**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.

Results for scalar and multidimensional systems will be

summarized.