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Thermal runaway in a non-local problem modelling Ohmic heating: Part I: Model derivation and some special cases

Published online by Cambridge University Press:  26 September 2008

A. A. Lacey
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK


We consider the non-local problem

which models the temperature when an electric current flows through a material with temperature dependent electrical resistivity f(u) > 0, subject to a fixed potential difference. It is found that for some special cases where f is decreasing and

so the problem can be scaled to make

then:(a) for λ < 8 there is a unique steady state which is globally asymptotically stable: (b) for λ = 8 there is no steady state and u is unbounded; (c) for λ > 8 there is no steady state and u blows up for all x, – 1 < x < 1.

Research Article
Copyright © Cambridge University Press 1995

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Chen, X. & Friedman, A. 1993 The thermistor problem for conductivity which vanishes at large temperatures. Quart. J. Appl. Math. 51, 101115.CrossRefGoogle Scholar
Cimatti, G. 1989 Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions. Quart. J. Appl. Math. 47, 117121.CrossRefGoogle Scholar
Cimatti, G. 1990 The stationary thermistor problem with a current limiting device. Proc. Roy. Soc. Edinburgh 116A, 7984.Google Scholar
Chafee, N. 1981 The electric ballast resistor: homogeneous and nonhomogeneous equilibria. In: Nonlinear differential equations: invariance stability and bifurcations (de Mottoni, P. & Salvadori, L., Eds.), pp. 97127. Academic Press, New York.CrossRefGoogle Scholar
Fowler, A. C., Frigaard, I. & Howison, S. D. 1992 Temperature surges in current-limiting circuit devices. SIAM J. Appl. Math. 52, 9981011.CrossRefGoogle Scholar
Fujita, H. 1969 On the nonlinear equations Δu + eu = 0 and υt = δυ + e υ. Bull. Am. Math. Soc. 75, 132135.CrossRefGoogle Scholar
Howison, S. D. 1989 A note on the thermistor problem in two space dimensions. Quart. J. Appl. Math. 47, 509512.CrossRefGoogle Scholar
Keller, H. B. & Cohen, D. S. 1967 Some positone problems suggested by nonlinear heat generation. J. Math. Mech. 16, 13611376.Google Scholar
Lacey, A. A. 1983 Mathematical analysis of thermal runaway for spatially inhomogeneous reactions. SIAM J. Appl. Math. 43, 13501366.CrossRefGoogle Scholar
Lacey, A. A. 1994 Thermal runaway in a non-local problem modelling Ohmic heating. II. General proof of blow-up and asymptotics of runaway. Euro. J. Appl. Math. To appear.Google Scholar
Sattinger, D. H. 1972 Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 21, 9791000.CrossRefGoogle Scholar
Xie, H. & Allegretto, W. 1991 Cx() solutions of a class of nonlinear degenerate elliptic systems arising in the thermistor problem. SIAM J. Math. Anal. 22, 14911499.CrossRefGoogle Scholar