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arXiv · 1511.00197

A Harnack inequality for the parabolic Allen-Cahn equation

Mihai Băileşteanu
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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.

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Mihai Băileşteanu. 2016-08-09. A Harnack inequality for the parabolic Allen-Cahn equation. https://arxiv.org/abs/1511.00197

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