We consider here the so called thermistor problem. The heat produced in a conductor by an electric current leads to the system:

u t -∇·k(u)∇u=σ(u)|∇ϕ| 2 ,∇·σ(u)∇ϕ=0inΩ×(0,T),u=0,ϕ=ϕ 0 onΓ×(0,T),u(·,0)=u 0 ·(1)

Here, Ω is a smooth bounded open set of ℝ n , Γ denotes the boundary, T is some positive given number, ϕ is the electrical potential, u the temperature inside the conductor, k(u)>0 the thermal conductivity and σ(u)>0 the electrical conductivity.

We show existence of a solution to (1), and focus on the question of uniqueness and on the problem of global existence or blow up.

Antontsev, S; Chipot, M (1992). *Some results on the thermistor problem.* In: Antontsev, S N; Hoffmann, K H; Khludnev, A M. Free boundary problems in continuum mechanics (Novosibirsk 1991). Basel: Birkhäuser, 47-57.

## Abstract

We consider here the so called thermistor problem. The heat produced in a conductor by an electric current leads to the system:

u t -∇·k(u)∇u=σ(u)|∇ϕ| 2 ,∇·σ(u)∇ϕ=0inΩ×(0,T),u=0,ϕ=ϕ 0 onΓ×(0,T),u(·,0)=u 0 ·(1)

Here, Ω is a smooth bounded open set of ℝ n , Γ denotes the boundary, T is some positive given number, ϕ is the electrical potential, u the temperature inside the conductor, k(u)>0 the thermal conductivity and σ(u)>0 the electrical conductivity.

We show existence of a solution to (1), and focus on the question of uniqueness and on the problem of global existence or blow up.

## Citations

## Altmetrics

## Additional indexing

Other titles: | Papers from the International Conference held in Novosibirsk, July 15--19, 1991. |
---|---|

Item Type: | Book Section, refereed, original work |

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | existence; uniqueness; blow up |

Language: | English |

Date: | 1992 |

Deposited On: | 27 Aug 2010 15:57 |

Last Modified: | 14 Sep 2016 13:39 |

Publisher: | Birkhäuser |

Series Name: | International Series of Numerical Mathematics. |

Number: | 106 |

ISSN: | 0373-3149 |

ISBN: | 3-7643-2784-7 |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=1229525 http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0817.35114 http://www.recherche-portal.ch/primo_library/libweb/action/search.do?fn=search&mode=Advanced&vid=ZAD&vl%28186672378UI0%29=isbn&vl%281UI0%29=contains&vl%28freeText0%29=3-7643-2784-7 |

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