Asymptotic behavior near transition fronts for equations of generalized Cahn–Hilliard form

We consider the asymptotic behavior of perturbations of standing wave solutions arising in evolu-tionary PDE of generalized Cahn–Hilliard form in one space dimension. Such equations are well known to arise in the study of spinodal decomposition, a phenomenon in which the rapid cooling of a homoge-neously mixed binary alloy causes separation to occur, resolving the mixture into its two components with their concentrations separated by sharp transition layers. Motivated by work of Bricmont, Kupiainen, and Taskinen [5], we regard the study of standing waves as an interesting step toward understanding the dynamics of these transitions. A critical feature of the Cahn–Hilliard equation 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 non-linear stability. Under the assumption of spectral stability, described in terms of an appropriate Evans function, we develop detailed asymptotics for perturbations from standing wave solutions, establishing phase-asymptotic orbital stability for initial perturbations decaying with appropriate algebraic rate.