A Harnack inequality for the parabolic Allen-Cahn equation

Mihai Băileşteanu

A Harnack inequality for the parabolic Allen-Cahn equation

Differential Geometry 2016-08-10 v2

Authors: Mihai Băileşteanu

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.

Keywords

parabolic Allen-Cahn equation differential Harnack inequality closed manifold standing wave solution gradient estimate elliptic equation

Cite

@article{arxiv.1511.00197,
  title  = {A Harnack inequality for the parabolic Allen-Cahn equation},
  author = {Mihai Băileşteanu},
  journal= {arXiv preprint arXiv:1511.00197},
  year   = {2016}
}

Comments

11 pages, 1 figure

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