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Thermal runaway in a non-local problem modelling Ohmic heating. Part II: General proof of blow-up and asymptotics of runaway

Published online by Cambridge University Press:  26 September 2008

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

We consider the non-local problem

It is found that for the case of decreasing f then: (i) for

there is a unique steady state which is globally asymptotically stable; (ii) for

then the problem can be scaled so that

in which case: (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. Some formal asymptotic estimates for the local behaviour of u as it blows up are obtained.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 6 , Issue 3 , June 1995 , pp. 201 - 224
Copyright
Copyright © Cambridge University Press 1995

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References

Cimatti, G. 1989 Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions. Quart. J. Appl. Math. 47, 117–121.CrossRefGoogle Scholar
Lacey, A. A. 1983 Mathematical analysis of thermal runaway for spatially inhomogeneous reactions. SIAM J. Appl. Math. 43, 1350–1366.Google Scholar
Lacey, A. A. 1994 Thermal runaway in a non-local problem modelling Ohmic heating. I. Model derivation and some special cases. Euro. J. Appl. Math. 6(2), 127–144.CrossRefGoogle Scholar