TY - JOUR T1 - Regularity Results for a Nonlinear Elliptic-Parabolic System with Oscillating Coefficients AU - Xu , Xiangsheng JO - Analysis in Theory and Applications VL - 4 SP - 541 EP - 556 PY - 2021 DA - 2021/11 SN - 37 DO - http://doi.org/10.4208/ata.OA-2020-0021 UR - https://global-sci.org/intro/article_detail/ata/19963.html KW - Oscillating coefficients, the thermistor problem, quadratic nonlinearity. AB -

In this paper we study the initial boundary value problem for the system $\mbox{div}(\sigma(u)\nabla\varphi)=0,$ $ u_t-\Delta u=\sigma(u)|\nabla\varphi|^2$. This problem is known as the thermistor problem which models the electrical heating of conductors. Our assumptions on $\sigma(u)$ leave open the possibility that  $\liminf_{u\rightarrow\infty}\sigma(u)=0$, while $\limsup_{u\rightarrow\infty}\sigma(u)$ is large. This means that $\sigma(u)$ can oscillate wildly between $0$ and a large positive number as $u\rightarrow \infty$. Thus our degeneracy is fundamentally different from the one that is present in porous medium type of equations. We obtain a weak solution $(u, \varphi)$ with $|\nabla \varphi|, |\nabla u|\in L^\infty$ by first establishing a uniform upper bound for $e^{\varepsilon u}$ for some small $\varepsilon$. This leads to an inequality in $\nabla\varphi$, from which the regularity result follows. This approach enables us to avoid first proving the Hölder continuity of $\varphi$ in the space variables, which would have required that the elliptic coefficient $\sigma(u)$ be an $A_2$ weight.  As it is known, the latter implies that $\ln\sigma(u)$ is "nearly bounded''.