  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.
