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

Wen Wang; Dapeng Xie; Hui Zhou

doi:10.3934/cpaa.2018093

Article Contents

2018, Volume 17, Issue 5: 1957-1974. Doi: 10.3934/cpaa.2018093

This issue Previous Article A quasilinear parabolic problem with a source term and a nonlocal absorption Next Article A blowup alternative result for fractional nonautonomous evolution equation of Volterra type

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

1.
School of Mathematics and Statistics, Hefei Normal University, Hefei 230601, China
2.
School of mathematical Science, University of Science and Technology of China, Hefei 230026, China

* Corresponding author: Wen Wang and Hui Zhou
* Corresponding author: Wen Wang and Hui Zhou

Received: August 2017

Revised: November 2017

Published: April 2018

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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Abstract

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.

Keywords:

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

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