  In this paper, by maximum principle and cutoff function, we investigate
gradient estimates for positive solutions to two nonlinear parabolic equations
under Ricci flow. The related Harnack inequalities are deduced. An result about
positive solutions on closed manifolds under Ricci flow is abtained. As
applications, gradient estimates and Harnack inequalities for positive
solutions to the heat equation under Ricci flow are derived. These results in
the paper can be regard as generalizing the gradient estimates of Li-Yau, J. Y.
Li, Hamilton and Li-Xu to the Ricci flow. Our results also improve the
estimates of S. P. Liu and J. Sun to the nonlinear parabolic equation under
Ricci flow.
