Asymptotic behavior near transition fronts for equations of Cahn--Hilliard type
Asymptotic behavior near transition fronts for equations of Cahn--Hilliard type
We consider the asymptotic behavior of perturbations
of standing wave solutions arising in evolutionary PDE of
Cahn--Hilliard type 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 homogeneously 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 \cite{BKT}, 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 nonlinear 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.
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