Gradient estimates and Liouville type theorems for Poisson equations

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).