# American Institute of Mathematical Sciences

September  2018, 17(5): 1957-1974. doi: 10.3934/cpaa.2018093

## 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

Received  August 2017 Revised  November 2017 Published  April 2018

Fund Project: 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).

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.
Citation: Wen Wang, Dapeng Xie, Hui Zhou. Local Aronson-Bénilan gradient estimates and Harnack inequality for the porous medium equation along Ricci flow. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1957-1974. doi: 10.3934/cpaa.2018093
##### References:
 [1] D. G. Aronson and P. Bénilan, Régularité des I'équatiomilieux poreux dans $R^n$, C. R. Acad. Sci. Paris. Sér. A-B, 288 (1979), A103-A105.   Google Scholar [2] M. Bailesteanu, X. D. Cao and A. Pulemotov, Gradient estimates for the heat equation under the Ricci flow, J. Funct. Anal., 258 (2010), 3517-3542.   Google Scholar [3] E. Calabi, An extension of E. Hopf's maximum principle with an application to Riemannian geometry, Duke Math. J., 25 (1958), 45-56.   Google Scholar [4] H. Cao and M. Zhu, Aronson-Bénilan estimates for the porous medium equation under the Ricci flow, Journal De Mathématiques Pures Et Appliqués, 104 (2015), 90-94.   Google Scholar [5] D. G. Chen and C. W. Xiong, Gradient estimates for doubly nonlinear diffusion equations, Nonlinear Anal., 112 (2015), 156-164.   Google Scholar [6] R. S. Hamilton, A matrix Harnack estimates for the heat equation, Comm. Anal. Geom., 1 (1993), 113-126.   Google Scholar [7] R. S. Hamilton, Three manifolds with positive Ricci cuevature, J. Differential Geom., 17 (1982), 255-306.   Google Scholar [8] G. Y. Huang, Z. J. Huang and H. Z. Li, Gradient estimates and differential Harnack inequalities for a nonlinear parabolic equation on Riemannian manifolds, Annals of Global Analysis & Geometry, 43 (2013), 209-232.   Google Scholar [9] S. Kuang and Q. S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow, J. Funct. Anal., 255 (2008), 1008-1023.   Google Scholar [10] P. Li and S. T. Yau, On the parabolic kernel of the Schröinger operator, Acta Math., 156 (1986), 153-201.   Google Scholar [11] J. Y. Li, Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds, J. Funct. Anal., 100 (1991), 233-256.   Google Scholar [12] J. Li and X. Xu, Defferential Harnack inequalities on Riemannian manifolds I: Linear heat equation, Adv. in Math., 226 (2011), 4456-4491.   Google Scholar [13] S. P. Liu, Gradient estimates for solutions of the heat equation under flow, Pacific J. of Math., 243 (2009), 165-179.   Google Scholar [14] X. D. Li, Hamiltons Harnack inequality and the W-entropy formula on cpmplete Riemannian manfolds, Stochastic Process. Appl., 126 (2016), 1264-1283.   Google Scholar [15] S. Z. Li and X. D. Li, On Harnack ineqlities for Witten Laplacian on Riemannian manifolds with supper Ricci flows, to appear in the Special Issue in honor of Prof. N. Mok's 60th birthday, Asian J. Math., 2017, https://arxiv.org/abs/1706.05304 Google Scholar [16] S. Z. Li and X. D. Li, Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds, J. Func. Anal., (2017). doi: 10.1016/j.jfa.2017.  Google Scholar [17] S. Z. Li and X. D. Li, Harnack inequality and W-entropy formual for Witten Laplacian on Riemannian manifolds with K-supper Perelman Ricci flow, preprint, https://arxiv.org/abs/1412.7034 Google Scholar [18] P. Lu, L. Ni, J. L. Vázquez and C. Villani, Local Aronson-Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds, J. Math. Pures Appl., 91 (2009), 1-19.   Google Scholar [19] L. Ma, L. Zhao and X. F. Song, Gradient estimate for the degenerate parabolic equation $u_{t}=Δ F(u)+H(u)$ on manifolds, J. Differential Equations, 224 (2008), 1157-1177.   Google Scholar [20] L. Shen, S. Yao, G. Zhang and X. Ren, Gradient estimate for porous medium equations under the Ricci flow, Appl. Math. J. Chinese Univ., 31 (2016), 481-490.   Google Scholar [21] J. Sun, Gradient estimates for positive solutions of the heat equation under geometric flow, Pacific J. Math., 253 (2011), 489-510.   Google Scholar [22] H. J. Sun, Higher eigenvalue estimates on riemannian manifolds with ricci curvature bounded below, Acta Math. Sinica (Chin. Ser.), 49 (2006), 539-548.   Google Scholar [23] W. Wang and P. Zhang, Some Gradient Estimates and Harnack Inequalities for Nonlinear Parabolic Equations on Riemannian Manifolds, Mathematische Nachrichten, 290 (2017), 1905-1917.   Google Scholar [24] Y. Z. Wang and W. Y. Chen, Gradient estimates and entropy monotonicity formula for doubly nonlinear diffusion equations on Riemannian manifolds, Math. Methods Appl. Sci., 37 (2014), 2772-2781.   Google Scholar [25] Y. Z. Wang and W. Y. Chen, Gradient estimates for weighted diffusion equations on smooth metric measure spaces. J. Math. (Wuhan), 33 (2013), 248-258. Google Scholar [26] Y. Z. Wang, E-entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds, submitted, 2017. Google Scholar [27] X. B. Zhu, Gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds, Nonlinear Analysis, 74 (2011), 5141-5146.   Google Scholar

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##### References:
 [1] D. G. Aronson and P. Bénilan, Régularité des I'équatiomilieux poreux dans $R^n$, C. R. Acad. Sci. Paris. Sér. A-B, 288 (1979), A103-A105.   Google Scholar [2] M. Bailesteanu, X. D. Cao and A. Pulemotov, Gradient estimates for the heat equation under the Ricci flow, J. Funct. Anal., 258 (2010), 3517-3542.   Google Scholar [3] E. Calabi, An extension of E. Hopf's maximum principle with an application to Riemannian geometry, Duke Math. J., 25 (1958), 45-56.   Google Scholar [4] H. Cao and M. Zhu, Aronson-Bénilan estimates for the porous medium equation under the Ricci flow, Journal De Mathématiques Pures Et Appliqués, 104 (2015), 90-94.   Google Scholar [5] D. G. Chen and C. W. Xiong, Gradient estimates for doubly nonlinear diffusion equations, Nonlinear Anal., 112 (2015), 156-164.   Google Scholar [6] R. S. Hamilton, A matrix Harnack estimates for the heat equation, Comm. Anal. Geom., 1 (1993), 113-126.   Google Scholar [7] R. S. Hamilton, Three manifolds with positive Ricci cuevature, J. Differential Geom., 17 (1982), 255-306.   Google Scholar [8] G. Y. Huang, Z. J. Huang and H. Z. Li, Gradient estimates and differential Harnack inequalities for a nonlinear parabolic equation on Riemannian manifolds, Annals of Global Analysis & Geometry, 43 (2013), 209-232.   Google Scholar [9] S. Kuang and Q. S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow, J. Funct. Anal., 255 (2008), 1008-1023.   Google Scholar [10] P. Li and S. T. Yau, On the parabolic kernel of the Schröinger operator, Acta Math., 156 (1986), 153-201.   Google Scholar [11] J. Y. Li, Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds, J. Funct. Anal., 100 (1991), 233-256.   Google Scholar [12] J. Li and X. Xu, Defferential Harnack inequalities on Riemannian manifolds I: Linear heat equation, Adv. in Math., 226 (2011), 4456-4491.   Google Scholar [13] S. P. Liu, Gradient estimates for solutions of the heat equation under flow, Pacific J. of Math., 243 (2009), 165-179.   Google Scholar [14] X. D. Li, Hamiltons Harnack inequality and the W-entropy formula on cpmplete Riemannian manfolds, Stochastic Process. Appl., 126 (2016), 1264-1283.   Google Scholar [15] S. Z. Li and X. D. Li, On Harnack ineqlities for Witten Laplacian on Riemannian manifolds with supper Ricci flows, to appear in the Special Issue in honor of Prof. N. Mok's 60th birthday, Asian J. Math., 2017, https://arxiv.org/abs/1706.05304 Google Scholar [16] S. Z. Li and X. D. Li, Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds, J. Func. Anal., (2017). doi: 10.1016/j.jfa.2017.  Google Scholar [17] S. Z. Li and X. D. Li, Harnack inequality and W-entropy formual for Witten Laplacian on Riemannian manifolds with K-supper Perelman Ricci flow, preprint, https://arxiv.org/abs/1412.7034 Google Scholar [18] P. Lu, L. Ni, J. L. Vázquez and C. Villani, Local Aronson-Bénilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds, J. Math. Pures Appl., 91 (2009), 1-19.   Google Scholar [19] L. Ma, L. Zhao and X. F. Song, Gradient estimate for the degenerate parabolic equation $u_{t}=Δ F(u)+H(u)$ on manifolds, J. Differential Equations, 224 (2008), 1157-1177.   Google Scholar [20] L. Shen, S. Yao, G. Zhang and X. Ren, Gradient estimate for porous medium equations under the Ricci flow, Appl. Math. J. Chinese Univ., 31 (2016), 481-490.   Google Scholar [21] J. Sun, Gradient estimates for positive solutions of the heat equation under geometric flow, Pacific J. Math., 253 (2011), 489-510.   Google Scholar [22] H. J. Sun, Higher eigenvalue estimates on riemannian manifolds with ricci curvature bounded below, Acta Math. Sinica (Chin. Ser.), 49 (2006), 539-548.   Google Scholar [23] W. Wang and P. Zhang, Some Gradient Estimates and Harnack Inequalities for Nonlinear Parabolic Equations on Riemannian Manifolds, Mathematische Nachrichten, 290 (2017), 1905-1917.   Google Scholar [24] Y. Z. Wang and W. Y. Chen, Gradient estimates and entropy monotonicity formula for doubly nonlinear diffusion equations on Riemannian manifolds, Math. Methods Appl. Sci., 37 (2014), 2772-2781.   Google Scholar [25] Y. Z. Wang and W. Y. Chen, Gradient estimates for weighted diffusion equations on smooth metric measure spaces. J. Math. (Wuhan), 33 (2013), 248-258. Google Scholar [26] Y. Z. Wang, E-entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds, submitted, 2017. Google Scholar [27] X. B. Zhu, Gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds, Nonlinear Analysis, 74 (2011), 5141-5146.   Google Scholar
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