A Pogorelov estimate and a Liouville type theorem to parabolic $k$-Hessian equations

TL;DR

This paper establishes Pogorelov estimates and Liouville theorems for parabolic $k$-Hessian equations, showing that solutions with certain convexity and growth conditions are necessarily quadratic in space and linear in time.

Contribution

It introduces new Pogorelov type estimates and Liouville theorems for solutions of parabolic $k$-Hessian equations under specific convexity and growth assumptions.

Findings

Solutions are linear in time plus quadratic in space.

Any $k+1$-convex-monotone solution with quadratic growth in initial data is of a specific polynomial form.

The results extend classical estimates to a parabolic $k$-Hessian context.

Abstract

We consider Pogorelov type estimates and Liouville type theorems to parabolic $k$-Hessian equations of the form $-u_t \sigma_k (D^2u) =1$ in $\mathbb{R}^n\times (-\infty, 0]$. We derive that any \textbf{$k+1$-convex-monotone} solution to $-u_t \sigma_k (D^2u) =1$ when $u(x,0)$ satisfies a quadratic growth and $0<m_1\le -u_t\le m_2$ must be a linear function of $t$ plus a quadratic polynomial of $x$.

Equations

$$-u_t{\sigma }_k(D^{2}u)=1,\text{in}{\mathbb{R}}^n\times (-\infty ,0].$$

$$\left\{\begin{array}[]{rl}-u_{t}\sigma_{k}(D^{2}u)=1,\ \textrm{in}\ \Omega,\\ u=0,\ \textrm{on}\ \partial\Omega,\end{array}\right.$$

$$\partial \Omega =(\overline{\Omega (t_0)}\times t_0)\cup \bigcup _{t\le 0}(\partial \Omega (t)\times \{t\}),$$

$${\sigma }_k(\lambda )=\sum _{1\le i_1<\cdots <i_k\le n}{\lambda }_{i_1}\cdots {\lambda }_{i_k}.$$

$${\Gamma }^k=\{\lambda =({\lambda }_1,\cdots ,{\lambda }_n)\in {\mathbb{R}}^n|{\sigma }_1(\lambda )>0,\cdots ,{\sigma }_k(\lambda )>0\}.$$

$$u(x)\ge b|x{|}^2-c,\text{mbox}for|x|\ge R.$$

$${\sigma }_k(D^{2}u)=f$$

$$(-u{)}^{\beta }\Delta u\le C,$$

$$\left\{\begin{array}[]{rl}-u_{t}\sigma_{2}(D^{2}u)=1,\ \textrm{in}\ \Omega,\\ u=0,\ \textrm{on}\ \partial\Omega.\end{array}\right.$$

$$(-u{)}^8\Delta u\le C,$$

$$\frac{{\sigma }_k(D^{2}u)}{{\sigma }_l(D^{2}u)}=1,(k>l).$$

$${\sigma }_2(D^{2}u)=1.$$

$$D^{2}u\ge [\delta -\sqrt{\frac{2}{n(n-1)}}]I,$$

$$u_t=(\det D^{2}u{)}^{1/n}.$$

$$-u_t\det D^{2}u=f,$$

$$\sigma_{l}(\lambda|i)=\sigma_{l}(\lambda)\big{|}_{\lambda_{i}=0}.$$

$$\sigma_{l}(\lambda|pq)=\sigma_{l}(\lambda)\big{|}_{\lambda_{p}=\lambda_{q}=0}.$$

$$\big{[}\frac{\sigma_{k}(\lambda)/C^{k}_{n}}{\sigma_{l}(\lambda)/C^{l}_{n}}\big{]}^{\frac{1}{k-l}}\leq\big{[}\frac{\sigma_{r}(\lambda)/C^{r}_{n}}{\sigma_{s}(\lambda)/C^{s}_{n}}\big{]}^{\frac{1}{r-s}}.$$

$${\kappa }_j{\sigma }_k^{jj,ii}+{\sigma }_k^{jj}\ge {\sigma }_k^{ii}.$$

$$C\ge {\sigma }_k(D^{2}u)\ge {\lambda }_1{\sigma }_{k-1}(\lambda |1)+{\sigma }_k(\lambda |1)\ge {\lambda }_1\cdots {\lambda }_k\ge {\lambda }_k^k.$$

$${\lambda }_n\ge -\sum _{i=2}^{n-1}{\lambda }_i\ge -C.$$

$${\kappa }_j{\sigma }_k^{jj,ii}+{\sigma }_k^{jj}$$

$$\frac{dg(A+tB)}{ds}\big{|}_{s=0}=\frac{\partial G}{\partial\mu_{p}}B_{pp},$$

$$\frac{d^{2}g(A+tB)}{ds^{2}}\big{|}_{s=0}=\sum_{p,q}\frac{\partial^{2}G}{\partial\mu_{p}\partial\mu_{q}}B_{pp}B_{qq}+\sum_{p\neq q}\frac{\frac{\partial G}{\partial\mu_{p}}-\frac{\partial G}{\partial\mu_{q}}}{\mu_{p}-\mu_{q}}B^{2}_{pq}.$$

$$\frac{dg(A+tB)}{ds}\big{|}_{s=0}=m\mu^{m-1}_{p}u_{ppi},$$

$$\frac{d^{2}g(A+tB)}{ds^{2}}\big{|}_{s=0}=m(m-1)\sum_{p}\mu_{p}^{m-2}u^{2}_{ppi}+m\sum_{p\neq q}\frac{\mu_{p}^{m-1}-\mu_{q}^{m-1}}{\mu_{p}-\mu_{q}}u^{2}_{pqi}.$$

$$-\sigma_{k}^{pp,qq}u_{pph}u_{qqh}+(1-\alpha+\frac{\alpha}{\delta})\frac{(\sigma_{k})^{2}_{h}}{\sigma_{k}}\geq\sigma_{k}(\alpha+1-\delta\alpha)\big{[}\frac{(\sigma_{\mu})_{h}}{\sigma_{\mu}}\big{]}^{2}-\frac{\sigma_{k}}{\sigma_{\mu}}\sigma_{\mu}^{pp,qq}u_{pph}u_{qqh}.$$

$$\sum _{i=2}^n{\sigma }_2^{ii}{\xi }_{ii}+{\sigma }_2^{11}{\xi }_{11}=\eta ,$$

$$-\sum _{i\ne j}{\xi }_{ii}{\xi }_{jj}\ge \frac{n-1}{2{\sigma }_2(W)}\frac{[2{\sigma }_2(W){\xi }_{11}-W_{11}\eta {]}^2}{[(n-1)W_{11}^2+2(n-2){\sigma }_2(W)]}-\frac{{\eta }^2}{2{\sigma }_2(W)}.$$

$$-\sum _{i\ne j}{u}_{iil}{u}_{jjl}$$

Full text

Mathematical Subject Classification (2010): Primary 35K55, Secondary 35B45.

Keywords: Pogorelov estimate, Liouville type theorem, parabolic $k$-Hessian equation, $k$-convex-monotone.

This research was supported by funds from Hubei Provincial Department of Education Key Projects D20171004.

1. Introduction

In this paper, we derive a Liouville type theorem for parabolic k-Hessian equations

[TABLE]

Namely, any $k+1$-convex-monotone solution of (1.1) with a quadratic growth and $0<m_{1}\leq-u_{t}\leq m_{2}$, must be a linear function of $t$ plus a quadratic polynomial of $x$.

To obtain the Liouville type theorem, the key points are Pogorelov estimates in our method. Thus, we consider the following equations

[TABLE]

where $D^{2}u\in\Gamma^{k+1}$, $0<m_{1}\leq-u_{t}\leq m_{2}$. Here $\Omega\subset\mathbb{R}^{n}\times(-\infty,0]$ is a bounded domain and $t\leq 0,$ $\Omega(t)=\{x\in\mathbb{R}^{n}|(x,t)\in\Omega\},$ $t_{0}=\inf\{t\leq 0|\Omega(t)\neq\emptyset\}$. The parabolic boundary $\partial\Omega$ is defined by

[TABLE]

where $\overline{\Omega(t_{0})}$ denotes the closure of $\Omega(t_{0})$ and $\partial\Omega(t)$ denotes the boundary of $\Omega(t)$. The k-th elementary symmetric polynomial is denoted by $\sigma_{k}$:

[TABLE]

$\sigma_{k}(D^{2}u)$ means $\sigma_{k}$ is applied to the eigenvalues of $D^{2}u$. Let $\Gamma^{k}$ be an open convex cone in $\mathbb{R}^{n}$:

[TABLE]

Here the function $u=u(x,t):\mathbb{R}^{n}\times(\infty,0]\rightarrow\mathbb{R}$ is said to be $k$-convex if the eigenvalues of $D^{2}u$ lie in $\Gamma^{k}$. Moreover, it is said to be $k$-convex-monotone if it is $k$ convex in $x$ and non-increasing in $t$. The quadratic growth means that there are some positive constants $b,c$ and sufficiently large $R$, such that,

[TABLE]

A priori estimates for elliptic $k$-Hessian equations

[TABLE]

have been studied intensively by many authors. In Chou-Wang [5], the authors got interior gradient and second order estimates when $f$ depends on $x,\ u$. Warren-Yuan [19] obtained $C^{2}$ interior estimates in the case of equations $\sigma_{2}(D^{2}u)=1$ in $\mathbb{R}^{3}$, which originated from special Lagrangian geometry. Guan-Qiu [8] established interior $C^{2}$ estimates for solutions of the prescribing scalar curvature equations and 2-Hessian equations under additional assumption that $\sigma_{3}(D^{2}u)>-A$ for some constant $A>0.$ The purely interior $C^{2}$ estimates for semi-convex solutions of above equation have been obtained by McGonagle-Song-Yuan [16] recently. For $k\geq 2$, Li-Ren-Wang [15] established Pogorelov estimates under the condition $k+1$-convex, when $f$ depends on $x,u,Du$.

Our paper is based on the work of Li-Ren-Wang [15]. Firstly, We extend the Pogorelov estimate from elliptic Hessian equations to parabolic Hessian equations. We have obtained the following Pogorelov type estimates.

Theorem 1.1**.**

Let $u$ be a $k+1$-convex-monotone solution of (1.2) satisfying $0<m_{1}\leq-u_{t}\leq m_{2}$. Then there exists a positive constant $\beta$ sufficiently large such that

[TABLE]

$C$* depends on the diameter of $\Omega(t)$, $m_{1}$, $m_{2}$, $k$ and $\sup|u|$.*

For $k=2$, we can decrease the power in (1.5) and improve the estimates as follows.

Theorem 1.2**.**

Let $u$ be a $3$-convex-monotone solution of the following equation (1.6) satisfying $0<m_{1}\leq-u_{t}\leq m_{2}$.

[TABLE]

Then

[TABLE]

$C$* depends on the diameter of $\Omega(t)$, $m_{1}$, $m_{2}$ and $\sup|u|$.*

These type of interior estimates are important for existence of isometric embedding of non-compact surfaces and for Liouville type theorems. There has been much activities on Liouville type theorems for elliptic $k$-Hessian equations. In 2003, Bao-Chen-Guan-Ji [2] studied the Liouville theorem to

[TABLE]

They proved that entire convex solutions of the equation (1.8) with a quadratic growth are quadratic polynomials. In 2010, Chang-Yuan [7]  considered

[TABLE]

and obtained that the entire solution to (1.9) is quadratic polynomial if

[TABLE]

where $\delta>0$. In 2016, Li-Ren-Wang [15] considered  $\sigma_{k}(D^{2}u)=1$ for general $k$. They obtained that global $k+1$-convex solutions with a quadratic growth are quadratic polynomials. Chen-Xiang [6] improved the condition from $k+1$-convex to $k$-convex for $k=2$ under $\sigma_{3}(D^{2}u)\geq-A$. Especially, for $n=3$, $\sigma_{3}(D^{2}u)\geq-A$ can be redundant. Then He-Sheng-Xiang [14] removed the condition $\sigma_{3}(D^{2}u)\geq-A$ for $2$-Hessian equations in general dimension $n$.

However, as far as we know, Liouville type theorems for parabolic fully nonlinear equations are known most for parabolic Monge-Ampère equations. Guti$\acute{e}$rrez-Huang [11] extended Theorem of J$\ddot{o}$rgens, Calabi, and Pogorelov to parabolic Monge-Ampère equations. Xiong-Bao [20] obtained Liouville theorems for

[TABLE]

Zhang-Bao-Wang [21] extend the theorem of Caffarelli and Li [4] to parabolic Monge-Ampère equation

[TABLE]

and obtain asymptotic behavior at infinity. And along the line of approach in their paper, other parabolic Monge-Ampère equations can be also treated. For general $k$, Nakamori S. and Takimoto K.[17] studied the bernstein type theorem for parabolic $k$-Hessian equations when the entire solution $u$ was convex-monotone. Recently, He-Pan-Xiang [13] prove that the $2$-convex-monotone solutions with $\sigma_{3}(D^{2}u)>-A$, $m_{1}\leq-u_{t}\leq m_{2}$ and a quadratic growth must be a linear function of $t$ plus a quadratic polynomial of $x$ when $k=2$.

Then using Theorem 1.1, we have established the following Liouville type theorem for parabolic $k$-Hessian equations.

Theorem 1.3**.**

Let $u$ be a $k+1$-convex-monotone solution of (1.1), $u(x,0)$ satisfying a quadratic growth, and $0<m_{1}\leq-u_{t}\leq m_{2}$. Then $u$ has the form $u(x,t)=-mt+p(x)$ where $m>0$ and $p(x)$ is a quadratic polynomial.

This paper is organized as follows. We start with some notations and Lemmas in section 2. In section 3 we prove a Pogorelov estimate for the $k+1$-convex-monotone solutions to parabolic $k$-Hessian equation (1.2). A Pogorelov estimate for the 2-convex-monotone solutions to parabolic 2-Hessian equation (1.6) is given in section 4. The proof of Liouville Theorem (Theorem1.3) is given in section 5.

2. Preliminaries

Throughout this paper, we use the Einstein summation convention and denote by $\lambda=(\lambda_{1},\cdots,\lambda_{n})$ the eigenvalues of $D^{2}u$. To begin this section, we introduce some notations.

Definition 2.1**.**

Let $\lambda=(\lambda_{1},\cdots,\lambda_{n})\in\mathbb{R}^{n}$.

(1)

[TABLE]

$\sigma_{k-1}(\lambda|i)$ is also denoted by $\sigma_{k}^{ii}$.

(2)

[TABLE]

$\sigma_{k-2}(\lambda|pq)$ is also denoted by $\sigma_{k}^{pp,qq}$.

The following Lemmas will be used in the proof for Pogorelov estimates.

Lemma 2.2**.**

(See [18]) Suppose $\lambda\in\Gamma^{k}$. For $0\leq l<k\leq n,0\leq s<r\leq n,k\geq r,l\geq s$, the following is the generalized Newton-MacLaurin inequality

[TABLE]

Lemma 2.3**.**

(See [15]) (1)Let $u$ be a $k+1$-convex fucntion, $\sigma_{k}(D^{2}u)\leq C$, $\lambda=(\lambda_{1},\cdots,\lambda_{n})$ be the eigenvalues of $D^{2}u$ with $\lambda_{1}\geq\cdots\geq\lambda_{n}$. Then there exists a positive constant $K_{0}$ such that $\lambda_{i}+K_{0}>0$.

(2)Assume there exists a positive constant $K_{0}$ such that $D^{2}u+K_{0}I>0$. Let $\lambda=(\lambda_{1},\cdots,\lambda_{n})$ be the eigenvalues of $D^{2}u$, $\kappa_{i}=\lambda_{i}+K_{0}>0.$ Then

[TABLE]

Proof.

(1)

[TABLE]

Since $\lambda\in\Gamma^{k}$, we have

[TABLE]

(2)

[TABLE]

Lemma 2.4**.**

(See [1])Let $B$ be a symmetric matrix, $A$ be a diagonal matrix, $G$ be a symmetric function of the eigenvalues of metrices. Let us denote by $\mu(B)$ the eigenvalues of $B$. Set $g(B)=G(\mu(B))$. Then

[TABLE]

and

[TABLE]

For our case, $G(\mu)=\sum_{j}\mu_{j}^{m}$, $A=D^{2}u(x_{0},t_{0})$, $B_{pq}=u_{pqi}$. Then

[TABLE]

and

[TABLE]

Lemma 2.5**.**

(See [9])Let $k>\mu$, $\alpha=\frac{1}{k-\mu}$. For any $\delta>0$, we have

[TABLE]

Lemma 2.6**.**

(See [3], [13]) Suppose that $W$ is diagonal, $W_{11}\geq\cdot\cdot\cdot\geq W_{nn}$ and the eigenvalues of $W$ lie in $\Gamma^{2}$. If $\xi_{ij}$ is symmetric and

[TABLE]

then

[TABLE]

For our case, let $W=D^{2}u$, $\sigma_{2}=-\frac{1}{u_{t}}$, $\xi_{ij}=u_{ijl}$, $\eta=\frac{u_{tl}}{u_{t}^{2}}$. Assume $\epsilon$ is a small positive constant and $u_{11}\geq\sqrt{\frac{2(n-2)}{(n-1)\epsilon m_{1}}}$. Then

[TABLE]

3. A Pogorelov estimate for the $k+1$-convex-monotone solutions to parabolic $k$-Hessian equations

In this section, we consider Pogorelov estimates for parabolic $k$-Hessian equations (1.2). We shall prove Theorem 1.1.

Since $u=0$ on $\partial\Omega$, we have $u\leq 0$ in $\Omega$ by the Comparison principle. By Lemma 2.3, there exists $K_{0}>0$ such that $D^{2}u+K_{0}I>0$. Take the test function

[TABLE]

where $P_{m}=\sum_{i}\kappa_{i}^{m}$, $\kappa_{i}=\lambda_{i}+K_{0}>0$. Constants $a$ and $m$ are positive constants to be determined later. Assume the maximum of $\phi$ is attained at $(x_{0},t_{0})$, $u_{ij}(x_{0},t_{0})$ is diagonal and $u_{11}(x_{0},t_{0})\geq\cdots\geq u_{nn}(x_{0},t_{0})$.

Then

[TABLE]

By Lemma 2.4, we obtain

[TABLE]

Moreover,

[TABLE]

Now differentiating equations (1.2), we obtain

[TABLE]

and

[TABLE]

Note that

[TABLE]

Then (3.5) implies that

[TABLE]

Then by (3.1)-(3.6), we have

[TABLE]

where

[TABLE]

We claim that:

Claim 3.1**.**

Suppose $u$ is the $k+1$-convex solution of (1.2) with $m_{1}\leq-u_{t}\leq m_{2}$. Then, either

[TABLE]

or

[TABLE]

It is obviously that (3.8) implies (1.5). If (3.9) holds, combining Lemma 2.2, we can obtain

[TABLE]

Then (1.5) is still holds and we completes the proof of Theorem 1.1.

Thus the proofs for (3.8) and (3.9) in Claim 3.1 are the remaining questions. By Lemma 2.3, we obtain

[TABLE]

and

[TABLE]

Moreover, by Cauthy inequality, we have

[TABLE]

It yields

[TABLE]

Therefore, by (3.10) and (3.11), we have

[TABLE]

We divide the proof into two cases: $i>1$ and $i=1$.

(A) $i>1$. In this case, we assert that

[TABLE]

We further divide case (A) into three subcases to prove the above assertion (3.13).

(A1) $\lambda_{i}\leq\lambda_{j}$, $\lambda_{i}\geq{K_{0}}{}$.

[TABLE]

Then combining $P_{m}\geq\kappa_{1}^{m}$, we have

[TABLE]

(A2) $\lambda_{i}\leq\lambda_{j}$, $\lambda_{i}<{K_{0}}{}$. Let $k\leq r\leq k+6$. We may assume $\kappa_{1}\geq\kappa_{i}^{r+2}$ and $\kappa_{1}^{k+1}\sigma_{k}^{jj}\geq\kappa_{1}^{k}\lambda_{1}\sigma_{k}^{11}\geq\kappa_{1}\kappa^{k-1}_{1}C\sigma_{k}\geq\sigma_{k}^{ii}$. Hence, for $2\leq r\leq 8$, we have

[TABLE]

where we have used $\kappa_{1}\geq\kappa_{i}^{r+2}$. Thus

[TABLE]

(A3) $\lambda_{i}\geq\lambda_{j}$.

[TABLE]

Combining $P_{m}\geq\kappa_{1}^{m}$, we have

[TABLE]

We choose $m\geq 9$. From the above three subcases, we obtain

[TABLE]

where we choose

[TABLE]

when $a>1$ and $m$ is sufficiently large.

(B) $i=1$. In this case, we shall prove that either (3.8) holds or

[TABLE]

Now set $\delta>0$ such that $\frac{1}{k-\mu+1}\leq\delta<1$. Here $\mu$ is an integer less than $k$. Then

[TABLE]

Note that, by (2.1) in Lemma 2.5, we have

[TABLE]

Besides

[TABLE]

Therefore, from (3.15) and (3.16) we have

[TABLE]

Combining (3.12) and (3.17), by direct calculation, the left hand side of (3.14) becomes

[TABLE]

where we have used $(m+3)(m-1)\geq(m+\frac{2}{a^{2}})^{2}$ for $a>\sqrt{2}$ and $m$ sufficiently large. Here

[TABLE]

[TABLE]

and

[TABLE]

Suppose that there exists $\delta_{k}>0$ such that $\frac{\lambda_{j}}{\lambda_{1}}\geq\delta_{k}\textrm{ for all }2\leq j\leq k$. Then

[TABLE]

and (3.8) holds. Therefore, we can focus on the otherwise situation and further divide case (B) into two subcases. For convenience, let us fix $a$ satisfying $a^{2}\geq\frac{8}{\alpha(1-\delta)}$.

(B1) Fix $\mu=1$ and $\delta_{1}=1$. Then we can find $\delta_{2}>0$ such that (3.14) holds when

[TABLE]

In fact, by direct calculation, we have

[TABLE]

if $\delta_{2}$ is small enough.

Note that for $k\geq j>1$,

[TABLE]

For $j\geq k+1$,

[TABLE]

Then combining (3.21), (3.22) and (3.23), (3.18) becomes

[TABLE]

if we choose $\delta_{2}$ sufficiently small. Then we have proved (3.14) when (3.20) holds.

(B2) Now we assert that we can further find constants $\delta_{3},\cdots,\delta_{k}$, such that (3.14) holds when

[TABLE]

and

[TABLE]

for some $\mu\in\{2,\cdots,k-1\}$.

To this end, we will prove it by induction. In other words, we assume (3.26) holds firstly. Then we shall find $\delta_{\mu+1}>0$ sufficiently small such that (3.14) holds provided we have (3.25). Since $\lambda\in\Gamma_{k+1}\subset\Gamma_{\mu+2}$, we have, for $a,b\leq\mu$,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Combining (3.27)-(3.30), by direct calculation, we have

[TABLE]

Moreover, by (3.31), we obtain

[TABLE]

if $\delta_{\mu+1}$ is sufficiently small. Besides,

[TABLE]

and

[TABLE]

From (3.32)-(3.34), if we choose $a$ sufficiently large and $\delta_{\mu+1}$ sufficiently small, $E_{1}$ becomes

[TABLE]

if $\delta_{\mu+1}$ is sufficiently small.

Note that for $k\geq j>\mu$,

[TABLE]

For $j\geq k+1$,

[TABLE]

Then combining (3.35)-(3.37) and (3.18) we have

[TABLE]

if we choose $\delta_{\mu+1}$ sufficiently small. Hence, combining (3.19), (3.18), (3.24) and (3.38), we have proved that either (3.8) or (3.14) holds. Besides, from (A1)-(A3), we have (3.13). Thus we have proved the Claim 3.1 and we complete the proof of Theorem 1.1.

4. A Pogorelov estimate for the 3-convex-monotone solutions to

parabolic 2-Hessian equations

In this section, we shall prove the Theorem 1.2. Since $u=0$ on $\partial\Omega$, we have $u\leq 0$ in $\Omega$ by the Comparison principle (see Theorem 17.1 in Page 443 of [10]). By Lemma 2.3, there exists $K_{0}>0$ such that $D^{2}u+K_{0}I>0$. Take the test function

[TABLE]

where $P_{m}=\sum_{i}\kappa_{i}^{m}$, $\kappa_{i}=\lambda_{i}+K_{0}>0$. Constants $a$ and $m$ are positive constants to be determined later. Assume the maximum of $\phi$ is attained at $(x_{0},t_{0})$, $u_{ij}(x_{0},t_{0})$ is diagonal and $u_{11}(x_{0},t_{0})\geq\cdots\geq u_{nn}(x_{0},t_{0})$.

Then

[TABLE]

By Lemma 2.4 we obtain

[TABLE]

Moreover,

[TABLE]

Now differentiating equation (1.2), we obtain

[TABLE]

and

[TABLE]

Note that

[TABLE]

Then (4.4) implies that

[TABLE]

Then by (4.1)-(4.5) and (2.2), we have

[TABLE]

where

[TABLE]

Now we assert that

[TABLE]

If (4.7) holds, we obtain

[TABLE]

Note that $\sigma_{2}^{11}u_{11}\geq c_{0}$. Thus (4.8) becomes

[TABLE]

where $N$ is sufficiently large. Then

[TABLE]

It completes the proof of Theorem 1.2.

Now the remaining question is the proof of assertion (4.7). Note that by Lemma 2.3, we obtain

[TABLE]

and

[TABLE]

Moreover, by Cauthy inequality, we have

[TABLE]

It yields

[TABLE]

Therefore, by (4.10) and (4.11) we have

[TABLE]

We divide the proof into two different cases: $i>1$ and $i=1$.

(A)$i>1$. In this case, we shall prove

[TABLE]

We further divide case (A) into three subcases.

(A1) $\lambda_{i}\leq\lambda_{j}$, $\lambda_{i}\geq{K_{0}}{}$.

[TABLE]

Then combining $P_{m}\geq\kappa_{1}^{m}$, we have

[TABLE]

(A2)$\lambda_{i}\leq\lambda_{j}$, $\lambda_{i}<{K_{0}}{}$. Let $2\leq r\leq 8$. We may assume $\kappa_{1}\geq\kappa_{i}^{r+2}$ and $\kappa_{1}^{3}\sigma_{2}^{jj}\geq\kappa_{1}^{2}\lambda_{1}\sigma_{2}^{11}\geq\kappa_{1}\kappa_{1}C\sigma_{2}\geq\sigma_{2}^{ii}$. Hence, for $2\leq r\leq 8$, we have

[TABLE]

Thus

[TABLE]

(A3)$\lambda_{i}\geq\lambda_{j}$.

[TABLE]

Combining $P_{m}\geq\kappa_{1}^{m}$, we have

[TABLE]

From the above three subcases, we have

[TABLE]

where we have used

[TABLE]

when $m$ is large and $a>1$.

(B)$i=1$. In this case, we shall prove that

[TABLE]

In fact, it is easy to see that

[TABLE]

where we have used

[TABLE]

Note that

[TABLE]

when $\epsilon$ is small. Then

[TABLE]

if $m$ is sufficiently large and $a^{2}>\frac{16}{5}$. From (4.12) and (4.13), we have proved the assertion (4.7). $\Box$

Remark 4.1*.*

We point out a fact. The power $8$ in (1.7) can be improved to any constant larger than $4$. Indeed, for any $a^{2}>4$, let $1<\eta<\frac{a^{2}}{4}$, $0<\epsilon<\frac{4(\eta-1)}{3+6\eta}$. Then we have

[TABLE]

So the argument is still valid.

5. Proof of the Liouville theorems

Proof of the Liouville theorem for parabolic k-Hessian equations: In this section, we give the proof of Theorem 1.3. The proof is classical.

Let $u$ be a solution of equation (1.1). Set

[TABLE]

and

[TABLE]

Then $v$ satisfies

[TABLE]

Note that

[TABLE]

and therefore

[TABLE]

Thus $\Omega_{R}(t)$ is bounded and

[TABLE]

By Theorem 1.1, it yields

[TABLE]

where $C$ is an absolutely constant. Besides, set $\widetilde{\Omega}_{R}=\{(x,t)|u(Rx,R^{2}t)-\frac{R^{2}}{2}\leq 0\}$. It is obviously that $-v\geq\frac{1}{2}$ in $\widetilde{\Omega}_{R}$. Thus

[TABLE]

It follows that

[TABLE]

in $\{(y,s)|u(y,s)\leq\frac{R^{2}}{2}\}$. Here $R$ is arbitrary. Furthermore, using Evans-Krylov theory (see [10]), we obtain

[TABLE]

It proves the theorem 1.3. $\Box$