Abstract
A local gradient estimate for positive solutions of porous medium equations on complete noncompact Riemannian manifolds under the Ricci flow is derived. Moreover, a global gradient estimate for such equations on compact Riemannian manifolds is also obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M Bailesteanu, XD Cao, A Pulemotov. Gradient estimates for the heat equation under the Ricci flow, J Funct Anal, 2010, 258: 3517–3542.
H D Cao, M Zhu. Aronson-Bénilan estimates for the porous medium equation under the Ricci flow, J Math Pures Appl, 2015, 104: 729–748.
Y Hu, ZM Qian, ZC Zhang. Gradient estimates for porous medium and fast diffusion equations via FBSDE approach, arXiv: 1206.1394, 2012.
G Y Huang, Z J Huang, HZ Li. Gradient estimates for the porous medium equations on Riemannian manifolds, J Geom Anal, 2013, 23: 1851–1875.
J F Li, XJ Xu. Differential Harnack inequalities on Riemannian manifolds I: linear heat equation, Adv Math, 2011, 226: 4456–4491.
P Li, S-T Yau. On the parabolic kernel of the Schrödinger operator, Acta Math, 1986, 156: 153–201.
S P Liu. Gradient estimates for solutions of the heat equation under Ricci flow, Pacific J Math, 2009, 243: 165–180.
P Lu, L Ni, J-L Vázquez, C Villani. Local Aronson-Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds, J Math Pures Appl, 2009, 91: 1–19.
B Q Ma, J Li. Gradient estimates of porous medium equations under the Ricci flow, J Geom, 2014, 105: 313–325.
Z M Qian, ZC Zhang. Local estimates for positive solutions of porous medium equations, arXiv: 1403.1821, 2014.
P Souplet, Q Zhang. Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds, Bull Lond Math Soc, 2006, 38: 1045–1053.
J-L Vázquez. Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Lecture Ser Math Appl, Vol 33, Oxford University Press, 2006.
J-L Vázquez. The Porous Medium Equation, Oxford Math Monogr, Clarendon Press, Oxford, 2007.
X B Zhu. Hamilton’s gradient estimates and Liouville theorems for porous medium equations on noncompact Riemannian manifolds, J Math Anal Appl, 2013, 402: 201–206.
X B Zhu. Local Aronson-Bénilan estimates for porous medium equations under Ricci flow, J Partial Differential Equations, 2011, 24: 324–333.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Science Foundation of China (11571361) and China Scholarship Council.
Rights and permissions
About this article
Cite this article
Shen, Lj., Yao, S., Zhang, Gy. et al. Gradient estimates for porous medium equations under the Ricci flow. Appl. Math. J. Chin. Univ. 31, 481–490 (2016). https://doi.org/10.1007/s11766-016-3368-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11766-016-3368-1