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

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