The thermistor problem for conductivity which vanishes at large temperature
Authors:
Xinfu Chen and Avner Friedman
Journal:
Quart. Appl. Math. 51 (1993), 101-115
MSC:
Primary 35R35; Secondary 35J45, 80A20
DOI:
https://doi.org/10.1090/qam/1205940
MathSciNet review:
MR1205940
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Abstract: The thermistor problem is modeled as a coupled system of nonlinear elliptic equations. When the conductivity coefficient $\sigma \left ( u \right )$ vanishes ($u$ = temperature) one of the equations becomes degenerate; this situation is considered in the present paper. We establish the existence of a weak solution and, under some special Dirichlet and Neumann boundary conditions, analyze the structure of the set $\left \{ {\sigma \left ( u \right ) = 0} \right \}$ and also prove uniqueness.
J. Bass, Thermoelasticity, McGraw-Hill Encyclopedia of Physics (S. P. Parker, ed.), McGraw-Hill, New York, 1982
- Giovanni Cimatti and Giovanni Prodi, Existence results for a nonlinear elliptic system modelling a temperature dependent electrical resistor, Ann. Mat. Pura Appl. (4) 152 (1988), 227–236. MR 980982, DOI https://doi.org/10.1007/BF01766151
- Giovanni Cimatti, A bound for the temperature in the thermistor problem, IMA J. Appl. Math. 40 (1988), no. 1, 15–22. MR 983747, DOI https://doi.org/10.1093/imamat/40.1.15
- Giovanni Cimatti, Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions, Quart. Appl. Math. 47 (1989), no. 1, 117–121. MR 987900, DOI https://doi.org/10.1090/S0033-569X-1989-0987900-9
H. Diesselhorst, Ueber das Probleme eines elektrisch erwärmten Leiters, Ann. Physics 1, 312–325 (1900)
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
- Sam Howison, A note on the thermistor problem in two space dimensions, Quart. Appl. Math. 47 (1989), no. 3, 509–512. MR 1012273, DOI https://doi.org/10.1090/qam/1012273
S. D. Howison, J. F. Rodrigues, and M. Shillor, Existence results for the problems of Joule heating of a resistor, to appear
F. J. Hyde, Thermistors, Iliffe Books, London, 1971
J. F. Llewellyn, The Physics of Electrical Contacts, Clarendon Press, Oxford, 1957
J. M. Young, Steady state Joule heating with temperature dependent conductivities, Appl. Sci. Res. 43, 55–65 (1986)
J. Bass, Thermoelasticity, McGraw-Hill Encyclopedia of Physics (S. P. Parker, ed.), McGraw-Hill, New York, 1982
G. Cimatti and G. Prodi, Existence results for a nonlinear elliptic system modelling a temperature dependent electrical resistor, Ann. Math. Pure Appl. 152, 227–236 (1988)
G. Cimatti, A bound for the temperature in the thermistor problem, IMA J. Appl. Math. 40, 15–22 (1988)
---, Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions, Quart. Appl. Math. 47, 117–121 (1989)
H. Diesselhorst, Ueber das Probleme eines elektrisch erwärmten Leiters, Ann. Physics 1, 312–325 (1900)
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, 1983
S. Howison, A note on the thermistor problem in two space dimensions, Quart. Appl. Math. 47, 509–512 (1989)
S. D. Howison, J. F. Rodrigues, and M. Shillor, Existence results for the problems of Joule heating of a resistor, to appear
F. J. Hyde, Thermistors, Iliffe Books, London, 1971
J. F. Llewellyn, The Physics of Electrical Contacts, Clarendon Press, Oxford, 1957
J. M. Young, Steady state Joule heating with temperature dependent conductivities, Appl. Sci. Res. 43, 55–65 (1986)
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© Copyright 1993
American Mathematical Society
