The Concavity Of P-Rényi Entropy Power For The Parabolic P-Laplace Type Equations And Applications

In this paper,we study the concavity of p-Rényi entropy power for different forms of parabolic p-Laplace type equations on Riemannian manifolds with the curvature-dimension conditions,then we derive some functional inequalities.As follows,Chapter 1,first of all we introduce the research status and motivation of this paper.Secondly,we put forward the major research purpose and the related concepts of this paper,finally we list the main research results.Chapter 2,we consider the parabolic p-Laplace type equation(?)tu=?pu(?)div(|?u|p-2?u).First,we give some necessary Lemmas.Then we show that the concavity of p-Rényi entropy power for the parabolic p-Laplace type equations on the compact Riemannian manifolds and Euclidean spaces with non-negative Ricci curvatures by the method of entropy production.We also prove the Lp-Gagliardo-Nirenberg inequalities and the improved forms.Chapter 3 is the generalization of Chapter 2.We consider the weighted parabolic p-Laplace type equation(?)tu=?p,fu=divf(|?u|p-2?u).We prove the concavity of p-Rényi entropy power on the weighted compact Riemannian manifolds with the curvature-dimension condition CD(-K,m).In the last chapter,we study the generalized parabolic p-Laplace type equation(?)tu=?pF(u).We study the variational formulae of the p-Rényi entropy power and p-Rényi entropy on the compact Riemannian manifolds with the curvature-dimension condition CD(K,n).In the end of the article,we derive the large-time behavior of p-Fisher information and the generalized Sobolev inequality by the Otto's geometric calculus.