Year: 2021
Author: Xiangsheng Xu
Analysis in Theory and Applications, Vol. 37 (2021), Iss. 4 : pp. 541–556
Abstract
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''.
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ata.OA-2020-0021
Analysis in Theory and Applications, Vol. 37 (2021), Iss. 4 : pp. 541–556
Published online: 2021-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 16
Keywords: Oscillating coefficients the thermistor problem quadratic nonlinearity.