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Allegretto, W.; Xie, H.

A non-local thermistor problem. (English) Zbl 0826.35120

Eur. J. Appl. Math. 6, No. 1, 83-94 (1995).
Summary: We consider the question of the existence/nonexistence of solutions for the nonlocal nonlinear elliptic system which models a thermistor driven by a current source. Specifically, we show that for small input current there exists a solution, while this will not in general be the case for a sufficiently large current. A feature of our estimates is that the conditions for non-existence are determined by local criteria on the domain and the coefficients. Our basic tools for existence involve truncation, \(L^{2, \mu}\) estimates and fixed point arguments. Nonexistence is obtained by averaging procedures and an application of Barta’s Inequality.

Cited in 6 Documents

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
35J60 Nonlinear elliptic equations
80A20 Heat and mass transfer, heat flow (MSC2010)

Keywords:

existence; non-existence; nonlocal nonlinear elliptic system; fixed point arguments; averaging procedures
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References:

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