Phase-asymptotic stability of transition front solutions in Cahn-Hilliard systems

I will discuss the asymptotic behavior of perturbations of transition front solutions arising in Cahn--Hilliard systems on $\mathbb{R}$. Such equations arise naturally in the study of phase separation processes, and systems describe cases in which three or more phases are possible. When a Cahn--Hilliard 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. 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.