Gradient estimates for a simple parabolic Lichnerowicz equation

Abstract

In this paper, we study the gradient estimates for positive solutions to the following nonlinear parabolic equation \begin{equation*} \frac{\partial u}{\partial t} = \triangle_{f}u+cu^{-\alpha} \end{equation*} on complete noncompact manifolds with Bakry--Émery Ricci curvature bounded below, where $\alpha$, $c$ are two real constants and $\alpha > 0$.