The existence of global weak solutions to the degenerate problem describing Joule's heating in a current and heat conductive medium is proved via the Galerkin method. The existence proof proceeds by a sequence of a priori estimates, which may be achieved by the standard method. Under suitable hypotheses on the electrical conductivity the boundedness in L_infinity (Q_T) of the absolute temperature of the medium is established by the method of Stampacchia. The paper is an extension of the work by Cimatti [3,4], Shi et al. [12] and Xu [16]. This extension consists of employing the equations for the electric field E, derived from Maxwell's equations, instead of the equation for the electric potential, with the appropriate modification in the energy balance equation.
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