Gradient estimates and Liouville type theorems for Poisson equations

TL;DR

This paper derives new gradient estimates for certain parabolic equations on smooth metric measure spaces and uses these to establish Liouville type theorems for positive or bounded solutions, extending understanding of solutions to equations like Fisher and Allen-Cahn.

Contribution

It provides novel gradient estimates for parabolic equations with specific nonlinearities and applies these to prove Liouville theorems in the context of smooth metric measure spaces.

Findings

New gradient estimates for parabolic equations on metric measure spaces.

Liouville theorems for solutions to Fisher and Allen-Cahn equations.

Extension of Liouville theorems to equations involving gradient Ricci solitons.

Abstract

In this paper, we will address to the following parabolic equation $$ u_t=\Delta_fu + F(u) $$ on a smooth metric measure space with Bakry-\'{E}mery curvature bounded from below. Here $F$ is a differentiable function defined in $\mathbb{R}$. Our motivation is originally inspired by gradient estimates of Allen-Cahn and Fisher equations (\cite{Bai17, CLPW17}). In this paper, we show new gradient estimates for these equations. As their applications, we obtain Liouville type theorems for positive or bounded solutions to the above equation when either $F=cu(1-u)$ (the Fisher equation) or; $F=-u^3+u$ (the Allen-Cahn equation); or $F=au\log u$ (the equation involving gradient Ricci solitons).