Generation, motion and thickness of transition layers for a nonlocal Allen-Cahn equation

TL;DR

This paper rigorously analyzes the behavior of a nonlocal Allen-Cahn equation as a small parameter approaches zero, focusing on interface generation, motion, and thickness, incorporating nonlocal effects and mean curvature motion.

Contribution

It provides a new rigorous analysis of interface dynamics and thickness estimates for a nonlocal Allen-Cahn equation, including nonlocal effects and stationary states.

Findings

Interface generation and motion are characterized rigorously.

A new estimate for interface thickness is derived.

Nonlocal effects influence stationary states and interface dynamics.

Abstract

We investigate the behavior, as a small parameter tends to zero, of a nonlocal Allen-Cahn equation. Given a rather general initial data, we perform a rigorous analysis of both the generation and the motion of interface, and obtain a new estimate for its thickness. The limit problem involves motion by mean curvature together with a nonlocal effect (which allows the possibility of nontrivial stationary state).