Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-18T10:39:16.603Z Has data issue: false hasContentIssue false
  • English
  • Français

Rapidly decreasing behaviour of solutions in nonlinear 3-D-thermoelasticity

Published online by Cambridge University Press:  17 April 2009

Song Jiang
Song Jiang
Affiliation:
Institut für Angewandte Mathematik, Wegelerstrasse 10, D-5300 Bonn 1, West Germany Department of Mathematics, Xi'an Jiaotong University, Xi'an, Shaanxi Province Peoples, Republic of China
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we study the asymptotic behaviour, as |x| → ∞, of solutions to the initial value problem in nonlinear three-dimensional thermoelasticity in some weighted Sobolev spaces. We show that under some conditions, solutions decrease fast for each t as x tends to infinity. We also consider the possible extension of the method presented in this paper to the initial boundary value problem in exterior domains.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 43 , Issue 1 , February 1991 , pp. 89 - 99
Copyright
Copyright © Australian Mathematical Society 1991

References

[l]Adams, R.A., Sobolev Spaces (Academic Press, New York, 1975).Google Scholar
[2]Carlson, D.E., ‘Linear Thermoelasticity’, in Hαndbuch der Physik VIα/2, pp. 297–345 (Springer-Verlag, Berlin, Heidelberg, New York, 1972).Google Scholar
[3]Chrzȩszczyk, A., ‘Some existence results in dynamical thermoelasticity; part I nonlinearcase’, Arch. Mech. 39 (1987), 605–617.Google Scholar
[4]Jiang, S., ‘Far field behavior of solutions to the equations of nonlinear 1-d-thermoelasticity’, Appl. Anal. 36 (1990), 25–35.Google Scholar
[5]Jiang, S., ‘Numerical solution for the Cauchy problem in nonlinear 1-d-thermoelasticity’, Computing 44 (1990), 147–158.Google Scholar
[6]Jiang, S. and Racke, R., ‘On some quasilinear hyperbolic-parabolic initial boundary value problems’, Math. Methods Appl. Sci. 12 (1990), 315–319.Google Scholar
[7]Kawashima, S., Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Thesis (Kyoto University, 1983).Google Scholar
[8]Racke, R., ‘On the Cauchy problem in nonlinear 3-d-thermoelasticity’, Math. Z. 203 (1990), 649–682.Google Scholar
[9]Slemrod, M., ‘Global existence, uniqueness and asymptotic stability of classical smooth solutions in one-dimensional non-linear thermoelasticity’, Arch. Rational Mech. Anal. 76 (1981), 97–133.Google Scholar
[10]Triebel, H., ‘Spaces of distributions with weights. Multipliers in Lp-spaces with weights’, Math. Nachr. 78 (1977), 339–355.Google Scholar
[11]Triebel, H., Theory of Function Spaces (Birkhäuser-Verlag, Basel, 1983).Google Scholar
[12]Tsutsumi, M., ‘Weighted Sobolev spaces and rapidly decreasing solutions of some nonlinear dispersive wave equations’, J. Differential Equations 42 (1981), 260–281.Google Scholar
[13]Vol'pert, A.I. and Hudjaev, S.I., ‘On the Cauchy problem for composite systems of nonlinear differential equations’, Math. USSR-Sb 16 (1972), 517–544.Google Scholar
[14]Zheng, S. and Shen, W., ‘Initial boundary value problems for quasilinear hyperbolic-parabolic coupled systems in higher dimensional spaces’, Chinese Ann. Math. Ser. B 4 (1983), 443–462.Google Scholar