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

Howard, Peter

Adv. Differ. Equ. 14, No. 1-2, 87-120 (2009).

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.

Reviewer: Lubomira Softova (Aversa)

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

Keywords:

Cahn-Hilliard equation; stationary solutions; stability; periodic solutions; pulse-type reversal solutions; monotonic transition waves