Gradient estimates for a nonlinear parabolic equation with potential under geometric flow

Abstract

Let $(M,g)$ be an dimensional complete Riemannian manifold. In this paper we prove local Li-Yau type gradient estimates for all positive solutions to the following nonlinear parabolic equation \begin{equation*} (\partial_t - Δ_g + \mathcal{R}) u(x, t) = - a u(x, t) \log u(x, t) \end{equation*} along the generalised geometric flow. Here $\mathcal{R}=\mathcal{R}(x,t)$ is a smooth potential function and $a$ is a constant. As an application we derived a global estimate and a space-time Harnack inequality.