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We study the dynamics of the one-dimensional $\varepsilon$-dependentCahn-Hilliard / Allen-Cahn equation within a neighborhood of an equilibrium of$N$ transition layers, that in general does not conserve mass. Two differentsettings are considered which differ in that, for the second, we impose amass-conservation constraint in place of one of the zero-mass flux boundaryconditions at $x=1$. Motivated by the study of Carr and Pego on the layeredmetastable patterns of Allen-Cahn in [10], and by this of Bates and Xun in [5]for the Cahn-Hilliard equation, we implement an $N$-dimensional, and amass-conservative $ N-1$-dimensional manifold respectively; therein, ametastable state with $N$ transition layers is approximated. We then determine,for both cases, the essential dynamics of the layers (ode systems with theequations of motion), expressed in terms of local coordinates relative to themanifold used. In particular, we estimate the spectrum of the linearizedCahn-Hilliard / Allen-Cahn operator, and specify wide families of$\varepsilon$-dependent weights $\delta(\varepsilon)$, $\mu(\varepsilon)$,acting at each part of the operator, for which the dynamics are stable and restexponentially small in $\varepsilon$. Our analysis enlightens the role of massconservation in the classification of the general mixed problem into two maincategories where the solution has a profile close to Allen-Cahn, or, when themass is conserved, close to the Cahn-Hilliard solution.

Layer dynamics for the one dimensional $\varepsilon$-dependent Cahn-Hilliard / Allen-Cahn equation