Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-17T15:49:15.542Z Has data issue: false hasContentIssue false
  • English
  • Français

A non-local thermistor problem

Published online by Cambridge University Press:  26 September 2008

W. Allegretto  and
H. Xie
W. Allegretto
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, CanadaT6G 2G1
H. Xie
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, CanadaT6G 2G1
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the question of the existence/non-existence of solutions for the non-local nonlinear elliptic system which models a thermistor driven by a current source. Specifically, we show that for small input current there exists a solution, while this will not in general be the case for a sufficiently large current. A feature of our estimates is that the conditions for non-existence are determined by local criteria on the domain and the coefficients. Our basic tools for existence involve truncation, L2.μ estimates and fixed point arguments. Non-existence is obtained by averaging procedures and an application of Barta's Inequality.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 6 , Issue 1 , February 1995 , pp. 83 - 94
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Macklen, E. D. 1979 Thermistors. Electrochemical Publications.Google Scholar
[2]Audet, S. & Middelhoek, S. 1989 Silicon Sensors. Academic Press.Google Scholar
[3]Kamins, T. 1988 Polycrystalline Silicon for Integrated Circuit Applications. Kluwer Academic.CrossRefGoogle Scholar
[4]Muller, R., Howe, R., Senturia, S., Smith, R. & White, R. 1991 Microsensors. IEEE Press.Google Scholar
[5]Cimatti, G. 1988 A bound for the temperature in the thermistor problem. IMAJ. Appl. Math. 40, 15–22.CrossRefGoogle Scholar
[6]Cimatti, G. 1991 On two problems of electrical heating of conductors. Quart. Appl. Math. XLIX (4), 729–740.CrossRefGoogle Scholar
[7]Cimatti, G. & Prodi, G. 1988 Existence results for a nonlinear elliptic system modelling a temperature dependent electrical resistor. Annali di Matematica 151–152, 227–236.Google Scholar
[8]Howison, S. D., Rodrigues, J. F. & Shillor, M.Stationary solutions to the thermistor problem. J. Math. Anal. Appl. (To appear.)Google Scholar
[9]Chen, X. Existence and regularity of solutions of a nonlinear degenerate elliptic system arising from a thermistor problem. (Preprint.)Google Scholar
[10]Cimatti, G. 1989 Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions. Quart. Appl. Math. XLVII, 117–121.CrossRefGoogle Scholar
[11]Xie, H. & Allegretto, W. 1991 Cα() solutions of a class of nonlinear degenerate elliptic systems arising in the thermistor problem. SIAM J. Math. Anal. 22, 1491–1499.CrossRefGoogle Scholar
[12]Xie, H.L2α(Ω) estimate to the mixed boundary value problem for second order elliptic equations and its applications in the thermistor problem. (Preprint.)Google Scholar
[13]Allegretto, W. & Xie, H. 1992 Existence of solutions for the time-dependent thermistor equations. IMA J. Appl. Math. 48, 271–281.Google Scholar
[14]Cimatti, G. Existence of weak solutions for the nonstationary problem of the Joule heating of a conductor. (To appear.)Google Scholar
[15]Fowler, A. & Howison, S. 1990 Temperature surges in thermistors. In: Proceedings of the 3rd European Conference on Mathematics in Industry (Manley, J. et al. , eds.), Kluwer Academic/B. G. Teubner Stuttgart, pp. 197–204.Google Scholar
[16]Shi, P., Shillor, M. & Xu, X. Existence of a solution to the Stefan problem with Joule's heating. (Preprint.)Google Scholar
[17]Young, J. H. 1986 Steady state Joule heating with temperature dependent conductivities. Appl. Scientific Res. 43, 55–65.CrossRefGoogle Scholar
[18]Chen, X. & Friedman, A. The thermistor problem with one-zero conductivity. IMA Preprint Series no. 793.Google Scholar
[19]Chen, X. & Friedman, A. The thermistor problem for conductivity which vanishes at large temperature. IMA Preprint Series no. 792.Google Scholar
[20]Cimatti, G. 1990 The stationary thermistor problem with a current limiting device. Proc. Royal Soc. Edinburgh 116A, 79–84.Google Scholar
[21]Diesselhorst, H. 1900 Uber das probleme eines elektrisch erwarmter leiters. Ann. Phy. 1, 312–325.CrossRefGoogle Scholar
[22]Troianiello, G. M. 1987 Elliptic Differential Equations and Obstacle Problems. Plenum Press.CrossRefGoogle Scholar
[23]Gilbarg, D. & Trudinger, N. S. 1983 Elliptic Partial Differential Equations of Second Order. Springer-Verlag.Google Scholar
[24]Protter, M. & Weinberger, H. 1967 Maximum Principles in Differential Equations. Prentice-Hall.Google Scholar