A Harnack inequality for the parabolic Allen-Cahn equation

Abstract

We prove a differential Harnack inequality for the solution of the parabolic Allen-Cahn equation $\frac{\partial f}{\partial t}=△f-(f^3-f)$ 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.

arXiv comment11 pages, 1 figure