## 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.

### MSC:

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