Slow Motion in One-Dimensional Cahn–Morral Systems

Abstract

In this paper one-dimensional Cahn–Morral systems, which are the multicomponent analogues of the Cahn–Hilliard model for phase separation and coarsening in binary mixtures, are studied. In particular, there is an examination of solutions that start with initial data close to the preferred phases except at finitely many transition points where the data has sharp transition layers, and it is shown that such solutions may evolve exponentially slowly, i.e., if $\varepsilon $ is the interaction length then there exists a constant C such that in $\exp ({C / \varepsilon })$ units of time the change in such a solution is $o(1)$. This corresponds to extremely slow coarsening of a multicomponent mixture after it has undergone fine-grained decomposition.