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UNIFORM BMO ESTIMATE OF PARABOLIC EQUATIONS AND GLOBAL WELL-POSEDNESS OF THE THERMISTOR PROBLEM

Part of: Equations of mathematical physics and other areas of application Parabolic equations and systems

Published online by Cambridge University Press:  02 December 2015

BUYANG LI  and
CHAOXIA YANG
BUYANG LI
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, PR China; [email protected]
CHAOXIA YANG
Affiliation:
College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, PR China; [email protected]

Abstract

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We prove global well-posedness of the time-dependent degenerate thermistor problem by establishing a uniform-in-time bounded mean ocsillation (BMO) estimate of inhomogeneous parabolic equations. Applying this estimate to the temperature equation, we derive a BMO bound of the temperature uniform with respect to time, which implies that the electric conductivity is an $A_{2}$ weight. The Hölder continuity of the electric potential is then proved by applying the De Giorgi–Nash–Moser estimate for degenerate elliptic equations with an $A_{2}$ coefficient. The uniqueness of the solution is proved based on the established regularity of the weak solution. Our results also imply the existence of a global classical solution when the initial and boundary data are smooth.

MSC classification

Primary: 35K20: Initial-boundary value problems for second-order parabolic equations
Secondary: 35Q99: None of the above, but in this section
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015 , e26
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

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