A Harnack inequality for the parabolic Allen-Cahn equation

TL;DR

This paper establishes a differential Harnack inequality for solutions of the parabolic Allen-Cahn equation on closed manifolds, leading to classical Harnack inequalities and insights into standing wave solutions.

Contribution

It introduces a novel differential Harnack inequality for the parabolic Allen-Cahn equation, connecting it with classical inequalities and gradient estimates.

Findings

Proves a differential Harnack inequality for the equation

Derives a classical Harnack inequality as a corollary

Provides a comparison with Modica's gradient estimate

Abstract

We prove a differential Harnack inequality for the solution of the parabolic Allen-Cahn equation $ \frac{\partial f}{\partial t}=\triangle f-(f^3-f)$ on a closed n-dimensional manifold. As a corollary we find a classical Harnack inequality. We also formally compare the standing wave solution to a gradient estimate of Modica from the 1980s for the elliptic equation.