El Hachimi, Abderrahmane; Ammi, Moulay Rachid Sidi Existence of weak solutions for the thermistor problem with degeneracy. (English) Zbl 1029.35059 Electron. J. Differ. Equ. 2002, Conf. 09, 127-137 (2002). 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 \((u,\varphi)\). Reviewer: Francisco José Pena Brage (Santiago de Compostela) Cited in 2 Documents MSC: 35D05 Existence of generalized solutions of PDE (MSC2000) 35Q60 PDEs in connection with optics and electromagnetic theory Keywords:coupled parabolic-elliptic system; regularized-truncated problem; regularization; theorem of Leray Schauder Citations:Zbl 0964.35005 PDFBibTeX XMLCite \textit{A. El Hachimi} and \textit{M. R. S. Ammi}, Electron. J. Differ. Equ. 2002, 127--137 (2002; Zbl 1029.35059) Full Text: EuDML EMIS