A Degenerate Stefan-Like Problem with Joule’s Heating

Abstract

This paper studies the system $({\partial / {\partial t}})\alpha (u) - {\operatorname{div}}\,a(\nabla u) \ni \sigma (u)|\nabla \varphi |^2 $, ${\operatorname{div}}\,(\sigma (u)\nabla \varphi ) = 0$ in a bounded domain of $\mathbb{R}^N $ coupled with initial and boundary conditions. Here, $\alpha $ is a maximal monotone graph in $\mathbb{R}$, a a monotone mapping from $\mathbb{R}^N $ to $\mathbb{R}^N $, and $\sigma $ a positive function on $\mathbb{R}$ with the limit of $\sigma (s)$ as $|s| \to \infty $ being zero. In the generality considered here, the problem may not always have a solution in the sense of distributions. Under certain assumptions on the data, an existence assertion is established for the problem that incorporates the new phenomena involved and, at the same time, retains the main feature of a classical weak solution.