### The Cahn-Hilliard equation with dynamic boundary conditions

#### Abstract

This paper is concerned with the following Cahn-Hilliard equation $\psi _t = \Delta \mu,$ where $\mu= -\Delta \psi -\psi +\psi ^3,$ subject on the boundary $\Gamma$ to the following dynamic boundary condition $\sigma _s \Delta _{||} \psi - \partial _{\nu} \psi + h_s -g_s \psi = \frac{1}{\Gamma _s} \psi _t$ and $\partial _{\nu} \mu =0,$ and the initial condition $\psi |_{t=0}= \psi _0.$ This problem was recently proposed by physicists to describe spinodal decomposition of binary mixtures where the effective interaction between the wall (i.e., the boundary $\Gamma$) and two mixture components are short-ranged. The global existence and uniqueness of solutions to this initial boundary value problem with highest-order boundary conditions is proved.

#### Article information

Source
Adv. Differential Equations Volume 8, Number 1 (2003), 83-110.

Dates
First available in Project Euclid: 19 December 2012

Mathematical Reviews number (MathSciNet)
MR1946559

Zentralblatt MATH identifier
1035.35050

#### Citation

Racke, Reinhard; Zheng, Songmu. The Cahn-Hilliard equation with dynamic boundary conditions. Adv. Differential Equations 8 (2003), no. 1, 83--110.https://projecteuclid.org/euclid.ade/1355926869