Gradient Estimates for a Class of Semilinear Parabolic Equations and Their Applications

Abstract

In this paper we address the following parabolic equation @@ on a smooth metric measure space with Bakry–Emery curvature bounded from below for F being a differentiable function defined on $\mathbb {R}$ . Our motivation is originally inspired by gradient estimates of Allen–Cahn and Fisher-KKP equations (Bǎilesteanu, M., Ann. Glob. Anal. Geom. 51, 367–378, 2017; Cao et al., Pac. J. Math. 290, 273–300, 2017). 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-KKP equation) or; F = −u3 + u (the Allen–Cahn equation); or $F=au\log u$ (the equation involving gradient Ricci solitons).