Metastability and layer dynamics for the hyperbolic relaxation of the Cahn-Hilliard equation

Abstract

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 manifold" ${\mathcal{M}}_0$ for such boundary value problem, that is we construct a narrow channel containing ${\mathcal{M}}_0$ 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.

38 pages, 1 figure

References in corpus (3)

• Slow motion for a hyperbolic variation of Allen-Cahn equation in one space dimension
• Metastable dynamics for hyperbolic variations of the Allen-Cahn equation
• Slow dynamics for the hyperbolic Cahn-Hilliard equation in one space dimension

Cited by in corpus (1)

• Metastable dynamics for a hyperbolic variant of the mass conserving Allen-Cahn equation in one space dimension