Gradient Estimation Of A Class Of Nonlinear Equations On Riemannian Manifolds

In this paper,we consider the following three problems:Gradient estimates for a class of parabolic equations;Gradient estimates for a class of elliptic equations in weighted spaces;Gradient estimates for a class of nonlinear equations of ?u+Cu?=0.In chapter one,we introduce the research background and current situation of the gradient estimation of positive solutions of partial differential equations on Riemannian manifolds,and some preliminary knowledge we need to use in the proof processBrighton[5]studied gradient estimation of positive f-harmonic functions in weighted measure spaces.Inspired by the ideas of Brighton in[5],in chapter two,we investigate gradient estimates for positive solutions to the following nonlinear parabolic equation by using Bochner formula and the aid of smooth cutoff functions ut=?u+aulogu+bu,where a,b are two real constants,on a given complete Riemannian manifold(Mn,g)and the Ricci flow(Mn,g(t)),respectively.We obtain Hamilton's gradient estimates of such equation and some Liouville type results in some appropriate conditionsIn chapter three,we consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a smooth metric measure space(M,g,e-f dv)?fu+aulogu+bu=0 where a,b are two real constants.Under the ?-Bakry-Emery Ricci curvature is bounded from below,we obtain a global gradient estimate which is not dependent on |?f|.In particular,we obtain that any bounded positive solution of the above equation must be constant under some suitable assumptions.In chapter four,we continue to consider gradient estimates for positive solutions to the following nonlinear elliptic equation?u+Cu?=0.where C?0.