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Existence for the thermoviscoelastic thermistor problem. (English) Zbl 1178.74033

Summary: 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. 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.

MSC:

74D05 Linear constitutive equations for materials with memory
74F05 Thermal effects in solid mechanics
74H20 Existence of solutions of dynamical problems in solid mechanics
74H30 Regularity of solutions of dynamical problems in solid mechanics
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