## Abstract

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.

Original language | English |
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Article number | e26 |

Pages (from-to) | 1-31 |

Number of pages | 31 |

Journal | Forum of Mathematics, Sigma |

Volume | 3 |

DOIs | |

Publication status | Published - 1 Jan 2015 |

## ASJC Scopus subject areas

- Analysis
- Theoretical Computer Science
- Algebra and Number Theory
- Statistics and Probability
- Mathematical Physics
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Mathematics