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 - \Delta_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.
Categories: math.DG