Spectral analysis of stationary solutions of the Cahn-Hilliard equation. (English) Zbl 1195.35048

The author studies the bounded non-constant stationary solutions for the Cahn-Hilliard equation in \(\mathbb R\)
\[ u_t=(M(u)(F'(u)-ku_{xx})_x)_x,\quad k>0,\;M\in C^2({\mathbb R}),\;F\in C^4({\mathbb R}). \]
It is studied the spectrum of the linear operator obtained upon linearization about each type of stationary solutions: periodic solutions, pulse-type reversal solutions, monotonic transition waves. It is established linear stability for all transition waves, linear instability for all reversal waves, and linear instability for a representative class of periodic waves.


35B35 Stability in context of PDEs
35B10 Periodic solutions to PDEs
35K59 Quasilinear parabolic equations
35K46 Initial value problems for higher-order parabolic systems