Asymptotic stability analysis for transition front solutions in Cahn–Hilliard systems

We consider the asymptotic behavior of perturbations of transition front solutions arising in Cahn–Hilliard systems on RR. 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.

► Spinodal decomposition in multicomponent alloys is described by Cahn–Hilliard systems. ► They admit transition front solutions corresponding to late-stage spinodal decomposition. ► We study stability of transition fronts for Cahn–Hilliard systems in one space dimension. ► Spectral stability defined in terms of an Evans function implies nonlinear stability.