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 ∂ f∂ t= f-(f3-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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