@Article{ATA-37-4,
author = {Xu, Xiangsheng},
title = {Regularity Results for a Nonlinear Elliptic-Parabolic System with Oscillating Coefficients},
journal = {Analysis in Theory and Applications},
year = {2021},
volume = {37},
number = {4},
pages = {541--556},
abstract = {<p style="text-align: justify;">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&nbsp; $\liminf_{u\rightarrow\infty}\sigma(u)=0$, 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.&nbsp; As it is known, the latter implies that $\ln\sigma(u)$ is &quot;nearly bounded&#39;&#39;.</p>},
issn = {1573-8175},
doi = {https://doi.org/10.4208/ata.OA-2020-0021},
url = {https://global-sci.com/article/73797/regularity-results-for-a-nonlinear-elliptic-parabolic-system-with-oscillating-coefficients}
}