Some results on the thermistor problem. (English) Zbl 0817.35114

Antontsev, S. N. (ed.) et al., Free boundary problems in continuum mechanics. International conference on free boundary problems in continuum mechanics, Novosibirsk, Russia, July 15-19, 1991. Basel: Birkhäuser. ISNM, Int. Ser. Numer. Math. 106, 47-57 (1992).
From the introduction: We consider here the so called thermistor problem. The heat produced in a conductor by an electric current leads to the system: \[ \begin{aligned} &u_ t- \nabla\cdot k(u) \nabla u= \sigma(u) |\nabla \varphi|^ 2, \quad \nabla\cdot \sigma(u) \nabla \varphi=0 \quad \text{in } \Omega\times (0,T),\\ &u=0, \quad \varphi= \varphi_ 0 \quad \text{on } \Gamma\times (0,T), \quad u(\cdot,0)= u_ 0. \end{aligned} \tag{1} \] Here, \(\Omega\) is a smooth bounded open set of \(\mathbb{R}^ n\), \(\Gamma\) denotes the boundary, \(T\) is some positive given number, \(\varphi\) is the electrical potential, \(u\) the temperature inside the conductor, \(k(u)>0\) the thermal conductivity and \(\sigma(u) >0\) 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.
For the entire collection see [Zbl 0807.00016].


35Q72 Other PDE from mechanics (MSC2000)
80A20 Heat and mass transfer, heat flow (MSC2010)
35K55 Nonlinear parabolic equations