Abstract
The existence of a weak solution to a dynamic model for a thermistor, which takes into account the thermoelastic properties of the device, is established. The model consists of a coupled system of the equations of dynamic thermoviscoelasticity, the heat equation with the Joule heating term, and the quasistatic charge conservation equation. The system is strongly nonlinear since the electrical conductivity is assumed to be temperature dependent, and the Joule heating term is quadratic in the gradient of the electric potential. The existence of a solution is obtained by considering a sequence of approximate time-retarded problems. After obtaining the necessary a priori estimates, a solution of the problem is found by passing to the approximation limit. The uniqueness of the solution remains an open problem.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
W. Allegreto, Y. Lin and S. Ma, Existence and long time behaviour of solutions to obstacle thermistor equations, Discrete Contin. Dynam. Systems, 8(2002), 757–780.
W. Allegreto, Y. Lin and A. Zhou, A box scheme for coupled systems resulting from microsensor thermistor problems, Dynam. Contin. Discrete. Impuls. Systems, 5(1999) 209–223.
W. Allegretto and H. Xie, A non-local thermistor problem, European J. Appl. Math., 6(1)(1995) 83–94.
K. T. Andrews, K. L. Kuttler and M. Shillor, On the dynamic behaviour of a thermoviscoelastic body in frictional contact with a rigid obstacle, European J. Appl. Math., 8(1997) 417–436.
S. N. Antontsev and M. Chipot, The thermistor problem: existence, smoothness, uniqueness, blowup, SIAM J. Math. Anal., 25(4)(1994) 1128–1156.
E. Bonetti and G. Bonfanti, Existence and uniqueness of the solution to a 3D thermoviscoelastic system, Electron. J. Differential Equations, 50 (2003) 1–15.
X. Chen, Existence and regularity of solutions of a nonlinear degenerate elliptic system arising from a thermistor problem, J. Partial Differential Equations, 7(1994) 19–34.
G. Cimatti, Remark on the existence and uniqueness for the thermistor problem under mixed boundary conditions, Quart. J. Appl. Math., 47(1989) 117–121.
G. Cimatti, Stability and multiplicity of solutions for the thermistor problem, Ann. Mat. Pura Appl., 181(2)(2002) 181–212.
E.A. Coddington and N. Levinson, “Theory of Ordinary Differential Equations,” McGraw-Hill, New York, 1955.
G. Duvaut and J.L. Lions, “Inequalities in Mechanics and Physics,” Springer, New-York, 1976.
M. T. González Montesinos and F. Ortegón Gallego, The stationary thermistor problem with degenerate thermal conductivity (in spanish). Proceedings of the Meeting of Andalusian Mathematicians, Vol. II (Sevilla, 2000), 519–527, Colecc. Abierta, 52, Univ. Sevilla Secr. Publ., Seville, 2001
J. R. Fernández, K. L. Kuttler, M.C. Muñiz and M. Shillor, A model and simulations of the thermoviscoelastic thermistor, in preparation.
R. Gariepy, M. Shillor and X. Xu, Existence of capacity solutions to a model for In Situ Vitrification, European J. Appl. Math., 9(9)(1998) 543–559.
S. D. Howison, A note on the thermistor problem in two space dimension, Quart. J. Appl. Math., 47(3)(1989) 509–512.
S. D. Howison, J. Rodrigues and M. Shillor, Stationary solutions to the thermistor problem, J. Math. Anal. Appl., 174(1993) 573–588.
S. Kutluay, A. R. Bahadir and A. Ozdeć, Various methods to the thermistor problem with a bulk electrical conductivity, Internat. J. Numer. Methods Engrg., 45(1)(1999) 1–12.
K. L. Kuttler and M. Shillor, Set-valued pseudomonotone maps and degenerate evolution inclusions, Comm. Contemp. Math., 1(1)(1999) 87–123.
A. A. Lacey, Thermal runaway in a nonlocal problem modelling Ohmic heating: Part I: Model derivation and some special cases, Euro. J. Appl. Math., 6(1995) 127–144.
J. L. Lions, “Quelques méthodes de résolution des problèmes aux limites non linéaires”, Dunod, Paris, 1969.
P. Shi, M. Shillor, and X. Xu, Existence of a solution to the Stefan problem with Joule’s heating, J. Differential Equations, 105(2)(1993) 239–263.
R. Showalter, Degenerate evolution equations and applications, Indiana Univ. Math. J., 23 (1973/74) 655–677.
X. Wu and X. Xu, Existence for the thermoelastic thermistor problem, J. Math. Anal. Appl., 319 (2006) 124–138.
H. Xie and W. Allegretto, C α(\( \bar \Omega \)) solutions of a class of nonlinear degenerate elliptic systems arising in the thermistor problem, SIAM J. Math. Anal., 22(6)(1991) 1491–1499.
X. Xu, The thermistor problem with conductivity vanishing for large temperature, Proc. Roy. Soc. Edinburgh, 124(1994) 1–21.
X. Xu, On the existence of bounded temperature in the thermistor problem with degeneracy, Nonlinear Anal., 42 (2000) 199–213.
X. Xu and M. Shillor, The Stefan problem with convection and Joule’s heating, Adv. Differential Equations, 2(4)(1997) 667–691.
S. Zhou and D. R. Westbrook, Numerical solutions of the thermistor equations, J. Comput. Appl. Math., 79(1)(1997) 101–118.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kuttler, K.L., Shillor, M. & Fernández, J.R. Existence for the thermoviscoelastic thermistor problem. Differ Equ Dyn Syst 16, 309–332 (2008). https://doi.org/10.1007/s12591-008-0017-z
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12591-008-0017-z