The Gradient Estimates And Related Problems Of Parabolic Equations On Riemannian Manifolds

The present thesis mainly investigate two problems in geometric analysis field.On one hand,we study the gradient estimates for positive solutions of parabolic equations on Riemannian manifolds.On the other hand,we construct the monotonic entropy of positive solutions to the conjugate heat equation coupled with the conformal Ricci flow on closed manifolds.Firstly,we establish the local elliptic gradient estimate for positive bounded solutions to a class of abstract nonlinear parabolic equation on complete Riemannian manifold with the ?-Bakry-(?)mery Ricci curvature bounded below;the local parabolic gradient estimate for positive solutions of the weighted heat equation on complete smooth metric measure spaces with integral Bakry-(?)mery Ricci curvature condition and the global elliptic gradient estimate for positive solutions of the heat equation on compact manifolds with integral Ricci curvature condition,respectively.As applications,we prove the parabolic Liouville theorems;analyze the existence of solutions to some nonlinear equations;Yamabe problem on complete Riemannian manifolds and extend the Cheeger-Colding's theory of splitting theorem,etc..Secondly,we obtain various differential Harnack estimates for positive solutions to the nonlinear backward heat equations with potentials on closed manifolds coupled with the Ricci-Bourguignon flow.Lastly,we construct the monotonic functional of positive solutions to the conjugate heat equation associated to the metrics evolving by the conformal Ricci flow on closed manifolds.In view of the above research contents,the work of this thesis can be detailed as follows:Chapter 1,we review the research background and recent progress of the gradient estimates with different curvature conditions on fixed or evolving manifolds,and the development of the monotonic entropies under different geometric flows.The main research work of this paper are given on the basis of these research backgrounds.Chapter 2,we study the gradient estimates for positive solutions to parabolic equations on fixed manifolds.In section 2.1,we obtain the local elliptic gradient estimate for positive solutions to a class of abstract nonlinear parabolic equations with ?-Bakry-(?)mery Ricci curvature bounded below.We also prove the parabolic Liouville theorems and the Yamabe problem on complete manifolds as applications.In section 2.2,we consider the local parabolic gradient estimate for positive solutions of the weighted heat equation on complete smooth metric measure spaces and the global elliptic gradient estimate for positive solutions of the heat equation on compact manifolds with integral curvature condition.As applications,we extend the Abresch-Gromoll excess estimate and some approximate properties between the parabolic approximations of the distance function and the distance function under integral Bakry-(?)mery Ricci curvature condition,which play an important role in extending the Cheeger-Colding's splitting theorem in Gromov-Hausdorff sense under integral curvature condition.Chapter 3,we derive the differential Harnack estimates for nonlinear backward heat equation with potentials on closed manifolds when the metric evolved along the Ricci-Bourguignon flow.In particular,we consider the nonlinear backward heat equation without potential and derive the Hamilton's gradient estimate,which shows that any positive solutions of the nonlinear backward heat equation can not blow up too fast.Chapter 4,we establish the monotonic entropy of positive solutions to the conjugate heat equation associated to the conformal Ricci flow on closed manifolds.Two methods are given to prove the monotonicity.We also give some applications in studying the volume non-collapsing theorem coupled with the conformal Ricci flow.