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A Harnack inequality for the parabolic Allen–Cahn equation

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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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Acknowledgements

The author would like to thank Prof. Xiaodong Cao for suggesting applying this method to the Allen–Cahn equation and for fruitful discussions.

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  1. Department of Mathematical Sciences, Central Connecticut State University, 120 Marcus White Hall, New Britain, CT, 06050, USA

    Mihai Băileşteanu

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  1. Mihai Băileşteanu
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Correspondence to Mihai Băileşteanu.

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Băileşteanu, M. A Harnack inequality for the parabolic Allen–Cahn equation. Ann Glob Anal Geom 51, 367–378 (2017). https://doi.org/10.1007/s10455-016-9540-2

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