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

TL;DR

This paper investigates the slow evolution of layered solutions in hyperbolic variants of the mass conserving Allen-Cahn equation, revealing exponentially slow layer dynamics and deriving an ODE system to describe this behavior.

Contribution

It introduces a hyperbolic version of the mass conserving Allen-Cahn equation and characterizes its metastable dynamics, including deriving an ODE system for layer motion.

Findings

Profiles with multiple transition layers evolve very slowly.

Derived an ODE system describing exponential layer motion.

Compared hyperbolic variants with classical Allen-Cahn and Cahn-Hilliard equations.

Abstract

In this paper, we consider some hyperbolic variants of the mass conserving Allen-Cahn equation, which is a nonlocal reaction-diffusion equation, introduced (as a simpler alternative to the Cahn-Hilliard equation) to describe phase separation in binary mixtures. In particular, we focus our attention on the metastable dynamics of some solutions to the equation in a bounded interval of the real line with homogeneous Neumann boundary conditions. It is shown that the evolution of profiles with $N+1$ transition layers is very slow and we derive a system of ODEs, which describes the exponentially slow motion of the layers. A comparison with the classical Allen-Cahn and Cahn-Hilliard equations and theirs hyperbolic variations is also performed.