Liouville theorems and Harnack inequalities for Allen-Cahn type equation

TL;DR

This paper establishes gradient estimates, Harnack inequalities, and Liouville theorems for Allen-Cahn and related nonlinear equations on Riemannian manifolds, extending classical results to broader curvature conditions.

Contribution

It introduces new gradient estimates and Liouville theorems for Allen-Cahn type equations under various curvature assumptions, generalizing previous results to a wider class of nonlinear equations.

Findings

Gradient estimates for Allen-Cahn solutions on manifolds with Ricci curvature bounds

Harnack inequalities and Liouville theorems derived from gradient estimates

Extension of results to equations with gradient terms under Bakry-Émery curvature

Abstract

We first give a logarithmic gradient estimate for positive solutions of Allen-Cahn equation on Riemannian manifolds with Ricci curvature bounded below. As its natural corallary, Harnack inequality and a Liouville theorem for classical positive solutions are obtained. Later, we consider similar estimate under integral curvature condition and generalize previous results to a class nonlinear equations which contain some classical elliptic equations such as Lane-Emden equation, static Whitehead-Newell equation and static Fisher-KPP equation. Last, we briefly generalize them to equation with gradient item under Bakry-\'{E}mery curvature condition.