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Volume 37, Issue 4
Regularity Results for a Nonlinear Elliptic-Parabolic System with Oscillating Coefficients

Xiangsheng Xu

Anal. Theory Appl., 37 (2021), pp. 541-556.

Published online: 2021-11

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  • 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''.

  • Keywords

Oscillating coefficients, the thermistor problem, quadratic nonlinearity.

  • AMS Subject Headings

35B45, 35B65, 35M33, 35Q92

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-37-541, author = {Xiangsheng and Xu and and 21371 and and Xiangsheng Xu}, 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 = {

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''.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-2020-0021}, url = {http://global-sci.org/intro/article_detail/ata/19963.html} }
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''.

Xiangsheng Xu. (1970). Regularity Results for a Nonlinear Elliptic-Parabolic System with Oscillating Coefficients. Analysis in Theory and Applications. 37 (4). 541-556. doi:10.4208/ata.OA-2020-0021
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