Some Gradient Estimates and Liouville Properties of the Fast Diffusion Equation on Riemannian Manifolds*

Citation:

Wen WANG,Rulong XIE,Pan ZHANG.Some Gradient Estimates and Liouville Properties of the Fast Diffusion Equation on Riemannian Manifolds*[J].Chinese Annals of Mathematics B,2021,42(4):529~550
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Authors:

Wen WANG; Rulong XIE;Pan ZHANG

Foundation:

National Natural Science Foundation of China (Nos. 11721101,11971026), the Natural Science Foundation of Anhui Province (Nos. 1908085QA04, 2008085QA08)and Natural Science Foundation of Education Committee of Anhui Province (Nos. KJ2017A454,KJ2019A0712, KJ2019A0713), Excellent Young Talents Foundation of Anhui Province (Nos. GXYQ2017048,GXYQ2017070, GXYQ2020049) and the research project of Hefei Normal University (No. 2020PT26).
Abstract: In the paper, the authors provide a new proof and derive some new elliptic type (Hamilton type) gradient estimates for fast diffusion equations on a complete noncompact Riemannian manifold with a fixed metric and along the Ricci flow by constructing a new auxiliary function. These results generalize earlier results in the literature. And some parabolic type Liouville theorems for ancient solutions are obtained.

Keywords:

Gradient estimate, Fast diffusion equation, Ricci flow, Liouville theorem

Classification:

58J35, 35K05, 53C21
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