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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Partial regularity of solutions to a class of degenerate systems
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by Xiangsheng Xu PDF
Trans. Amer. Math. Soc. 349 (1997), 1973-1992 Request permission

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
  • 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).
  • © Copyright 1997 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 349 (1997), 1973-1992
  • MSC (1991): Primary 35B65, 35K65
  • DOI: https://doi.org/10.1090/S0002-9947-97-01734-0
  • MathSciNet review: 1373648