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Existence and uniqueness for the nonstationary problem of the electrical heating of a conductor due to the Joule-Thomson effect. (English) Zbl 0801.35008

Summary: Existence of a weak solution is established for the initial-boundary value problem for the system \[ {\partial\over \partial t} u- \text{div}(\Theta(u)\nabla u)+ \sigma(u)\alpha(u)\nabla u\nabla v= \sigma(u)|\nabla v|^ 2,\quad \text{div}(\sigma(u)\nabla v)= 0. \] The question of uniqueness is also considered in some special cases.

MSC:

35D05 Existence of generalized solutions of PDE (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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