Linear Stability for Transition Front Solutions in Multidimensional Cahn–Hilliard Systems

Abstract

We consider linear stability for planar transition front solutions \(\bar{u} (x_1)\) arising in multidimensional (i.e., \(x\in \mathbb {R}^n\)) Cahn–Hilliard systems. In previous work the author has established that the linear operator obtained from linearization about the transition front has (after Fourier transform in the transverse variable \(\tilde{x} = (x_2, x_3, \dots , x_n) \mapsto \xi \)) a leading eigenvalue that moves into the stable (Re \({\uplambda }< 0\)) half-plane at rate \(|\xi |^3\). This constitutes precisely the type of borderline case that has been effectively analyzed by the pointwise semigroup methods of Zumbrun and Howard, and we follow that approach here. In particular, the approach can be viewed as a three-step process including: (1) characterization of the spectrum of the linearized operator; (2) derivation of linear semigroup estimates, typically encoded into estimates on an appropriate Green’s function; and (3) implementation of an iterative process to accommodate nonlinearities. In this paper we address Step (2) in the case of multidimensional Cahn–Hilliard systems.