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Some results on the thermistor problem


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.

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.

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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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