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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 $$\sigma$$.

##### MSC:
 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)
##### Keywords:
blow up; electrical degeneracy; capacity solution
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