Li-Yau-Hamilton Estimates And Applications

The estimation of Li-Yau-Hamilton is an important research direction in geometric analysis and have been widely used in various fields,which receive more attention and research from many experts and scholars.Based on the existing work,this thesis mainly investigate the matrix Li-Yau-Hamilton estimate of nonlinear heat equation,the Harnack inequality for positive solutions of nonlinear heat equations on manifolds,as well as the Li-Yau type gradient estimate for positive solution of nonlinear heat equation under integral curvature condition.As an application,this thesis also discuss the monotonicity of parabolic frequency functional on Riemannian manifolds.Some new results are obtained,which generalize and enrich the previous work.The whole thesis is divided into six chapters.The specific work is as follows.Chapter 1 introduces the research background and current situation of the fractional boundary value problems,as well as the basic definitions and theorems required in this thesis.Chapter 2 concern with the matrix Li-Yau-Hamilton estimates for nonlinear heat equations.Firstly,we derive such an estimate on a K(?)hler manifold with a fixed K(?)hler metric.Then we consider the estimate on K(?)hler manifolds with K(?)hler metrics evolving under the rescaled K(?)hler-Ricci flow.Both of the estimates can be generalized to constrained cases.Chapter 3 discuss the differential Harnack inequalities for nonlinear parabolic equations with potential and nonlinear backward parabolic equations under the K(?)hlerRicci flow and rescaled K(?)hler-Ricci flow.Chapter 4 consider a Li-Yau gradient estimate for positive solution to the nonlinear heat equation with Neumann boundary conditions on a compact Riemannian submanifold with boundary,and satisfying the integral Ricci curvature assumption.Chapter 5 investigate the monotonicity of parabolic frequency for weighted Laplace heat equation on Riemannian manifolds.Finally,the main results and the following research work are given.