The Gradient Estimates For Fast Diffusion Equations

In this thesis we will mainly research gradient estimates for positive solutions of fast diffusion equationut= ?um, 0 < m < 1on Riemannian manifolds evolving under the Ricci flow. Fast diffusion equation is short for FDE. As a nonlinear partial differential equation, prior estimate is an important aspect of approaching existence,uniqueness and regularity of its solutions. Gradient estimate plays an important role of prior estimate.The investigation of gradient estimates is a long history. It is firstly explored by Li and Yau, and then generalized by other mathematicians. R.S. Hamilton got a new estimate again in 1993, which is called for Hamilton’s gradient estimate. Gradient estimate is mainly grouped by Hamilton’s estimate and Li-Yau’s estimate, which is also called by space gradient estimate and space-time gradient estimate.There are few researches on FDE on manifolds. We mainly investigate local space estimate, local space-time estimate, global space-time estimate and Harnack inequalities for FDE on manifolds.We try to generalize partial conclusions of Bailesteanu-Cao-Pulemotov, which is about heat equation, to FDE. Then check whether our conclusions are consistent with those of Bailesteanu-Cao-Pulemotov when the parameter m becomes more easier(m â†' 1). We don’t investigate global space estimate about FDE on manifolds. There are many kinds of nonlinear FDE and we just research one of them.