## Advances in Differential Equations

### Spectral analysis of stationary solutions of the Cahn--Hilliard equation

Peter Howard

#### Abstract

For the Cahn--Hilliard equation on $\mathbb{R}$, there are precisely three types of bounded non-constant stationary solutions: periodic solutions, pulse-type reversal solutions, and monotonic transition waves. We study the spectrum of the linear operator obtained upon linearization about each of these waves, establishing linear stability for all transition waves, linear instability for all reversal waves, and linear instability for a representative class of periodic waves. For the case of transitions, the author has shown in previous work that linear stability implies nonlinear stability, and so nonlinear (phase-asymptotic) stability is established for such waves.

#### Article information

Source
Adv. Differential Equations, Volume 14, Number 1/2 (2009), 87-120.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867279

Mathematical Reviews number (MathSciNet)
MR2478930

Zentralblatt MATH identifier
1195.35048

Subjects
Primary: 35B10: Periodic solutions 35B35: Stability

#### Citation

Howard, Peter. Spectral analysis of stationary solutions of the Cahn--Hilliard equation. Adv. Differential Equations 14 (2009), no. 1/2, 87--120. https://projecteuclid.org/euclid.ade/1355867279