Asymptotic behavior near planar transition fronts for the Cahn-Hilliard equation

We consider the asymptotic behavior of perturbations of planar wave solutions arising in the Cahn-Hilliard equation in space dimensions d≥2. Such equations are well known to arise in the study of spinodal decomposition, a phenomenon in which rapid cooling of a homogeneously mixed binary alloy causes separation to occur, resolving the mixture into regions in which one component or the other is dominant, with these regions separated by steep transition layers. A critical feature of the Cahn-Hilliard equation in one space dimension is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis, a feature that is known to complicate the step from spectral to nonlinear stability. The analysis of planar waves in multiple space dimensions is further complicated by the fact that the leading eigenvalue for this linearized operator (leading in the case of stability) moves into the negative-real half plane with cubic scaling λ∼|ξ|3, where ξ∈Rd−1 denotes a Fourier variable associated with spatial components transverse to the planar wave. Under the assumption of spectral stability, described in terms of an appropriate Evans function, we develop detailed asymptotics for perturbations from planar wave solutions, establishing asymptotic stability for initial perturbations decaying at an appropriate algebraic rate in an L1 norm of the transverse variables.