LOCAL ARONSON-B´ENILAN GRADIENT ESTIMATES AND HARNACK INEQUALITY FOR THE POROUS MEDIUM EQUATION ALONG RICCI FLOW

. In this paper, we prove some new local Aronson-B´enilan type gradient estimates for positive solutions of the porous medium equation u t = ∆ 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-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 = ∆u 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] generalized 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 in [5,19,27]. In 2016, Li in [14] proved an improved version of the dimension free Hamilton Harnack inequality for the heat equationfor the Witten Laplacian on complete weighted Riemannian manifolds. Recently, in [16] the Hamilton differential Harnack inequality for Witten Laplacian on Riemannian manifolds was proved.
In 1979, Aronson and Bénilan [1] obtained a famous second order differential inequality for all positive solutions of (1) on the Euclidean space R n with m > 1 − 2 n . In 2009, Lu, Ni, Vázquez and Villani in [18] studied the PME on manifolds, and obtained the result below.
Theorem A (Lu, Ni, Vázquez and Villani). Let (M n , g) be an n-dimensional complete Riemannian manifold with Ric(B p (2R)) ≥ −K, K > 0. Assume that u is a positive solution to (1). Let v = m m−1 u m−1 and M = max Bp(2R)×[0,T ] v. Then for any α > 1, we have n(m−1)+2 and the constant Cdepends only on n. Moreover, when R → ∞, the following gradient estimate on complete noncompact Riemannian manifold (M n , g) can be deduced: Huang, Huang and Li in [8] generalized the results of Lu, Ni, Vázquez and Villani, and obtained Li-Yau type, Hamilton type and Li-Xu type gradient estimates. Wang and Chen [24,25,26] proved Perelam type W-entropy monotonicity formula and various differential Harnack inequalities for porous medium equation on compact Riemannian manifolds. Recently, above some results had been generalized to the Ricci flow.
Remark 1. If α(t) is bounded, then there exist a constant C 3 such that γ α−1 ≤ C 3 and γα 4 α−1 ≤ C 3 . In this case, the two inequalities (3) and (4) are the same. Let us give some special functions to illustrate Theorem 1.1 holds for different circumstances and we leave the detailed calculation to the appendix in section 4. Then 2. Hamilton type: Then 3. Li-Xu type: Then where α(t) is bounded uniformly. 4. Linear Li-Xu type: Then The local estimates above imply global estimates.
The following Harnack inequality can be derived from Corollary 1.

Preliminary
which is equivalent to the following form: Lemma 2.1. Assume that (M n , g(x, t)) satisfies the hypotheses of Theorem 1.1. We introduce the differential operator Proof. Simple calculation shows and where we use the fact that Combining (8) and (9), we have Since Substituting above identity and the following identity On the other hand, similar calculations show where we utilize the formula (11) above (14).
By utilizing Bochner's formula, we have From (14) and (15), we obtain By utilizing (13) and (16), we have It is not difficult to calculate that We deduce from (18) and (19) that Besides, From (20) and (21), we have Substituting (22) into (17), we arrive at Further, applying Young's inequality to (23), we conclude the result. We complete the proof of Lemma 2.1.
3. Proof of main results. In this section, we will prove our main results.
Proof of Theorem 1.1. Now let ϕ(r) be a C 2 function on [0, ∞) such that where C is an absolute constant. Let define by where ρ(x, t) = d(x, x 0 , t). By using the maximum principle, the argument of Calabi [3] allows us to suppose that the function φ(x, t) with support in B 2R,T , is C 2 at the maximum point. By utilizing the Laplacian comparison theorem, we deduce that where C is a constant depending on n.
For any 0 ≤ T 1 ≤ T , let H = φG and (x 1 , t 1 ) be the point in B 2R,T1 at which G attain its maximum value. We can suppose that the value is positive, because otherwise the proof is trivial. Then at the point (x 1 , t 1 ), we infer By the evolution formula of the geodesic length under the Ricci flow [4], we calculate where γ t1 is the geodesic connecting x and x 0 under the metric g(t 1 ), S is the unite tangent vector to γ t1 , and ds is the element of the arc length. Hence, by applying (33) and (34), we have Further using the inequality Hence, we deduce that Combine (31), (32) and (35), we have This inequality becomes For the inequality If γ is nondecreasing which satisfies the system Recall that α(t) and γ(t) are non-decreasing and t 1 < T 1 . Hence, we have φG(x, T 1 ) ≤(φG)(x 1 , t 1 ) Hence, we have for φ ≡ 1 on B R,T , If γ is nondecreasing which satisfies the system Recall that α(t) and γ(t) are non-decreasing and t 1 < T 1 . Hence, we have Hence, we have for φ ≡ 1 on B R,T , Because T 1 is arbitrary, so the conclusion is valid.