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On the effects of thermal degeneracy in the thermistor problem. (English) Zbl 1055.35069
The paper under review is concerned with the following problem: \[ \frac{\partial u}{\partial t} -\text{div} \big(K(u)\nabla u\big) = \sigma(u) |\nabla\varphi| ^2 \quad\text{in } \,\Omega_T = \Omega\times(0,T),\tag{1} \] \[ \text{div} (\sigma(u)\nabla\varphi) = 0 \quad\text{in } \Omega_T, \tag{2} \] \[ u= u_0 \quad\text{on }(\Omega\times\{0\})\cup(\partial\Omega\times (0,T)),\tag{3} \] \[ \varphi=\varphi_0 \quad \text{on } \Gamma_D\times (0,T),\qquad \frac{\partial\varphi}{\partial\nu}=0 \quad\text{on } \Gamma_N\times (0,T), \tag{4} \] where \(\Omega\subset\mathbb{R}^n\) is a bounded domain with smooth boundary \(\partial\Omega,\; \Gamma_N=\partial\Omega\smallsetminus \overline{\Gamma}_0\) and \(K\), \(\sigma\) and \(u_0,\; \varphi_0\) are known functions. The unknown functions \(u\) and \(\varphi\) represent the temperature and the electrical potential, respectively, of a conductor.
The aim of the paper is to study (1)–(4) in the case where both \(K(s)\) and \(\sigma(s)\) may tend to zero when \(s\to \infty\). The basic assumptions on the data are as follows: \[ K \text{ and } \sigma \text{ are continuous and positive}, \tag{5} \] \[ u_0\in W^{1,\infty}(\Omega_T), \quad \varphi_0\in L^\infty(0,T; W^{1,\infty}(\Omega)).\tag{6} \] The author proves the following results on the existence of weak solutions to (1)-(4):
A) existence of a bounded weak solution to (1)–(4) for each \(T>0\) under (5), (6) and additional conditions on \(K\) and \(\sigma\);
B) existence of a capacity solution to (1)–(4) under (5), (6) and \[ \sigma\leq M=\text{ const},\quad \int^{+\infty}_0 K(s) \,ds=+\infty \] and two results on the blow up behaviour of \(\int^{u(x,t)}_0 K(s)\,ds\).
The proofs of these results are based on approximating weak solutions to (1)–(4) by solutions to a system with regularized functions \(K\) and \(\sigma\).

35K65 Degenerate parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
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