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Stationary solutions of an equation modelling Ohmic heating. (English) Zbl 0803.35072

Summary: We study questions of multiplicity and stability of stationary solutions of the nonlocal reaction-diffusion equation \(u_ t = u_{xx} + \lambda f(u)/(a + \int^ 1_ 0 f(u)dx)^ 2\) which arises in the theory of electrical devices with temperature-dependent resistivity and where \(f(u)\), which is taken to be a strictly positive function, represents the temperature-dependent resistivity. We also prove that solutions exist for all positive time and must enter a bounded region as \(t\) goes to infinity.

MSC:

35K57 Reaction-diffusion equations
80A99 Thermodynamics and heat transfer
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References:

[1] Lacey, A., Thermal runaway in a nonlocal problem modelling ohmic heating (1993), (preprint) · Zbl 0849.35058
[2] Fowler, A. C.; Frigaard, I.; Howison, S. D., Temperature surges in current-limiting circuit devices, SIAM J. App. Math., 52, 4, 998-1011 (1992) · Zbl 0800.80001
[3] Freitas, P., Some problems in nonlocal reaction-diffusion equations, (Ph.D. Thesis (1994), Heriot-Watt University: Heriot-Watt University Edinburgh)
[4] Grinfeld, M.; Furter, J. E.; Eilbeck, J. C., A monotonicity theorem and its application to stationary solutions of the phase field model, IMA J. Appl. Math., 49, 61-72 (1992) · Zbl 0796.35008
[5] Grinfeld, M.; Novick-Cohen, A., Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proc. Royal Soc. Edinburgh A (1994), (to appear) · Zbl 0828.34007
[6] Fujita, H., On the nonlinear equations \(Δu + e^u = 0\) and \(∂v / ∂t = Δv + e^v \), Bull. Amer. Math. Soc., 75, 132-135 (1969) · Zbl 0216.12101
[7] Lions, P. L., On the existence of positive solutions of semilinear elliptic equations, SIAM Review, 24, 441-467 (1993) · Zbl 0511.35033
[8] Freitas, P., A nonlocal Sturm-Liouville eigenvalue problem, Proc. Royal Soc. Edinburgh A (1994), (to appear) · Zbl 0798.34033
[9] Schaaf, R., Global Solution Branches of Two Point Boundary Value Problems, (Lect. Notes in Math. (1990), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0780.34010
[10] Smoller, J., Shock Waves and Reaction Diffusion Equations (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0508.35002
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