The thermistor problem for conductivity which vanishes at large temperature

Abstract

The thermistor problem is modeled as a coupled system of nonlinear elliptic equations. When the conductivity coefficient σ ( u ) \sigma \left ( u \right ) vanishes ( u u = temperature) one of the equations becomes degenerate; this situation is considered in the present paper. We establish the existence of a weak solution and, under some special Dirichlet and Neumann boundary conditions, analyze the structure of the set { σ ( u ) = 0 } \left \{ {\sigma \left ( u \right ) = 0} \right \} and also prove uniqueness.