Abstract
For a semilinear system of equations of thermoelasticity, we establish a theorem on the existence and uniqueness of global solutions in a multidimensional space under the condition that the initial data are sufficiently small. We also obtain estimates for the decrease of solutions as time increases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Slemrod, “Global existence, uniqueness, and asymptotic stability of classical smooth solutions in one-dimensional nonlinear ther- moelasticity,” Arch. Ration. Mech. Anal, 76, No. 2, 97–133 (1981).
R. Racke, “Initial boundary-value problems in thermoelasticity,” Lect. Notes Math., 1357, 341–358 (1988).
R. Racke, Y. Shibata, and S. Zheng, “Global solvability and exponential stability in one-dimensional nonlinear thermoelasticity,” Quart. Appl. Math., 51, No. 4, 751–763 (1993).
S. Kawashima and M. Okada, “Smooth global solutions for the one-dimensional equations in magnetohydrodynamics,” Proc. Jpn. Acad. Ser. A, Math. Sci., 53, 384–387 (1982).
S. Zheng and W. Shen, “Global solutions to the Cauchy problem for quasilinear hyperbolic parabolic coupled systems,” Sci. Sinica. Ser. A, 30, 1133–1149 (1987).
W. J. Hrusa and M. A. Tarabek, “On smooth solutions of the Cauchy problem in one-dimensional nonlinear thermoelasticity,” Quart. Appl. Math., 47, 631–644 (1989).
D. Hoff and K. Zumbrun, “Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow,” Indiana Univ. Math. J., 44, No. 2, 603–676 (1995).
V. G. Maz’ya, Sobolev Spaces [in Russian] Leningrad University, Leningrad 1985.
M. Reed and B. Simon, Methods of Modem Mathematical Physics [Russian translation], Vol. 2, Mir, Moscow (1978).
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin 1981.
Ph. Clement, H. J. A. M. Heijmans, S. Angenent, C. J. van Duijn, and B. de Pagter, One-Parameter Semigroups, North-Holland, Amsterdam 1987.
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction [Russian translation] Mir, Moscow 1980.
O. V. Besov, V. P. IT in, and S. M. Nikol’skii, Integral Representations of Functions and Imbedding Theorems [in Russian] Nauka, Moscow 1975.
J. A. Goldstein, Semigroups of Linear Operators and Applications [Russian translation] Vyshcha Shkola, Kiev 1989.
M. Reed, “Abstract nonlinear wave equations,” Lect. Notes Math., 507 (1976).
I. Segal, “Nonlinear semigroups,” Ann. Math., 78, 339–364 (1963).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Botsenyuk, O.M. Global small solutions of the cauchy problem for a semilinear system of equations of thermoelasticity. Ukr Math J 52, 499–512 (2000). https://doi.org/10.1007/BF02515393
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02515393