Asymptotic stability of transition fronts in Cahn-Hilliard systems

Abstract

We consider the asymptotic behavior of perturbations of transition front solutions arising in Cahn-Hilliard systems on R. Such equations arise naturally in the study of phase separation, 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 cases such as this, nonlinear stability cannot be concluded from classical semigroup considerations and a more refined development is appropriate. Our main result asserts that spectral stability - a necessary condition for stability, defined in terms of an appropriate Evans function - implies nonlinear stability.