Metastability and Layer Dynamics for the Hyperbolic Relaxation of the Cahn-Hilliard Equation

The goal of this paper is to accurately describe the metastable dynamics of the solutions to the hyperbolic relaxation of the Cahn-Hilliard equation in a bounded interval of the real line, subject to homogeneous Neumann boundary conditions. We prove the existence of an approximately invariant manifoldfor such boundary value problem, that is we construct a narrow channel containing and satisfying the following property: a solution starting from the channel evolves very slowly and leaves the channel only after an exponentially long time. Moreover, in the channel the solution has a transition layer structure and we derive a system of ODEs, which accurately describes the slow dynamics of the layers. A comparison with the layer dynamics of the classic Cahn-Hilliard equation is also performed.