Stationary Solutions to the Thermistor Problem

A mathematical model for the description of heat conduction in a conductor in the presence of Joule heating is considered. The problem is to solve a coupled system of partial differential equations consisting of the heat equation with Joule heating as a source, and the current conservation equation with temperature dependent electrical conductivity. In the steady state this is a coupled system of elliptic equations, subject to general boundary conditions. The existence of a weak solution is proved using Schauder′s fixed point theorem together with elliptic estimates. Uniqueness of the solution is proved for sufficiently small data. A nonlocal nonlinear boundary condition is considered as well and existence of a weak solution to this problem is shown together with a sufficient condition for the existence of more than one solution.