A novel high-order symmetric and energy-preserving continuous-stage Runge-Kutta-Nyström Fourier pseudo-spectral scheme for solving the two-dimensional nonlinear wave equation

1.   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 $\nabla$ and $\Delta$ 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 $\infty$-Bakry-Émery Ricci curvature and $m$-Bakry-Émery Ricci curvature as follows ^[6,20]

where $m \geq n$ is a constant, ${\rm Ric}$ is the Ricci curvature of $M$ and $\mathcal{L}_{V}$ denotes the Lie derivative along the direction $V$. In particular, we use the convention that $m = n$ if and only if $V\equiv0$. 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 $M\times[0, T]$ with $m$-Bakry-Émery Ricci tensor bounded below, where $q(x, t)$ is a function on $M\times[0, T]$ 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 $u^{\frac{1}{3}}$ instead of $\ln u$, we derive localised Hamilton type gradient estimates for $\frac{|\nabla u|}{\sqrt{u}}$. Most previous studies cited in the paper give the gradient estimates for $\frac{|\nabla u|}{u}$. The main theorems are below.

Theorem 1.1. Let $(M^n, g)$ be a complete Riemannian manifold with

for some constant $K_{1} > 0$ in $B(\overline{x}, \rho)$, some fixed point $\overline{x}$ in $M$ and some fixed radius $\rho$. Assume that there exists a constant $D_{1} > 0$ such that $u\in(0, D_{1}]$ is a smooth solution to the general parabolic Eq $(1.4)$ in $Q_{2\rho, T_{1}-T_{0}} = B(\overline{x}, 2\rho)\times [T_{0}, T_{1}]$, where $T_{1} > T_{0}$. Then there exists a universal constant $c(n)$ that depends only on $n$ so that

in $Q_{\frac{\rho}{2}, T_{1}-T_{0}}$ with $t\neq T_{0}$.

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 ${\rm Ric}_{V}\geq-(m-1)K_{1}$ for some constant $K_{1} > 0$, and notice that ${\rm Ric}_{V}\geq-(m-1)K_{1}$ is weak than ${\rm Ric}_{V}^{m}\geq-(m-1)K_{1}$. 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 $V = \nabla f$, 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

for some constant $K_{1} > 0$ in $B(\overline{x}, \rho)$, some fixed point $\overline{x}$ in $M$ and some fixed radius $\rho$. Assume that $u$ is a positive smooth solution to the equation

in $Q_{2\rho, T_{1}-T_{0}} = B(\overline{x}, 2\rho)\times [T_{0}, T_{1}]$, where $T_{1} > T_{0}$.

(1) When $a > 0$, assuming that $1 \leq u \leq D_{1}$ in $Q_{2\rho, T_{1}-T_{0}}$, there exists a constant $c = c(n)$ such that

in $Q_{\frac{\rho}{2}, T_{1}-T_{0}}$ with $t\neq T_{0}$.

(2) When $a < 0$, assuming that $0 < u \leq D_{1}$ in $Q_{2\rho, T_{1}-T_{0}}$, there exists a constant $c = c(n)$ such that

in $Q_{\frac{\rho}{2}, T_{1}-T_{0}}$ with $t\neq T_{0}$.

Using the corollary, we get the following Liouville type result.

Corollary 1.3. Let $(M^n, g)$ be a complete Riemannian manifold with ${\rm Ric}_{V}^{m}\geq-(m-1)K_{1}$ 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.

(1) When $a > 0$, assume that $1 \leq u \leq D_{1}$. If $a = (m-1)K_{1}$, then $u\equiv 1$.

(2) When $a < 0$, assume that $0 < u \leq \exp\{-2+\frac{2(m-1)K_{1}}{a}\}$, then $u$ does not exist.

Remark 1.2. When $V = 0$, $\Delta_{V}$ and ${\rm Ric}_{V}^{m}$ become $\Delta$ and ${\rm Ric}$, respectively. It is clear that Corollary 1.2–1.3 generalize Theorem 1.3 and Corollary 1.1 in ^[14].

We can obtain a global estimate from Theorem 1.1 by taking $\rho\to0$.

Corollary 1.4. Let $(M^n, g)$ be a complete Riemannian manifold with ${\rm Ric}_{V}^{m}\geq-(m-1)K_{1}$ for some constant $K_{1} > 0$. $u$ is a positive smooth solution to the general parabolic Eq $(1.4)$ on $M^{n}\times[T_{0}, T_{1}]$. Suppose that $u\leq D_{1}$ on $M^{n}\times[T_{0}, T_{1}]$. We also suppose that

on $M^{n}\times[T_{0}, T_{1}]$. Then there exists a universal constant $c$ that depends only on $n$ so that

in $M^{n}\times[T_{0}, T_{1}]$ with $t\neq T_{0}$.

Let $A(u) = a(u(x, t))^{\beta}$ in Corollary 1.4, we obtain Hamilton type gradient estimates for bounded positive solutions of the equation

Corollary 1.5. Let $(M^n, g)$ be a complete Riemannian manifold with ${\rm Ric}_{V}^{m}\geq-(m-1)K_{1}$ for some constant $K_{1} > 0$. $u$ is a positive smooth solution to $(1.10)$ on $M^{n}\times[T_{0}, T_{1}]$. Suppose that $u\leq D_{1}$ on $M^{n}\times[T_{0}, T_{1}]$. We also suppose that

on $M^{n}\times[T_{0}, T_{1}]$. Then in $M^{n}\times[T_{0}, T_{1}]$ with $t\neq T_{0}$, there exists a universal constant $c$ that depends only on $n$ so that

where

In the next part, our result concerns gradient estimates for positive solutions of

on $(M^n, g(t))$ with the metric evolving under the geometric flow:

where $\Delta_{t}$ depends on $t$ and it denotes the Laplacian of $g(t)$, and ${\rm S}(t)$ 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).

Theorem 1.6. Let $(M^n, g(t))_{t\in[0, T]}$ be a complete solution to the geometric flow $(1.13)$ on $M^n$ with

for some $K_{2}$, $K_{3} > 0$ on $Q_{\rho, T} = B(\overline{x}, \rho)\times [0, T]$. Assume that there exists a constant $L_{1} > 0$ such that $u\in(0, L_{1}]$ is a smooth solution to the general parabolic Eq $(1.12)$ in $Q_{2\rho, T}$. Then there exists a universal constant $c(n)$ that depends only on $n$ so that

in $Q_{\frac{\rho}{2}, T}$.

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 $p$, $q\in C^{2, 1} (M^{n}\times[0, T])$, $b$ is a positive constant and $a, \alpha$ 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.

2.   Gradient estimates for (1.4): Proof of Theorem 1.1

2.1.   Basic lemmas

We first give some notations for the convenience of writing throughout the paper. Let $h : = u^{\frac{1}{3}}$ and $\widehat{A}(h) : = \frac{A(u)}{3u^{\frac{2}{3}}}$. Then $\widehat{A}_{h} = A'(u)-\frac{2A(u)}{3u}$. To prove Theorem 1.1 we need two basic lemmas. First, we derive the following lemma.

Lemma 2.1. Let $(M^n, g)$ be a complete Riemannian manifold with ${\rm Ric}_{V}^{m}\geq-(m-1)K_{1}$ for some constant $K_{1} > 0$. $u$ is a positive smooth solution to the general parabolic Eq $(1.4)$ in $Q_{2\rho, T_{1}-T_{0}}$. If $h : = u^{\frac{1}{3}}$, and $\mu: = h\cdot|\nabla h|^{2}$, then we have

Proof. Since $h : = u^{\frac{1}{3}}$, by a simple computation, we can derive the following equation from (1.4):

By direct computations, we have

and

By the following fact:

it yields

Plugging (2.4) into (2.3), we have

The partial derivative of $\mu$ with respect to $t$ is given by

It follows from (2.2), (2.5) and (2.6) that

Note that

Therefore,

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]).

Lemma 2.2. Fix $T_{0}$, $T_{1}\in \mathbb{R}$ and $T_{0} < T_{1}$. Given $\tau\in (T_{0}, T_{1}]$, there exists a smooth function $\overline{\Psi}: [0, +\infty)\times[T_{0}, T_{1}]\to \mathbb{R}$ satisfying the following requirements.

1). The support of $\overline{\Psi}(s, t)$ is a subset of $[0, \rho]\times[T_{0}, T_{1}]$, and $0\leq\overline{\Psi}\leq1$ in $[0, \rho]\times[T_{0}, T_{1}]$.

2). The equalities $\overline{\Psi}(s, t) = 1$ in $[0, \frac{\rho}{2}]\times[\tau, T_{1}]$ and $\frac{\partial \overline{\Psi}}{\partial s}(s, t) = 0$ in $[0, \frac{\rho}{2}]\times[T_{0}, T_{1}]$.

3). The estimate $|\partial_{t}\overline{\Psi}|\leq \frac{\overline{C}}{\tau-T_{0}}\overline{\Psi}^{\frac{1}{2}}$ is satisfied on $[0, +\infty]\times[T_{0}, T_{1}]$ for some $\overline{C} > 0$, and $\overline{\Psi}(s, T_{0}) = 0$ for all $s\in[0, +\infty)$.

4). The inequalities $-\frac{C_{b}\overline{\Psi}^{b}}{\rho}\leq\frac{\partial \overline{\Psi}}{\partial s}\leq0$ and $|\frac{\partial^{2} \overline{\Psi}}{\partial s^{2}}|\leq \frac{C_{b}\overline{\Psi}^{b}}{\rho^{2}}$ hold on $[0, +\infty]\times[T_{0}, T_{1}]$ for every $b\in(0, 1)$ with some constant $C_{b}$ that depends on $b$.

Throughout this section, we employ the cut-off function $\Psi : M^{n}\times[T_{0}, T_{1}] \to\mathbb{R}$ by

where $r(x) : = d(x, \overline{x})$ is the distance function from some fixed point $\overline{x} \in M^{n}$.

2.2.   Proof of Theorem 1.1

From Lemma 2.1, we have

For fixed $\tau\in (T_{0}, T_{1}]$, let $(x_{1}, t_{1})$ be a maximum point for $\Psi\mu$ in $Q_{\rho, \tau-T_{0}}$. Obviously at $(x_{1}, t_{1})$, we have the following facts: $\nabla(\Psi\mu) = 0$, $\Delta_{V}(\Psi\mu)\leq0$, and $\partial_{t}(\Psi\mu)\geq0$. It follows from (2.10) that at such point

In other words, we have just proved that

at $(x_{1}, t_{1})$.

Next, to realize the theorem, it suffices to bound each term on the right-hand side of (2.12). To deal with $\Delta_{V}\Psi(x_{1}, t_{1})$, we divide the arguments into two cases.

Case 1. If $d(\overline{x}, x_{1}) < \frac{\rho}{2}$, then it follows from Lemma 2.2 that $\Psi(x, t) = 1$ around $(x_{1}, t_{1})$ in the space direction. Therefore, $\Delta_{V}\Psi(x_{1}, t_{1}) = 0$.

Case 2. Suppose that $d(\overline{x}, x_{1})\geq\frac{\rho}{2}$. Since ${\rm Ric}_{V}^{m}\geq-(m-1)K_{1}$, 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, $0 \leq\Psi\leq1$ and Cauchy's inequality.

On the other hand, by Young's inequality and Lemma 2.2, we obtain

and

Now, we plug (2.13)–(2.17) into (2.12) to get

The finally, since $\Psi(x, \tau) = 1$ in $B(\overline{x}, \frac{\rho}{2})$, it follows from (2.18) that

Since $\tau\in(T_{0}, T_{1}]$ is arbitrary and $\mu = \frac{|\nabla u|^{2}}{9u}$, we have

in $Q_{\frac{\rho}{2}, T_{1}-T_{0}}$. We complete the proof.

2.3.   Proof of Corollary 1.3 and Corollary 1.5

Proof of Corollary 1.3.

(1) When $a > 0$, for $a = (m-1)K_{1}$, using the inequality (1.7), we have

Letting $\rho\to +\infty$, $t \to +\infty$ in (2.21), we get $u$ is a constant. Using $\Delta_{V}u-au \ln u = 0$, we get $u = 1$.

(2) When $a < 0$, for $D_{1} = \exp\{-2+\frac{2(m-1)K_{1}}{a}\}$, using the inequality (1.8), we have

Letting $\rho\to +\infty$, $t \to +\infty$ in (2.22), we get $u$ is a constant. Using $\Delta_{V}u-au \ln u = 0$, we get $u = 1$, but $0 < u \leq D_{1} = \exp\{-2+\frac{2(m-1)K_{1}}{a}\} < 1$. So $u$ does not exist.

Proof of Corollary 1.5.

Let

From Corollary 1.4, we just have to compute $\Lambda$. 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.

3.1.   Basic lemmas

We first derive a general evolution equation under the geometric flow.

Lemma 3.1. (^[21]) Suppose the metric evolves by $(1.13)$. Then for any smooth function $f$, we have

Next, we derive the following lemma in the same fashion of Lemma 2.1.

Lemma 3.2. Let $(M^{n}, g(t))_{t\in[0, T]}$ be a complete solution to the geometric flow $(1.13)$ and $u$ be a smooth positive solution to the nonlinear parabolic Eq $(1.12)$ in $Q_{2\rho, T}$. Suppose that there exists positive constants $K_{2}$ and $K_{3}$, such that

in $Q_{\rho, T}$. If $h : = u^{\frac{1}{3}}$, and $\mu: = h\cdot|\nabla h|^{2}$, then in $Q_{\rho, T}$, we have

Proof. Since $u$ is a solution to the nonlinear parabolic Eq (1.12), the function $h = u^{\frac{1}{3}}$ satisfies

As in the proof of Lemma 2.1, we have that

On the other hand, by the equation $\partial_{t}g(t) = 2{\rm S}(t)$, we have

Therefore,

where we used (2.8), the assumption on bound of ${\rm Ric+S}$ and

The proof is complete.

Finally, we employ the cut-off function $\Psi : M^{n}\times[T_{0}, T_{1}] \to\mathbb{R}$, with $T_{0} = 0$, $T_{1} = T$, by

where $r(x, t) : = d_{g(t)}(x, \overline{x})$ is the distance function from some fixed point $\overline{x} \in M^{n}$ with respect to the metric $g(t)$.

3.2.   Proof of Theorem 1.6

To prove Theorem 1.6, we follows the same procedure used previously to prove Theorem 1.1; hence in view of Lemma 3.2,

For fixed $\tau\in (0, T]$, let $(x_{2}, t_{2})$ be a maximum point for $\Psi\mu$ in $Q_{\rho, \tau}: = B(\overline{x}, \rho)\times [0, \tau]\subset Q_{\rho, T}$. It follows from (3.5) that at such point

At $(x_{2}, t_{2})$, making use of (3.6), we further obtain

Next, we need to bound each term on the right-hand side of (3.7). To deal with $\Delta_{t}\Psi(x_{2}, t_{2})$, we divide the arguments into two cases:

Case 1: $r(x_{2}, t_{2}) < \frac{\rho}{2}$. In this case, from Lemma 2.2, it follows that $\Psi(x, t) = 1$ around $(x_{2}, t_{2})$ in the space direction. Therefore, $\Delta_{t_{2}}\Psi(x_{2}, t_{2}) = 0$.

Case 2: $r(x_{2}, t_{2})\geq\frac{\rho}{2}$. Since ${\rm Ric}_{g(t)}\geq-(n-1)K_{2}$, the Laplace comparison theorem (see ^[17]) implies that

Then, it follows that

at $(x_{2}, t_{2})$, which agrees with Case 1.

Next, we estimate $\partial_{t}\Psi$. $\forall x\in B(\overline{x}, \rho)$, let $\gamma: [0, a]\to M^{n}$ be a minimal geodesic connecting $x$ and $\overline{x}$ at time $t\in[0, T]$. Then, we have

Thus, together with Lemma 2.2, we find that

Therefore, we have

at $(x_{2}, t_{2})$ for some universal constant $c > 0$ that depends only on $n$. By similar computations as in the proof of Theorem 1.1, we arrive at

and

Plugging (3.9)–(3.13) into (3.6), we get

Note that $\Psi(x, \tau) = 1$ when $d(x, \overline{x}) < \frac{\rho}{2}$, it follows from (2.18) that

Since $\tau\in(0, T]$ is arbitrary and $\mu = \frac{|\nabla u|^{2}}{9u}$, we have

in $Q_{\frac{\rho}{2}, T}$.

We complete the proof.

Remark 3.1. We also obtain the corresponding applications similar to Corollary 1.4 and Corollary 1.5, which we will not write them down here.

Acknowledgments

The author would like to thank the referee for helpful suggestion which makes the paper more readable. This work was supported by NSFC (Nos. 11971415, 11901500), Henan Province Science Foundation for Youths (No.212300410235), and the Key Scientific Research Program in Universities of Henan Province (No.21A110021) and Nanhu Scholars Program for Young Scholars of XYNU (No. 2019).

Conflict of interest

The author declares no conflict of interest.