Gradient Estimates For The Nonlinear Diffusion Equations With P-Laplacian On Riemannian Manifolds

Gradient estimate is an important research topic in stochastic analysis and geometric analysis.In this thesis,we study the Li-Yau type gradient estimates and Hamilton type gradient estimates of three nonlinear diffusion equations with p-Laplacian on compact Rie-mannian manifolds.As applications,the corresponding Harnack inequalities are derived.Specific research contents are as follows:(1)We consider the following nonlinear reaction-diffusion equation ut=?pu?+cuq,where p>1,? and q are constants satisfying certain conditions.Firstly,we introduce the linearized operator of p-Laplace operator and its parabolic operator,furthermore we construct the auxiliary function.Using the maximum principle and p-Bochner formula,under the condition of non-negative Ricci curvature,we derive Li-Yau type gradient estimate and Hamilton type gradient estimate for the positive solution of NRDE.As applications,the corresponding Harnack inequalities are obtained.(2)We derive gradient estimates for the positive solutions to weighted nonlinear reaction-diffusion equation on compact weighted Riemannian manifold with curvature dimension condition CD(0,N).(3)We study gradient estimate for more generalized nonlinear diffusion equation#12 By using similar methods to above,we derive Li-Yau type gradient estimate and Hamilton type gradient estimate of the equation with nonnegative Ricci curvature when F(u)satisfies certain conditions.