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The thermistor problem with conductivity vanishing for large temperature. (English) Zbl 0807.35143
The author considers the elliptic-parabolic system of the form: ${\partial \over \partial t} u = \Delta u + \sigma (u) | \nabla \varphi |^ 2, \quad \text{div} \bigl( \sigma (u) \nabla \varphi \bigr) = 0 \quad \text{in} \quad Q_ T \equiv \Omega \times (0,T) \tag{1}$ where $$\Omega$$ is the bounded domain in $$\mathbb{R}^ N$$ with smooth boundary $$\partial \Omega$$ and $$T$$ a fixed positive number. The boundary and initial conditions are: $u(x,t) = 0 \text{ on } S_ T \equiv \partial \Omega \times (0,T), \quad \varphi (x,t) = \varphi_ 0 (x,t) \text{ on } S_ T, \quad u(x,0) = u_ 0(x) \text{ on } \Omega. \tag{2}$ Recently the problems of the kind have received a lot of attention. The associated steady-state system coupled with various boundary conditions has been studied by many authors. In their studies they assume that $$\sigma$$ is continuous and there exist two positive numbers $$C<M$$ such that $$C \leq \sigma(s) \leq M$$ for all $$s \in \mathbb{R}$$. It was shown, for instance, that these two conditions are sufficient for the existence of weak solution [P. Shi, M. Shillor and X. Xu, J. Differ. Equations 105, 239-263 (1993)].
But a new difficulty arises when $$\sigma (s)$$ satisfies $$\sigma (s) = 0$$ for $$s \geq a$$, $$a>0$$ and $$0 < \sigma (s) < M$$ for $$- \infty < s < a$$, $$M>0$$, and exactly the case is considered in the paper under review. The second equation in the system degenerates at points where $$u$$ is equal to $$a$$. One way of avoiding this difficulty is to try to derive a priori estimates which can ensure that $$u$$ stays away from $$a$$. This has been done in the stationary case where $$a = \infty$$. [H. Xie and W. Allegretto, SIAM J. Math. Anal. 22, No, 6, 1491-1499 (1991; Zbl 0744.35016)]. The same attempt fails here. One cannot rule out the possibility that $$u=a$$ somewhere in $$Q_ T$$. As a result, no a priori bound for $$\nabla \varphi$$ in any $$[L^ p(Q_ T)]^ N$$, $$p \geq 1$$, is possible, and we are forced to look for a solution in the space $$L^ 2 (0,T; W_ 0^{1,2} (\Omega)) \times L^ p (Q_ T)$$. This gives rise to the possibility that $$\nabla \varphi$$ is only a distribution. As a result, the term $$\sigma (u) \nabla \varphi$$ in the second equation appears as a product of an $$L^ P$$ function and a distribution, which is meaningless in the framework of the distribution theory. The notion of a capacity solution which was introduced by the author earlier [X. Xu, Commun. Partial Differ. Equations 18, No. 1-2, 199-213 (1993); SIAM J. Math. Anal. 23, No. 6, 1417-1438 (1992; Zbl 0768.35081)] is fruitfully employed in this work also. It turns out that this notion of a solution is just general enough to encompass the new phenomena involved.
Reviewer: I.E.Tralle (Minsk)

##### MSC:
 35Q60 PDEs in connection with optics and electromagnetic theory 35M10 PDEs of mixed type 35J70 Degenerate elliptic equations
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##### References:
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