Existence for the thermoelastic thermistor problem

We study a new model for the thermistor problem that consists of a system of dynamic thermoelasticity equations of displacement and a stationary charge conservation equation of electrical current. The heat source generated by Joule heating is quadratic in the gradient of the electrical potential. This system is nonlinear and degenerate since the electrical conductivity is assumed to be temperature dependent and vanishes at large temperatures. We establish the existence of a very weak solution, the so-called “capacity solution” for the model. The proof is based on regularization and time retarding.