The Exterior Problem of Parabolic Hessian Quotient Equations

Abstract

In this paper, we investigate the exterior problem of parabolic Hessian quotient equations. By utilizing Perron’s method, we establish the existence of viscosity solutions that exhibit generalized asymptotic behavior at infinity. The main approach we adopt involves constructing sub- and supersolutions to handle the non-constant term on the right-hand side of the equation.

1. Introduction

In this paper, we study the solutions to asymptotic behavior in regard to the exterior problem of the parabolic Hessian quotient equation, which is given by

$$-u_t{\displaystyle \frac{{\sigma }_k\left(\lambda \left(D^{2}u\right)\right)}{{\sigma }_l\left(\lambda \left(D^{2}u\right)\right)}}=f(x,t),$$

(1)

in ${\mathbb{R}}_{-}^{N+1}={\mathbb{R}}^N\times (-\infty ,0]$, where $3\le N,0\le l<k\le N$, which always holds in this paper. The Hessian operator is defined as

$${\sigma }_m\left(\lambda \left(D^{2}u\right)\right)={\sigma }_m\left(\lambda \right)=\sum _{1\le l_1<\cdots <l_m\le N}{\lambda }_{l_1}\cdots {\lambda }_{l_m}(1\le m\le N),$$

and $\lambda =\lambda \left(D^{2}u\right)$ is the eigenvalue vector $({\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_N)$ of the Hessian matrix $D^{2}u$. Furthermore, we define ${\sigma }_0\left(\lambda \right)=1.$

Fully nonlinear equations such as (1) are adept at capturing the intrinsic characteristics of certain geometric and physical phenomena. As such, they have significant applications across various scientific and technological fields, including geometric analysis (such as studying curvature flows, Ricci flows, etc.); optimal transport theory (such as addressing optimal transformation problems between probability measures); image processing (being utilized for image denoising and edge detection); and economics (modeling the dynamic evolution processes of economic systems). Therefore, (1) has been investigated widely. Specifically, (1) reduces to the parabolic Hessian equation $-u_t{\sigma }_k\left(\lambda \left(D^{2}u\right)\right)=f$ if $l=0$. Furthermore, when $k=N$ and $l=0$, Equation (1) transforms into the parabolic Monge–Ampère equation $-u_{t}det{D}^{2}u=f$.

For the solutions exhibiting asymptotic behavior in the exterior problem, in 2003 Caffarelli-Li [1] established pioneering work regarding the Monge–Ampère equation $det{D}^{2}u=1$, and there have been many works examining fully nonlinear elliptic and parabolic equations. Bao-Li-Zhang [2] in 2015 discussed the Monge–Ampère equation $det{D}^{2}u=f$ in regard to f being a perturbation of 1 at infinity. Dai-Bao [3] and Dai [4], respectively, in 2011 studied the Hessian equation ${\sigma }_k\left(\lambda \left(D^{2}u\right)\right)=1$ and Hessian quotient equation ${\displaystyle \frac{{\sigma }_k\left(\lambda \left(D^{2}u\right)\right)}{{\sigma }_l\left(\lambda \left(D^{2}u\right)\right)}}=1$ and extended the results of the Monge–Ampère equation. Bao-Li-Li [5] in 2014 made further progress in terms of the Hessian equation and obtained the solutions with the generalized asymptotic behavior. Dai [6] in 2014 and Zhou-Bao [7], respectively, in 2022 discussed the parabolic Monge–Ampère equation $u_t-logdet{D}^{2}u=1$ and parabolic Hessian equation $-u_t{\sigma }_k\left(\lambda \left(D^{2}u\right)\right)=1$. Recently, Zhao-Dai [8] discussed the parabolic Hessian quotient Hessian $-u_t{\displaystyle \frac{{\sigma }_k\left(\lambda \left(D^{2}u\right)\right)}{{\sigma }_l\left(\lambda \left(D^{2}u\right)\right)}}=1$ and discovered the existence of solutions which have generalized asymptotic behavior

$$\underset{-t+{\left|x\right|}^2\to \infty }{lim\; sup}\left({(-t+|x|}^2{)}^{{\displaystyle \frac{1}{2}}{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}-1}\left|u(x,t)-\left({\displaystyle \frac{1}{2}}{x}^{\prime }Qx+\varsigma ·x+c-t\right)\right|\right)<\infty ,$$

(2)

where ${\alpha }_k$ and ${\underline{\alpha }}_l$ are defined in Section 2 and Q is a matrix. Zhou [9] obtained the results for $Q=I$, with I being the identity matrix. Motivated by the results in [2,8], in this paper, we study (1), with f being a perturbation of 1 at infinity, that is, f satisfies (10). The difference between this article and reference [8] lies in the distinct functions on the right-hand side of the equation. Furthermore, this paper constructs supersolutions by utilizing solutions for the ordinary differential equations, and this approach is more general than that presented in [8].

The solutions of Monge–Ampère equation, which have asymptotic behavior, are tightly linked with the Jörgens–Calabi–Pogorelov results ([10,11,12]), stated as follows: if the classical solution u is convex and satisfies

$$det{D}^{2}u=1\mathrm{in}{\mathbb{R}}^N,$$

then u is a quadratic polynomial. Similar results can be found in [13,14,15,16]. Likewise, the exterior problem of parabolic Monge–Ampère equations is tightly linked with the parabolic Jörgens–Calabi–Pogorelov theorem ([17]), stated as follows: if $C^{4,2}\left({\mathbb{R}}_{-}^{N+1}\right)$ solution $u(x,t)$ is parabolically convex and satisfies

$$-u_{t}det{D}^{2}u=1\mathrm{in}{\mathbb{R}}_{-}^{N+1},$$

(3)

and $-y_1\le u_t(x,t)\le -y_2$ for constants $y_1,y_2>0$, then $u(x,t)$ is composed of $-ct$ plus a convex quadratic polynomial with the constant $c>0$. See also Xiong-Bao [18], Wang-Bao [19], Zhang-Bao-Wang [20] etc.

Let $P\left(x\right)$ be a strictly convex and twice continuously differentiable function, ensuring that the domain

$$\mathrm{\Omega }=\left\{\right(x,t\left)\right|P\left(x\right)<t\le 0\}$$

is bounded and nonempty. Set the parabolic boundary ${\partial }_p\mathrm{\Omega }=\{(x,t)\mid P\left(x\right)=t\}$. Let $\varpi =\varpi (x,t)\in C^0({\mathbb{R}}_{-}^{N+1}\backslash \overline{\mathrm{\Omega }})$. In this work, we aim to investigate the exterior problem defined by

$$\begin{array}{cc}\hfill & {\displaystyle -u_t\frac{{\sigma }_k\left(\lambda \left(D^{2}u\right)\right)}{{\sigma }_l\left(\lambda \left(D^{2}u\right)\right)}=f(x,t)\mathrm{in}{\mathbb{R}}_{-}^{N+1}\stackrel{ø}{¯}\mathrm{\Omega },}\end{array}$$

(4)

$$\begin{array}{cc}\hfill & u=\varpi \mathrm{on}{\partial }_p\mathrm{\Omega }.\end{array}$$

(5)

The method for constructing supersolutions developed in this paper is also applicable to the studying of exterior problems for other types of equations.

The remainder of this paper is divided into two sections. Several preliminaries are provided in the first section. The second section presents the main result and the related evidence.

2. Preliminaries

Basic notations:

(i)

$u\in C^{i,j}({\mathbb{R}}_{-}^{N+1}\backslash \overline{\mathrm{\Omega }})$: u has continuous derivatives up to order i in the spatial variable $x\in {\mathbb{R}}^N$ and up to order j in the time variable $t\in \mathbb{R}$.

(ii)

$u\in C^0({\mathbb{R}}_{-}^{N+1}\backslash \overline{\mathrm{\Omega }})$: u is continuous in x and t.

We introduce the symmetric cone ${\mathrm{\Gamma }}_k$ as follows:

$${\mathrm{\Gamma }}_k=\left\{\lambda |\lambda \in {\mathbb{R}}^{N}with{\sigma }_m\left(\lambda \right)>0\mathrm{for}m=1,\cdots ,k\right\}.$$

The function u is $k-$convex for x, which means that $\lambda \left(D^{2}u(x,t)\right)\in {\mathrm{\Gamma }}_k$. The function u is parabolically $k-$convex, which means that u is non-increasing with respect to t and $k-$convex for x.

For any $(\widehat{p},\widehat{o})\in {\mathbb{R}}^{N+1}$ and $r>0$, define

$$G_r(\widehat{p},\widehat{o}):=\left\{(x,t)\left|\right|x-\widehat{p}| <r,\widehat{o}-r^2<t\le \widehat{o}\right\}.$$

The following definitions can be referred to [8].

Definition 1.

Let $u\in C^0({\mathbb{R}}_{-}^{N+1}\backslash \mathrm{\Omega })$. If for all $(\widehat{p},\widehat{o})\in {\mathbb{R}}_{-}^{N+1}\stackrel{ø}{¯}\mathrm{\Omega }$ and all parabolically $k-$convex function $F\in C^{2,1}\left(G_r(\widehat{p},\widehat{o})\right)$ satisfying

$$u(\widehat{p},\widehat{o})=F(\widehat{p},\widehat{o}),u(x,t)\ge (\le )F(x,t),for any(x,t)\in G_r(\widehat{p},\widehat{o})\subset {\mathbb{R}}_{-}^{N+1}\stackrel{ø}{¯}\mathrm{\Omega },$$

we can obtain at $(\widehat{p},\widehat{o})$,

$$-F_t{\displaystyle \frac{{\sigma }_k\left(\lambda \left(D^{2}F\right)\right)}{{\sigma }_l\left(\lambda \left(D^{2}F\right)\right)}}\le (\ge )f(\widehat{p},\widehat{o}),$$

then u is called a viscosity supersolution (respectively, subsolution ) to (4).

For a subsolution, F does not require parabolic k-convexity.

If $u\in C^0\left({\mathbb{R}}_{-}^{N+1}\stackrel{ø}{¯}\mathrm{\Omega }\right)$ serves as both a viscosity supersolution and a viscosity subsolution, it is termed a viscosity solution of (4).

Definition 2.

When u is a viscosity supersolution (subsolution) of (4) and satisfies $u\ge (\le )\varpi (x,t)$ on the parabolic boundary ${\partial }_p\mathrm{\Omega }$, it is designated as a viscosity supersolution (subsolution) of the problem (4) and (5).

If $u\in C^0\left(\overline{{\mathbb{R}}_{-}^{N+1}\backslash \mathrm{\Omega }}\right)$ is a viscosity solution of (4) and meets the condition $u(x,t)=\varpi (x,t)$ on the parabolic boundary ${\partial }_p\mathrm{\Omega }$, it is designated as a viscosity solution of (4) and (5).

Let ${\mathcal{Q}}_{k,l}$ denote the set of $N\times N$ positive definite, real symmetric matrices Q which meet ${\displaystyle \frac{{\sigma }_k\left(\lambda \left(Q\right)\right)}{{\sigma }_l\left(\lambda \left(Q\right)\right)}}=1.$

Definition 3.

Let the matrix $Q=diag\left(q_1,q_2,\cdots ,q_N\right)\in {\mathcal{Q}}_{k,l}$ and let

$$\rho =\sqrt{-2t+x^{\prime }Qx}=\sqrt{-2t+\sum _{n=1}^N{q}_n{x}_n^2}.$$

(6)

If $u=u\left(\rho \right)$, then u is referred to as a parabolically generalized symmetric function over Q.

Let $q:=\left(q_1,q_2,\cdots ,q_N\right)\in {\mathrm{\Gamma }}_k$ such that ${\sigma }_k\left(q\right)={\sigma }_l\left(q\right)$. Define

$${\alpha }_k={\alpha }_k\left(q\right)=\underset{{\displaystyle \genfrac{}{}{0pt}{}{x\in {\mathbb{R}}^N\underset{¯}{\{}0\}}{t\in (-\infty ,0]}}}{sup}{\displaystyle \frac{{\sum }_{n=1}^N{\sigma }_{k-1;n}\left(q\right)q_n^2{x}_n^2}{{\sigma }_k\left(q\right)(-2t+{\sum }_{n=1}^N{q}_n{x}_n^2)}},$$

(7)

and

$${\underline{\alpha }}_l={\underline{\alpha }}_l\left(q\right)=\underset{{\displaystyle \genfrac{}{}{0pt}{}{x\in {\mathbb{R}}^N\underset{¯}{\{}0\}}{t\in (-\infty ,0]}}}{inf}{\displaystyle \frac{{\sum }_{n=1}^N{\sigma }_{l-1;n}\left(q\right)q_n^2{x}_n^2}{{\sigma }_l\left(q\right)(-2t+{\sum }_{n=1}^N{q}_n{x}_n^2)}}.$$

(8)

According to Lemma A1, then

$${\displaystyle \frac{k-l}{N}}\le {\overline{\alpha }}_k-{\underline{\alpha }}_l<{\overline{\alpha }}_k\le 1.$$

Suppose that $f\in C^0({\mathbb{R}}^N\times (-\infty ,0])$ satisfies

$$0<\underset{{\mathbb{R}}^N\times (-\infty ,0]}{inf}f\le \underset{{\mathbb{R}}^N\times (-\infty ,0]}{sup}f<+\infty ,$$

(9)

and for some constant $2<\eta$,

$$f(x,t)=1+O\left({\rho }^{-\eta }\right),\mathrm{if} {\left|x\right|}^2-t\to +\infty .$$

(10)

Based on the conditions imposed on f, we can find functions $\overline{f},\underline{f}\in C^0\left([0,+\infty )\right)$ which satisfy the requirements that for $(x,t)\in {\mathbb{R}}^N\times (-\infty ,0],$

$$+\infty >\underset{\rho \in [0,+\infty )}{sup}f\left(\rho \right)\ge \overline{f}\left(\rho \right)\ge f(x,t)\ge \underline{f}\left(\rho \right)\ge \underset{\rho \in [0,+\infty )}{inf}\underline{f}\left(\rho \right)>0,$$

and

$$\overline{f}\left(\rho \right)\ge 1\ge \underline{f}\left(\rho \right).$$

(11)

Moreover, we can assume that $\underline{f}\left(\rho \right)$ is strictly increasing in $\rho$ and, for some, $C_1$, ${\varsigma }_0>0$,

$$\overline{f}\left(\rho \right)=1+C_1{\rho }^{-\eta },\rho >{\varsigma }_0,$$

(12)

and

$$\underline{f}\left(\rho \right)=1-C_1{\rho }^{-\eta },\rho >{\varsigma }_0.$$

Theorem 1.

Consider the problem

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{\xi {\left(\rho \right)}^k+{\alpha }_k\rho \xi {\left(\rho \right)}^{k-1}{\xi }^{\prime }\left(\rho \right)}{\xi {\left(\rho \right)}^l+{\underline{\alpha }}_l\rho \xi {\left(\rho \right)}^{l-1}{\xi }^{\prime }\left(\rho \right)}}=f\left(\rho \right),\rho >1,\hfill \\ \xi \left(1\right)=\theta ,\hfill \\ \xi {\left(\rho \right)}^k+{\alpha }_k\rho \xi {\left(\rho \right)}^{k-1}{\xi }^{\prime }\left(\rho \right)>0,\hfill \end{array}\right.\end{array}$$

(13)

where $q=(q_1,\cdots ,q_N),q_1\le \cdots \le q_N,\mathrm{diag}(q_1,\cdots ,q_N)\in {\mathcal{Q}}_{k,l}$ and $\theta \ge 1.$ On the interval $[1,+\infty )$, (13) possesses a unique smooth solution $\xi =\xi (\rho ,\theta )$ that satisfies

(i) 

$f^{{\displaystyle \frac{1}{k-l}}}\left(\rho \right)\le \xi \left(\rho \right)=\xi (\rho ,\theta )\le \theta ,{\partial }_{\rho }\xi (\rho ,\theta )\le 0$ for $\rho \ge 1.$

(ii) 

$\xi (\rho ,\theta )$ is continuous, ${\partial }_{\theta }\xi (\rho ,\theta )>0$ and

$$\underset{\theta \to +\infty }{lim}\xi (\rho ,\theta )=+\infty ,\rho \ge 1.$$

Proof. 

The proof is detailed in [21]. □

Let $Q=diag\left(q_1,q_2,\cdots ,q_N\right)\in {\mathcal{Q}}_{k,l}$. By the definition of $\rho$, then

$${\displaystyle \frac{\partial \rho }{\partial t}}=-{\displaystyle \frac{1}{\rho }},$$

$${\displaystyle \frac{\partial \rho }{\partial x_n}}={\displaystyle \frac{q_n{x}_n}{\rho }},$$

$${\displaystyle \frac{{\partial }^2\rho }{\partial x_n\partial x_j}}={\displaystyle \frac{q_n}{\rho }}{\delta }_{nj}-{\displaystyle \frac{q_n{x}_n}{{\rho }^2}}{\displaystyle \frac{q_j{x}_j}{\rho }}.$$

Let $H(x,t)=H\left(\rho \right)\in C^{2,1}({\mathbb{R}}^{N+1}\backslash \left\{0\right\})$ be a parabolically generalized symmetric function. For indices n and j ranging from 1 to N, then

$$H_t=H^{\prime }\left(\rho \right){\displaystyle \frac{\partial \rho }{\partial t}},$$

$$H_n=H^{\prime }\left(\rho \right){\displaystyle \frac{\partial \rho }{\partial x_n}},$$

$$\begin{array}{c}\hfill H_{nj}=H^{\prime }\left(\rho \right){\displaystyle \frac{{\partial }^2\rho }{\partial x_n\partial x_j}}+H^{″}\left(\rho \right){\displaystyle \frac{\partial \rho }{\partial x_n}}{\displaystyle \frac{\partial \rho }{\partial x_j}}.\end{array}$$

So

$$\begin{array}{c}\hfill H_{nj}={\displaystyle \frac{H^{\prime }\left(\rho \right)}{\rho }}{q}_n{\delta }_{nj}+{\displaystyle \frac{H^{\prime \prime }\left(\rho \right)-\frac{H^{\prime }\left(\rho \right)}{\rho }}{{\rho }^2}}\left(q_n{x}_n\right)\left(q_j{x}_j\right).\end{array}$$

(14)

Theorem 2.

Suppose that $\gamma >1,{\tau }_1\in \mathbb{R}$, ξ satisfy (13) and define

$$\begin{array}{c}\hfill {\displaystyle \varphi (x,t)=\varphi \left(\rho \right)={\varphi }_{{\tau }_1,\gamma ,\theta }\left(\rho \right)={\tau }_1+{\int }_{\gamma }^{\rho }\mu \xi (\mu ,\theta )d\mu ,\rho \ge \gamma .}\end{array}$$

(15)

Then ϕ has parabolical $k-$convexity and satisfies

$${\displaystyle -{\varphi }_t\frac{{\sigma }_k\left(\lambda \left(D^2\varphi \right)\right)}{{\sigma }_l\left(\lambda \left(D^2\varphi \right)\right)}\ge f(x,t).}$$

Proof. 

Clearly,

$${\varphi }^{\prime }\left(\rho \right)=\rho \xi \left(\rho \right),$$

$${\varphi }^{″}\left(\rho \right)=\xi \left(\rho \right)+\rho {\xi }^{\prime }\left(\rho \right).$$

So

$$\xi \left(\rho \right)={\displaystyle \frac{{\varphi }^{\prime }\left(\rho \right)}{\rho }},$$

$${\xi }^{\prime }\left(\rho \right)={\displaystyle \frac{{\varphi }^{″}\left(\rho \right)-\frac{{\varphi }^{\prime }\left(\rho \right)}{\rho }}{\rho }}.$$

On the other hand,

$${\varphi }_t=-\xi \left(\rho \right).$$

Therefore by (14), we know that

$$\begin{array}{c}\hfill D^2\varphi ={\left(\xi \left(\rho \right)q_n{\delta }_{nj}+{\displaystyle \frac{{\xi }^{\prime }\left(\rho \right)}{\rho }}\left(q_n{x}_n\right)\left(q_j{x}_j\right)\right)}_{N\times N}.\end{array}$$

By Proposition A1, we can know that

$${\sigma }_k\left(\lambda \left(D^2\varphi \right)\right)=\xi {\left(\rho \right)}^k{\sigma }_k\left(q\right)+\xi {\left(\rho \right)}^{k-1}{\displaystyle \frac{{\xi }^{\prime }\left(\rho \right)}{\rho }}\sum _{n=1}^N{\sigma }_{k-1;n}\left(q\right){\left(q_n{x}_n\right)}^2.$$

Since ${\xi }^{\prime }\left(\rho \right)\le 0$, then for $1\le m\le k,$

$$\begin{array}{cc}\hfill {\sigma }_m\left(\lambda \left(D^2\varphi \right)\right)& ={\sigma }_m\left(q\right)\xi {\left(\rho \right)}^m+{\displaystyle \frac{{\xi }^{\prime }\left(\rho \right)}{\rho }}\xi {\left(\rho \right)}^{m-1}\sum _{n=1}^N{\sigma }_{m-1;n}\left(q\right){\left(q_n{x}_n\right)}^2\hfill \\ \hfill & ={\sigma }_m\left(q\right)\left[\xi {\left(\rho \right)}^m+\rho {\xi }^{\prime }\left(\rho \right)\xi {\left(\rho \right)}^{m-1}{\displaystyle \frac{{\displaystyle \sum _{n=1}^N{\sigma }_{m-1;n}\left(q\right)q_n^2{x}_n^2}}{{\displaystyle {\sigma }_m\left(q\right)(-2t+\sum _{n=1}^N{q}_n^2{x}_n^2)}}}\right]\hfill \\ \hfill & \ge {\sigma }_m\left(q\right)\xi {\left(\rho \right)}^{m-1}\left[\xi \left(\rho \right)+{\alpha }_m\left(q\right)\rho {\xi }^{\prime }\left(\rho \right)\right].\hfill \end{array}$$

(16)

From the equation in (13), we have

$$\begin{array}{c}\hfill {\xi }^{\prime }=-{\displaystyle \frac{1}{\rho }}·{\displaystyle \frac{\xi }{{\overline{\alpha }}_k}}·{\displaystyle \frac{{\xi }^{k-l}-f}{{\xi }^{k-l}-f\frac{{\underline{\alpha }}_l}{{\overline{\alpha }}_k}}}.\end{array}$$

Since $k>l$, then, by Lemma A1, ${\underline{\alpha }}_l\le {\underline{\alpha }}_k\le {\displaystyle \frac{k}{n}}\le {\alpha }_k$, so $-{\displaystyle \frac{{\underline{\alpha }}_l}{{\alpha }_k}}\ge -1.$ By Theorem 1, we know that

$$0\le {\displaystyle \frac{{\xi }^{k-l}-f}{{\xi }^{k-l}-f\frac{{\underline{\alpha }}_l}{{\alpha }_k}}}<1\le {\displaystyle \frac{{\alpha }_k}{{\overline{\alpha }}_m}},$$

which implies that

$$\xi +{\alpha }_m\rho {\xi }^{\prime }>0.$$

So, by (16), $\varphi$ has parabolical $k-$convexity.

Additionally,

$$\begin{array}{cc}\hfill -{\varphi }_t{\displaystyle \frac{{\sigma }_k\left(\lambda \left(D^2\varphi \right)\right)}{{\sigma }_l\left(\lambda \left(D^2\varphi \right)\right)}}& =\xi \left(\rho \right){\displaystyle \frac{{\displaystyle {\sigma }_k\left(q\right)\xi {\left(\rho \right)}^k+\xi {\left(\rho \right)}^{k-1}\frac{{\xi }^{\prime }\left(\rho \right)}{\rho }\sum _{n=1}^N{\sigma }_{k-1;n}\left(q\right){\left(q_n{x}_n\right)}^2}}{{\displaystyle {\sigma }_l\left(q\right)\xi {\left(\rho \right)}^l+\xi {\left(\rho \right)}^{l-1}\frac{{\xi }^{\prime }\left(\rho \right)}{\rho }\sum _{n=1}^N{\sigma }_{l-1;n}\left(q\right){\left(q_n{x}_n\right)}^2}}}\hfill \\ \hfill & =\xi \left(\rho \right){\displaystyle \frac{{\sigma }_k\left(q\right)\left[{\displaystyle \xi {\left(\rho \right)}^k+\rho \xi {\left(\rho \right)}^{k-1}\frac{{\xi }^{\prime }\left(\rho \right)}{{\rho }^2}\sum _{n=1}^N\frac{{\sigma }_{k-1;n}\left(q\right)}{{\sigma }_k\left(q\right)}{q}_n^2{x}_n^2}\right]}{{\sigma }_l\left(q\right)\left[{\displaystyle \xi {\left(\rho \right)}^l+\rho \xi {\left(\rho \right)}^{l-1}\frac{{\xi }^{\prime }\left(\rho \right)}{{\rho }^2}\sum _{n=1}^N\frac{{\sigma }_{l-1;n}\left(q\right)}{{\sigma }_l\left(q\right)}{q}_n^2{x}_n^2}\right]}}\hfill \\ \hfill & =\xi \left(\rho \right){\displaystyle \frac{{\sigma }_k\left(q\right)\left[\xi {\left(\rho \right)}^k+\rho \xi {\left(\rho \right)}^{k-1}{\xi }^{\prime }\left(\rho \right)\frac{{\displaystyle \sum _{n=1}^N{\sigma }_{k-1;n}\left(q\right)q_n^2{x}_n^2}}{{\sigma }_k\left(q\right)\left({\displaystyle -2t+\sum _{n=1}^N{q}_n^2{x}_n^2}\right)}\right]}{{\sigma }_l\left(q\right)\left[\xi {\left(\rho \right)}^l+\rho \xi {\left(\rho \right)}^{l-1}{\xi }^{\prime }\left(\rho \right)\frac{{\displaystyle \sum _{n=1}^N{\sigma }_{l-1;n}\left(q\right)q_n^2{x}_n^2}}{{\sigma }_l\left(q\right)\left({\displaystyle -2t+\sum _{n=1}^N{q}_n^2{x}_n^2}\right)}\right]}}.\hfill \end{array}$$

Since, by Theorem 1, $\xi \left(\rho \right)\ge f^{{\displaystyle \frac{1}{k-l}}}\left(\rho \right)\ge 1$ and ${\xi }^{\prime }\left(\rho \right)\le 0,$ therefore, by the definitions of ${\alpha }_k$ and ${\underline{\alpha }}_l$, then

$$\begin{array}{cc}\hfill -{\varphi }_t{\displaystyle \frac{{\sigma }_k\left(\lambda \left(D^2\varphi \right)\right)}{{\sigma }_l\left(\lambda \left(D^2\varphi \right)\right)}}& \ge {\displaystyle \frac{\xi {\left(\rho \right)}^k+\rho \xi {\left(\rho \right)}^{k-1}{\xi }^{\prime }\left(\rho \right){\alpha }_k}{\xi {\left(\rho \right)}^l+\rho \xi {\left(\rho \right)}^{l-1}{\xi }^{\prime }\left(\rho \right){\underline{\alpha }}_l}}\hfill \\ \hfill & =f\left(\rho \right)\ge f(x,t).\hfill \end{array}$$

(17)

□

Theorem 3.

As $\rho \to \infty ,$

$$\varphi \left(\rho \right)={\displaystyle \frac{1}{2}}{\rho }^2+{\underline{\phi }}_{{\tau }_1,\gamma }\left(\theta \right)+\left\{\begin{array}{c}O\left({\rho }^{2-min\{\eta ,{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}\}}\right),if\eta \ne {\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}},\hfill \\ O({\rho }^{2-{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}}ln\rho ),if\eta ={\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}},\hfill \end{array}\right.$$

(18)

where

$${\displaystyle {\underline{\phi }}_{{\tau }_1,\gamma }\left(\theta \right)={\tau }_1-\frac{1}{2}{\gamma }^2+{\int }_{\gamma }^{+\infty }\mu [\xi (\mu ,\theta )-1]d\mu \to +\infty ,as\theta \to +\infty ,}$$

(19)

and for fixed $\theta ,{\int }_{\gamma }^{+\infty }\mu [\xi (\mu ,\theta )-1]d\mu$ is bounded.

Proof. 

The proof is detailed in [21]. □

Theorem 4.

Let $Q\in {\mathcal{Q}}_{k,l},q:=(q_1,q_2,\cdots ,q_N):=\lambda \left(Q\right),0<q_1\le q_2\le \cdots \le q_N$, ${\displaystyle \frac{{\underline{\alpha }}_l}{{\alpha }_k}}\underline{f}\left(1\right)<{\mathrm{\Theta }}^{k-l}<\underline{f}\left(1\right)$. Then the problem

$$\left\{\begin{array}{c}{\displaystyle \frac{\Xi {\left(\rho \right)}^k+{\alpha }_k\rho \Xi {\left(\rho \right)}^{k-1}{\Xi }^{\prime }\left(\rho \right)}{\Xi {\left(\rho \right)}^l+{\underline{\alpha }}_l\rho \Xi {\left(\rho \right)}^{l-1}{\Xi }^{\prime }\left(\rho \right)}}=\underline{f}\left(\rho \right),\rho >1,\hfill \\ \Xi \left(1\right)=\mathrm{\Theta },\hfill \end{array}\right.$$

(20)

has a smooth solution $\Xi \left(\rho \right)=\Xi (\rho ,\mathrm{\Theta })$ on $[1,+\infty )$ satisfying

(i) 

${\displaystyle \frac{{\underline{\alpha }}_l}{{\alpha }_k}}\underline{f}\left(\rho \right)<{\Xi }^{k-l}(\rho ,\mathrm{\Theta })<\underline{f}\left(\rho \right),{\partial }_{\rho }\Xi (\rho ,\mathrm{\Theta })\ge 0$ for $\rho \ge 1$.

(ii) 

$\Xi (\rho ,\mathrm{\Theta })$ is continuous, ${\partial }_{\mathrm{\Theta }}\Xi (\rho ,\mathrm{\Theta })>0$.

Proof. 

The proof is detailed in [21]. □

Let $Q\in {\mathcal{Q}}_{k,l},q:=(q_1,q_2,\cdots ,q_N):=\lambda \left(Q\right)$. Set ${\mathrm{\Omega }}_A=\left\{(x,t)|{\sum }_{n=1}^N{q}_n{x}_n^2-A^2<2t\le 0\right\}.$ Let ${\mathrm{\Omega }}_{A_1}\subset \subset \mathrm{\Omega }\subset \subset {\mathrm{\Omega }}_{A_2}$, where $A_1<1<A_2$.

Define

$$\mathrm{\Phi }(x,t):=\mathrm{\Phi }\left(\rho \right):={\mathrm{\Phi }}_{{\tau }_2,A_1,\mathrm{\Theta }}\left(\rho \right):={\tau }_2+{\int }_{A_1}^{\rho }\mu \Xi (\mu ,\mathrm{\Theta })d\mu ,\forall (x,t)\in {\mathbb{R}}_{-}^{N+1}\backslash {\mathrm{\Omega }}_{A_1},$$

where ${\tau }_2$ is any constant.

Theorem 5.

Φ has parabolical $k-$convexity and satisfies

$${\displaystyle -{\mathrm{\Phi }}_t\frac{{\sigma }_k\left(\lambda \left(D^2\mathrm{\Phi }\right)\right)}{{\sigma }_l\left(\lambda \left(D^2\mathrm{\Phi }\right)\right)}\le f(x,t).}$$

Proof. 

By Proposition A1, we have that for $j=1$, ⋯, k,

$$\begin{array}{c}\hfill {\sigma }_j\left(\lambda \left(D^2\mathrm{\Phi }\right)\right)={\sigma }_j\left(q\right)\Xi {\left(\rho \right)}^j+{\displaystyle \frac{{\Xi }^{\prime }\left(\rho \right)}{\rho }}\Xi {\left(\rho \right)}^{j-1}\sum _{i=1}^n{\sigma }_{j-1;i}\left(q\right)q_i^2{x}_i^2\ge 0.\end{array}$$

Here, we utilize the fact that ${\Xi }^{\prime }\ge 0$, as established in Theorem 4 (i). Furthermore, from (20), it follows that, for any $(x,t)\in {\mathbb{R}}_{-}^{N+1}\backslash {\mathrm{\Omega }}_{A_1}$,

$$\begin{array}{cc}\hfill -{\mathrm{\Phi }}_t{\displaystyle \frac{{\sigma }_k\left(\lambda \left(D^2\mathrm{\Phi }\right)\right)}{{\sigma }_l\left(\lambda \left(D^2\mathrm{\Phi }\right)\right)}}=& \Xi \left(\rho \right){\displaystyle \frac{\Xi {\left(\rho \right)}^k{\sigma }_k\left(q\right)+\Xi {\left(\rho \right)}^{k-1}\frac{{\Xi }^{\prime }\left(\rho \right)}{\rho }{\sum }_{i=1}^n{\sigma }_{k-1;i}\left(q\right)q_i^2{x}_i^2}{\Xi {\left(\rho \right)}^l{\sigma }_l\left(q\right)+\Xi {\left(\rho \right)}^{l-1}\frac{{\Xi }^{\prime }\left(\rho \right)}{\rho }{\sum }_{i=1}^n{\sigma }_{l-1;i}\left(q\right)q_i^2{x}_i^2}}\hfill \\ \hfill \le & {\displaystyle \frac{\Xi {\left(\rho \right)}^k+{\alpha }_k\rho \Xi {\left(\rho \right)}^{k-1}{\Xi }^{\prime }}{\Xi {\left(\rho \right)}^l+{\underline{\alpha }}_l\rho \Xi {\left(\rho \right)}^{l-1}{\Xi }^{\prime }}}\hfill \\ \hfill =& \underline{f}\left(\rho \right)\le f(x,t).\hfill \end{array}$$

(21)

□

Theorem 6.

As $\rho \to \infty ,$

$$\mathrm{\Phi }\left(\rho \right)={\displaystyle \frac{1}{2}}{\rho }^2+{\phi }_{{\tau }_2,A_1}\left(\mathrm{\Theta }\right)+\left\{\begin{array}{c}O\left({\rho }^{2-min\{\eta ,{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}\}}\right),if\eta \ne {\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}},\hfill \\ O({\rho }^{2-{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}}ln\rho ),if\eta ={\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}},\hfill \end{array}\right.$$

(22)

where

$${\phi }_{{\tau }_2,A_1}\left(\mathrm{\Theta }\right):={\tau }_2-{\displaystyle \frac{1}{2}}{A}_1^2+{\int }_{A_1}^{+\infty }\mu [\Xi (\mu ,\mathrm{\Theta })-1]d\mu .$$

Furthermore, for a fixed Θ, the infinite integral ${\int }_{A_1}^{+\infty }\mu [\Xi (\mu ,\mathrm{\Theta })-1]d\mu$ is bounded.

Proof. 

The proof is detailed in [21]. □

3. Principal Result

Our principal result is as follows:

Theorem 7.

Suppose ${\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}>2$ and $\varpi \in C^{2,2}\left(\mathrm{\Omega }\right)$. For any $Q\in {\mathcal{Q}}_{k,l},\varsigma \in {\mathbb{R}}^N$ and constant c, there is then a constant $\beta =\beta (\mathrm{\Omega },|\left|\varpi \right|{|}_{C^{2,2}\left(\mathrm{\Omega }\right)},N,Q,\varsigma )$ such that, for all $c>\beta$, the problem presented in (4) and (5) admits a unique viscosity solution $u\in C^0\left(\overline{{\mathbb{R}}_{-}^{N+1}\backslash \mathrm{\Omega }}\right)$, which satisfies that if $\eta \ne {\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}},$

$$\underset{-t+{\left|x\right|}^2\to \infty }{lim\; sup}\left({(-t+|x|}^2{)}^{{\displaystyle \frac{1}{2}}min\left\{\eta ,{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}\right\}-1}\left|u(x,t)-\left(-t+{\displaystyle \frac{1}{2}}{x}^{\prime }Qx+\varsigma ·x+c\right)\right|\right)<\infty .$$

(23)

if $\eta ={\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}},$

$$\underset{-t+{\left|x\right|}^2\to \infty }{lim\; sup}\left({(-t+|x|}^2{)}^{{\displaystyle \frac{1}{2}}{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}-1}(ln(-t+{\left|x\right|}^2{\left)\right)}^{-1}\left|u(x,t)-\left(-t+{\displaystyle \frac{1}{2}}{x}^{\prime }Qx+\varsigma ·x+c\right)\right|\right)<\infty .$$

(24)

Proof. 

The proof scheme of Theorem 7 is comprised of the following three steps.

• Step 1. Formulation of subsolutions and supersolutions

Due to Lemma A4, for any $Q\in {\mathcal{Q}}_{k,l}$ and $(\overline{h},\overline{s})\in {\partial }_p\mathrm{\Omega }$, there exist positive constants $c_0,C_0$ as well as a vector $\overline{z}(\overline{h},\overline{s})\in {\mathbb{R}}^N$ with $|\overline{z}(\overline{h},\overline{s})|\le C_0$ such that

$$\varpi >U_{\overline{h},\overline{s}}on{\partial }_p\mathrm{\Omega }\backslash \left\{(\overline{h},\overline{s})\right\},$$

(25)

where

$$U_{\overline{h},\overline{s}}=U_{\overline{h},\overline{s}}(x,t)=\varpi (\overline{h},\overline{s})-\overline{c}(t-\overline{s})+{\displaystyle \frac{1}{2}}{(x-\overline{z})}^{\prime }Q(x-\overline{z})-{\displaystyle \frac{1}{2}}{(\overline{h}-\overline{z})}^{\prime }Q(\overline{h}-\overline{z}),(x,t)\in {\mathbb{R}}_{-}^{N+1}.$$

Choose $\overline{c}>max\left\{{\displaystyle c_0,\underset{{\mathbb{R}}^N\times (-\infty ,0]}{sup}f}\right\}$, then in ${\mathbb{R}}_{-}^{N+1}$,

$$-{\left(U_{\overline{h},\overline{s}}\right)}_t{\displaystyle \frac{{\sigma }_k\left(\lambda \left(D^2{U}_{\overline{h},\overline{s}}\right)\right)}{{\sigma }_l\left(\lambda \left(D^2{U}_{\overline{h},\overline{s}}\right)\right)}}=\overline{c}\ge f(x,t).$$

In ${\mathbb{R}}_{-}^{N+1}$, let

$$U(x,t)=\underset{(\overline{h},\overline{s})\in {\partial }_p\mathrm{\Omega }}{sup}{U}_{\overline{h},\overline{s}}(x,t),$$

then

$$U=\varpi \mathrm{on}{\partial }_p\mathrm{\Omega },$$

(26)

and

$$-U_t{\displaystyle \frac{{\sigma }_k\left(\lambda \left(D^{2}U\right)\right)}{{\sigma }_l\left(\lambda \left(D^{2}U\right)\right)}}\ge f\mathrm{in}{\mathbb{R}}_{-}^{N+1},$$

(27)

in the sense of viscosity.

For simplicity, we can assume that, without loss of generality, Q is in diagonal form and $\varsigma =0$. For positive constants $K,\theta$ to be determined, let

$${\displaystyle W(x,t)=K+{\int }_{A_1}^{\rho }\mu \Xi (\mu ,\mathrm{\Theta })d\mu \mathrm{in}{\mathbb{R}}_{-}^{N+1}\backslash \mathrm{\Omega },}$$

and

$$\underline{W}(x,t)={\int }_{A_2}^{\rho }\mu \xi (\mu ,\theta )d\mu +\underset{{\mathrm{\Omega }}_{A_2}}{inf}U\mathrm{in}{\mathbb{R}}_{-}^{N+1}.$$

(28)

Then, clearly,

$$\underline{W}(x,t)\le {\int }_{A_2}^{A_2}\mu \xi (\mu ,\theta )d\mu +\underset{{\mathrm{\Omega }}_{A_2}}{inf}U\le U(x,t)\mathrm{on}{\partial }_p\mathrm{\Omega }.$$

(29)

Let $A_3:=1+A_2$. Given that $\xi$ increases with $\theta$, we can choose ${\theta }_1$ and $K_1$ to be sufficiently large enough to make the next three inequalities correct; meanwhile, for $\theta >{\theta }_1,K>K_1$,

$$\underline{W}(x,t)={\int }_{A_2}^{A_3}\mu \xi (\mu ,\theta )d\mu +\underset{{\mathrm{\Omega }}_{A_2}}{inf}U\ge U(x,t)\mathrm{on}{\partial }_p{\mathrm{\Omega }}_{A_3},$$

(30)

$${\displaystyle W=K+{\int }_{A_1}^{A_3}\mu \Xi (\mu ,\mathrm{\Theta })d\mu \ge U\mathrm{on}{\partial }_p{\mathrm{\Omega }}_{A_3},}$$

(31)

$$W=K\ge U\ge \underline{W}\mathrm{on}{\partial }_p{\mathrm{\Omega }}_{A_1}.$$

(32)

By Theorems 2 and 5, $\underline{W}$ and $W$ are parabolically $k-$convex. In addition, by (17) and (21), we can understand that

$$\begin{array}{c}\hfill -{\underline{W}}_t{\displaystyle \frac{{\sigma }_k\left(\lambda \left(D^2\underline{W}\right)\right)}{{\sigma }_l\left(\lambda \left(D^2\underline{W}\right)\right)}}\ge f(x,t),\end{array}$$

and

$$\begin{array}{c}\hfill -W_t{\displaystyle \frac{{\sigma }_k\left(\lambda \left(D^{2}W\right)\right)}{{\sigma }_l\left(\lambda \left(D^{2}W\right)\right)}}\le f(x,t).\end{array}$$

So $\underline{W}$ and $W$ serve as a subsolution and a supersolution, respectively, of (4).

By Theorems 3 and 6, as $-t+|x{|}^2\to +\infty ,$

$$\begin{array}{c}\hfill \underline{W}=-t+{\displaystyle \frac{1}{2}}\sum _{n=1}^N{q}_n{x}_n^2+\underline{\phi }(\theta )+\left\{\begin{array}{c}O((-t+{\left|x\right|}^2{)}^{2-min\{\eta ,{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}\}}),if\eta \ne {\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}},\hfill \\ O((-t+{\left|x\right|}^2{)}^{2-{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}}{ln(-t+|x|}^2)),if\eta ={\displaystyle \frac{k-l}{{\overline{\alpha }}_k-{\underline{\alpha }}_l}},\hfill \end{array}\right.\end{array}$$

where

$${\displaystyle \underline{\phi }(\theta )=\underset{{\mathrm{\Omega }}_{A_2}}{inf}U-\frac{1}{2}{A}_2^2+{\int }_{A_2}^{+\infty }\mu [\xi (\mu ,\theta )-1]d\mu ,}$$

and

$$\begin{array}{c}\hfill W=-t+{\displaystyle \frac{1}{2}}\sum _{n=1}^N{q}_n{x}_n^2+\overline{\phi }(K)+\left\{\begin{array}{c}O((-t+{\left|x\right|}^2{)}^{2-min\{\eta ,{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}\}}),if\eta \ne {\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}},\hfill \\ O((-t+{\left|x\right|}^2{)}^{2-{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}}{ln(-t+|x|}^2)),if\eta ={\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}},\hfill \end{array}\right.\end{array}$$

where

$${\displaystyle \phi \left(K\right)=K-\frac{1}{2}{A}_1^2+{\int }_{A_1}^{+\infty }\mu [\Xi (\mu ,\mathrm{\Theta })-1]d\mu .}$$

Obviously, $\underline{\phi }\left(\theta \right)$ increases on the interval $(0,+\infty )$, and there is

$$\underset{\theta \to \infty }{lim}\underline{\phi }\left(\theta \right)=+\infty .$$

Let $\beta =max\{\phi \left(K_1\right),\underline{\phi }\left({\theta }_1\right)\}.$ For all $c>\beta$, it follows that $\theta =\theta \left(c\right),K=K\left(c\right)$ exist and satisfy $\underline{\phi }\left(\theta \left(c\right)\right)=c,\phi \left(K\left(c\right)\right)=c$. Thus, as $-t+|x{|}^2\to +\infty$, it can be inferred that

$$\begin{array}{c}\hfill \underline{W}=-t+{\displaystyle \frac{1}{2}}\sum _{n=1}^N{q}_n{x}_n^2+c+\left\{\begin{array}{c}O((-t+{\left|x\right|}^2{)}^{2-min\{\eta ,{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}\}}),if\eta \ne {\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}},\hfill \\ O((-t+{\left|x\right|}^2{)}^{2-{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}}{ln(-t+|x|}^2)),if\eta ={\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}},\hfill \end{array}\right.\end{array}$$

(33)

and

$$\begin{array}{c}\hfill W=-t+{\displaystyle \frac{1}{2}}\sum _{n=1}^N{q}_n{x}_n^2+c+\left\{\begin{array}{c}O((-t+{\left|x\right|}^2{)}^{2-min\{\eta ,{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}\}}),if\eta \ne {\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}},\hfill \\ O((-t+{\left|x\right|}^2{)}^{2-{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}}{ln(-t+|x|}^2)),if\eta ={\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}.\hfill \end{array}\right.\end{array}$$

(34)

Define, for all $c>\beta$,

$$\omega =\left\{\begin{array}{c}max\left\{U,\underline{W}\right\},in{\mathrm{\Omega }}_{A_3}\backslash {\mathrm{\Omega }}_{A_1},\hfill \\ \underline{W},in{\mathbb{R}}_{-}^{N+1}\backslash {\mathrm{\Omega }}_{A_3}.\hfill \end{array}\right.$$

• Step 2. Proof of $\mathbf{W}\ge \mathbf{\omega }$

From Lemma A3,

$$W\ge \underline{W}in{\mathbb{R}}_{-}^{N+1}\backslash {\mathrm{\Omega }}_{A_1}.$$

(35)

Through (30), $\omega \in C^0\left({\mathbb{R}}_{-}^{N+1}\backslash \mathrm{\Omega }\right)$. By (27), it follows that $\omega$ meets

$$-{\omega }_t{\displaystyle \frac{{\sigma }_k\left(\lambda \left(D^2\omega \right)\right)}{{\sigma }_l\left(\lambda \left(D^2\omega \right)\right)}}\ge f,\mathrm{in}{\mathbb{R}}_{-}^{N+1}\backslash \mathrm{\Omega }$$

(36)

in the viscosity sense. By (29), (26),

$$\omega =U=\varpi \mathrm{on}{\partial }_p\mathrm{\Omega }.$$

(37)

Thus, in the viscosity sense, $\omega$ satisfies (4) and (5). Due to (33), as $|x{|}^2-t\to +\infty$, it can be inferred that

$$\begin{array}{c}\hfill \omega =-t+{\displaystyle \frac{1}{2}}\sum _{n=1}^N{q}_n{x}_n^2+c+\left\{\begin{array}{c}O\left(\right(-t+{\left|x\right|}^2{)}^{2-min\{\eta ,{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}\}}),\mathrm{if}\eta \ne {\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}},\hfill \\ O\left(\right(-t+{\left|x\right|}^2{)}^{2-{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}}{ln(-t+|x|}^2\left)\right),\mathrm{if}\eta ={\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}},\hfill \end{array}\right.\end{array}$$

(38)

In addition, by (27), (31), (32), and Lemma A3,

$$W\ge U\mathrm{in}{\mathrm{\Omega }}_{A_3}\backslash {\mathrm{\Omega }}_{A_1}.$$

And then, in combining (35), we can infer

$$W\ge \omega \mathrm{in}{\mathbb{R}}_{-}^{N+1}\backslash \mathrm{\Omega }.$$

(39)

• Step 3. Proof of uniqueness and existence

Proving uniqueness:

Let u and v be two viscosity solutions to (4), (5), (23), and (24). Then, for any $\epsilon >0$, there exists a positive constant $A_4$ such that $\mathrm{\Omega }\subset {\mathrm{\Omega }}_{A_4}$, and

$$\epsilon +u\ge vin{\mathbb{R}}_{-}^{N+1}\backslash {\mathrm{\Omega }}_{A_4}.$$

By Lemma A3, in ${\mathrm{\Omega }}_{A_4}\backslash \mathrm{\Omega }$, there holds $\epsilon +u\ge v$. This implies that $\epsilon +u\ge v$ throughout ${\mathbb{R}}_{-}^{N+1}\backslash \mathrm{\Omega }$. As $\epsilon$ approaches zero, we conclude that $u\ge v$ in ${\mathbb{R}}_{-}^{N+1}\backslash \mathrm{\Omega }$. Using a similar argument, we can also show that $u\le v$ in ${\mathbb{R}}_{-}^{N+1}\backslash \mathrm{\Omega }$. Thus, it follows that $v=u$ in ${\mathbb{R}}_{-}^{N+1}\backslash \mathrm{\Omega }$.

Proving existence:

We have successfully developed a subsolution $\omega$ and a supersolution $W$ for Equation (4) in the viscosity sense, ensuring that $W\ge \omega$.

Let the set $\mathcal{B}$ be defined as the collection of functions Y that are subsolutions of the problem (4), (5) in the viscosity sense and satisfy the inequality

$$W\ge Yin{\mathbb{R}}_{-}^{N+1}\backslash \mathrm{\Omega }.$$

Since $\omega \in \mathcal{B}$, $\mathcal{B}\ne \varnothing$. Define in ${\mathbb{R}}_{-}^{N+1}\backslash \mathrm{\Omega }$,

$$u_c=sup\left\{Y|Y\in \mathcal{B}\right\}.$$

Then, we can have

$$W\ge u_{c}in{\mathbb{R}}_{-}^{N+1}\backslash \mathrm{\Omega }.$$

(40)

Because $\omega \in \mathcal{B}$, by (38), when $-t+|x{|}^2\to +\infty$,

$$\begin{array}{c}\hfill u_c=-t+{\displaystyle \frac{1}{2}}\sum _{n=1}^N{q}_n{x}_n^2+c+\left\{\begin{array}{c}O((-t+{\left|x\right|}^2{)}^{2-min\{\eta ,{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}\}}),\mathrm{if}\eta \ne {\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}},\hfill \\ O((-t+{\left|x\right|}^2{)}^{2-{\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}}}{ln(-t+|x|}^2)),\mathrm{if}\eta ={\displaystyle \frac{k-l}{{\alpha }_k-{\underline{\alpha }}_l}},\hfill \end{array}\right.\end{array}$$

(41)

For any $(\varrho ,\vartheta )\in {\partial }_p\mathrm{\Omega }$, using (37), it can be inferred that

$$\varpi (\varrho ,\vartheta )=\underset{(x,t)\to (\varrho ,\vartheta )}{lim}\omega (x,t)\le \underset{(x,t)\to (\varrho ,\vartheta )}{liminf}{u}_c(x,t).$$

Additionally, we can prove

$$\underset{(x,t)\to (\varrho ,\vartheta )}{limsup}{u}_c(x,t)\le \varpi (\varrho ,\vartheta ).$$

In fact, for every $Y\in \mathcal{B}$,

$$\left\{\begin{array}{c}-Y_t+\mathrm{\Delta }Y\ge 0,\mathrm{in}{\mathrm{\Omega }}_{A_2}\backslash \mathrm{\Omega },\hfill \\ Y\le \varpi ,\mathrm{on}{\partial }_p\mathrm{\Omega },\hfill \\ Y\le {\displaystyle \underset{{\partial }_p{\mathrm{\Omega }}_{A_2}}{sup}}W=:\Lambda ,\mathrm{on}{\partial }_p{\mathrm{\Omega }}_{A_2},\hfill \end{array}\right.$$

hold in the sense of viscosity. Choose $Z\in C^{2,1}\left({\mathrm{\Omega }}_{A_2}\backslash \mathrm{\Omega }\right)\cap C^0\left(\overline{{\mathrm{\Omega }}_{A_2}}\backslash \mathrm{\Omega }\right)$ to meet

$$\left\{\begin{array}{c}-Z_t+\mathrm{\Delta }Z=0,\mathrm{in}{\mathrm{\Omega }}_{A_2}\backslash \mathrm{\Omega },\hfill \\ Z=\varpi ,\mathrm{on}{\partial }_p\mathrm{\Omega },\hfill \\ Z=\Lambda ,\mathrm{on}{\partial }_p{\mathrm{\Omega }}_{A_2}.\hfill \end{array}\right.$$

By applying the comparison principle, it is straightforward to show that $Y\le Z$ in $\overline{{\mathrm{\Omega }}_{A_2}}\backslash \mathrm{\Omega }$. Consequently, we obtain $u_c\le Z$, $(x,t)\in \overline{{\mathrm{\Omega }}_{A_2}}\backslash \mathrm{\Omega }$. Therefore,

$$\varpi (\varrho ,\vartheta )=\underset{(x,t)\to (\varrho ,\vartheta )}{lim}Z(x,t)\ge \underset{(x,t)\to (\varrho ,\vartheta )}{limsup}{u}_c(x,t).$$

Finally, in order to show that $u_c$ satisfies (4) in the viscosity sense, we can adopt the methods described in [6].

Then, Theorem 7 is proved. □

Remark 1.

In fact, the supersolution can be $W=-t+{\displaystyle \frac{1}{2}}{x}^{\prime }Qx$. However, the form of the supersolution in this paper is more suitable for $f=f_0\left(\rho \right)+O\left({\rho }^{-\eta }\right).$

4. Conclusions and Future Directions

This manuscript focuses on the viscosity solutions with generalized asymptotic properties for ${\displaystyle -u_t\frac{{\sigma }_k\left(\lambda \left(D^{2}u\right)\right)}{{\sigma }_l\left(\lambda \left(D^{2}u\right)\right)}=f}$ in the exterior domain. By utilizing Perron’s method, the uniqueness and existence of these generalized parabolically symmetric solutions are established. The proof of this primary result is divided into three steps, with Theorems 1 and 4 playing crucial roles.

Additionally, it is intriguing to explore the generalized asymptotic behavior of the equation ${\displaystyle -u_t\frac{{\sigma }_k\left(\lambda \left(D^{2}u\right)\right)}{{\sigma }_l\left(\lambda \left(D^{2}u\right)\right)}=f}$, where f represents a perturbation around $f_0\left(\rho \right)$. This extension provides deeper insights into the behavior of such equations under varying conditions.

In summary, parabolic Hessian quotient equations constitute an active branch of partial differential equations. Their theoretical development has not only advanced pure mathematics but has also provided powerful tools for solving practical problems.