Title: Phase-asymptotic stability of transition front solutions in Cahn-Hilliard
equations and systems
Abstract:
I will discuss the asymptotic behavior of perturbations of transition
front solutions arising in Cahn--Hilliard equations and systems on $\mathbb{R}$
and $\mathbb{R}^n$. Such equations arise naturally in the study of phase
separation processes, where a two-phase process can often be modeled by a
Cahn-Hilliard equation, while a process with more than two phases can be modeled
by a Cahn-Hilliard system.
When a Cahn--Hilliard equation or system is linearized about
a transition front solution, the linearized operator has an
eigenvalue at 0 (due to shift invariance), which is not separated
from essential spectrum. In many cases, it's possible to verify that the
remaining spectrum lies on the negative real axis, so that stability is entirely
determined by the nature of this leading eigenvalue. Working primarily in the
case of a single equation on $\mathbb{R}$, I will discuss the nature of this
leading eigenvalue and also the verification
that spectral stability---defined in terms of an appropriate Evans
function---implies phase-asymptotic stability.