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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Partial regularity of solutions
to a class of degenerate systems


Author: Xiangsheng Xu
Journal: Trans. Amer. Math. Soc. 349 (1997), 1973-1992
MSC (1991): Primary 35B65, 35K65
MathSciNet review: 1373648
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Abstract: We consider the system $\displaystyle \frac{\partial u }{\partial t}-\Delta u=\sigma \left ( u\right ) \left | \nabla \varphi \right | ^2$, $ \mathrm {div}\left ( \sigma \left ( u\right ) \nabla \varphi \right ) =0$ in $Q_T\equiv \Omega \times \left ( 0,T\right ] $ coupled with suitable initial-boundary conditions, where $\Omega $ is a bounded domain in $ \mathbf {R}^N$ with smooth boundary and $\sigma \left ( u\right ) $ 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 $Q_T\backslash Q_0$ is empty, and $u$ is essentially bounded on $Q_T\backslash Q_0$.


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Additional Information

Xiangsheng Xu
Affiliation: Department of Mathematics and Statistics, Mississippi State University, Mississippi State, Mississippi 39762
Email: xxu@math.msstate.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01734-0
PII: S 0002-9947(97)01734-0
Keywords: Partial regularity, degenerate systems
Received by editor(s): September 26, 1994
Received by editor(s) in revised form: November 27, 1995
Additional Notes: This work was supported in part by an NSF grant (DMS942448).
Article copyright: © Copyright 1997 American Mathematical Society