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

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 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 in time 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.