J. R. Fern\'andez, K. L. Kuttler and Meir Shillor,

The existence of a weak solution to a dynamic model for a thermistor,
which takes into account the thermoelastic properties of the device, is
established. The model consists of a coupled system of the equations of
dynamic thermoviscoelasticity, the heat equation with the Joule heating
term, and the quasistatic charge conservation equation. The system is
strongly nonlinear since the electrical conductivity is assumed to be
temperature dependent, and the Joule heating term is quadratic in the
gradient of the electric potential. The existence of a solution is obtained
by considering a sequence of approximate time-retarded problems for
which the theory of set-valued pseudomonotone operators is applied. Then,
after obtaining the necessary a priori estimates, a solution of the problem
is found by passing to the approximation limit. The uniqueness of the solution
remains an open problem.