COMMUNICATIONS ON doi:10.3934/cpaa.2018093 PURE AND APPLIED ANALYSIS Volume 17, Number 5, September 2018 pp. 1957–1974 LOCAL ARONSON-B ´ ENILAN GRADIENT ESTIMATES AND HARNACK INEQUALITY FOR THE POROUS MEDIUM EQUATION ALONG RICCI FLOW Wen Wang ∗ 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 Dapeng Xie and Hui Zhou ∗ School of Mathematics and Statistics, Hefei Normal University Hefei 230601, China (Communicated by Feng-Yu Wang) Abstract. In this paper, we prove some new local Aronson-B´ enilan type gra- dient estimates for positive solutions of the porous medium equation ut =Δu 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-V´ azquez-Villani and Huang-Huang-Li to the Ricci flow. 1. Introduction and main results. In this paper, we mainly derive the parabolic version of gradient estimates and Harnack inequality for positive solutions to the porous medium equation (PME for short) u t =Δu m , m> 1 (1) along Ricci flow. Let (M n ,g) be a complete Riemannian manifold. Li and Yau [10] established a famous gradient estimate for positive solutions to the heat equation. In 1991, Li in [11] deduced gradient estimates and Harnack inequalities for positive solutions to some nonlinear parabolic equation on M × [0, ∞). In 1993, Hamilton in [6] gener- alized the constant α in Li and Yau’s result to the function α(t)= e 2Kt (see (5) for details). In 2006, Sun [22] also proved gradient estimates with different α. In 2011, Li and Xu in [12] further generalized Li and Yau’s result, and found two new functions α(t). Recently, the first author and Zhang in [23] further generalized Li and Xu’s results to the nonlinear parabolic equation. Related results can be found 2000 Mathematics Subject Classification. Primary: 58J35, 35K05, 53C21. Key words and phrases. Porous medium equation, gradient estimate, Harnack inequality. 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). * Corresponding author: Wen Wang and Hui Zhou. 1957