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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
Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK

Abstract

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.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1995

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