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.
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Acknowledgements
We are grateful to Professors Jiayu Li and Xiaobao Zhu for their support and encouragement. The first author would like to thank Professor Qi S Zhang for introducing this topic in the summer course at USTC. We appreciate the referees for the valuable suggestions and the very careful reading of the original manuscript.
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This work was supported by the 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).
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Wang, W., Xie, R. & Zhang, P. Some Gradient Estimates and Liouville Properties of the Fast Diffusion Equation on Riemannian Manifolds. Chin. Ann. Math. Ser. B 42, 529–550 (2021). https://doi.org/10.1007/s11401-021-0276-3
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DOI: https://doi.org/10.1007/s11401-021-0276-3