Gradient estimates and Liouville type theorems for Poisson equations

2018/03/20 by Nguyen Thac Dung, Dung, Nguyen Thac, Nguyễn Ngọc Khánh +1
Mathematics · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1803.07251

Abstract

In this paper, we will address to the following parabolic equation u_t=Δ_fu + F(u) on a smooth metric measure space with Bakry-Émery curvature bounded from below. Here F is a differentiable function defined in ℝ. Our motivation is originally inspired by gradient estimates of Allen-Cahn and Fisher equations (\citeBai17, 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=-u³+u (the Allen-Cahn equation); or F=aulog u (the equation involving gradient Ricci solitons).

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