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The thermistor problem with conductivity vanishing for large temperature

Published online by Cambridge University Press:  14 November 2011

Xiangsheng Xu
Xiangsheng Xu
Affiliation:
Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701, U.S.A
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Abstract

We consider the system (∂/∂t)u = ∆u + σ(u)|∇φ|2, div (σ(u)∇φ) = 0 in a bounded region of ℝN coupled with initial and boundary conditions, where σ(s) ∈ C(ℝ) is nonnegative and σ(u) = 0 if and only if u ≧ a for some a > 0. Owing to the degeneracy involved, solutions of the problem display new phenomena that cannot be incorporated into the classical weak formulation. The notion of a capacity solution introduced in [14,15] is employed to study the problem. It turns out that this notion of a solution is just general enough to encompass the new phenomena involved.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 1 , 1994 , pp. 1 - 21
Copyright
Copyright © Royal Society of Edinburgh 1994

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