Abstract
In this paper we study gradient estimates for the positive solutions of the porous medium equation:
where m>1, which is a nonlinear version of the heat equation. We derive local gradient estimates of the Li–Yau type for positive solutions of porous medium equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequalities are obtained. In particular, our results improve the ones of Lu, Ni, Vázquez, and Villani (in J. Math. Pures Appl. 91:1–19, 2009). Moreover, our results recover the ones of Davies (in Cambridge Tracts Math vol. 92, 1989), Hamilton (in Comm. Anal. Geom. 1:113–125, 1993) and Li and Xu (in Adv. Math. 226:4456–4491, 2011).
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The authors would like to thank the referee for some helpful suggestions, making our paper more readable.
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Communicated by Jiaping Wang.
The research of the first author is supported by NSFC grant (No. 11001076) and NSF of Henan Provincial Education Department (No. 2010A110008). The research of the third author is supported by NSFC grant (No. 10971110).
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Huang, G., Huang, Z. & Li, H. Gradient Estimates for the Porous Medium Equations on Riemannian Manifolds. J Geom Anal 23, 1851–1875 (2013). https://doi.org/10.1007/s12220-012-9310-8
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DOI: https://doi.org/10.1007/s12220-012-9310-8