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Remarks on the equilibrium theory for the Cahn-Hilliard equation in one space dimension. (English) Zbl 0722.35082

Reaction-diffusion equations, Proc. Symp. Year, Edinburgh/UK 1987-1988, 75-93 (1990).
[For the entire collection see Zbl 0713.00008.]
The Cahn-Hilliard equation in one space dimension equipped with no flux boundary conditions is given by \[ (CH)\quad u_ t=(-\epsilon^ 2u_{xx}+W'(u))_{xx},\quad -1<x<1;\quad u_ x=u_{xxx}=0\quad at\quad x=-1,1. \] Equation (CH) is one of the most celebrated models devised to study the phenomena of separation and coarsening for a melted binary alloy under conditions of constant temperature, where the unknown function u is the concentration of one of the components. In this article we are concerned with the equilibrium theory for (CH), i.e., the study of the stable time independent solutions of (CH). In particular we study the behaviour of these solutions for \(\epsilon\) small.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
92E20 Classical flows, reactions, etc. in chemistry
35B25 Singular perturbations in context of PDEs

Citations:

Zbl 0713.00008