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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 $$(u,\varphi)$$.

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