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The Penrose-Fife-type equations: Counting the one-dimensional stationary solutions. (English) Zbl 0857.35055

Summary: A method for counting the solutions for Penrose-Fife-type phase field equations is derived. The method used is similar to that developed recently for obtaining a precise count for the number of solutions for the Cahn-Hilliard equation [M. Grinfeld and the first author, Proc. R., Soc. Edinb., Sect. A 125, No. 2, 351-370 (1995; Zbl 0828.34007)], and is based on the derivation of an extended system of Picard-Fuchs equations as well as on estimates obtained by the first author and L. A. Peletier [Proc. R., Soc. Edinb., Sect. A 123, No. 6, 1071-1098 (1993; Zbl 0818.35127)]. The Penrose-Fife-type phase field equations represent a thermodynamically consistent model for phase separation of a conserved order parameter (typically concentration) in binary systems in which latent heat effects are important in the phase separation process.

MSC:

35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35K55 Nonlinear parabolic equations

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References:

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