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Local Aronson-Bénilan gradient estimates and Harnack inequality for the porous medium equation along Ricci flow

  • * Corresponding author: Wen Wang and Hui Zhou

    * Corresponding author: Wen Wang and Hui Zhou 
The first author is supported by the Higher School outstanding young talent support project of Anhui province in 2017 (gxyq2017048), the Higher School Natural Science Foundation of Anhui Province (KJ2017A937), the Young Foundtion of Hefei Normal University (2017QN41, 2017QN44) and the Natural Science Foundation of Anhui Province (1708085MA16).
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  • In this paper, we prove some new local Aronson-Bénilan type gradient estimates for positive solutions of the porous medium equation

    $u_{t}=Δ u^{m}, m>1$

    coupled with Ricci flow, assuming that the Ricci curvature is bounded. As application, the related Harnack inequality is derived. Our results generalize known results. These results may be regarded as the generalizations of the gradient estimates of Lu-Ni-Vázquez-Villani and Huang-Huang-Li to the Ricci flow.

    Mathematics Subject Classification: Primary: 58J35, 35K05, 53C21.

    Citation:

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