Some new gradient estimates for two nonlinear parabolic equations under Ricci flow

TL;DR

This paper derives new gradient estimates and Harnack inequalities for positive solutions to nonlinear parabolic equations under Ricci flow, generalizing and improving previous results in the field.

Contribution

It introduces novel gradient estimates and Harnack inequalities for nonlinear parabolic equations under Ricci flow, extending classical results to this geometric setting.

Findings

Gradient estimates for solutions under Ricci flow

Harnack inequalities for positive solutions

Generalization of Li-Yau and Hamilton estimates

Abstract

In this paper, by maximum principle and cutoff function, we investigate gradient estimates for positive solutions to two nonlinear parabolic equations under Ricci flow. The related Harnack inequalities are deduced. An result about positive solutions on closed manifolds under Ricci flow is abtained. As applications, gradient estimates and Harnack inequalities for positive solutions to the heat equation under Ricci flow are derived. These results in the paper can be regard as generalizing the gradient estimates of Li-Yau, J. Y. Li, Hamilton and Li-Xu to the Ricci flow. Our results also improve the estimates of S. P. Liu and J. Sun to the nonlinear parabolic equation under Ricci flow.

Equations

$$u_t=\Delta u$$

$$\underset{B_{2R}}{\sup }\left(\frac{|\nabla u{|}^2}{u^2}-\alpha \frac{u_t}{u}\right)\le \frac{C{\alpha }^2}{R^2}\left(\frac{{\alpha }^2}{{\alpha }^2-1}+\sqrt{K}R\right)+\frac{n{\alpha }^{2}K}{\alpha -1}+\frac{n{\alpha }^2}{2t}.$$

$$\frac{|\nabla u{|}^2}{u^2}-\alpha \frac{u_t}{u}\le \frac{n{\alpha }^{2}k}{2(\alpha -1)}+\frac{n{\alpha }^2}{2t}.$$

$$\left(\Delta -\frac{\partial }{\partial t}\right)u(x,t)+h(x,t)u^{\alpha }(x,t)=0$$

$${\partial }_{t}u(x,t)=\Delta u(x,t)+h(x,t)u^l(x,t)$$

$${\partial }_{t}u(x,t)=\Delta u(x,t)+au(x,t)\log u(x,t)$$

$$\frac{\partial g(t)}{\partial t}=-2R_{ij},(x,t)\in M^n\times [0,T].$$

$$u_t=\Delta u^m,m>1$$

$$u_t=\Delta u^m,0<m<1$$

$$\{\begin{array}{r}\frac{2\phi }{n}-2\alpha K\ge (\frac{2\phi }{n}-{\alpha }^{\prime })\frac{1}{\alpha },\\ \frac{2\phi }{n}-{\alpha }^{\prime }>0,\\ \frac{{\phi }^2}{n}+\alpha {\phi }^{\prime }\ge 0.\end{array}$$

$$\frac{{\gamma }^{\prime }}{\gamma }-(\frac{2\phi }{n}-{\alpha }^{\prime })\frac{1}{\alpha }\le 0.$$

$$\frac{|\nabla u{|}^2}{u^2}-\alpha \frac{u_t}{u}+\alpha h(x,t)u^{l-1}$$

$$\le C{\alpha }^2\left(\frac{1}{R^2}+\frac{\sqrt{K}}{R}+K\right)+\frac{C{n}^2{\alpha }^4}{R^2\gamma }+n^{\frac{3}{2}}{\alpha }^{2}K$$

$$+\alpha \sqrt{n{\overline{u}}_1{\delta }_3}+n{\alpha }^2{\overline{u}}_1{\delta }_1+\sqrt{\frac{2-l}{2}}{\alpha }^{\frac{3}{2}}\sqrt{n{\overline{u}}_1{\delta }_2}+\alpha \phi .$$

$$\frac{|\nabla u{|}^2}{u^2}-\alpha \frac{u_t}{u}+\alpha h(x,t)u^{l-1}$$

$$\le C{\alpha }^2\left(\frac{1}{R^2}+\frac{\sqrt{K}}{R}+K\right)+\frac{C{n}^2{\alpha }^4}{R^2\gamma }+n^{\frac{3}{2}}{\alpha }^{2}K$$

$$+\alpha \sqrt{n{\overline{u}}_1{\delta }_3}+n{\alpha }^2{\overline{u}}_1{\delta }_1+\sqrt{\frac{2-l}{2}}{\alpha }^{\frac{3}{2}}\sqrt{n{\overline{u}}_1{\delta }_2}+\alpha \phi ,$$

$${\overline{u}}_1:=\underset{B_{2R,T}}{\max }{u}^{l-1},{\delta }_1:=\underset{B_{2R,T}}{\max }h(x,t).$$

$$\frac{|\nabla u{|}^2}{u^2}-\alpha \frac{u_t}{u}+\alpha h(x,t)u^{l-1}$$

$$\le C{\alpha }^2\left(\frac{1}{R^2}+\frac{\sqrt{K}}{R}+K\right)+\frac{C{n}^2{\alpha }^4}{R^2\gamma }+n^{\frac{3}{2}}{\alpha }^{2}K+n{\alpha }^2(l-1){\delta }_1{\overline{u}}_2$$

$$+\alpha \sqrt{\frac{n(l\alpha -1){\overline{u}}_2{\delta }_2}{l-1}}+{\alpha }^{\frac{3}{2}}\sqrt{n(l-1){\delta }_1\phi }+{\alpha }^{\frac{3}{2}}\sqrt{n{\delta }_3{\overline{u}}_2}+\alpha \phi .$$

$$\frac{|\nabla u{|}^2}{u^2}-\alpha \frac{u_t}{u}+\alpha h(x,t)u^{l-1}$$

$$\le C{\alpha }^2\left(\frac{1}{R^2}+\frac{\sqrt{K}}{R}+K\right)+\frac{C{n}^2{\alpha }^4}{R^2\gamma }+n^{\frac{3}{2}}{\alpha }^{2}K+n{\alpha }^2(l-1){\delta }_1{\overline{u}}_2$$

$$+\alpha \sqrt{\frac{n(l\alpha -1){\overline{u}}_2{\delta }_2}{l-1}}+{\alpha }^{\frac{3}{2}}\sqrt{n(l-1){\delta }_1\phi }+{\alpha }^{\frac{3}{2}}\sqrt{n{\delta }_3{\overline{u}}_2}+\alpha \phi ,$$

$${\overline{u}}_2:=\underset{B_{2R,T}}{\max }{u}^{l-1},{\delta }_1:=\underset{B_{2R,T}}{\max }h(x,t).$$

$$\alpha (t)=constant,\phi (t)=\frac{\alpha n}{t}+\frac{nK{\alpha }^2}{\alpha -1},\gamma (t)=t^{\theta }with0<\theta \le 2.$$

$$\frac{|\nabla u{|}^2}{u^2}-\alpha \frac{u_t}{u}+\alpha h(x,t)u^{l-1}$$

$$\le C{\alpha }^2\left[\frac{1}{R^2}(1+\sqrt{K}R)+\frac{{\alpha }^2}{\alpha -1}\frac{1}{R^2}+K\right]+\alpha \phi$$

$$+n^{\frac{3}{2}}{\alpha }^{2}K+\alpha \sqrt{n{\overline{u}}_1{\delta }_3}+n{\alpha }^2{\overline{u}}_1{\delta }_1+\sqrt{\frac{2-l}{2}}{\alpha }^{\frac{3}{2}}\sqrt{n{\overline{u}}_1{\delta }_2}.$$

$$\frac{|\nabla u{|}^2}{u^2}-\alpha \frac{u_t}{u}+\alpha h(x,t)u^{l-1}$$

$$\le C{\alpha }^2\left[\frac{1}{R^2}(1+\sqrt{K}R)+\frac{{\alpha }^2}{\alpha -1}\frac{1}{R^2}+K\right]+\alpha \phi$$

$$+n^{\frac{3}{2}}{\alpha }^{2}K+n{\alpha }^2(l-1){\delta }_1{\overline{u}}_2+\alpha \sqrt{\frac{n(l\alpha -1){\overline{u}}_2{\delta }_2}{l-1}}$$

$$+{\alpha }^{\frac{3}{2}}\sqrt{n(l-1){\delta }_1\phi }+{\alpha }^{\frac{3}{2}}\sqrt{n{\delta }_3{\overline{u}}_2}.$$

$$\alpha (t)=e^{2Kt},\phi (t)=\frac{n}{t}{e}^{4Kt},\gamma (t)=t{e}^{2Kt}.$$

$$\frac{|\nabla u{|}^2}{u^2}-\alpha \frac{u_t}{u}+\alpha h(x,t)u^{l-1}$$

$$\le C{\alpha }^2\left[\frac{1}{R^2}(1+\sqrt{K}R)+K\right]+\frac{C{\alpha }^4}{R^{2}t{e}^{2Kt}}+\alpha \phi$$

$$+n^{\frac{3}{2}}{\alpha }^{2}K+\alpha \sqrt{n{\overline{u}}_1{\delta }_3}+n{\alpha }^2{\overline{u}}_1{\delta }_1+\sqrt{\frac{2-l}{2}}\alpha \sqrt{n{\overline{u}}_1{\delta }_2}.$$

$$\frac{|\nabla u{|}^2}{u^2}-\alpha \frac{u_t}{u}+\alpha h(x,t)u^{l-1}$$

Full text

Corresponding author: Wen Wang, E-mail: wwen2014@mail.ustc.edu.cn

This paper was typeset using AmS-LaTeX

1. Introduction

Beginning with the pioneering work of Li and Yau [14], gradient estimates are also known as differential Harnack inequalities, which have tremendous impact in geometric analysis, as shown for example in [14, 15, 16]. Moreover, both have very important applications in singularity analysis. In perelman’s geometrization conjecture [22, 23] on the poincaré conjecture, a differential Harnack inequality played an important role.

Next, we simply introduce research progress associated with this article.

Let $(M^{n},g)$ be a complete Riemannian manifold. Li and Yau [14] established a famous gradient estimate for positive solutions to the following heat equation

[TABLE]

on $(M^{n},g)$, which is described as

Theorem A (Li-Yau [14]) Let $(M^{n},g)$ be a complete Riemannian manifold. Suppose that on the ball $B_{2R}$, $Ricci(B_{2R})\geq-K$. Then for any $\alpha>1$,

[TABLE]

In general, on a complete Riemannian manifold, if $Ricci(M)\geq-k$, by letting $R\rightarrow\infty$ in (1.2), one inferred

[TABLE]

In $1991$, Li [15] generalized Li and Yau’s estimates to the nonlinear parabolic equation

[TABLE]

on $(M^{n},g)$. In $1993$, Hamilton in [8] generalized the constant $\alpha$ of Li and Yau’s result to the function $\alpha(t)=e^{2Kt}$. In $2006$, Sun [27] also obtained a gradient estimate of different coefficient. In $2011$, Li and Xu in [17] further promoted Li and Yau’s result, and found two new functions $\alpha(t)$. Recently, first author and Zhang in [28] further generalized Li and Xu’s results to the nonlinear parabolic equation  (1.4). Related results can be found in [5, 11, 32].

In this paper, we investigate the two nonlinear parabolic equations

[TABLE]

and

[TABLE]

under Ricci flow, where the function $h(x,t)\geq 0$ is defined on $M^{n}\times[0,T]$, which is $C^{2}$ in the first variable and $C^{1}$ in the second variable, $T$ is a positive constant and $l,a\in\mathbb{R}$, respectively.

Recently, there are a number of studies on Ricci flow on manifolds by R. Hamilton [9, 10] and others, because the Ricci flow is a powerful tool in analyzing the structure of manifolds. Assume $M^{n}$ is an $n$-dimensional manifold without boundary, and let $(M^{n},g(t))_{t\in[0,T]}$ be an $n$-dimensional complete manifold with metric $g(t)$ evolving by the Ricci flow

[TABLE]

In 2008, Kuang and Zhang [11] proved a gradient estimate for positive solutions to the conjugate heat equation under Ricci flow on a closed manifold. In 2009, Liu [18] derived a gradient estimate for positive solutions to the heat equation under Ricci flow. Afterwards, Sun[26] generalized Liu’s results to general geometric flow. In 2010, Bailesteanu, Cao and Pulemotov [1] established some gradient estimates for positive solutions to the heat equation under Ricci flow. In 2016, Li and Zhu [19] generalized J. Y. Li’s [15] estimates under Ricci flow. Recently, Cao and Zhu [3] derived some Aronson and Bénilan estimates for porous medium equation

[TABLE]

under Ricci flow. Li, Bai and Zhang [13] studied fast diffusion equation

[TABLE]

under the Ricci flow. Zhao and Fang [31] generalized Yang’s result [30] to the Ricci flow.

Firstly, we introduce three $C^{1}$ functions $\alpha(t)$, $\varphi(t)$ and $\gamma(t)$ $:(0,+\infty)\rightarrow(0,+\infty)$. Suppose that three $C^{1}$ functions $\alpha(t)$, $\varphi(t)$ and $\gamma(t)$ satisfy the following conditions:

$(C1)$  $\alpha(t)>1$, $\varphi(t)$ and $\gamma(t)$.

$(C2)$  $\alpha(t)$ and $\varphi(t)$ satisfy the following system

[TABLE]

$(C3)$  $\gamma(t)$ satisfies

[TABLE]

$(C4)$  $\gamma(t)$ is non-decreasing, and $\alpha(t)$ is also non-decreasing or is bounded uniformly.

This paper is organized as follows: We prove gradient estimates for the equation  (1.5) in Section $2$ and gradient estimates for the equation  (1.6) in Section $3$. We derive related Harnack inequalities in Section $4$. As special case, we deduce gradient estimates and Harnack inequality to the heat equation in section $5$. Detailed calculation of some specific functions $\alpha(t)$, $\varphi(t)$ and $\gamma(t)$ are given in section $6$.

2. Gradient estimates for the equation  (1.5)

In this section, we will derive some new gradient estimates for positive solutions to equation  (1.5) under the Ricci flow.

2.1. Main results

We state our results as follows.

Theorem 2.1**.**

Let $(M^{n},g(t))_{t\in[0,T]}$ be a complete solution to the Ricci flow  (1.7). Assume that $|\mathrm{Ric}(x,t)|\leq K$ for some $K>0$ and all $t\in[0,T]$. Suppose that there exist three functions $\alpha(t)$, $\varphi(t)$ and $\gamma(t)$ satisfy conditions (C1), (C2), (C3) and (C4).

Given $x_{0}\in M^{n}$ and $R>0$, let $u$ be a positive solution of the equation  (1.5) in the cube $B_{2R,T}:=\{(x,t)|d(x,x_{0},t)\leq 2R,0\leq t\leq T\}$. Let $h(x,t)$ be a function defined on $M^{n}\times[0,T]$ which is $C^{1}$ in $t$ and $C^{2}$ in $x$, satisfying $|\nabla h|^{2}\leq\delta_{2}h$ and $\Delta h\geq-\delta_{3}$ on $B_{2R,T}$ for some positive constants $\delta_{2}$ and $\delta_{3}$.

$(1)$* $l\leq 1$. If $\frac{\gamma\alpha^{4}}{\alpha-1}\leq C_{1}$ for some constant $C_{1}$, then*

[TABLE]

If $\frac{\gamma}{\alpha-1}\leq C_{2}$ for some constant $C_{2}$, then

[TABLE]

where $C$ is a positive constant depending only on $n$ and set

[TABLE]

$(2)$* $l>1$. If $\frac{\gamma\alpha^{4}}{\alpha-1}\leq C_{1}$ for some constant $C_{1}$, then*

[TABLE]

If $\frac{\gamma}{\alpha-1}\leq C_{2}$ for some constant $C_{2}$, then

[TABLE]

where $C$ is a positive constant depending only on $n$ and set

[TABLE]

Let us list some examples to illustrate the Theorem $2.1$ holds for different circumstances and see appendix in section $6$ for detailed calculation process.

Corollary 2.1**.**

Suppose that $(M^{n},g(t))_{t\in[0,T]}$ satisfies the hypotheses of Theorem $2.1$. Then the following special estimates are valid.

1. Li-Yau type:

[TABLE]

If $l\leq 1$, then

[TABLE]

If $l>1$, then

[TABLE]

2. Hamilton type:

[TABLE]

If $l\leq 1$, then

[TABLE]

If $l>1$, then

[TABLE]

3. Li-Xu type:

[TABLE]

If $l\leq 1$, then

[TABLE]

If $l>1$, then

[TABLE]

where $\alpha(t)$ is bounded uniformly.

4. Linear Li-Xu type:

[TABLE]

If $l\leq 1$, then

[TABLE]

If $l>1$, then

[TABLE]

Remark 2.1*.*

The above results can be regard as generalizing the gradient estimates of Li-Yau [14], J. Y. Li [15], Hamilton [8] and Li-Xu [17] to the Ricci flow. Our results also generalize the estimates of S. P. Liu [18] and J. Sun [26] to the nonlinear parabolic equation under the Ricci flow.

The local estimates in Theorem $2.1$ imply global estimates.

Corollary 2.2**.**

Let $(M^{n},g(t))_{t\in[0,T]}$ be a complete solution to the Ricci flow  (1.7). Assume that $|\mathrm{Ric}(x,t)|\leq K$ for some $K>0$ and all $(x,t)\in M^{n}\times[0,T]$. Let $u(x,t)$ be a positive solution to equation  (1.5) on $M^{n}\times[0,T]$. Let $h(x,t)$ be a function defined on $M^{n}\times[0,T]$ which is $C^{1}$ in $t$ and $C^{2}$ in $x$, satisfying $|\nabla h|^{2}\leq\delta_{2}h$ and $\Delta h\geq-\delta_{3}$ on $M^{n}\times[0,T]$ for some positive constants $\delta_{2}$ and $\delta_{3}$.

If $l\leq 1$ and for $(x,t)\in M^{n}\times(0,T]$, then

[TABLE]

where where $C$ is a positive constant depending only on $n$ and set

[TABLE]

If $l>1$ and for $(x,t)\in M^{n}\times(0,T]$, then

[TABLE]

where where $C$ is a positive constant depending only on $n$ and set

[TABLE]

We can derive a gradient estimate for an any positive solution to the following nonlinear parabolic equation under the Ricci flow on a closed manifold without any curvature conditions. The method of the proof is inspired by Hamilton [10], Shi [23] and Liu [18].

Theorem 2.2**.**

Let $(M^{n},g(x,t))_{t\in[0,T]}$ be a solution to the Ricci flow  (1.7) on a closed manifold. If $u$ is a positive solution to equation

[TABLE]

where $h(t)$ is a $C^{1}$ function and $h(t)\leq 0$. Then for $l\geq 1$, we have

[TABLE]

2.2. Auxilliary lemma

To prove main results, we need a lemma.

Let $f=\ln u$. Then

[TABLE]

Let $F=|\nabla f|^{2}-\alpha f_{t}+\alpha hu^{l-1}-\alpha\varphi$, where $\alpha=\alpha(t)>1$ and $\varphi=\varphi(t)>0$.

Lemma 2.1**.**

Suppose that $(M^{n},g(t))_{t\in[0,T]}$ satisfies the hypotheses of Theorem $2.1$. We also assume that $\alpha(t)>1$ and $\varphi(t)>0$ satisfy the following system

[TABLE]

and $\alpha(t)$ is non-decreasing. Then

[TABLE]

Proof.

By directly computing, we have

[TABLE]

where we have used Bochner’s formula and

[TABLE]

Applying Young’s inequality

[TABLE]

we conclude for $|R_{ij}|\leq K$,

[TABLE]

On the other hand, we infer

[TABLE]

We follow from  (2.5) and  (2.6),

[TABLE]

By using the formula

[TABLE]

we obtain

[TABLE]

Applying the following two equations

[TABLE]

to  (2.7), we have

[TABLE]

Further applying unit matrix $(\delta_{ij})_{n\times n}$ and  (2.8), we derive

[TABLE]

Applying  (2.2), we have

[TABLE]

Therefore,  (2.4) is derived from  (2.3) and  (2.9). The proof is complete. ∎

2.3. Proof of Theorem $2.1$ and $2.2$

In this section, we will prove the Theorem $2.1$ and $2.2$.

Proof of Theorem $2.1$.

Let $G=\gamma(t)F$ and $\gamma(t)>0$ be non-decreasing. Then

[TABLE]

Now let $\varphi(r)$ be a $C^{2}$ function on $[0,\infty)$ such that

[TABLE]

and

[TABLE]

where $C$ is an absolute constant. Let define by

[TABLE]

where $\rho(x,t)=d(x,x_{0},t)$. By using maximum principle, the argument of Calabi  [2] allows us to suppose that the function $\phi(x,t)$ with support in $B_{2R,T}$, is $C^{2}$ at the maximum point. By utilizing the Laplacian theorem, we deduce that

[TABLE]

For any $0\leq T_{1}\leq T$, let $H=\phi G$ and $(x_{1},t_{1})$ be the point in $B_{2R,T_{1}}$ at which $H$ attains 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

[TABLE]

By the evolution formula of the geodesic length under the Ricci flow  [6], we calculate

[TABLE]

where $\gamma_{t_{1}}$ is the geodesic connecting $x$ and $x_{0}$ under the metric $g(t_{1})$, $S$ is the unite tangent vector to $\gamma_{t_{1}}$, and $ds$ is the element of the arc length.

All the following computations are at the point $(x_{1},t_{1})$. It is not difficult to find that

[TABLE]

and

[TABLE]

To obtain main results, two cases will be shown.

Case $1$ $l\leq 1$.

From (2.14), we have $\Delta f\leq 0$. Then by substituting it into  (2.3), we obtain

[TABLE]

where we drop one term $h\gamma\alpha(l-1)^{2}u^{l-1}|\nabla f|^{2}$. Using  (2.13), we infer

[TABLE]

Multiply $\phi$ to inequality  (2.3), we have

[TABLE]

Using the Cauchy inequality

[TABLE]

[TABLE]

we conclude

[TABLE]

where we use the fact that $(\alpha-1)(l-1)+(\alpha-1)+\alpha(1-l)\leq\alpha(3-2l)-1$. Further using the inequality $Ax^{2}+Bx\geq-\frac{B^{2}}{4A}$ with $A>0$, we have

[TABLE]

and

[TABLE]

Substituting above two inequalities into  (2.3), we deduce that

[TABLE]

Applying  (2.11), we infer

[TABLE]

For the inequality $Ax^{2}-Bx\leq C$, one has $x\leq\frac{B}{A}+\left(\frac{C}{A}\right)^{\frac{1}{2}}$, where $A,B,C>0$. By using this inequality to  (2.17) and then we arrive at

[TABLE]

If $\gamma$ is nondecreasing which satisfies the system

[TABLE]

Recall that $\alpha(t)$ and $\gamma(t)$ are non-decreasing and $t_{1}<T_{1}$. Hence, we have

[TABLE]

Hence, we have for $\phi\equiv 1$ on $B_{R,T}$,

[TABLE]

If $\gamma$ is nondecreasing which satisfies the system

[TABLE]

Recall that $\alpha(t)$ and $\gamma(t)$ are non-decreasing and $t_{1}<T_{1}$. Hence, we have

[TABLE]

Hence, we have for $\phi\equiv 1$ on $B_{R,T}$,

[TABLE]

Because $T_{1}$ is arbitrary in $0<T_{1}<T$ and $\alpha(3-2\alpha)-1\leq\alpha-1$ and $\alpha-1+\alpha(1-l)\leq\alpha(2-l)$. Thus the conclusion is valid.

Case $2$ $l>1$.

Substituting  (2.14) into  (2.3), we have

[TABLE]

Using  (2.13), we infer

[TABLE]

Multiply $\phi$ to (2.20), we have

[TABLE]

where we drop one term $\frac{\phi^{2}(\alpha-1)^{2}\gamma}{n\alpha^{2}}|\nabla f|^{4}$.

Further using the inequality $Ax^{2}+Bx\geq-\frac{B^{2}}{4A}$ with $A>0$, we have

[TABLE]

and

[TABLE]

Substituting above two inequalities into (2.21), we deduce that

[TABLE]

Applying  (2.11), we infer

[TABLE]

For the inequality $Ax^{2}-2Bx\leq C$, one has $x\leq\frac{2B}{A}+\left(\frac{C}{A}\right)^{\frac{1}{2}}$, where $A,B,C>0$. By using this inequality to  (2.22) and then we arrive at

[TABLE]

If $\gamma$ is nondecreasing which satisfies the system

[TABLE]

Recall that $\alpha(t)$ and $\gamma(t)$ are non-decreasing and $t_{1}<T_{1}$. Hence, we have

[TABLE]

Hence, we have for $\phi\equiv 1$ on $B_{R,T}$,

[TABLE]

If $\gamma$ is nondecreasing which satisfies the system

[TABLE]

Recall that $\alpha(t)$ and $\gamma(t)$ are non-decreasing and $t_{1}<T_{1}$. Hence, we have

[TABLE]

Hence, we have for $\phi\equiv 1$ on $B_{R,T}$,

[TABLE]

Because $T_{1}$ is arbitrary in $0<T_{1}<T$, the conclusion is valid.

∎

Proof of Theorem $2.2$.

Since $u_{t}=\nabla u+h(x,t)u^{l}$, we have

[TABLE]

Applying Bochner’s formula, above equation becomes

[TABLE]

Besides,

[TABLE]

Let $\overline{F}=t|\nabla u|^{2}+Xu^{2}$, where $X$ is a constant to be decided. Then combining  (2.25) with  (2.27), we obtain

[TABLE]

Selecting $X=\frac{1}{2}$ and using maximum principle, we infer

[TABLE]

which implies the theorem is valid. ∎

3. Gradient estimates for the equation $(1.6)$

Recalled that $(M^{n},g(t))$ is called a gradient Ricci soliton if there is a smooth function $f$ on $M^{n}$ such that for some constant $c\in\mathbb{R}$, which satisfies

[TABLE]

where $D^{2}f$ is the Hessian of $f$. Let $u=e^{f}$, after some computation applying (3.1) as done in [21], we get

[TABLE]

for some constant $A_{0}$, where $n$ is the dimension of $M^{n}$. In [21], Ma proved the local gradient estimate of positive solutions to the equation

[TABLE]

where $a>0$ and $b\in\mathbb{R}$ are constants for complete noncaompact manifolds with a fixed metric and curvature locally bounded below. In [31], Yang generalized Ma’s result and derived a local gradient estimates for positive solutions to the equation

[TABLE]

where $a,b\in\mathbb{R}$ are constants for complete noncompact manifolds with a fixed metric and curvature locally bounded below. Replacing $u$ by $e^{b/a}u$, above equation becomes

[TABLE]

One can find in [29, 30] some related results for equation  (3.2) on manifolds.

In this section, we consider the nonlinear parabolic equation  (1.6) under the Ricci flow.

3.1. Main results

Our main results state as follows.

Theorem 3.1**.**

Let $(M^{n},g(t))_{t\in[0,T]}$ be a complete solution to the Ricci flow  (1.7). Assume that $|\mathrm{Ric}(x,t)|\leq K$ for some $K>0$ and all $t\in[0,T]$. Suppose that there exist three functions $\alpha(t)$, $\varphi(t)$ and $\gamma(t)$ satisfy the following conditions (C1), (C2), (C3) and (C4).

Given $x_{0}\in M$ and $R>0$, let $u$ be a positive solution of the nonlinear parabolic equation

[TABLE]

in the cube $B_{2R,T}:=\{(x,t)|d(x,x_{0},t)\leq 2R,0\leq t\leq T\}$, where $a$ is a constant.

(1) For $a\leq 0$. If $\frac{\gamma\alpha^{4}}{\alpha-1}\leq C_{1}$ for some constant $C_{1}$, then

[TABLE]

If $\frac{\gamma}{\alpha-1}\leq C_{2}$ for some constant $C_{2}$, then

[TABLE]

(2) For $a>0$. If $\frac{\gamma\alpha^{4}}{\alpha-1}\leq C_{1}$ for some constant $C_{1}$, then

[TABLE]

If $\frac{\gamma}{\alpha-1}\leq C_{2}$ for some constant $C_{2}$, then

[TABLE]

where $C$ is a constant.

Corollary 3.1**.**

Let $(M^{n},g(t))_{t\in[0,T]}$ be a complete solution to the Ricci flow  (1.7). Assume that $|\mathrm{Ric}(x,t)|\leq K$ for some $K>0$ and all $t\in[0,T]$. Given $x_{0}\in M$ and $R>0$, let $u$ be a positive solution of the nonlinear parabolic equation  (1.6) in the cube $B_{2R,T}:=\{(x,t)|d(x,x_{0},t)\leq 2R,0\leq t\leq T$}. Then the following special estimates are valid.

1. Li-Yau type:

[TABLE]

If $a\leq 0$, then

[TABLE]

If $a>0$, then

[TABLE]

2. Hamilton type:

[TABLE]

If $a\leq 0$, then

[TABLE]

If $a>0$, then

[TABLE]

3. Li-Xu type:

[TABLE]

If $a\leq 0$, then

[TABLE]

If $a>0$, then

[TABLE]

4. Linear Li-Xu type:

[TABLE]

If $a\leq 0$, then

[TABLE]

If $a>0$, then

[TABLE]

The local estimates above imply global estimates.

Corollary 3.2**.**

Let $(M^{n},g(0))$ be a complete noncompact Riemannian manifold without boundary, and asuume $g(t)$ evolves by Ricci flow in such a way that $|\mathrm{Ric}|\leq K$ for $t\in[0,T]$. Let $u(x,t)$ be a positive solution to the equation  (1.6). If $l\in\mathbb{R}$ and for $(x,t)\in M^{n}\times(0,T]$, then

[TABLE]

Remark 3.1*.*

The above results may be regard as generalizing the gradient estimates of Yang [30] to the Ricci flow.

3.2. Auxiliary lemma

To prove the theorem $3.1$, the following a lemma is needed.

Let $f=\log u$. Then

[TABLE]

Let $F=|\nabla f|^{2}-\alpha f_{t}+\alpha af-\alpha\varphi$, where $\alpha=\alpha(t)$ and $\varphi=\varphi(t)$. Then

[TABLE]

Lemma 3.1**.**

We assume that $\alpha(t)>1$ and $\varphi(t)>0$ satisfy the following system (2.3). Then

[TABLE]

Proof.

A computation is shown that

[TABLE]

and

[TABLE]

We follow that from  (3.6) and  (3.7)

[TABLE]

Further, by utilizing the unit matrix $(\delta_{ij})_{n\times n}$ and $(3.8)$, we obtain

[TABLE]

∎

3.3. The proof of Theorem

In this section, we will prove Theorem $3.1$.

Proof of Theorem $3.1$.

Let $G=\gamma(t)F$ and $\gamma(t)>0$ be non-decreasing. Then

[TABLE]

Now, let $\varphi(r)$ be a $C^{2}$ function on $[0,\infty)$ such that

[TABLE]

and

[TABLE]

where $C$ is an absolute constant. Define by

[TABLE]

where $\rho(x,t)=d(x,x_{0},t)$. By using maximum principle, the argument of Calabi [2] allows us to suppose that the function $\phi(x,t)$ with support in $B_{2R,T}$, is $C^{2}$ at the maximum point. By utilizing the Laplacian theorem, we deduce that

[TABLE]

For any $0\leq T_{1}\leq T$, let $H=\phi G$ and $(x_{1},t_{1})$ be the point in $B_{2R,T_{1}}$ at which $H$ attains its maximum value. We can suppose that $H$ is positive, because otherwise the proof is trivial. Then at the point $(x_{1},t_{1})$, we infer

[TABLE]

By the evolution formula of the geodesic length under the Ricci flow [6], we calculate

[TABLE]

where $\gamma_{t_{1}}$ is the geodesic connecting $x$ and $x_{0}$ under the metric $g(t_{1})$, $S$ is the unite tangent vector to $\gamma_{t_{1}}$, and $ds$ is the element of the arc length.

All the following computations are at the point $(x_{1},t_{1})$. Since

[TABLE]

and

[TABLE]

Case 1  $a\leq 0$. Combining  (3.14) with  (3.10), we have

[TABLE]

Using  (3.12) and  (3.13), we infer

[TABLE]

Multiply $\phi$ to $(3.15)$, we have

[TABLE]

We use the fact

[TABLE]

and

[TABLE]

to  (3.16), we deduce that

[TABLE]

For the inequality $Ax^{2}-2Bx\leq C$, one has $x\leq\frac{2B}{A}+\left(\frac{C}{A}\right)^{\frac{1}{2}}$, where $A,B,C>0$. Hence, we infer

[TABLE]

If $\gamma$ is nondecreasing which satisfies the system

[TABLE]

Recall that $\alpha(t)$ and $\gamma(t)$ are non-decreasing and $t_{1}<T_{1}$. Hence, we have

[TABLE]

Hence, we have for $\phi\equiv 1$ on $B_{R,T}$,

[TABLE]

If $\gamma$ is nondecreasing which satisfies the system

[TABLE]

Recall that $\alpha(t)$ and $\gamma(t)$ are non-decreasing and $t_{1}<T_{1}$. Hence, we have

[TABLE]

Hence, we have for $\phi\equiv 1$ on $B_{R,T}$,

[TABLE]

Because $T_{1}$ is arbitrary in $0<T_{1}<T$, the conclusion is valid.

Case 2  $a\geq 0$. It is not difficult to find $\Delta f\leq-\frac{F}{\alpha}$ form  (3.14). Then, we have from  (3.10)

[TABLE]

Using  (3.13) and  (3.13), we infer

[TABLE]

Multiply $\phi$, we have

[TABLE]

where we drop the term $\frac{\phi^{2}(\alpha-1)^{2}\gamma}{n\alpha^{2}}|\nabla f|^{4}$. We use the fact

[TABLE]

to  (3.17), we deduce that

[TABLE]

For the inequality $Ax^{2}-2Bx\leq C$, one has $x\leq\frac{2B}{A}+\left(\frac{C}{A}\right)^{\frac{1}{2}}$, where $A,B,C>0$.

[TABLE]

If $\gamma$ is nondecreasing which satisfies the system

[TABLE]

Recall that $\alpha(t)$ and $\gamma(t)$ are non-decreasing and $t_{1}<T_{1}$. Hence, we have

[TABLE]

Hence, we have for $\phi\equiv 1$ on $B_{R,T}$,

[TABLE]

If $\gamma$ is nondecreasing which satisfies the system

[TABLE]

Recall that $\alpha(t)$ and $\gamma(t)$ are non-decreasing and $t_{1}<T_{1}$. Hence, we have

[TABLE]

Hence, we have for $\phi\equiv 1$ on $B_{R,T}$,

[TABLE]

Because $T_{1}$ is arbitrary in $0<T_{1}<T$, the conclusion is valid. This proof is complete.

∎

4. Harnack Inequalities

In this section, as application of main theorems, some Harnack inequalities are derived.

Theorem 4.1**.**

Let $(M^{n},g(x,t))_{t\in[0,T]}$ be a complete solution to the Ricci flow (1.7). Suppose that $|\mathrm{Ric}|\leq K$ for some $K>0$, and all $(x,t)\in M^{n}\times[0,T]$. Assume that $u(x,t)$ is a positive solution for  (1.6). Let $h(x,t)$ be a function defined on $M^{n}\times[0,T]$ which is $C^{1}$ in $t$ and $C^{2}$ in $x$, satisfying $|\nabla h|^{2}\leq\delta_{2}h$ and $\Delta h\geq-\delta_{3}$ on $M^{n}\times[0,T]$ for some positive constants $\delta_{2}$ and $\delta_{3}$. Then for all $(x_{1},t_{1})\in M^{n}\times(0,T)$ and $(x_{2},t_{2})\in M^{n}\times(0,T)$ such that $t_{1}<t_{2}$, we have

[TABLE]

where

[TABLE]

Proof.

Firstly, the estimate in Corollary $2.2$ can be written as

[TABLE]

where $\mu_{1}=\sqrt{\overline{u}_{1}\delta_{3}}+\overline{u}_{1}\delta_{1}+\sqrt{(2-l)\overline{u}_{1}\delta_{2}}$ and $\mu_{2}=(l-1)\overline{u}_{2}\delta_{1}+\sqrt{\overline{u}_{2}\delta_{3}}$.

Now we only prove the conclusion for $l\leq 1$.

Define $l(s)=\log(\gamma(s),(1-s)t_{2}+st_{1})$. Obviously, we infer that $l(0)=\log u(y,t_{2})$ and $l(1)=\log u(x,t_{1})$. Direct calculation shows

[TABLE]

Integrating above inequality over $\gamma(s)$, we obtain

[TABLE]

The proof is complete. ∎

We also derive an Harnack inequality for the equation (1.6). The proof is similar to Theorem $4.1$, so we omit it.

Theorem 4.2**.**

Let $(M^{n},g(x,t))_{t\in[0,T]}$ be a complete solution to the Ricci flow (1.7). Suppose that $|\mathrm{Ric}|\leq K$ for some $K>0$, and all $(x,t)\in M^{n}\times[0,T]$. Assume that $u(x,t)$ is a positive solution for  (1.6). Then for all $(x_{1},t_{1})\in M^{n}\times(0,T)$ and $(x_{2},t_{2})\in M^{n}\times(0,T)$ such that $t_{1}<t_{2}$, we have

[TABLE]

5. Application to heat equation

According to Theorem $2.1$ and Theorem $3.1$, we derive corresponding gradient estimates and Harnack inequalities to the heat equation under Ricci flow

Theorem 5.1**.**

Let $(M^{n},g(t))_{t\in[0,T]}$ be a complete solution to the Ricci flow  (1.7). Assume that $|\mathrm{Ric}(x,t)|\leq K$ for some $K>0$ and all $t\in[0,T]$. Suppose that there exist three functions $\alpha(t)$, $\varphi(t)$ and $\gamma(t)$ satisfy the following conditions (C1), (C2), (C3) and (C4).

Given $x_{0}\in M$ and $R>0$, let $u(x,t)$ be a positive solution of the heat equation

[TABLE]

in the cube $B_{2R,T}:=\{(x,t)|d(x,x_{0},t)\leq 2R,0\leq t\leq T\}$, where $c$ is a constant.

If $\frac{\gamma\alpha^{4}}{\alpha-1}\leq C_{1}$ for some constant $C_{1}$, then

[TABLE]

If $\frac{\gamma}{\alpha-1}\leq C_{2}$ for some constant $C_{2}$, then

[TABLE]

where $C$ is a constant.

Corollary 5.1**.**

Let $(M^{n},g(t))_{t\in[0,T]}$ be a complete solution to the Ricci flow  (1.7). Assume that $|\mathrm{Ric}(x,t)|\leq K$ for some $K>0$ and all $t\in[0,T]$. Given $x_{0}\in M$ and $R>0$, let $u(x,t)$ be a positive solution of the heat equation  (5.1) in the cube $B_{2R,T}:=\{(x,t)|d(x,x_{0},t)\leq 2R,0\leq t\leq T$}. Then the following special estimates are valid.

1. Li-Yau type:

[TABLE]

[TABLE]

2. Hamilton type:

[TABLE]

[TABLE]

3. Li-Xu type:

[TABLE]

[TABLE]

4. Linear Li-Xu type:

[TABLE]

[TABLE]

Let $R\rightarrow\infty$, a global estimate is derived.

Corollary 5.2**.**

Let $(M^{n},g(t))_{t\in[0,T]}$ be a complete solution to the Ricci flow  (1.7). Assume that $|\mathrm{Ric}(x,t)|\leq K$ for some $K>0$ and all $t\in[0,T]$. Suppose that there exist three functions $\alpha(t)$, $\varphi(t)$ and $\gamma(t)$ satisfy the following conditions (C1), (C2), (C3) and (C4).

Given $x_{0}\in M$ and $R>0$, let $u(x,t)$ be a positive solution of the heat equation $(5.2)$ in the cube $M^{n}\times[0,T]$. Then

[TABLE]

where $C$ is a constant.

Using theorem $4.1$, we derive a Harnack inequality.

Corollary 5.3**.**

(Harnack Inequality) Let $(M^{n},g(t))_{t\in[0,T]}$ be a complete solution to the Ricci flow (1.7). Suppose that $|\mathrm{Ric}|\leq K$ for some $K>0$, and all $(x,t)\in M^{n}\times[0,T]$. Assume that $u(x,t)$ is a positive solution for  (5.1). Then for all $(x_{1},t_{1})\in M^{n}\times(0,T)$ and $(x_{2},t_{2})\in M^{n}\times(0,T)$ such that $t_{1}<t_{2}$, we have

[TABLE]

6. Appendix

We will check some special functions $\alpha(t)>1$, $\varphi(t)>0$ and $\gamma(t)>0$ satisfy the following two systems

[TABLE]

and

[TABLE]

Besides, $\alpha(t)$ and $\gamma(t)$ are non-decreasing.

$(1)$ Let $\alpha(t)=1+2Kt$, $\varphi(t)=\frac{n}{t}+nK(1+2Kt+\mu Kt)$ $(\mu\geq\frac{1}{4})$ and $\gamma(t)=Kt$.

One can has

[TABLE]

Hence, $\alpha(t)=1+2Kt$, $\varphi(t)=\frac{n}{t}+nK(1+2Kt+\mu Kt)$ $(0<\mu\leq\frac{1}{4})$ satisfy system (6.1).

On the other hand, one has

[TABLE]

and $\frac{\gamma}{\alpha-1}=\frac{1}{2}$. So, (6.2) is also satisfied.

$(2)$ $\alpha(t)=e^{2Kt}$, $\varphi(t)=\frac{n}{t}e^{4Kt}$ and $\gamma(t)=te^{2Kt}$, where $(0<Kt\leq 1)$ . Direct calculation gives

[TABLE]

Hence, $\alpha(t)=e^{2Kt}$ and $\varphi(t)=\frac{n}{t}e^{4Kt}$ satisfy system (6.1).

Besides, we have

[TABLE]

and as $t\rightarrow 0^{+}$, $\frac{\gamma}{\alpha-1}=\frac{te^{2Kt}}{e^{2Kt}-1}\rightarrow\frac{1}{2K}$. This implies $\frac{\gamma}{\alpha-1}\leq C$. So, (6.2) is also satisfied.

$(3)$ $\alpha(t)=1+\frac{\sinh(Kt)\cosh(Kt)-Kt}{\sinh^{2}(Kt)}$, $\varphi(t)=2nK[1+\coth(Kt)]$ and $\gamma(t)=\tanh(Kt)$. Direct calculation gives

[TABLE]

Let $f(x)=e^{4x}-3e^{2x}+2+4x$ with $x\leq 0$. Obviously, $f(0)=0$ and

[TABLE]

Then we get $f(x)>0$ for $x>0$. Hence, we have

[TABLE]

Hence, $\alpha(t)=1+\frac{\sinh(Kt)\cosh(Kt)-Kt}{\sinh^{2}(Kt)}$ and $\varphi(t)=2nK[1+\coth(Kt)]$ satisfy system (6.1).

On the other hand, as $t\rightarrow 0$, we have $\frac{\gamma\alpha^{4}}{\alpha-1}\rightarrow 2$; $\frac{\gamma\alpha^{4}}{\alpha-1}\rightarrow 1$ for $t\rightarrow\infty$. These imply $\frac{\gamma\alpha^{4}}{\alpha-1}\leq C$, here $C$ is a universal constant.

Besides, we have

[TABLE]

So, (6.2) is also satisfied.

$(4)$ $\alpha(t)=constant$, $\varphi(t)=\frac{\alpha n}{t}+\frac{nK\alpha^{2}}{\alpha-1}$ and $\gamma(t)=t^{\theta}$ with $0<\theta\leq 2$. Direct calculation gives

[TABLE]

Hence, $\alpha(t)=constant$, and $\varphi(t)=\frac{\alpha n}{t}+\frac{nK\alpha^{2}}{\alpha-1}$ satisfy system (6.1).

On the other hand, we have

[TABLE]

So, (6.2) is also satisfied.

7. Acknowledgement

We are grateful to Professor Jiayu Li for his encouragement. We also thank Professor Qi S Zhang for introduction of this problem in the summer course.