Harnack estimates for nonlinear parabolic equations under the Ricci flow

In this paper, we consider first the Li-Yau Harnack estimates for a nonlinear parabolic equation $\partial_{t}u=\Delta_{t}u-qu
-au(\ln u)^{\alpha}$ under the Ricci flow, where $\alpha>0$ is a constant. To extend these estimates to a more general situation, in the second part, we consider the gradient estimates for a positive solution of the nonlinear parabolic equation $\partial _{t}u=\Delta _{t}u+hu^{p}$ on a Riemannian manifold whose metrics evolve under the geometric flow $\partial _{t}g(t)=-2S_{g(t)}$.
To obtain these estimates, we introduce a quantity $\underline{\boldsymbol{S}}$ along the flow which measures whether
the tensor $S_{ij}$ satisfies the second contracted Bianchi identity. Under conditions
on ${\rm Ric}_{g(t)}, S_{g(t)}$, and $\underline{\boldsymbol{S}}$, we obtain the gradient estimates.