Hamilton-Souplet-Zhang's gradient estimates for two types of nonlinear parabolic equations under the Ricci flow

In this paper, we consider gradient estimates for two type of nonlinear parabolic equations under the Ricci flow: one is the equation $$u_t=\Delta u+au\log u+bu$$ with $a,b$ two real constants, the other is $$u_t=\Delta u+\lambda u^{\alpha}$$ with $\lambda,\alpha$ two real constants. By a suitable scaling for the above two equations, we obtain Hamilton-Souplet-Zhang type gradient estimates.


Introduction
Since the nonlinear parabolic equation u t = ∆u + au log u + bu (1.1) on a given Riemannian manifold is related to gradient Ricci solitons which are self-similar solutions to the Ricci flow, many attentions are paid to the study on gradient estimates for the equation (1.1), for example, see [5-7, 9, 10,12,17]. Here a, b in (1.1) are two real constants. Clearly, a heat equation is a special case of (1.1) when a = b = 0. Hence many known results on heat equations are generalized to the nonlinear parabolic equation (1.1). For gradient estimates of solutions to (1.1) of Li-Yau type, Davies type, Hamilton type and Li-Xu type on a given Riemannian manifold, we refer to [2,5,7,17] and the references therein. In a recent paper [3], Dung and Khanh obtained Hamilton-Souplet-Zhang type gradient estimates on a given Riemannian manifold for (1.1). On a family of Riemannian metrics g(t) evolving by the Ricci flow Hsu in [8] obtained Li-Yau type gradient estimates of (1.1).
In [15], generalizing Hamilton's estimate in [4], Souplet and Zhang proved 1 Theorem A [15]. Let (M n , g) be an n-dimensional Riemannian manifold with Ric(M n ) ≥ −k, where k is a non-negative constant. Suppose that u is a positive solution to the equation where the constant C depends only on the dimension n.
The key to prove Theorem A of Souplet and Zhang is the scaling u → u = u/A. After this scaling, (1.2) becomes the following heat equation with respect toũ:ũ t = ∆ũ (1.5) since the heat equation is linear. Under this case, we obtain that 0 <ũ ≤ 1. Inspired by this method, in this paper, we also adopt the similar scaling method by u →ũ = u/A to study the nonlinear parabolic equation (1.1). By the scaling, we can derive from (1.1) the following analogous equation: with |Ric| ≤ k for some positive constant k and u ≤ A. Then there exists a constant C depending only on the dimension of M such that The study to Li-Yau type estimates of the following nonlinear parabolic equation where λ, α are two real constants, can be traced back to Li [13]. Later, for 0 < α < 1, Zhu in [21] obtained Hamilton-Souplet-Zhang type gradient estimates of (1.9) on a given Riemannian manifold. On gradient estimates of the elliptic case of (1.9), see [18,20]. A natural subject is to study Hamilton-Souplet-Zhang type gradient estimates of the nonlinear equation (1.9) under the Ricci flow. Our second result is the following: with |Ric| ≤ k for some positive constant k and u ≤ B. Then there exists a constant C depending only on the dimension of M such that 1) if α ≥ 1, then Remark 1.2. There are many studies on gradient estimates of the heat equation (1.2) under geometric flows, we refer to [14,16] and among others.

Proof of Theorem 1.1
In order to prove our Theorem 1.1, we first give a lemma which will play an important role in the proof.
Proof. Under the scaling u →ũ = u/A, we have 0 <ũ ≤ 1. From (1.1), we obtain thatũ satisfies the following equatioñ Then we have By the definition of w, we have On the other hand, where, in the second equality, we used the Ricci formula: By the formulas (2.5) and (2.6), we arrive at (2.7) Note that (2.8) Therefore, (2.7) can be written as Then, the desired estimate (2.1) follows. and 0 ≤ ψ ≤ 1. 2) ψ is decreasing as a radial function in the spatial variables.

Proof of Theorem 1.2
As in the proof of Theorem 1.1, we first give a key lemma: Proof. By the scaling u →ũ = u/B, we have 0 <ũ ≤ 1. Therefore, we obtain from (1.9) thatũ satisfies u t = ∆ũ +λũ α , (3.2) whereλ = λB α−1 . Let f = logũ ≤ 0 and Then, the function f satisfies It follows from (3.3) that Similarly, by the Ricci formula, we obtain Thus, we derive from (3.6) and (3.7) (3.8) Using the relationship (3.8) can be written as where ε = ε(f ) is a function depending on f which will be determined. Applying the inequality (3.12) Thus, the desired estimate (3.1) is attained.