Invariant manifolds for metastable patterns in ut = ε2uxx—f(u)

Abstract

We consider the above equation on the interval 0 ≦ x ≦ 1 subject to Neumann boundary conditions with f(u) = F′(u) where F is a double well energy density function with equal minima. Our previous work [3] proved the existence and persistence of very slowly evolving patterns (metastable states) in solutions with two-phase initial data. Here we characterise these metastable states in terms of the global unstable manifolds of equilibria, as conjectured by Fusco and Hale [6].