The thermistor problem with conductivity vanishing for large temperature.

*(English)*Zbl 0807.35143The 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.

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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\textit{X. Xu}, Proc. R. Soc. Edinb., Sect. A, Math. 124, No. 1, 1--21 (1994; Zbl 0807.35143)

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##### References:

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