Hamilton type gradient estimates for a general type of nonlinear parabolic equations on Riemannian manifolds

In this paper, we prove Hamilton type gradient estimates for positive solutions to a general type of nonlinear parabolic equation concerning V-Laplacian: (∆V − q(x, t) − ∂t)u(x, t) = A(u(x, t)) on complete Riemannian manifold (with fixed metric). When V = 0 and the metric evolves under the geometric flow, we also derive some Hamilton type gradient estimates. Finally, as applications, we obtain some Liouville type theorems of some specific parabolic equations.


Introduction
Gradient estimates are very powerful tools in geometric analysis. In 1970s, Cheng-Yau [3] proved a local version of Yau's gradient estimate (see [25]) for the harmonic function on manifolds. In [16], Li and Yau introduced a gradient estimate for positive solutions of the following parabolic equation, which was known as the well-known Li-Yau gradient estimate and it is the main ingredient in the proof of Harnack-type inequalities. In [10], Hamilton proved an elliptic type gradient estimate for heat equations on compact Riemannian manifolds, which was known as the Hamilton's gradient estimate and it was later generalized to the noncompact case by Kotschwar [15]. The Hamilton's gradient estimate is useful for proving monotonicity formulas (see [9]). In [22], Souplet and Zhang derived a localized Cheng-Yau type estimate for the heat equation by adding a logarithmic correction term, which is called the Souplet-Zhang's gradient estimate. After the above work, there is a rich literature on extensions of the Li-Yau gradient estimate, Hamilton's gradient estimate and Souplet-Zhang's gradient estimate to diverse settings and evolution equations. We only cite [1,8,11,12,18,19,24,28,31] here and one may find more references therein. An important generalization of the Laplacian is the following diffusion operator on a Riemannian manifold (M, g) of dimension n, where V is a smooth vector field on M. Here ∇ and ∆ are the Levi-Civita connenction and Laplacian with respect to metric g, respectively. The V-Laplacian can be considered as a special case of V-harmonic maps introduced in [5]. Recall that on a complete Riemannian manifold (M, g), we can define the ∞-Bakry-Émery Ricci curvature and m-Bakry-Émery Ricci curvature as follows [6,20] where m ≥ n is a constant, Ric is the Ricci curvature of M and L V denotes the Lie derivative along the direction V. In particular, we use the convention that m = n if and only if V ≡ 0. There have been plenty of gradient estimates obtained not only for the heat equation, but more generally, for other nonlinear equations concerning the V-Laplacian on manifolds, for example, [4,13,20,27,32]. In [7], Chen and Zhao proved Li-Yau type gradient estimates and Souplet-Zhang type gradient estimates for positive solutions to a general parabolic equation on M×[0, T ] with m-Bakry-Émery Ricci tensor bounded below, where q(x, t) is a function on M×[0, T ] of C 2 in x-variables and C 1 in t-variable, and A(u) is a function of C 2 in u. In the present paper, by studying the evolution of quantity u 1 3 instead of ln u, we derive localised Hamilton type gradient estimates for |∇u| √ u . Most previous studies cited in the paper give the gradient estimates for |∇u| u . The main theorems are below. Theorem 1.1. Let (M n , g) be a complete Riemannian manifold with , some fixed point x in M and some fixed radius ρ. Assume that there exists a constant D 1 > 0 such that u ∈ (0, D 1 ] is a smooth solution to the general parabolic Eq Then there exists a universal constant c(n) that depends only on n so that Remark 1.1. Hamilton [10] first got this gradient estimate for the heat equation on a compact manifold. We also have Hamilton type estimates if we assume that Ric V ≥ −(m − 1)K 1 for some constant K 1 > 0, and notice that Ric V ≥ −(m − 1)K 1 is weak than Ric m V ≥ −(m − 1)K 1 . Since we do not have a good enough V-Laplacian comparison for general smooth vector field V, we need the condition that |V| is bounded in this case. Nevertheless, when V = ∇ f , we can use the method given in [23] to obtain all results in this paper, without assuming that |V| is bounded.
If q = 0 and A(u) = au ln u, where a is a constant, then following the proof of Theorem 1.1 we have Corollary 1.2. Let (M n , g) be a complete Riemannian manifold with , some fixed point x in M and some fixed radius ρ. Assume that u is a positive smooth solution to the equation in Q ρ 2 ,T 1 −T 0 with t T 0 . Using the corollary, we get the following Liouville type result. Corollary 1.3. Let (M n , g) be a complete Riemannian manifold with Ric m V ≥ −(m − 1)K 1 for some constant K 1 > 0. Assume that u is a positive and bounded solution to the Eq (1.6) and u is independent of time.
We can obtain a global estimate from Theorem 1.1 by taking ρ → 0.
We also suppose that Then there exists a universal constant c that depends only on n so that Let A(u) = a(u(x, t)) β in Corollary 1.4, we obtain Hamilton type gradient estimates for bounded positive solutions of the equation (1.10) there exists a universal constant c that depends only on n so that In the next part, our result concerns gradient estimates for positive solutions of ) with the metric evolving under the geometric flow: where ∆ t depends on t and it denotes the Laplacian of g(t), and S(t) is a symmetric (0, 2)-tensor field on (M n , g(t)). In [31], Zhao proved localised Li-Yau type gradient estimates and Souplet-Zhang type gradient estimates for positive solutions of (1.12) under the geometric flow (1.13). In this paper, we have the following localised Hamilton type gradient estimates for positive solutions to the general parabolic Eq (1.12) under the geometric flow (1.13).
Assume that there exists a constant L 1 > 0 such that u ∈ (0, L 1 ] is a smooth solution to the general parabolic Eq (1.12) in Q 2ρ,T . Then there exists a universal constant c(n) that depends only on n so that (1.14) in Q ρ 2 ,T . Remark 1.3. Recently, some Hamilton type estimates have been achieved to positive solutions of under the Ricci flow in [26], and for under the Yamabe flow in [29], where p, q ∈ C 2,1 (M n × [0, T ]), b is a positive constant and a, α are real constants. Our results generalize many previous well-known gradient estimate results.
The paper is organized as follows. In Section 2, we provide a proof of Theorem 1.1 and a proof of Corollary 1.3 and Corollary 1.5. In Section 3, we study gradient estimates of (1.12) under the geometric flow (1.13) and give a proof of Theorem 1.6.

Basic lemmas
We first give some notations for the convenience of writing throughout the paper. Let h := u .
3u . To prove Theorem 1.1 we need two basic lemmas. First, we derive the following lemma.
Proof. Since h := u 1 3 , by a simple computation, we can derive the following equation from (1.4): By direct computations, we have By the following fact: it yields The partial derivative of µ with respect to t is given by (2.6) It follows from (2.2), (2.5) and (2.6) that which is the desired estimate.
The following cut-off function will be used in the proof of Theorem 1.1 (see [2,16,22,30]).

4). The inequalities −
for every b ∈ (0, 1) with some constant C b that depends on b.
Throughout this section, we employ the cut-off function Ψ : where r(x) := d(x, x) is the distance function from some fixed point x ∈ M n .
Case 2. Suppose that d(x, x 1 ) ≥ ρ 2 . Since Ric m V ≥ −(m − 1)K 1 , we can apply the generalized Laplace comparison theorem (see Corollary 3.2 in [20]) to get Using the generalized Laplace comparison theorem and Lemma 2.2, we have at (x 1 , t 1 ), which agrees with Case 1. Therefore, we have for some universal constant c > 0. Here we used Lemma 2.2, 0 ≤ Ψ ≤ 1 and Cauchy's inequality.

Proof of Corollary 1.3 and Corollary 1.5
Proof of Corollary 1.3.
Proof of Corollary 1.5.
From Corollary 1.4, we just have to compute Λ. By the definition, we have 3. Gradient estimates for (1.12) under geometric flow: Proof of Theorem 1.6 In this section, we consider positive solutions of the nonlinear parabolic Eq (1.12) on (M n , g) with the metric evolving under the geometric flow (1.13). To prove Theorem 1.6, we follow the procedure used in the proof of Theorem 1.1.

Basic lemmas
We first derive a general evolution equation under the geometric flow.
Next, we derive the following lemma in the same fashion of Lemma 2.1. Ric g(t) ≥ −K 2 g(t), S g(t) g(t) ≤ K 3 in Q ρ,T . If h := u 1 3 , and µ := h · |∇h| 2 , then in Q ρ,T , we have (3.1) Proof. Since u is a solution to the nonlinear parabolic Eq (1.12), the function h = u 1 3 satisfies As in the proof of Lemma 2.1, we have that On the other hand, by the equation ∂ t g(t) = 2S(t), we have where we used (2.8), the assumption on bound of Ric + S and The proof is complete.
Finally, we employ the cut-off function Ψ : where r(x, t) := d g(t) (x, x) is the distance function from some fixed point x ∈ M n with respect to the metric g(t).
Then, it follows that at (x 2 , t 2 ), which agrees with Case 1.