Gradient Estimates And Harnack Inequalities To Solutions For Some Equations Along Ricci Flow

In this thesis,we investigate the gradient estimates and Harnack inequalities for positive solutions of some classes of nonlinear parabolic equations along Ricci flow.My thesis consists of the following results.(1)Generalize Li-Yau's gradient estimate of the heat equation on manifolds to the nonlinear parabolic equation under Ricci flow,and establish the related Harnack inequalities.(2)Generalize the constant ? of Li-Yau gradient estimate to the function ?(t)which satisfy given systems;(3)As special case of the system,Li-Yau type,Hamilton type,gradient estimates are derived.(4)Consider the nonlinear parabolic equation on Riemannian manifolds.For a<0 and a>1,we prove Hamilton's elliptic type gradient estimates and Liouville type theorem,which generalize Zhu's results for 0<?<1.In addition,we also derive Li-Yau type gradient estimates and Harnack inequalities,which generalize Li and Xu's results to the nonlinear parabolic equalition.(5)We obtain the Hamilton type gradient estimate to the porous medium type equation on Riemannian manifolds,and simplify the result of Souplet and Zhang to the heat equation.