Stationary solutions to the thermistor problem. (English) Zbl 0787.35033

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.


35J65 Nonlinear boundary value problems for linear elliptic equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35Q80 Applications of PDE in areas other than physics (MSC2000)
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