an:00269921
Zbl 0787.35033
Howison, S. D.; Rodrigues, J. F.; Shillor, M.
Stationary solutions to the thermistor problem
EN
J. Math. Anal. Appl. 174, No. 2, 573-588 (1993).
00013684
1993
j
35J65 35J55 35Q80
quasilinear elliptic system; existence of weak solutions; uniqueness for small data; mixed boundary conditions; electric device (thermistor)
The authors consider the nonlinear problem
\[
-\nabla \cdot(k(u)\nabla u)=\sigma(u) | \nabla \varphi |^ 2,\quad \nabla \cdot(\sigma(u)\nabla \varphi)= 0 \tag{*}
\]
in a bounded domain \(\Omega \subset \mathbb{R}^ n\), \(n \geq 2\), with Lipschitz boundary and with general mixed boundary conditions, which is a mathematical model for the description of the steady state distribution of the temperature \(u\) and the electrical potential \(\varphi\) in an electric device (thermistor) whose electrical properties are temperature dependent.
The existence of a weak solution of \((*)\) for general boundary conditions is proved under the sole assumption that \(\sigma\) is continuous and uniformly positive. Uniqueness of the solution is also proved for sufficiently small data. Then, a new nonlocal condition which relates to the way the device is connected to the rest of the electrical circuit is analyzed, and sufficient conditions for the existence of a weak solution for such a problem are derived. Finally, a sufficient condition for non- uniqueness of the problem in one space dimension is given.
I.Zino (St.Peterburg)