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Existence of weak solutions for the thermistor problem with degeneracy. (English) Zbl 1029.35059
This paper is devoted to the study of coupled parabolic-elliptic system of partial differential equations related to the thermistor problem. More precisely, authors research the existence of solutions of \[ \partial_t u-\triangle \theta(u)=\sigma(u)|\nabla\varphi|^2, \qquad \nabla\cdot(\sigma(u)\nabla\varphi)=0 \] with mixed Dirichlet, Robin conditions on \(u\) and Dirichlet, homogeneous Neumann conditions on \(\varphi\). Their goal is to prove existence of weak solutions when \(\theta\) is a continuous non decreasing function from \(\mathbb{R}\) to \(\mathbb{R}\), with \(\theta(0)=0\) and \(\sigma\) is a real positive continuous function. This extends a result of X. Xu [Nonlinear Anal., Theory Methods Appl. 42, 199-213 (2000; Zbl 0964.35005)]. The solution is obtained as a limit of a sequence of weak solutions of some regularized-truncated problem associated with the previous equations. Since the Dirichlet condition on \(u\) is supposed to be bounded, it is also true for the weak solution \((u,\varphi)\).

35D05 Existence of generalized solutions of PDE (MSC2000)
35Q60 PDEs in connection with optics and electromagnetic theory
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