Motion of interfaces for a damped hyperbolic Allen-Cahn equation

TL;DR

This paper investigates the interface motion in a damped hyperbolic Allen-Cahn equation, showing that under certain conditions, the interface moves by mean curvature as the diffusion parameter approaches zero, similar to the classical parabolic case.

Contribution

It extends the understanding of interface dynamics to the hyperbolic Allen-Cahn equation, especially for radially symmetric solutions, demonstrating mean curvature motion in this framework.

Findings

Interface moves by mean curvature as epsilon approaches zero.

Differences between hyperbolic and parabolic cases are analyzed.

Radially symmetric solutions exhibit similar interface behavior to classical models.

Abstract

Consider the Allen-Cahn equation $u_t=\varepsilon^2\Delta u-F'(u)$, where $F$ is a double well potential with wells of equal depth, located at $\pm1$. There are a lot of papers devoted to the study of the limiting behavior of the solutions as the diffusion coefficient $\varepsilon\to0^+$, and it is well known that, if the initial datum $u(\cdot,0)$ takes the values $+1$ and $-1$ in the regions $\Omega_+$ and $\Omega_-$, then the "interface" connecting $\Omega_+$ and $\Omega_-$ moves with normal velocity equal to the sum of its principal curvatures, i.e. the interface moves by mean curvature flow. This paper concerns with the motion of the inteface for a damped hyperbolic Allen-Cahn equation, in a bounded domain of $\mathbb{R}^n$, for $n=2$ or $n=3$. In particular, we focus the attention on radially simmetric solutions, studying in detail the differences with the classic parabolic case, and we prove that, under appropriate assumptions on the initial data $u(\cdot,0)$ and $u_t(\cdot,0)$, the interface moves by mean curvature as $\varepsilon\to0^+$ also in the hyperbolic framework.