On a contact problem in thermoelasticity with second sound
Author:
Jan Sprenger
Journal:
Quart. Appl. Math. 67 (2009), 601-615
MSC (2000):
Primary 35Q99
DOI:
https://doi.org/10.1090/S0033-569X-09-01132-7
Published electronically:
July 29, 2009
MathSciNet review:
2588226
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We investigate the existence and stability of a thermoelastic contact problem with second sound. Previous results established the existence and stability of a solution of the corresponding classical system in the case of radial symmetry. However, recent works have shown that sometimes stability can be lost when the classical Fourier heat conduction is substituted by Cattaneo’s Law. We show that also in this case this substitution does indeed lead to a loss in regularity that proves to be a major problem prohibiting the transfer of the existence proof for the classical problem to the problem with second sound, leaving the existence of a solution an open question. We then prove that, if a viscoelastic term is added to the equations, providing additional regularity, then existence and exponential stability (the second, as can be expected, only in the case of radial symmetry) will follow.
References
- K. A. Ames and L. E. Payne, Uniqueness and continuous dependence of solutions to a multidimensional thermoelastic contact problem, J. Elasticity 34 (1994), no. 2, 139–148. MR 1288855, DOI https://doi.org/10.1007/BF00041189
- C. M. Elliott and Qi Tang, A dynamic contact problem in thermoelasticity, Nonlinear Anal. 23 (1994), no. 7, 883–898. MR 1302150, DOI https://doi.org/10.1016/0362-546X%2894%2990126-0
- Hongjun Gao and Jaime E. Muñoz Rivera, Global existence and decay for the semilinear thermoelastic contact problem, J. Differential Equations 186 (2002), no. 1, 52–68. MR 1941092, DOI https://doi.org/10.1016/S0022-0396%2802%2900016-5
- Jong Uhn Kim, A boundary thin obstacle problem for a wave equation, Comm. Partial Differential Equations 14 (1989), no. 8-9, 1011–1026. MR 1017060, DOI https://doi.org/10.1080/03605308908820640
- Jaime E. Muñoz Rivera and Song Jiang, The thermoelastic and viscoelastic contact of two rods, J. Math. Anal. Appl. 217 (1998), no. 2, 423–458. MR 1492098, DOI https://doi.org/10.1006/jmaa.1997.5717
- Jaime E. Muñoz Rivera and Reinhard Racke, Multidimensional contact problems in thermoelasticity, SIAM J. Appl. Math. 58 (1998), no. 4, 1307–1337. MR 1628692, DOI https://doi.org/10.1137/S003613999631306X
- H. Ferñandez Sare and R. Racke, On the stability of damped Timoshenko systems - Cattaneo versus Fourier law, Archive Rational Mech. Anal. (to appear) (2007).
- Peter Shi and Meir Shillor, Existence of a solution to the $N$-dimensional problem of thermoelastic contact, Comm. Partial Differential Equations 17 (1992), no. 9-10, 1597–1618. MR 1187623, DOI https://doi.org/10.1080/03605309208820897
References
- K. A. Ames and L. E. Payne, Uniqueness and continuous dependence of solutions to a multidimensional thermoelastic contact problem, J. Elasticity 34 (1994), 139–148. MR 1288855 (95f:73004)
- C. M. Elliott and Q. Tang, A dynamic contact problem in thermoelasticity, Nonlinear Analysis 23 (1994), 883–898. MR 1302150 (95i:73013)
- H. Gao and J. Muñoz Rivera, Global existence and decay for the semilinear thermoelastic contact problem, J. Differential Equations 186 (2002), 52–68. MR 1941092 (2004a:74010)
- J.U. Kim, A boundary thin obstacle problem for a wave equation, Comm. Partial Differential Equations 14 (1989), 1011–1026. MR 1017060 (91a:35121)
- J. Muñoz Rivera and S. Jiang, The thermoelastic and viscoelastic contact of two rods, J. Math. Anal. Appl. 217 (1998), 423–458. MR 1492098 (98m:73076)
- J. Muñoz Rivera and R. Racke, Multidimensional contact problems in thermoelasticity, SIAM J. Appl. Math. 58 (1998), 1307–1337. MR 1628692 (99f:73015)
- H. Ferñandez Sare and R. Racke, On the stability of damped Timoshenko systems - Cattaneo versus Fourier law, Archive Rational Mech. Anal. (to appear) (2007).
- P. Shi and M. Shillor, Existence of a solution to the $N$-dimensional problem of thermoelastic contact, Comm. Partial Differential Equations 17 (1992), 1597–1618. MR 1187623 (93i:73052)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2000):
35Q99
Retrieve articles in all journals
with MSC (2000):
35Q99
Additional Information
Jan Sprenger
Affiliation:
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria
Email:
jsprenger@asc.tuwien.ac.at
Received by editor(s):
March 18, 2008
Published electronically:
July 29, 2009
Additional Notes:
The author would like to thank Professor Dr. Racke for helpful suggestions and the opportunity to work on this topic.
Article copyright:
© Copyright 2009
Brown University
The copyright for this article reverts to public domain 28 years after publication.
