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Partial regularity of solutions to a class of degenerate systems
Authors:Xiangsheng Xu
Affiliation:Department of Mathematics and Statistics, Mississippi State University, Mississippi State, Mississippi 39762
Abstract:
We consider the system $displaystyle frac{partial u }{partial t}-Delta u=sigma left ( uright ) left | nabla varphi right | ^2$, $ mathrm {div}left ( sigma left ( uright ) nabla varphi right ) =0$ in $Q_Tequiv Omega times left ( 0,Tright ] $ coupled with suitable initial-boundary conditions, where $Omega $ is a bounded domain in $ mathbf {R}^N$ with smooth boundary and $sigma left ( uright ) $ is a continuous and positive function of $u$. Our main result is that under some conditions on $sigma $ there exists a relatively open subset $Q_0$ of $Q_T$ such that $u$ is locally Hölder continuous on $Q_0$, the interior of $Q_Tbackslash Q_0$ is empty, and $u$ is essentially bounded on $Q_Tbackslash Q_0$.

Keywords:Partial regularity   degenerate systems
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