## Advances in Differential Equations

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

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

Reinhard Racke and Songmu Zheng

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355926869

**Mathematical Reviews number (MathSciNet)**

MR1946559

**Zentralblatt MATH identifier**

1035.35050

**Subjects**

Primary: 35K55: Nonlinear parabolic equations

Secondary: 35K20: Initial-boundary value problems for second-order parabolic equations 74N20: Dynamics of phase boundaries

#### 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