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

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.