Gradient Estimates For Nonlinear Parabolic Equations

This paper mainly studies the gradient estimates for parabolic partial differential equations on manifolds.It consists of two main research objects.One is the matrix gradient estimate for the restricted heat equations on complete K?hler manifold,this result extends the relevant results of Cao and Ni.The other one is gradient estimate for a class of nonlinear parabolic partial differential equation on a Riemann manifold evolving under the Ricci flow,this result extends the relevant results of A.Abolarinwa.The details of each chapter are as follow.In Chapter 1,we introduce the relevant background and status of gradient estimates,then we give the contents to be researched in this paper.In Sections 3 and 4,we introduce the basic knowledge of the Riemann manifold and K?hler manifold.In Chapter 2,we first introduce some lemmas to be used,then we prove a class of matrix Li-Yau-Hamilton estimate for heat equation on complete K?hler manifold.In Chapter 3,we give the gradient estimate for positive solution of (?) and corresponding Harnack inequality.