. Rational Mech. Anal. 76(1981), 97–133.CrossRefGoogle Scholar
20Zheng, S. M.. Initial boundary value problems for quasilinear hyperbolic-parabolic coupled systems in higher dimensional spaces. Chinese Ann. Math. Ser. B 4f(4) (1983), 443–462.Google Scholar
21Zheng, S. M. and Shen, W. X.. Global solutions to the Cauchy problem of quasilinear hyperbolic parapolic coupled systems. sci. Sinica Ser. A 30 (1987), 1133–1149.Google Scholar