Slow motion for a hyperbolic variation of Allen-Cahn equation in one space dimension

TL;DR

This paper proves that solutions to a hyperbolic Allen-Cahn equation in one dimension exhibit metastable behavior similar to the classical case, with initial velocity influencing transition dynamics, supported by numerical experiments.

Contribution

It extends metastability analysis to a hyperbolic Allen-Cahn variation, highlighting the impact of initial velocity on transition creation or elimination.

Findings

Solutions maintain initial transition count over long time scales.

Initial velocity can create or eliminate transitions.

Numerical experiments confirm theoretical predictions.

Abstract

The aim of this paper is to prove that, for specific initial data $(u_0,u_1)$ and with homogeneous Neumann boundary conditions, the solution of the IBVP for a hyperbolic variation of Allen-Cahn equation on the interval $[a,b]$ shares the well-known dynamical metastability valid for the classical parabolic case. In particular, using the "energy approach" proposed by Bronsard and Kohn [8], if $\varepsilon\ll 1$ is the diffusion coefficient, we show that in a time scale of order $\varepsilon^{-k}$ nothing happens and the solution maintains the same number of transitions of its initial datum $u_0$. The novelty consists mainly in the role of the initial velocity $u_1$, which may create or eliminate transitions in later times. Numerical experiments are also provided in the particular case of the Allen-Cahn equation with relaxation.