A Harnack inequality for the parabolic Allen–Cahn equation

Abstract

Abstract. We prove a differential Harnack inequality for the solution of the parabolic Allen-Cahn equation ∂f ∂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.

A Differential Harnack Inequality for the Newell-Whitehead-Segel Equation

“…for some function v : R n → R (see [1]). Traveling wave solutions to the Newell-Whitehead equation are used to model traveling wave convection in binary fluids, and other forms of oscillatory instability (see [5]).…”

Gradient estimates for bounded solutions of semilinear elliptic equations and the Allen-Cahn equation on manifolds

“…Recently, the authors in [9] and [23] have respectively considered Harnack and gradient estimates on parabolic and elliptic Allen-Cahn equations and obtained interesting results. The line of approach in this paper is different from those in [9] and [23], though our results can be compared with that of [23].…”

“…Recently, the authors in [9] and [23] have respectively considered Harnack and gradient estimates on parabolic and elliptic Allen-Cahn equations and obtained interesting results. The line of approach in this paper is different from those in [9] and [23], though our results can be compared with that of [23]. The adopted methodology towards obtaining the gradient estimate in this paper follows closely the program introduced by Brighton [11] where the classical Bochner formula will be applied to the power of solution in contrast to Yau's idea [33] of using logarithm of positive solution to derive some initial inequality.…”

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“…which is called the parabolic Allen-Cahn equation. A Harnack inequality for this equation was studied in [1]. The gradient estimates for the elliptic Allen-Cahn equation on…”