Some Results on the Thermistor Problem

Abstract

We would like to consider here the so called Thermistor problem. The heat produced is a conductor by an electric current leads system:

$$\left\{ {\begin{array}{*{20}{c}} {{{u}_{t}} - \nabla \cdot k\left( u \right)\nabla u = \sigma \left( u \right){{{\left| {\nabla \varphi } \right|}}^{2}} in \Omega \times \left( {0,T} \right), \left( {1.1.1} \right)} \hfill \\ {u = 0 on \Gamma \times \left( {0,T} \right), u\left( { \cdot ,0} \right) = {{u}_{0}}, \left( {1.1.2} \right)} \hfill \\ {\nabla \cdot \sigma \left( u \right)\nabla \varphi = 0 in \Omega \times \left( {0,T} \right), \left( {1.1.3} \right)} \hfill \\ {\varphi = {{\varphi }_{0}} on \Gamma \times \left( {0,T} \right). \left( {1.1.4} \right)} \hfill \\ \end{array} } \right. $$

(1.1)

Here, Ω is a smooth bounded open set of R ⁿ, Γ denotes its boundary,Т is some positive given number,φ is the electrical potential, the thermal conductivity and σ(u) >0 the electrical conductivity. Of course the physical situation is when n = 3 and Ω is the spatial domain occupied by the body that we consider and which is assumed to conduct both heat and electricity. However, the mathematical results are worth to be considered for any n≥1.