Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Asymptotic behavior of discontinuous solutions in 3-D thermoelasticity with second sound

Authors: Reinhard Racke and Ya-Guang Wang
Journal: Quart. Appl. Math. 66 (2008), 707-724
MSC (2000): Primary 35B40, 74F05
Published electronically: October 7, 2008
MathSciNet review: 2465141
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Abstract: This paper is devoted to the study of the Cauchy problem for linear and semilinear thermoelastic systems with second sound in three space dimensions with discontinuous initial data. Due to Cattaneo's law, replacing Fourier's law for heat conduction, the thermoelastic system with second sound is hyperbolic. We investigate the behavior of discontinuous solutions as the relaxation parameter tends to zero, which corresponds to a formal convergence of the system to the hyperbolic-parabolic type of classical thermoelasticity. By studying expansions with respect to the relaxation parameter of the jumps of the potential part of the system on the evolving characteristic surfaces, we obtain that the jump of the temperature goes to zero while the jumps of the heat flux and the gradient of the potential part of the elastic wave are propagated along the characteristic curves of the elastic fields when the relaxation parameter goes to zero. An interesting phenomenon is when time goes to infinity: the behavior will depend on the mean curvature of the initial surface of discontinuity. These jumps decay exponentially when time goes to infinity, more rapidly for a smaller heat conductive coefficient in linear problems and in nonlinear problems when certain growth conditions are imposed on the nonlinear functions.

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Additional Information

Reinhard Racke
Affiliation: Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany
Email: reinhard.racke@uni-konstanz.de

Ya-Guang Wang
Affiliation: Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
Email: ygwang@sjtu.edu.cn

DOI: https://doi.org/10.1090/S0033-569X-08-01121-2
Keywords: Hyperbolic thermoelasticity, semilinear, discontinuous solutions, asymptotic behavior, curvature
Received by editor(s): May 20, 2007
Published electronically: October 7, 2008
Article copyright: © Copyright 2008 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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