Monotone Positive Radial Solution of Double Index Logarithm Parabolic Equations

Abstract

This article mainly studies the double index logarithmic nonlinear fractional $g\text{-}$Laplacian parabolic equations with the Marchaud fractional time derivatives ${\displaystyle {\partial }_t^{\alpha }}$. Compared with the classical direct moving plane method, in order to overcome the challenges posed by the double non-locality of space-time and the nonlinearity of the fractional $g\text{-}$Laplacian, we establish the unbounded narrow domain principle, which provides a starting point for the moving plane method. Meanwhile, for the purpose of eliminating the assumptions of boundedness on the solutions, the averaging effects of a non-local operator are established; then, these averaging effects are applied twice to ensure that the plane can be continuously moved toward infinity. Based on the above, the monotonicity of a positive solution for the above fractional $g\text{-}$Laplacian parabolic equations is studied.

1. Introduction

In this paper, we mainly study the double index logarithmic fractional $g\text{-}$Laplacian parabolic equations with the Marchaud fractional time derivatives

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_t^{\alpha }}u(z,t)+{(-{\Delta }_g)}^{s}u(z,t)=ln\left|u^q\left(z,t\right)+1\right|+ln\left|u^p\left(z,t\right)+1\right|,\hfill & \mathrm{in}{\displaystyle {\mathbb{R}}_{+}^n}\times \mathbb{R},\hfill \\ u(z,t)=0,\hfill & \mathrm{in}\left({\mathbb{R}}^n\backslash {\displaystyle {\mathbb{R}}_{+}^n}\right)\times \mathbb{R},\hfill \end{array}\right.$$

(1)

where ${\displaystyle {\mathbb{R}}_{+}^n}:=\{z\in {\mathbb{R}}^n\mid z_1>0\}$ represents the right half space and $p,q\ge 1$.

The Marchaud fractional time derivative ${\displaystyle {\partial }_t^{\alpha }}$ is defined as

$${\displaystyle {\partial }_t^{\alpha }}u(z,t):=C_{\alpha }{\displaystyle {\int }_{-\infty }^t}\frac{u\left(z,t\right)-u\left(z,\tau \right)}{{\left(t-\tau \right)}^{1+\alpha }}d\tau ,\mathrm{for}\mathrm{order}\alpha \in (0,1),$$

(2)

where $C_{\alpha }=\frac{\alpha }{\Gamma \left(1-\alpha \right)}$ represents the normalized positive constant. To make sense of the integral in Equation (2), let $u(z,t)\in C^1(\mathbb{R})\times {\displaystyle L_{\alpha }^{-}}(\mathbb{R})$, where

$${\displaystyle L_{\alpha }^{-}}(\mathbb{R}):=\{u\in {\displaystyle L_{loc}^1}(\mathbb{R})|{\displaystyle {\int }_{-\infty }^t}\frac{\left|u(z,\rho )\right|}{1+{\left|\rho \right|}^{1+\alpha }}d\rho <+\infty \mathrm{for}\mathrm{any}t\in \mathbb{R}\}.$$

The Marchaud fractional time derivative was introduced by Marchaud in 1927 [1]. The time non-locality explains the historical dependence introduced in dynamics due to abnormally large waiting times. The introduction of the Marchaud fractional time derivative can better describe some complex phenomena in the real world, such as nonlinear effects of media and memory effects. The background and properties of the Marchaud fractional time derivative can be referred to in [2,3,4]. This fractional time derivative is widely used in various fields. In finance, a fractional time derivative can be used to solve the optimal portfolio problem of investors [5]. In continuum mechanics, a fractional operator has a clear mechanical explanation by the definition of fractional derivatives [6]. In physical phenomena, it is used to describe magneto-thermoelastic heat conduction [7].

Among the non-local nonlinear operators with non-standard growth that occur naturally in fractional Orlicz–Sobolev spaces, the most notable of which is the fractional $g\text{-}$Laplacian. The spatial non-locality of the fractional $g\text{-}$Laplacian explains that the behavior of a point in the system is affected by a distant position in space; that is, there are non-local effects in the system. More background on the fractional $g\text{-}$Laplacian can be found in references [8,9]. The fractional $g\text{-}$Laplacian ${(-{\Delta }_g)}^s$ has received increasing attention in recent years because it can simulate the non-power behavior of non-local problems. For some interesting results, we can refer to [10,11,12,13] and the references therein.

Define fractional $g\text{-}$Laplacian ${(-{\Delta }_g)}^s$ [14],

$${(-{\Delta }_g)}^{s}u(z,t):=\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}g\left(\frac{u\left(z,t\right)-u\left(\vartheta ,t\right)}{{\left|z-\vartheta \right|}^s}\right)\frac{d\vartheta }{{\left|z-\vartheta \right|}^{n+s}},$$

(3)

where $\mathrm{P}.\mathrm{V}.$ represents the integral principal value and $g=G^{\prime }$ corresponds to the derivative of a Young function, G. That is,

$$G(t)={\displaystyle {\int }_0^t}g(\tau )d\tau .$$

The properties of g are as follows:

(1)

$g(t)>0\mathrm{for}t>0;$

(2)

$g(a)+g(b)\ge cg(a+b),\mathrm{where}c>0\mathrm{is}\mathrm{a}\mathrm{constant};$

(3)

$g(a)-g(b)\ge dg(a-b),\mathrm{where}d>0\mathrm{is}\mathrm{a}\mathrm{constant};$

(4)

$g\mathrm{is}\mathrm{nondecreasing}\mathrm{on}(0,\infty );$

(5)

$g(-t)=-g(t);$

(6)

$g^{\prime }>0,\mathrm{since}G\mathrm{is}\mathrm{convex}.$

Because of the non-locality of the fractional $g\text{-}$Laplacian ${(-{\Delta }_g)}^s$, the behavior of u at infinity needs to be properly controlled when dealing with the operator. We will define

$$L_g({\mathbb{R}}^n):=\left\{u\in {\displaystyle L_{loc}^1}\left({\mathbb{R}}^n\right)\mid {\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}g\left(\frac{\left|u(z,t)\right|}{1+{\left|z\right|}^s}\right)\frac{dz}{1+{\left|z\right|}^{n+s}}<\infty \right\}.$$

This guarantees that the operator is well defined. In addition, when $G(t)=t^2$, this corresponds to the fractional Laplacian. When $G(t)=t^p$, it is the fractional $p\text{-}$Laplacian.

The moving plane method introduced by Alexandrof is mainly used to study local elliptic and parabolic equations. But due to the non-local property of the fractional Laplace operator, the traditional moving plane method is not suitable for pseudo-differential equations containing a fractional Laplace operator. In order to resolve this dilemma, Caffarelli and Silvestr [15] developed an extended method for converting non-local questions into local questions in high-dimensional spaces, which has been successfully applied to equations with fractional Laplacian (see [16,17,18] and the references therein). Alternatively, by transforming a given pseudo-differential equation into its equivalent integral equation, the properties of the solutions are studied by applying the moving plane method in the integral form and the regularity lifting. The results of this method can be referred to in [19,20,21,22]. However, when employing the extension method or the corresponding integral method, it is necessary to add some additional conditions or restrictions. After that, in [23], Chen, Li, and Li made a further breakthrough in this field by introducing a method of moving the plane directly, thereby eliminating these limitations and simplifying the proof process. Subsequently, this effective direct method has been extensively applied in analyzing the symmetries, monotonicity, and nonexistence of various elliptic equations and systems (see [24,25,26,27,28] and the references therein). But due to the difficulty caused by double non-locality in space-time, the study of the geometric properties of the solutions for space-time fractional nonlinear parabolic equations is very scarce. Until 2023, Chen and Li studied the monotonicity of the solutions for dual fractional nonlinear parabolic equations by using the direct moving plane method in [29]. Inspired by the above ideas, we will investigate a monotone positive radial solution of the double index logarithmic nonlinear fractional $g\text{-}$Laplacian parabolic Equations (1) with the Marchaud fractional time derivatives ${\displaystyle {\partial }_t^{\alpha }}$. We successfully address the challenges arising from the double non-locality of space-time and the nonlinearity of the fractional $g\text{-}$Laplacian in this equation.

In contrast to the previous approach of taking limits along a subsequence of $\{w_{{\Lambda }_k}\}$, we utilize the method of average effects to eliminate the assumption of boundedness on the solution. We believe that this method will become a valuable tool in studying unbounded solution sequences. The structure of this article is as follows: In Section 2, we mainly prove the narrow region principle of the antisymmetric function and some maximum principles, which provides a starting point for the moving plane method. In Section 3, for the purpose of eliminating the assumptions of boundedness on the solutions, the averaging effects of the non-local operator are established. In Section 4, the main result of this paper is proved by using the direct moving plane method; that is, the positive solution of Equation (1) is strictly increasing in the $z_1\text{-}$direction for any $t\in \mathbb{R}$.

Notations. The $z_1\text{-}$direction can be any direction.

$$T_{\Lambda }:=\{z=(z_1,z^{{ }^{\prime }})\in {\mathbb{R}}^n\mid z_1=\Lambda \mathrm{for}\Lambda \in \mathbb{R}\}$$

is the moving planes.

$${\Sigma }_{\Lambda }:=\{z\in {\mathbb{R}}^n\mid z_1<\Lambda \}\mathrm{and}{\Omega }_{\Lambda }:=\{z\in {\displaystyle {\mathbb{R}}_{+}^n}\mid z_1<\Lambda \}$$

are the regions to the left of the hyperplane $T_{\Lambda }$ in ${\mathbb{R}}^n$ and in ${\displaystyle {\mathbb{R}}_{+}^n}$, respectively.

$$z^{\Lambda }:=(2\Lambda -z_1,z_2,\cdots ,z_n)$$

is the reflection point of z about $T_{\Lambda }$. $u(z,t)$ is a solution of (1) and $u_{\Lambda }(z,t):=u(z^{\Lambda },t)$.

Denote

$$w_{\Lambda }(z,t):=u_{\Lambda }(z,t)-u(z,t).$$

2. Maximum Principle

In this section, we mainly prove the following four theorems: the four theorems are the narrow region principle (Theorem 1) and maximum principle (Theorem 2) of an antisymmetric function on an unbounded domain, and the maximum principle (Theorem 3) and maximum principle of an antisymmetric function (Theorem 4) on a bounded domain. From this point on, C represents a constant that may differ between each line, and only the related dependencies are explained later. And $C_i$ is the positive constant throughout the article.

Theorem 1.

Suppose that Ω is an unbounded narrow region contained within $\{z\in {\Sigma }_{\Lambda }|\Lambda -2l<z_1<\Lambda \}$ for some small l and

$$w_{\Lambda }\in ({\displaystyle C_{loc}^{1,1}}(\Omega )\cap L_g({\mathbb{R}}^n))\times (C^1(\mathbb{R})\cap {\displaystyle L_{\alpha }^{-}}(\mathbb{R}))$$

is lower semi-continuous with respect to z on $\overline{\Omega }$.

If

$$w_{\Lambda }(z,t)\ge {-C(1+|z|}^{\nu })forsome0<\nu <2s$$

(4)

and

$$\left\{\begin{array}{c}{\displaystyle {\partial }_t^{\alpha }}{w}_{\Lambda }(z,t)+{(-{\Delta }_g)}^s{u}_{\Lambda }(z,t)-{(-{\Delta }_g)}^{s}u(z,t)=\left(\frac{q{\displaystyle {\xi }_1^{q-1}}\left(z,t\right)}{{\displaystyle {\xi }_1^q}\left(z,t\right)+1}+\frac{p{\displaystyle {\xi }_2^{p-1}}\left(z,t\right)}{{\displaystyle {\xi }_2^p}\left(z,t\right)+1}\right)w_{\Lambda }(z,t),\hfill \\ (z,t)\in \Omega \times \mathbb{R},\hfill \\ w_{\Lambda }(z,t)\ge 0,(z,t)\in ({\Sigma }_{\Lambda }\backslash \Omega )\times \mathbb{R},\hfill \\ w_{\Lambda }(z,t)=-w_{\Lambda }(z^{\Lambda },t),(z,t)\in {\Sigma }_{\Lambda }\times \mathbb{R}.\hfill \end{array}\right.$$

(5)

where ${\xi }_1(z,t)$ and ${\xi }_2(z,t)$ fall in-between $u_{\Lambda }(z,t)$ and $u(z,t)$, then

$$w_{\Lambda }(z,t)\ge 0,in{\Sigma }_{\Lambda }\times \mathbb{R},$$

(6)

for l small enough. Moreover, if $w_{\Lambda }(z^0,t_0)=0$ for some point $(z^0,t_0)\in \Omega \times \mathbb{R}$, then

$$w_{\Lambda }(z,t)\equiv 0,in{\mathbb{R}}^n\times (-\infty ,t_0].$$

(7)

Proof. 

To obtain Equation (6), we will use proof by contradiction. Since condition (4) may cause $w_{\Lambda }(z,t)$ to reach negative infinity when $|x|\to \infty$, then $w_{\Lambda }(z,t)$ may not reach the minimum in z. To solve this difficulty, define

$$h(z):=\left[{\displaystyle {\left(1-\frac{{\left(z_1-\left(\Lambda -l\right)\right)}^2}{l^2}\right)}_{+}^s}+1\right]{\left(1+{\left|z^{\prime }\right|}^2\right)}^{\frac{\theta }{2}}$$

for some $\nu <\theta <2s$. Hence, we obtain

$${\displaystyle \underset{\left|z\right|\to \infty }{lim}}{\overline{w}}_{\Lambda }\left(z,t\right):={\displaystyle \underset{\left|z\right|\to \infty }{lim}}\frac{w_{\Lambda }\left(z,t\right)}{h\left(z\right)}\ge 0.$$

(8)

Assume that there exists a point $z\in \Omega$ such that ${\overline{w}}_{\Lambda }\left(z,t\right)<0$ for every fixed $t\in \mathbb{R}$; then, there must be $z\left(t\right)\in \Omega$ such that

$${\overline{w}}_{\Lambda }\left(z\left(t\right),t\right)={\displaystyle \underset{z\in \Omega }{min}}{\overline{w}}_{\Lambda }\left(z,t\right)<0.$$

(9)

By Equation (4), $\nu <\theta$, and the definition of ${\overline{w}}_{\Lambda }\left(z,t\right)$, it follows that ${\overline{w}}_{\Lambda }\left(z\left(t\right),t\right)$ is bounded.

Therefore, if Equation (6) does not hold, there must exist a constant $m>0$ such that

$${\displaystyle \underset{\Omega \times \mathbb{R}}{inf}}{\overline{w}}_{\Lambda }(z,t)={\displaystyle \underset{\mathbb{R}}{inf}}{\overline{w}}_{\Lambda }(z\left(t\right),t)=-m<0.$$

(10)

This means that there is a sequence, $\{(z^k,t_k)\}\subset \Omega \times \mathbb{R}$, and it holds

$${\overline{w}}_{\Lambda }(z^k,t_k)=-m_k\to -m\mathrm{as}k\to \infty .$$

Let ${\epsilon }_k:=m-m_k$. It is obvious that ${\epsilon }_k>0$ and ${\epsilon }_k\to 0$ as $k\to \infty$.

Because the minimum of ${\overline{w}}_{\Lambda }(z(t),t)$ may not be reached when $t\in \mathbb{R}$, to address this challenge, let

$$v_k(z,t):={\overline{w}}_{\Lambda }(z,t)-{\epsilon }_k{\eta }_k(t),$$

where

$${\eta }_k(t)=\eta (t-t_k)\in {\displaystyle C_0^{\infty }}(-2+t_k,2+t_k),$$

represents a smooth cut-off function, satisfying

$${\eta }_k(t)\left\{\begin{array}{ccc}=1,\hfill & \hfill & t\in [-1+t_k,1+t_k],\hfill \\ \in [0,1],\hfill & \hfill & t\in (-2+t_k,-1+t_k)\cup (1+t_k,2+t_k),\hfill \\ =0,\hfill & \hfill & t\notin (-2+t_k,2+t_k).\hfill \end{array}\right.$$

We have

$$v_k(z^k,t_k)={\overline{w}}_{\Lambda }(z^k,t_k)-{\epsilon }_k=-m_k-m+m_k=-m.$$

Consequently, there exists $({\overline{z}}^k,{\overline{t}}_k)\in \Omega \times (-2+t_k,2+t_k)$ such that

$$-m-{\epsilon }_k\le v_k({\overline{z}}^k,{\overline{t}}_k)={\displaystyle \underset{\Omega \times \mathbb{R}}{inf}}{v}_k(z,t)\le -m.$$

(11)

It follows that

$$-m\le {\overline{w}}_{\Lambda }({\overline{z}}^k,{\overline{t}}_k)\le -m+{\epsilon }_k=-m_k<0.$$

(12)

By direct calculation, we have

$${\displaystyle {\partial }_t^{\alpha }}{v}_k({\overline{z}}^k,{\overline{t}}_k)=C_{\alpha }{\displaystyle {\int }_{-\infty }^{{\overline{t}}_k}}\frac{v_k({\overline{z}}^k,{\overline{t}}_k)-v_k({\overline{z}}^k,\tau )}{{({\overline{t}}_k-\tau )}^{1+\alpha }}d\tau \le 0.$$

By Lemma 5.1 in [29] and the definition of $v_k(z,t)$, it follows that

$${\displaystyle {\partial }_t^{\alpha }}{\overline{w}}_{\Lambda }({\overline{z}}^k,{\overline{t}}_k)\le {\epsilon }_k{\displaystyle {\partial }_t^{\alpha }}{\eta }_k({\overline{t}}_k)\le C{\epsilon }_k,$$

(13)

here, C is a positive constant. In turn, by the properties of g, and Equations (5), (9), and (12), $|{\overline{z}}^k-\vartheta |<|{\overline{z}}^k-{\vartheta }^{\Lambda }|$ and $h(\vartheta )>h({\vartheta }^{\Lambda })$ for $\vartheta \in {\Sigma }_{\Lambda }$, and we have

$$\begin{array}{cc}\hfill & {(-{\Delta }_g)}^s{u}_{\Lambda }({\overline{z}}^k,{\overline{t}}_k)-{(-{\Delta }_g)}^{s}u({\overline{z}}^k,{\overline{t}}_k)\hfill \\ \hfill =& \mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}\left[g\left(\frac{u_{\Lambda }({\overline{z}}^k,{\overline{t}}_k)-u_{\Lambda }(\vartheta ,{\overline{t}}_k)}{|{\overline{z}}^k{-\vartheta |}^s}\right)-g\left(\frac{u({\overline{z}}^k,{\overline{t}}_k)-u(\vartheta ,{\overline{t}}_k)}{|{\overline{z}}^k{-\vartheta |}^s}\right)\right]\frac{d\vartheta }{|{\overline{z}}^k{-\vartheta |}^{n+s}}\hfill \\ \hfill =& \mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}\frac{g^{\prime }\left(\xi \left(\vartheta ,{\overline{t}}_k\right)\right)\left(w_{\Lambda }\left({\overline{z}}^k,{\overline{t}}_k\right)-w_{\Lambda }\left(\vartheta ,{\overline{t}}_k\right)\right)}{{\left|{\overline{z}}^k-\vartheta \right|}^{n+2s}}d\vartheta \hfill \\ \hfill =& \mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}\frac{g^{\prime }\left(\xi \left(\vartheta ,{\overline{t}}_k\right)\right)\left({\overline{w}}_{\Lambda }\left({\overline{z}}^k,{\overline{t}}_k\right)h\left({\overline{z}}^k\right)-{\overline{w}}_{\Lambda }\left({\overline{z}}^k,{\overline{t}}_k\right)h\left(\vartheta \right)+{\overline{w}}_{\Lambda }\left({\overline{z}}^k,{\overline{t}}_k\right)h\left(\vartheta \right)-{\overline{w}}_{\Lambda }\left(\vartheta ,{\overline{t}}_k\right)h\left(\vartheta \right)\right)}{{\left|{\overline{z}}^k-\vartheta \right|}^{n+2s}}d\vartheta \hfill \\ \hfill =& C_1\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}\frac{h\left(\vartheta \right)\left({\overline{w}}_{\Lambda }\left({\overline{z}}^k,{\overline{t}}_k\right)-{\overline{w}}_{\Lambda }\left(\vartheta ,{\overline{t}}_k\right)\right)}{{\left|{\overline{z}}^k-\vartheta \right|}^{n+2s}}d\vartheta +C_2{\overline{w}}_{\Lambda }({\overline{z}}^k,{\overline{t}}_k){(-\Delta )}^{s}h({\overline{z}}^k)\hfill \\ \hfill \le & C_1{\displaystyle {\int }_{{\Sigma }_{\Lambda }}^{ }}\frac{h\left(\vartheta \right){\overline{w}}_{\Lambda }\left({\overline{z}}^k,{\overline{t}}_k\right)-w_{\Lambda }\left(\vartheta ,{\overline{t}}_k\right)}{{\left|{\overline{z}}^k-{\vartheta }^{\Lambda }\right|}^{n+2s}}d\vartheta +C_1{\displaystyle {\int }_{{\Sigma }_{\Lambda }}^{ }}\frac{h\left({\vartheta }^{\Lambda }\right){\overline{w}}_{\Lambda }\left({\overline{z}}^k,{\overline{t}}_k\right)+w_{\Lambda }\left(\vartheta ,{\overline{t}}_k\right)}{{\left|{\overline{z}}^k-{\vartheta }^{\Lambda }\right|}^{n+2s}}d\vartheta \hfill \\ \hfill & +C_2{\overline{w}}_{\Lambda }({\overline{z}}^k,{\overline{t}}_k){(-\Delta )}^{s}h({\overline{z}}^k)\hfill \\ \hfill \le & C_1{\displaystyle {\int }_{{\Sigma }_{\Lambda }}^{ }}\frac{2h\left({\vartheta }^{\Lambda }\right){\overline{w}}_{\Lambda }\left({\overline{z}}^k,{\overline{t}}_k\right)}{{\left|{\overline{z}}^k-{\vartheta }^{\Lambda }\right|}^{n+2s}}d\vartheta +C_2{\overline{w}}_{\Lambda }({\overline{z}}^k,{\overline{t}}_k){(-\Delta )}^{s}h({\overline{z}}^k)\hfill \\ \hfill \le & C_2{\overline{w}}_{\Lambda }({\overline{z}}^k,{\overline{t}}_k){(-\Delta )}^{s}h({\overline{z}}^k),\hfill \end{array}$$

where

$$\frac{u_{\Lambda }({\overline{z}}^k,{\overline{t}}_k)-u_{\Lambda }(\vartheta ,{\overline{t}}_k)}{|{\overline{z}}^k{-\vartheta |}^s}<\xi \left(\vartheta ,{\overline{t}}_k\right)<\frac{u({\overline{z}}^k,{\overline{t}}_k)-u(\vartheta ,{\overline{t}}_k)}{|{\overline{z}}^k{-\vartheta |}^s}$$

and $C_1,C_2>0$. Then, by Lemma 2.1 in [30], we can show that there is a positive constant $C_3$ such that

$$\frac{{(-\Delta )}^{s}h(z)}{h(z)}\ge \frac{C_3}{l^{2s}}\mathrm{for}\mathrm{all}\Lambda -2l<z_1<\Lambda \mathrm{with}\mathrm{sufficently}\mathrm{small}l.$$

We have

$${(-{\Delta }_g)}^s{u}_{\Lambda }({\overline{z}}^k,{\overline{t}}_k)-{(-{\Delta }_g)}^{s}u({\overline{z}}^k,{\overline{t}}_k)\le \frac{C_2{C}_3}{l^{2s}}h({\overline{z}}^k){\overline{w}}_{\Lambda }({\overline{z}}^k,{\overline{t}}_k).$$

(14)

Combining Equations (5), (13), and (14) and the boundedness of $\left(\frac{q{\displaystyle {\xi }_1^{q-1}}\left(z,t\right)}{{\displaystyle {\xi }_1^q}\left(z,t\right)+1}+\frac{p{\displaystyle {\xi }_2^{p-1}}\left(z,t\right)}{{\displaystyle {\xi }_2^p}\left(z,t\right)+1}\right)$ from above, we obtain

$$\begin{array}{cc}\hfill -\frac{C_2{C}_3}{l^{2s}}{m}_{k}h({\overline{z}}^k)& \ge \frac{C_2{C}_3}{l^{2s}}h({\overline{z}}^k){\overline{w}}_{\Lambda }({\overline{z}}^k,{\overline{t}}_k)\hfill \\ \hfill & \ge {(-{\Delta }_g)}^s{u}_{\Lambda }({\overline{z}}^k,{\overline{t}}_k)-{(-{\Delta }_g)}^{s}u({\overline{z}}^k,{\overline{t}}_k)\hfill \\ \hfill & =-{\displaystyle {\partial }_t^{\alpha }}{w}_{\Lambda }({\overline{z}}^k,{\overline{t}}_k)+\left(\frac{q{\displaystyle {\xi }_1^{q-1}}\left({\overline{z}}^k,{\overline{t}}_k\right)}{{\displaystyle {\xi }_1^q}\left({\overline{z}}^k,{\overline{t}}_k\right)+1}+\frac{p{\displaystyle {\xi }_2^{p-1}}\left({\overline{z}}^k,{\overline{t}}_k\right)}{{\displaystyle {\xi }_2^p}\left({\overline{z}}^k,{\overline{t}}_k\right)+1}\right)w_{\Lambda }({\overline{z}}^k,{\overline{t}}_k)\hfill \\ \hfill & =-h({\overline{z}}^k){\displaystyle {\partial }_t^{\alpha }}{\overline{w}}_{\Lambda }({\overline{z}}^k,{\overline{t}}_k)+\left(\frac{q{\displaystyle {\xi }_1^{q-1}}\left({\overline{z}}^k,{\overline{t}}_k\right)}{{\displaystyle {\xi }_1^q}\left({\overline{z}}^k,{\overline{t}}_k\right)+1}+\frac{p{\displaystyle {\xi }_2^{p-1}}\left({\overline{z}}^k,{\overline{t}}_k\right)}{{\displaystyle {\xi }_2^p}\left({\overline{z}}^k,{\overline{t}}_k\right)+1}\right)h({\overline{z}}^k){\overline{w}}_{\Lambda }({\overline{z}}^k,{\overline{t}}_k)\hfill \\ \hfill & \ge -C{\epsilon }_{k}h({\overline{z}}^k)-Cmh({\overline{z}}^k).\hfill \end{array}$$

We deduce that

$$\frac{C_2{C}_3}{l^{2s}}\leftarrow \frac{C_2{C}_3}{l^{2s}}\frac{m_k}{m}\le \frac{C{\epsilon }_k}{m}+C\to C\mathrm{as}k\to \infty .$$

We obtain the contradiction for l that is small enough. Then, Equation (6) is verified.

Next, we prove Equation (7). As a consequence of Equation (6), we can conclude that

$$w_{\Lambda }(z^0,t_0)={\displaystyle \underset{{\Sigma }_{\Lambda }\times \mathbb{R}}{min}}{w}_{\Lambda }(z,t)=0.$$

If $w_{\Lambda }(z,t_0)\not\equiv 0$ in ${\Sigma }_{\Lambda }$, by calculation, we obtain

$${\displaystyle {\partial }_t^{\alpha }}{w}_{\Lambda }(z^0,t_0)=-C_{\alpha }{\displaystyle {\int }_{-\infty }^{t_0}}\frac{w_{\Lambda }(z^0,\tau )}{{(t_0-\tau )}^{1+\alpha }}d\tau \le 0$$

and

$$\begin{array}{cc}\hfill & {(-{\Delta }_g)}^s{u}_{\Lambda }(z^0,t_0)-{(-{\Delta }_g)}^{s}u(z^0,t_0)\hfill \\ \hfill =& \mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}\left[g\left(\frac{u_{\Lambda }(z^0,t_0)-u_{\Lambda }(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)-g\left(\frac{u(z^0,t_0)-u(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)\right]\frac{d\vartheta }{|z^0{-\vartheta |}^{n+s}}\hfill \\ \hfill =& \mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\Sigma }_{\Lambda }}^{ }}\left[g\left(\frac{u_{\Lambda }(z^0,t_0)-u_{\Lambda }(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)-g\left(\frac{u(z^0,t_0)-u(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)\right]\frac{d\vartheta }{|z^0{-\vartheta |}^{n+s}}\hfill \\ \hfill & +{\displaystyle {\int }_{{\Sigma }_{\Lambda }}^{ }}\left[g\left(\frac{u_{\Lambda }(z^0,t_0)-u(\vartheta ,t_0)}{|z^0-{\vartheta }^{\Lambda }{|}^s}\right)-g\left(\frac{u(z^0,t_0)-u_{\Lambda }(\vartheta ,t_0)}{|z^0-{\vartheta }^{\Lambda }{|}^s}\right)\right]\frac{d\vartheta }{|z^0-{\vartheta }^{\Lambda }{|}^{n+s}}\hfill \\ \hfill =:& I_1+I_2,\hfill \end{array}$$

here,

$$\begin{array}{cc}\hfill I_1:=& {\displaystyle {\int }_{{\Sigma }_{\Lambda }}^{ }}\left[g\left(\frac{\left(u_{\Lambda }(z^0,t_0)-u_{\Lambda }(\vartheta ,t_0)\right)}{|z^0{-\vartheta |}^s}\right)-g\left(\frac{\left(u(z^0,t_0)-u(\vartheta ,t_0)\right)}{|z^0{-\vartheta |}^s}\right)\right]\hfill \\ \hfill & \left(\frac{1}{|z^0{-\vartheta |}^{n+s}}-\frac{1}{|z^0-{\vartheta }^{\Lambda }{|}^{n+s}}\right)d\vartheta \hfill \end{array}$$

(15)

and

$$\begin{array}{cc}\hfill I_2:=& {\displaystyle {\int }_{{\Sigma }_{\Lambda }}^{ }}\left[g\left(\frac{u_{\Lambda }(z^0,t_0)-u(\vartheta ,t_0)}{|z^0-{\vartheta }^{\Lambda }{|}^s}\right)-g\left(\frac{u(z^0,t_0)-u_{\Lambda }(\vartheta ,t_0)}{|z^0-{\vartheta }^{\Lambda }{|}^s}\right)\right.\hfill \\ \hfill & \left.+g\left(\frac{u_{\Lambda }(z^0,t_0)-u_{\Lambda }(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)-g\left(\frac{u(z^0,t_0)-u(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)\right]\frac{d\vartheta }{{\left|z^0-{\vartheta }^{\Lambda }\right|}^{n+s}}.\hfill \end{array}$$

(16)

Because of

$$\frac{1}{|z^0{-\vartheta |}^{n+\alpha }}-\frac{1}{|z^0-{\vartheta }^{\Lambda }{|}^{n+\alpha }}>0$$

and

$$w_{\Lambda }(z^0,t_0)-w_{\Lambda }(\vartheta ,t_0)=u_{\Lambda }(z^0,t_0)-u_{\Lambda }(\vartheta ,t_0)-(u(z^0,t_0)-u(\vartheta ,t_0))<0,$$

we obtain

$$g\left(\frac{\left(u_{\Lambda }(z^0,t_0)-u_{\Lambda }(\vartheta ,t_0)\right)}{|z^0{-\vartheta |}^s}\right)-g\left(\frac{\left(u(z^0,t_0)-u(\vartheta ,t_0)\right)}{|z^0{-\vartheta |}^s}\right)<0,$$

it follows that $I_1<0$. For the other term, we have

$$I_2\le w_{\Lambda }(z^0,t_0){\displaystyle {\int }_{{\Sigma }_{\Lambda }}^{ }}\frac{g^{\prime }({\zeta }_1(\vartheta ,t))+g^{\prime }({\zeta }_2(\vartheta ,t))}{|z^0-{\vartheta }^{\Lambda }{|}^{n+2s}}d\vartheta =0,$$

with

$${\zeta }_1(\vartheta ,t)\mathrm{between}\frac{u_{\Lambda }(z^0,t_0)-u(\vartheta ,t_0)}{|z^0-{\vartheta }^{\Lambda }{|}^s}\mathrm{and}\frac{u(z^0,t_0)-u_{\Lambda }(\vartheta ,t_0)}{|z^0-{\vartheta }^{\Lambda }{|}^s}$$

and

$${\zeta }_2(\vartheta ,t)\mathrm{between}\frac{u_{\Lambda }(z^0,t_0)-u_{\Lambda }(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\mathrm{and}\frac{u(z^0,t_0)-u(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}.$$

Combining $I_1<0$ and $I_2=0$, we obtain

$${(-{\Delta }_g)}^s{u}_{\Lambda }(z^0,t_0)-{(-{\Delta }_g)}^{s}u(z^0,t_0)<0,$$

then

$${\displaystyle {\partial }_t^{\alpha }}{w}_{\Lambda }(z^0,t_0)+{(-{\Delta }_g)}^s{u}_{\Lambda }(z^0,t_0)-{(-{\Delta }_g)}^{s}u(z^0,t_0)<0.$$

This contradiction aligns with Equation (5). It holds that $w_{\Lambda }(z,t_0)\equiv 0$ in ${\Sigma }_{\Lambda }$. In addition, by the antisymmetry of $w_{\Lambda }(z,t)$ in z, we have

$$w_{\Lambda }(z,t_0)\equiv 0\mathrm{in}{\mathbb{R}}^n.$$

Therefore, for $z\in {\Sigma }_{\Lambda }$ such that $w_{\Lambda }(z,t)\not\equiv 0$ in $(-\infty ,t_0)$, by employing similar estimates as above, we obtain

$${\displaystyle {\partial }_t^{\alpha }}{w}_{\Lambda }(z,t_0)=-C_{\alpha }{\displaystyle {\int }_{-\infty }^{t_0}}\frac{w_{\Lambda }(z,\tau )}{{(t_0-\tau )}^{1+\alpha }}d\tau <0$$

and

$${(-{\Delta }_g)}^s{u}_{\Lambda }(z,t_0)-{(-{\Delta }_g)}^{s}u(z,t_0)=0,$$

which also means that

$${\displaystyle {\partial }_t^{\alpha }}{w}_{\Lambda }(z,t_0)+{(-{\Delta }_g)}^s{u}_{\Lambda }(z,t_0)-{(-{\Delta }_g)}^{s}u(z,t_0)<0.$$

This contradiction aligns with Equation (5). So, $w_{\Lambda }(z,t)\equiv 0$ in ${\Sigma }_{\Lambda }\times (-\infty ,t_0]$. Again, through the antisymmetry of $w_{\Lambda }(z,t)$ for z, we can conclude that

$$w_{\Lambda }(z,t)\equiv 0\mathrm{in}{\mathbb{R}}^n\times (-\infty ,t_0].$$

This completes the proof.  □

Theorem 2.

Assume that $\Omega \subset {\Sigma }_{\Lambda }$ is an unbounded domain of finite width in the direction of $z_1$ and

$$w_{\Lambda }\in ({\displaystyle C_{loc}^{1,1}}(\Omega )\cap L_g({\mathbb{R}}^n))\times (C^1(\mathbb{R})\cap {\displaystyle L_{\alpha }^{-}}(\mathbb{R}))$$

is lower semi-continuous with respect to z on $\overline{\Omega }$.

If

$$w_{\Lambda }(z,t)\ge {-C(1+|z|}^{\nu })forsome0<\nu <2s,$$

and

$$\left\{\begin{array}{c}{\displaystyle {\partial }_t^{\alpha }}{w}_{\Lambda }(z,t)+{(-{\Delta }_g)}^s{u}_{\Lambda }(z,t)-{(-{\Delta }_g)}^{s}u(z,t)=\left(\frac{q{\displaystyle {\xi }_1^{q-1}}\left(z,t\right)}{{\displaystyle {\xi }_1^q}\left(z,t\right)+1}+\frac{p{\displaystyle {\xi }_2^{p-1}}\left(z,t\right)}{{\displaystyle {\xi }_2^p}\left(z,t\right)+1}\right)w_{\Lambda }(z,t),\hfill \\ in\in \Omega \times \mathbb{R},\hfill \\ w_{\Lambda }(z,t)\ge 0,in({\Sigma }_{\Lambda }\backslash \Omega )\times \mathbb{R},\hfill \\ w_{\Lambda }(z,t)=-w_{\Lambda }(z^{\Lambda },t),in{\Sigma }_{\Lambda }\times \mathbb{R}.\hfill \end{array}\right.$$

(17)

where ${\xi }_1(z,t)$ and ${\xi }_2(z,t)$ fall in-between $u_{\Lambda }(z,t)$ and $u(z,t)$,

then

$$w_{\Lambda }(z,t)\ge 0for(z,t)\in {\Sigma }_{\Lambda }\times \mathbb{R}.$$

(18)

Moreover, if $w_{\Lambda }(z^0,t_0)=0$ for a point, $(z^0,t_0)\in \Omega \times \mathbb{R}$, then

$$w_{\Lambda }(z,t)\equiv 0for(z,t)\in {\mathbb{R}}^n\times (-\infty ,t_0].$$

Proof. 

Because $\Omega$ is an unbounded domain of finite width in the direction of $z_1$, assume that $\Omega$ is contained in $\{z\in {\Sigma }_{\Lambda }|\Lambda -2a<z_1<\Lambda \}$ for some $a>0$. Here, we choose the auxiliary functions

$$h(z):=\left[{\displaystyle {\left(1-\frac{{\left(z_1-\left(\Lambda -a\right)\right)}^2}{a^2}\right)}_{+}^s}+1\right]{\left(1+{\left|b{z}^{\prime }\right|}^2\right)}^{\frac{\theta }{2}}$$

for some $\nu <\theta <2s$. Here, b is a small enough positive constant and depends on a. For any $z\in \Omega$ and some constant $C_3>0$, we have

$$\frac{{(-\Delta )}^{s}h(z)}{h(z)}\ge \frac{C_3}{l^{2s}}.$$

It is clear from the calculations that we can obtain $\frac{q{\displaystyle {\xi }_1^{q-1}}\left(z,t\right)}{{\displaystyle {\xi }_1^q}\left(z,t\right)+1}+\frac{p{\displaystyle {\xi }_2^{p-1}}\left(z,t\right)}{{\displaystyle {\xi }_2^p}\left(z,t\right)+1}\le p+q$. Similar to the notation and computation in Theorem 1 proofs, if Equation (18) is false, we can finally derive

$$\frac{C_2{C}_3}{a^{2s}}\leftarrow \frac{C_2{C}_3}{a^{2s}}\frac{m_k}{m}\le \frac{C{\epsilon }_k}{m}+p+q\to p+q,$$

as $k\to \infty$. By the arbitrariness of a, take $a^{2s}<\frac{C_2{C}_3}{p+q}$; then, it is a contradiction. Therefore, the validity of Equation (18) is confirmed. The proof is complete.  □

Theorem 3.

Assume that $\Omega \subset {\mathbb{R}}^n$ is a bounded domain, $[t_1,t_2]\subset \mathbb{R}$ is a finite interval, and

$$u(z,t)\in ({\displaystyle C_{loc}^{1,1}}(\Omega )\cap L_g({\mathbb{R}}^n))\times (C^1([t_1,t_2])\cap {\displaystyle L_{\alpha }^{-}}(\mathbb{R}))$$

is lower semi-continuous with respect to z on $\overline{\Omega }$. If

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_t^{\alpha }}u(z,t)+{(-{\Delta }_g)}^{s}u(z,t)\ge 0,\hfill & (z,t)\in \Omega \times (t_1,t_2],\hfill \\ u(z,t)\ge 0,\hfill & (z,t)\in {\Omega }^c\times (t_1,t_2],\hfill \\ u(z,t)\ge 0,\hfill & (z,t)\in \Omega \times (-\infty ,t_1],\hfill \end{array}\right.$$

(19)

then $u(z,t)\ge 0$ in $\Omega \times (t_1,t_2]$.

Proof. 

If the conclusion is invalid, there exists $(z^0,t_0)\in \Omega \times (t_1,t_2]$ such that

$$u(z^0,t_0)={\displaystyle \underset{\Omega \times (t_1,t_2]}{min}}u(z,t)<0.$$

By Equation (19), we can obtain

$$\begin{array}{cc}\hfill & {\displaystyle {\partial }_t^{\alpha }}u(z^0,t_0)+{(-{\Delta }_g)}^{s}u(z^0,t_0)\hfill \\ \hfill =& C_{\alpha }{\displaystyle {\int }_{-\infty }^{t_0}}\frac{u(z^0,t_0)-u(z^0,\tau )}{{(t_0-\tau )}^{1+\alpha }}d\tau +\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}g\left(\frac{u(z^0,t_0)-u(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)\frac{d\vartheta }{|z^0{-\vartheta |}^{n+s}}\hfill \\ \hfill =& C_{\alpha }{\displaystyle {\int }_{-\infty }^{t_1}}\frac{u(z^0,t_0)-u(z^0,\tau )}{{(t_0-\tau )}^{1+\alpha }}d\tau +C_{\alpha }{\displaystyle {\int }_{t_1}^{t_0}}\frac{u(z^0,t_0)-u(z^0,\tau )}{{(t_0-\tau )}^{1+\alpha }}d\tau \hfill \\ \hfill & +\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{\Omega }^{ }}g\left(\frac{u(z^0,t_0)-u(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)\frac{d\vartheta }{|z^0{-\vartheta |}^{n+s}}+{\displaystyle {\int }_{{\Omega }^c}^{ }}g\left(\frac{u(z^0,t_0)-u(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)\frac{d\vartheta }{|z^0{-\vartheta |}^{n+s}}\hfill \\ \hfill <& 0.\hfill \end{array}$$

This is inconsistent with Equation (19). So, the proof is completed.  □

Theorem 4.

Assume that $\Omega \subset {\Sigma }_{\Lambda }$ is a bounded domain, $[t_1,t_2]\subset \mathbb{R}$ is a finite interval, and

$$w_{\Lambda }\in ({\displaystyle C_{loc}^{1,1}}(\Omega )\cap L_g({\mathbb{R}}^n))\times (C^1([t_1,t_2])\cap {\displaystyle L_{\alpha }^{-}}(\mathbb{R}))$$

is lower semi-continuous for z in $\overline{\Omega }$. If

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_t^{\alpha }}{w}_{\Lambda }(z,t)+{(-{\Delta }_g)}^s{u}_{\Lambda }(z,t)-{(-{\Delta }_g)}^{s}u(z,t)\ge 0,\hfill & (z,t)\in \Omega \times (t_1,t_2],\hfill \\ w_{\Lambda }(z,t)\ge 0,\hfill & (z,t)\in ({\Sigma }_{\Lambda }\backslash \Omega )\times (t_1,t_2],\hfill \\ w_{\Lambda }(z,t)\ge 0,\hfill & (z,t)\in \Omega \times (-\infty ,t_1],\hfill \\ w_{\Lambda }(z,t)=-w_{\Lambda }(z^{\Lambda },t),\hfill & (z,t)\in {\Sigma }_{\Lambda }\times \mathbb{R},\hfill \end{array}\right.$$

(20)

then $w_{\Lambda }(z,t)\ge 0$ in $\Omega \times (t_1,t_2]$.

Proof. 

If the conclusion is not valid, there exists $(z^0,t_0)\in \Omega \times (t_1,t_2]$ such that

$$w_{\Lambda }(z^0,t_0)={\displaystyle \underset{{\Sigma }_{\Lambda }\times (t_1,t_2]}{min}}{w}_{\Lambda }(z,t)<0.$$

By the antisymmetry of $w_{\Lambda }(z,t)$ in z and (2.17), we have

$$\begin{array}{cc}\hfill & {\displaystyle {\partial }_t^{\alpha }}{w}_{\Lambda }(z^0,t_0)+{(-{\Delta }_g)}^s{u}_{\Lambda }(z^0,t_0)-{(-{\Delta }_g)}^{s}u(z^0,t_0)\hfill \\ \hfill =& C_{\alpha }{\displaystyle {\int }_{-\infty }^{t_0}}\frac{w_{\Lambda }(z^0,t_0)-w_{\Lambda }(z^0,\tau )}{{(t_0-\tau )}^{1+\alpha }}d\tau +\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}\left[g\left(\frac{u_{\Lambda }(z^0,t_0)-u_{\Lambda }(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)\right.\hfill \\ \hfill & \left.-g\left(\frac{u(z^0,t_0)-u(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)\right]\frac{d\vartheta }{|z^0{-\vartheta |}^{n+s}}\hfill \\ \hfill =& C_{\alpha }{\displaystyle {\int }_{-\infty }^{t_1}}\frac{w_{\Lambda }(z^0,t_0)-w_{\Lambda }(z^0,\tau )}{{(t_0-\tau )}^{1+\alpha }}d\tau +C_{\alpha }{\displaystyle {\int }_{t_1}^{t_0}}\frac{w_{\Lambda }(z^0,t_0)-w_{\Lambda }(z^0,\tau )}{{(t_0-\tau )}^{1+\alpha }}d\tau \hfill \\ \hfill & +\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\Sigma }_{\Lambda }}^{ }}\left[g\left(\frac{u_{\Lambda }(z^0,t_0)-u_{\Lambda }(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)-g\left(\frac{u(z^0,t_0)-u(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)\right]\frac{d\vartheta }{|z^0{-\vartheta |}^{n+s}}\hfill \\ \hfill & +\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\displaystyle {\Sigma }_{\Lambda }^c}}^{ }}\left[g\left(\frac{u_{\Lambda }(z^0,t_0)-u_{\Lambda }(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)-g\left(\frac{u(z^0,t_0)-u(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)\right]\frac{d\vartheta }{|z^0{-\vartheta |}^{n+s}}\hfill \\ \hfill <& \mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\Sigma }_{\Lambda }}^{ }}\left[g\left(\frac{u_{\Lambda }(z^0,t_0)-u_{\Lambda }(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)-g\left(\frac{u(z^0,t_0)-u(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)\right]\frac{d\vartheta }{|z^0{-\vartheta |}^{n+s}}\hfill \\ \hfill & +\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\Sigma }_{\Lambda }}^{ }}\left[g\left(\frac{u_{\Lambda }(z^0,t_0)-u(\vartheta ,t_0)}{|z^0-{\vartheta }^{\Lambda }{|}^s}\right)-g\left(\frac{u(z^0,t_0)-u_{\Lambda }(\vartheta ,t_0)}{|z^0-{\vartheta }^{\Lambda }{|}^s}\right)\right]\frac{d\vartheta }{|z^0-{\vartheta }^{\Lambda }{|}^{n+s}}\hfill \\ \hfill =& I_1+I_2\hfill \\ \hfill <& 0,\hfill \end{array}$$

with $I_1$ and $I_2$ being Equations (15) and (16). Since $w_{\Lambda }(z^0,t_0)<0$, we obtain $I_1<0$ and $I_2<0$.

Obviously, the above inequality is contradictory to inequality (20). Then, we have successfully completed the proof.  □

3. Averaging Effects

In this section, to prove our main results, we introduce averaging effects (Theorem 5) and averaging effects of antisymmetric functions (Theorem 6) for double non-local operators ${\displaystyle {\partial }_t^{\alpha }}+{(-{\Delta }_g)}^s$.

Theorem 5.

Let $D\subset {\mathbb{R}}^n$. For any $z^0\in {\mathbb{R}}^n$ and some $t_0\in \mathbb{R}$, assume that there exists a radius $r>0$, satisfying $B_r(z^0)\cap \overline{D}=⌀$ as shown in Figure 1, and

$$u(z,t)\ge C_0>0for(z,y)\in D\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}].$$

(21)

Figure 1. The positional relationship between the region D and the ball $B_r(z^0)$ in ${\mathbb{R}}^n$.

Suppose that

$$u(z,t)\in ({\displaystyle C_{loc}^{1,1}}(B_r(z^0))\cap L_g({\mathbb{R}}^n))\times (C^1([t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}])\cap {\displaystyle L_{\alpha }^{-}}(\mathbb{R}))$$

is lower semi-continuous for z in $\overline{B_r(z^0)}$ and satisfies

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_t^{\alpha }}u(z,t)+{(-{\Delta }_g)}^{s}u(z,t)\ge -\epsilon ,\hfill & in{B}_r(z^0)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}],\hfill \\ u(z,t)\ge 0,\hfill & in{\displaystyle B_r^c}(z^0)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}],\hfill \\ u(z,t)\ge 0,\hfill & in{B}_r(z^0)\times (-\infty ,t_0-r^{\frac{2s}{\alpha }}],\hfill \end{array}\right.$$

(22)

for some small enough positive constant ε. Consequently, there exists a positive constant $C_1$ such that

$$u(z^0,t_0)\ge C_1>0.$$

Proof. 

By constructing a sub-solution, we can derive a lower bound estimation. Let

$$\Psi (z,t):=\phi (z)\eta (t)=C{\displaystyle {\left(1-{\left|\frac{z-z^0}{r}\right|}^2\right)}_{+}^s}\eta (t).$$

Here, $\eta (t)$ represents a smooth cut-off function, satisfying

$$\eta (t)\left\{\begin{array}{ccc}=1,\hfill & \hfill & t\in [-\frac{r^{\frac{2s}{\alpha }}}{2}+t_0,\frac{r^{\frac{2s}{\alpha }}}{2}+t_0],\hfill \\ \in [0,1],\hfill & \hfill & t\in (-r^{\frac{2s}{\alpha }}+t_0,-\frac{r^{\frac{2s}{\alpha }}}{2}+t_0)\cup (\frac{r^{\frac{2s}{\alpha }}}{2}+t_0,r^{\frac{2s}{\alpha }}+t_0),\hfill \\ =0,\hfill & \hfill & t\notin (-r^{\frac{2s}{\alpha }}+t_0,r^{\frac{2s}{\alpha }}+t_0).\hfill \end{array}\right.$$

By choosing a suitable positive constant C, it follows that

$$\left\{\begin{array}{cc}{(-\Delta )}^s\phi (z)=\frac{1}{r^{2s}},\hfill & \mathrm{in}{B}_r(z^0),\hfill \\ \phi (z)=0,\hfill & \mathrm{in}{\displaystyle B_r^c}(z^0).\hfill \end{array}\right.$$

(23)

Let

$${\displaystyle \underline{u}}(z,t):=u(z,t){\chi }_D(z)+\delta \Psi (z,t),$$

where

$${\chi }_D(z)=\left\{\begin{array}{ccc}1,\hfill & \hfill & z\in D,\hfill \\ 0,\hfill & \hfill & z\notin D,\hfill \end{array}\right.$$

and $\delta$ is a positive constant that will be determined at a later time.

Next, we will prove that ${\displaystyle \underline{u}}(z,t)$ is a sub-solution of $u(z,t)$ in $B_r(z^0)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}]$. Combining Equations (21)–(23), $B_r(z^0)\cap \overline{D}=⌀$, the properties of g, and Corollary 5.2 in [29], for $(z,t)\in B_r(z^0)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}]$, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle {\partial }_t^{\alpha }}\left(u\left(z,t\right)-{\displaystyle \underline{u}}\left(z,t\right)\right)+{(-{\Delta }_g)}^{s}u(z,t)-{(-{\Delta }_g)}^s{\displaystyle \underline{u}}(z,t)\hfill \\ \hfill \ge & -\epsilon -\delta \phi (z){\displaystyle {\partial }_t^{\alpha }}\eta (t)-\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}g\left(\frac{\delta \Psi \left(z,t\right)-{\displaystyle \underline{u}}\left(\vartheta ,t\right)}{{\left|z-\vartheta \right|}^s}\right)\frac{1}{{\left|z-\vartheta \right|}^{n+s}}d\vartheta \hfill \\ \hfill \ge & -\epsilon -\delta \phi (z){\displaystyle {\partial }_t^{\alpha }}\eta (t)-{(-{\Delta }_g)}^s(\delta \Psi (z,t))\hfill \\ \hfill & +\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_D^{ }}\left[g\left(\frac{\delta \Psi \left(z,t\right)}{{\left|z-\vartheta \right|}^s}\right)-g\left(\frac{\delta \Psi \left(z,t\right)-u\left(\vartheta ,t\right)}{{\left|z-\vartheta \right|}^s}\right)\right]\frac{1}{{\left|z-\vartheta \right|}^{n+s}}d\vartheta \hfill \\ \hfill \ge & -\epsilon -\delta \phi (z){\displaystyle {\partial }_t^{\alpha }}\eta (t)-C_2\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}\left[g\left(\frac{\delta \Psi \left(z,t\right)}{{\left|z-\vartheta \right|}^s}\right)-g\left(\frac{\delta \Psi \left(\vartheta ,t\right)}{{\left|z-\vartheta \right|}^s}\right)\right]\frac{1}{{\left|z-\vartheta \right|}^{n+s}}d\vartheta \hfill \\ \hfill & +{\displaystyle {\int }_D^{ }}\frac{g^{\prime }\left({\zeta }_1(\vartheta ,t)\right)u\left(\vartheta ,t\right)}{{\left|z-\vartheta \right|}^{n+2s}}d\vartheta \hfill \\ \hfill \ge & -\epsilon -\delta \phi (z){\displaystyle {\partial }_t^{\alpha }}\eta (t)-C_2\delta \eta (t)\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}\frac{g^{\prime }\left({\zeta }_2(\vartheta ,t)\right)(\phi (z)-\phi (\vartheta ))}{{\left|z-\vartheta \right|}^{n+2s}}d\vartheta +C_3\hfill \\ \hfill \ge & -\epsilon -\frac{\delta C_4}{r^{2s}}+C_3,\hfill \end{array}$$

where

$${\zeta }_1(\vartheta ,t)\mathrm{between}\frac{\delta \Psi \left(z,t\right)}{{\left|z-\vartheta \right|}^s}\mathrm{and}\frac{\delta \Psi \left(z,t\right)-u\left(\vartheta ,t\right)}{{\left|z-\vartheta \right|}^s}$$

and

$${\zeta }_2(\vartheta ,t)\mathrm{between}\frac{\delta \Psi \left(z,t\right)}{{\left|z-\vartheta \right|}^s}\mathrm{and}\frac{\delta \Psi \left(\vartheta ,t\right)}{{\left|z-\vartheta \right|}^s}.$$

By choosing $\epsilon =\frac{C_3}{2}$ and $\delta =\frac{C_3{r}^{2s}}{2C_4}$, we can obtain

$${\displaystyle {\partial }_t^{\alpha }}\left(u\left(z,t\right)-{\displaystyle \underline{u}}\left(z,t\right)\right)+{(-{\Delta }_g)}^{s}u(z,t)-{(-{\Delta }_g)}^s{\displaystyle \underline{u}}(z,t)\ge 0,(z,t)\in B_r(z^0)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}].$$

Due to Equation (22) and the definition of $\Phi (z,t)$, we can deduce that

$$u\left(z,t\right)-{\displaystyle \underline{u}}\left(z,t\right)=u(z,t)-u(z,t){\chi }_D(z)\ge 0\mathrm{in}{\displaystyle B_r^c}(z^0)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}],$$

and

$$u\left(z,t\right)-{\displaystyle \underline{u}}\left(z,t\right)=u(z,t)\ge 0\mathrm{in}{B}_r(z^0)\times (-\infty ,t_0-r^{\frac{2s}{\alpha }}].$$

Anyway, we have successfully acquired

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_t^{\alpha }}\left(u\left(z,t\right)-{\displaystyle \underline{u}}\left(z,t\right)\right)+{(-{\Delta }_g)}^{s}u(z,t)-{(-{\Delta }_g)}^s{\displaystyle \underline{u}}(z,t)\ge 0,\hfill & \mathrm{in}{B}_r(z^0)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}],\hfill \\ u\left(z,t\right)-{\displaystyle \underline{u}}\left(z,t\right)\ge 0,\hfill & \mathrm{in}{\displaystyle B_r^c}(z^0)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}],\hfill \\ u\left(z,t\right)-{\displaystyle \underline{u}}\left(z,t\right)\ge 0,\hfill & \mathrm{in}{B}_r(z^0)\times (-\infty ,t_0-r^{\frac{2s}{\alpha }}].\hfill \end{array}\right.$$

From Theorem 3, we have

$$u\left(z,t\right)\ge {\displaystyle \underline{u}}(z,t)\mathrm{in}{B}_r(z^0)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}].$$

Consequently, it can be inferred that

$$u(z^0,t_0)\ge {\displaystyle \underline{u}}(z^0,t_0)=\delta \phi (z^0)\eta (t_0)=C\delta =:C_1>0.$$

Therefore, we successfully prove Theorem 5.  □

Theorem 6.

Let $D\subset {\Sigma }_{\Lambda }$. For any $z^0\in {\Sigma }_{\Lambda }$ and some $t_0\in \mathbb{R}$, assume that there exists a ball, $B_r(z^0)\subset {\Sigma }_{\Lambda }$, satisfying $B_r(z^0)\cap \overline{D}=⌀$, $r\le \frac{dist(z^0,T_{\Lambda })}{2}$ as shown in Figure 2, and

$$w_{\Lambda }(z,t)\ge C_0>0inD\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}].$$

(24)

Figure 2. The positional relationship between the region D and the ball $B_r(z^0)$ in ${\Sigma }_{\Lambda }$.

Suppose that

$$w_{\Lambda }(z,t)\in ({\displaystyle C_{loc}^{1,1}}(B_r(z^0))\cap L_g({\mathbb{R}}^n))\times (C^1([t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}])\cap {\displaystyle L_{\alpha }^{-}}(\mathbb{R}))$$

is lower semi-continuous for z in $\overline{B_r(z^0)}$ and satisfies

$$\left\{\begin{array}{c}{\displaystyle {\partial }_t^{\alpha }}{w}_{\Lambda }(z,t)+{(-{\Delta }_g)}^s{u}_{\Lambda }(z,t)-{(-{\Delta }_g)}^{s}u(z,t)\ge -\epsilon ,(z,t)\in B_r(z^0)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}],\hfill \\ w_{\Lambda }(z,t)\ge 0,(z,t)\in \left({\Sigma }_{\Lambda }\backslash B_r(z^0)\right)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}],\hfill \\ w_{\Lambda }(z,t)\ge 0,(z,t)\in B_r(z^0)\times (-\infty ,t_0-r^{\frac{2s}{\alpha }}],\hfill \\ w_{\Lambda }(z,t)=-w_{\Lambda }(z^{\Lambda },t),(z,t)\in {\Sigma }_{\Lambda }\times \mathbb{R},\hfill \end{array}\right.$$

(25)

for some small enough positive constant ε. Consequently, there exists a positive constant $C_1$ such that

$$w_{\Lambda }(z^0,t_0)\ge C_1>0.$$

Proof. 

In the process of proving this theorem, the most important step is to construct a sub-solution for $w_{\Lambda }(z,t)$. Let

$$\phi (z)={\displaystyle {\left(1-{\left|\frac{z-z^0}{r}\right|}^2\right)}_{+}^s}\mathrm{and}{\phi }_{\Lambda }(z)={\displaystyle {\left(1-{\left|\frac{z^{\Lambda }-z^0}{r}\right|}^2\right)}_{+}^s}.$$

It is easy to obtain an antisymmetric function with respect to the plane $T_{\Lambda }$.

$$\Phi (z):=\phi (z)-{\phi }_{\Lambda }(z).$$

Denote $\eta (t)\in {\displaystyle C_0^{\infty }}\left(t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}\right)$, where $\eta (t)$ represents a smooth cut-off function, satisfying

$$\eta (t)\left\{\begin{array}{ccc}=1,\hfill & \hfill & t\in [-\frac{r^{\frac{2s}{\alpha }}}{2}+t_0,\frac{r^{\frac{2s}{\alpha }}}{2}+t_0],\hfill \\ \in [0,1],\hfill & \hfill & t\in (-r^{\frac{2s}{\alpha }}+t_0,-\frac{r^{\frac{2s}{\alpha }}}{2}+t_0)\cup (\frac{r^{\frac{2s}{\alpha }}}{2}+t_0,r^{\frac{2s}{\alpha }}+t_0),\hfill \\ =0,\hfill & \hfill & t\notin (-r^{\frac{2s}{\alpha }}+t_0,r^{\frac{2s}{\alpha }}+t_0).\hfill \end{array}\right.$$

Let

$${{\displaystyle \underline{w}}}_{\Lambda }(z,t):=w_{\Lambda }(z,t){\chi }_{D\cup D^{\Lambda }}(z)+\delta \Phi (z)\eta (t),$$

where

$${\chi }_{D\cup D^{\Lambda }}(z)=\left\{\begin{array}{ccc}1,\hfill & \hfill & z\in D\cup D^{\Lambda },\hfill \\ 0,\hfill & \hfill & z\notin D\cup D^{\Lambda },\hfill \end{array}\right.$$

the domain $D^{\Lambda }$ is a reflection of the domain D with respect to the plane $T_{\Lambda }$, and $\delta$ is a positive constant that will be determined later.

Next, we will prove that ${{\displaystyle \underline{w}}}_{\Lambda }(z,t)$ is a sub-solution of $w_{\Lambda }(z,t)$ in $B_r(z^0)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}]$. This is for $(z,t)\in B_r(z^0)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}]$. By Equations (23)–(25), Corollary 5.2 in [29], the properties of g, and $r\le \frac{dist(z^0,T_{\Lambda })}{2}$, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle {\partial }_t^{\alpha }}\left(w_{\Lambda }\left(z,t\right)-{{\displaystyle \underline{w}}}_{\Lambda }\left(z,t\right)\right)+{(-{\Delta }_g)}^s{u}_{\Lambda }(z,t)-{(-{\Delta }_g)}^{s}u(z,t)-{(-{\Delta }_g)}^s{{\displaystyle \underline{w}}}_{\Lambda }(z,t)\hfill \\ \hfill \ge & -\epsilon -\delta \Phi (z){\displaystyle {\partial }_t^{\alpha }}\eta (t)-\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}g\left(\frac{\delta \Phi \left(z\right)\eta \left(t\right)-{{\displaystyle \underline{w}}}_{\Lambda }\left(\vartheta ,t\right)}{{\left|z-\vartheta \right|}^s}\right)\frac{1}{{\left|z-\vartheta \right|}^{n+s}}d\vartheta \hfill \\ \hfill \ge & -\epsilon -\delta \Phi (z){\displaystyle {\partial }_t^{\alpha }}\eta (t)-{(-{\Delta }_g)}^s(\delta \Phi (z)\eta (t))\hfill \\ \hfill & +{\displaystyle {\int }_{D\cup D^{\Lambda }}^{ }}\left[g\left(\frac{\delta \Phi \left(z\right)\eta \left(t\right)}{{\left|z-\vartheta \right|}^s}\right)-g\left(\frac{\delta \Phi \left(z\right)\eta \left(t\right)-w_{\Lambda }\left(\vartheta ,t\right)}{{\left|z-\vartheta \right|}^s}\right)\right]\frac{1}{{\left|z-\vartheta \right|}^{n+s}}d\vartheta \hfill \\ \hfill \ge & -\epsilon -\delta \Phi (z){\displaystyle {\partial }_t^{\alpha }}\eta (t)-\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}g\left(\frac{\delta \Phi (z)\eta (t)-\delta \Phi (\vartheta )\eta (t)}{{\left|z-\vartheta \right|}^s}\right)\frac{1}{{\left|z-\vartheta \right|}^{n+s}}d\vartheta \hfill \\ \hfill & +{\displaystyle {\int }_{D\cup D^{\Lambda }}^{ }}\frac{g^{\prime }\left({\zeta }_1\left(\vartheta ,t\right)\right)w_{\Lambda }\left(\vartheta ,t\right)}{{\left|z-\vartheta \right|}^{n+2s}}d\vartheta \hfill \\ \hfill \ge & -\epsilon -\delta \Phi (z){\displaystyle {\partial }_t^{\alpha }}\eta (t)-C_2\delta \eta (t)\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}\frac{g^{\prime }\left({\zeta }_2(\vartheta ,t)\right)(\Phi (z)-\Phi (\vartheta ))}{{\left|z-\vartheta \right|}^{n+2s}}d\vartheta \hfill \\ \hfill & +C_3{\displaystyle {\int }_D^{ }}\left[\frac{1}{{\left|z-\vartheta \right|}^{n+2s}}-\frac{1}{{\left|z-{\vartheta }^{\Lambda }\right|}^{n+2s}}\right]d\vartheta \hfill \\ \hfill \ge & -\epsilon -\frac{\delta C_4}{r^{2s}}-\delta \eta (t){\displaystyle {\int }_{B_r({(z^0)}^{\Lambda })}^{ }}\frac{g^{\prime }\left({\zeta }_2(\vartheta ,t)\right){\phi }_{\Lambda }(\vartheta )}{{\left|z-\vartheta \right|}^{n+2s}}d\vartheta +C_5\hfill \\ \hfill \ge & -\epsilon -\frac{\delta C_6}{r^{2s}}+C_5,\hfill \end{array}$$

where

$${\zeta }_1\left(\vartheta ,t\right)\mathrm{between}\frac{\delta \Phi \left(z\right)\eta \left(t\right)}{{\left|z-\vartheta \right|}^s}\mathrm{and}\frac{\delta \Phi \left(z\right)\eta \left(t\right)-w_{\Lambda }\left(\vartheta ,t\right)}{{\left|z-\vartheta \right|}^s}$$

and

$${\zeta }_2\left(\vartheta ,t\right)\mathrm{between}\frac{\delta \Phi (z)\eta (t)}{{\left|z-\vartheta \right|}^s}\mathrm{and}\frac{\delta \Phi (\vartheta )\eta (t)}{{\left|z-\vartheta \right|}^s}.$$

By selecting $\epsilon =\frac{C_5}{2}$ and $\delta =\frac{C_5{r}^{2s}}{2C_6}$, for $(z,t)\in B_r(z^0)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}]$, we obtain

$${\displaystyle {\partial }_t^{\alpha }}\left(w_{\Lambda }\left(z,t\right)-{{\displaystyle \underline{w}}}_{\Lambda }\left(z,t\right)\right)+{(-{\Delta }_g)}^s{u}_{\Lambda }(z,t)-{(-{\Delta }_g)}^{s}u(z,t)-{(-{\Delta }_g)}^s{{\displaystyle \underline{w}}}_{\Lambda }(z,t)\ge 0.$$

By Equation (25), we obtain

$$w_{\Lambda }\left(z,t\right)-{{\displaystyle \underline{w}}}_{\Lambda }\left(z,t\right)=w_{\Lambda }\left(z,t\right)-w_{\Lambda }\left(z,t\right){\chi }_{D\cup D^{\Lambda }}(z)\ge 0\mathrm{in}\left({\Sigma }_{\Lambda }\backslash B_r(z^0)\right)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}].$$

and

$$w_{\Lambda }\left(z,t\right)-{{\displaystyle \underline{w}}}_{\Lambda }\left(z,t\right)=w_{\Lambda }\left(z,t\right)\ge 0\mathrm{in}{B}_r(z^0)\times (-\infty ,t_0-r^{\frac{2s}{\alpha }}].$$

Anyway, we have successfully acquired

$$\left\{\begin{array}{c}{\displaystyle {\partial }_t^{\alpha }}\left(w_{\Lambda }\left(z,t\right)-{{\displaystyle \underline{w}}}_{\Lambda }\left(z,t\right)\right)+{(-{\Delta }_g)}^s{u}_{\Lambda }(z,t)-{(-{\Delta }_g)}^{s}u(z,t)-{(-{\Delta }_g)}^s{{\displaystyle \underline{w}}}_{\Lambda }(z,t)\ge 0,\hfill \\ \begin{array}{cc}& \mathrm{in}{B}_r(z^0)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}],\hfill \\ w_{\Lambda }\left(z,t\right)-{{\displaystyle \underline{w}}}_{\Lambda }\left(z,t\right)\ge 0,\hfill & \mathrm{in}\left({\Sigma }_{\Lambda }\backslash B_r(z^0)\right)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}],\hfill \\ w_{\Lambda }\left(z,t\right)-{{\displaystyle \underline{w}}}_{\Lambda }\left(z,t\right)\ge 0,\hfill & \mathrm{in}{B}_r(z^0)\times (-\infty ,t_0-r^{\frac{2s}{\alpha }}],\hfill \\ w_{\Lambda }\left(z,t\right)-{{\displaystyle \underline{w}}}_{\Lambda }\left(z,t\right)=-\left(w_{\Lambda }\left(z^{\Lambda },t\right)-{{\displaystyle \underline{w}}}_{\Lambda }\left(z^{\Lambda },t\right)\right),\hfill & \mathrm{in}{\Sigma }_{\Lambda }\times \mathbb{R}.\hfill \end{array}\end{array}\right.$$

From Theorem 4, we obtain

$$w_{\Lambda }(z,t)\ge {{\displaystyle \underline{w}}}_{\Lambda }\left(z,t\right)\mathrm{in}{B}_r(z^0)\times (t_0-r^{\frac{2s}{\alpha }},t_0+r^{\frac{2s}{\alpha }}].$$

Consequently, it can be inferred that

$$w_{\Lambda }(z^0,t_0)\ge {{\displaystyle \underline{w}}}_{\Lambda }(z^0,t_0)=\delta \phi (z^0)\eta (t_0)=\delta =:C_1>0.$$

□

Remark 1.

The average effect shows that the positiveness of the solution of the fractional time-diffusion equation in some region D will spread to any other region B that does not intersect D. The average effect is influenced by the distance between the two regions, and the shorter the distance, the more significant the effect.

4. Application of Direct Moving Plane Method

In this section, in combination with the maximum principles and the average effects described above, the monotonicity of the positive solution for Equation (26) in half space is established by the direct moving plane method. Consider

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_t^{\alpha }}u(z,t)+{(-{\Delta }_g)}^{s}u(z,t)=ln\left|u^q\left(z,t\right)+1\right|+ln\left|u^p\left(z,t\right)+1\right|,\hfill & \mathrm{in}{\displaystyle {\mathbb{R}}_{+}^n}\times \mathbb{R},\hfill \\ u(z,t)=0,\hfill & \mathrm{in}\left({\mathbb{R}}^n\backslash {\displaystyle {\mathbb{R}}_{+}^n}\right)\times \mathbb{R}.\hfill \end{array}\right.$$

(26)

where ${\displaystyle {\mathbb{R}}_{+}^n}:=\{z\in {\mathbb{R}}^n\mid z_1>0\}$ represents the right half space and $p,q\ge 1$.

Theorem 7.

Assume that $u(z,t)\in ({\displaystyle C_{loc}^{1,1}}({\displaystyle {\mathbb{R}}_{+}^n})\cap L_g({\mathbb{R}}^n))\times (C^1(\mathbb{R})\cap {\displaystyle L_{\alpha }^{-}}(\mathbb{R}))$ is a positive solution of Equation (26) and $u(z,t)$ is uniformly continuous in z. If

$$u(z,t)\le C(1+|z{|}^{\nu })forsome0<\nu <2s,$$

then the positive solution $u(z,t)$ of Equation (26) is strictly increasing in half space along the $z_1$-direction for any $t\in \mathbb{R}$.

Proof. 

By performing a direct calculation, it follows that

$$\left\{\begin{array}{l}{\displaystyle {\partial }_t^{\alpha }}{w}_{\Lambda }(z,t)+{(-{\Delta }_g)}^s{u}_{\Lambda }(z,t)-{(-{\Delta }_g)}^{s}u(z,t)=\left(\frac{q{\displaystyle {\xi }_1^{q-1}}\left(z,t\right)}{{\displaystyle {\xi }_1^q}\left(z,t\right)+1}+\frac{p{\displaystyle {\xi }_2^{p-1}}\left(z,t\right)}{{\displaystyle {\xi }_2^p}\left(z,t\right)+1}\right)w_{\Lambda }(z,t),\hfill \\ \begin{array}{cc}& \hfill (z,t)\in {\Omega }_{\Lambda }\times \mathbb{R},\\ w_{\Lambda }(z,t)\ge 0,\hfill & \hfill (z,t)\in ({\Sigma }_{\Lambda }\backslash {\Omega }_{\Lambda })\times \mathbb{R},\\ w_{\Lambda }(z,t)=-w_{\Lambda }(z^{\Lambda },t),\hfill & \hfill (z,t)\in {\Sigma }_{\Lambda }\times \mathbb{R}.\end{array}\end{array}\right.$$

(27)

where ${\xi }_1(z,t)$ and ${\xi }_2(z,t)$ fall in-between $u_{\Lambda }(z,t)$ and $u(z,t)$. To obtain our result, we just need to show that

$$w_{\Lambda }(z,t)>0\mathrm{in}{\Omega }_{\Lambda }\times \mathbb{R},$$

for any $\Lambda >0$. The proof will be divided into three distinct steps.

Step 1: By starting from $z_1=0$ and moving the plane $T_{\Lambda }$ toward the right along the $z_1$-axis. With the assumption of Theorem 7, we can apply Theorem 1 to Equation (27) for $\Lambda >0$ small enough; therefore, we have

$$w_{\Lambda }(z,t)\ge 0\mathrm{for}(z,t)\in {\Omega }_{\Lambda }\times \mathbb{R}.$$

(28)

Obviously, when $\Lambda >0$ is small enough, ${\Omega }_{\Lambda }$ is a narrow region. The starting point of the moving plane $T_{\Lambda }$ is provided by inequality (28).

Step 2: We proceed with the continuous movement of the plane $T_{\Lambda }$ along the $z_1$-axis toward the right, ensuring that inequality (28) remains valid until the plane reaches its limiting position. Let

$${\Lambda }_0=sup\{\Lambda |w_{\mu }(z,t)\ge 0,(z,t)\in {\Sigma }_{\mu }\times \mathbb{R}\mathrm{for}\mathrm{any}\mu \le \Lambda \}.$$

Next, our goal is to prove

$${\Lambda }_0=+\infty .$$

(29)

Otherwise, if $0<{\Lambda }_0<+\infty$, according to its definition, we can find a sequence, ${\Lambda }_k$, with ${\Lambda }_k>{\Lambda }_0$ such that ${\Lambda }_k\to {\Lambda }_0$ when $k\to \infty$; then, we have

$${\displaystyle {\Sigma }_{{\Lambda }_k}^{-}}\times \mathbb{R}:=\{(z,t)\in {\Sigma }_{{\Lambda }_k}\times \mathbb{R}|w_{{\Lambda }_k}(z,t)<0\}$$

that is nonempty and ${\displaystyle \underset{{\Sigma }_{{\Lambda }_k}\times \mathbb{R}}{inf}}{w}_{{\Lambda }_k}(z,t)<0$. First of all, we need to demonstrate that

$${\displaystyle \underset{{\Sigma }_{{\Lambda }_k}\times \mathbb{R}}{inf}}{w}_{{\Lambda }_k}(z,t)\to 0\mathrm{as}k\to \infty .$$

(30)

If this assumption does not hold, it follows that there is a positive constant M, satisfying

$${\displaystyle \underset{{\Sigma }_{{\Lambda }_k}\times \mathbb{R}}{inf}}{w}_{{\Lambda }_k}(z,t)<-M<0.$$

Thus, we can find a sequence, $\{(z^k,t_k)\}\subset {\Sigma }_{{\Lambda }_k}\times \mathbb{R}$, such that

$$w_{{\Lambda }_k}(z^k,t_k)\le -M<0.$$

(31)

It follows that either $z^k$ is between $T_{{\Lambda }_0}$ and $T_{{\Lambda }_k}$ or $z^k\in {\Omega }_{{\Lambda }_0}$. When $z^k$ is between $T_{{\Lambda }_0}$ and $T_{{\Lambda }_k}$, then combining ${\Lambda }_k\to {\Lambda }_0$ as $k\to \infty$ and the uniform continuity of $u(z,t)$ in z, we can obtain

$$w_{{\Lambda }_k}(z^k,t_k)=u_{{\Lambda }_k}(z^k,t_k)-u(z^k,t_k)\to 0\mathrm{as}k\to \infty .$$

This contradicts Equation (31).

When $z^k\in {\Omega }_{{\Lambda }_0}$, we have

$$w_{{\Lambda }_k}(z^k,t_k)-w_{{\Lambda }_0}(z^k,t_k)=u_{{\Lambda }_k}(z^k,t_k)-u_{{\Lambda }_0}(z^k,t_k)\to 0\mathrm{as}k\to \infty .$$

However, by Equation (31) and $w_{{\Lambda }_0}(z^k,t_k)\ge 0$, we derive

$$w_{{\Lambda }_k}(z^k,t_k)-w_{{\Lambda }_0}(z^k,t_k)\le -M<0.$$

This is a contradiction. So, Equation (30) is valid, and we infer that

$${\displaystyle \underset{{\Sigma }_{{\Lambda }_k}\times \mathbb{R}}{inf}}{w}_{{\Lambda }_k}(z,t)=:-m_k\to 0\mathrm{as}k\to \infty .$$

(32)

In the following, denote

$$h_k:={\displaystyle \underset{{\displaystyle {\Sigma }_{{\Lambda }_k}^{-}}\times \mathbb{R}}{sup}}\left(\frac{q{\displaystyle {\xi }_1^{q-1}}\left(z,t\right)}{{\displaystyle {\xi }_1^q}\left(z,t\right)+1}+\frac{p{\displaystyle {\xi }_2^{p-1}}\left(z,t\right)}{{\displaystyle {\xi }_2^p}\left(z,t\right)+1}\right),$$

with ${\xi }_1(z,t)$ and ${\xi }_2(z,t)$ falling in-between $u_{{\Lambda }_k}(z,t)$ and $u(z,t)$. There are two possible cases that arise from the above analysis.

Case 1. If $h_k\le {\epsilon }_k\to 0$ as $k\to \infty$, let $\Lambda ={\Lambda }_k$, and by applying Theorem 2 to problem (27), it follows that

$$w_{{\Lambda }_k}(z,t)\ge 0\mathrm{in}{\Sigma }_{{\Lambda }_k}\times \mathbb{R}$$

for a large enough k. This is inconsistent with the definition of ${\Lambda }_k$, hence the contradiction.

Case 2. If $h_k\nrightarrow 0$ when $k\to \infty$, it implies that there exist ${\delta }_0>0$ and a subsequence of $\{h_k\}$ (which we will still denote as $\{h_k\}$), satisfying $h_k\ge {\delta }_0>0$. Since ${(ln\left|u^q\left(z,t\right)+1\right|+ln\left|u^p\left(z,t\right)+1\right|)}^{\prime }=0$ when $u(z,t)=0$ and regarding Equation (32), we can infer that there exist ${\epsilon }_0>0$ along with a sequence, $\{(z^k,t_k)\}\subset {\displaystyle {\Sigma }_{{\Lambda }_k}^{-}}\times \mathbb{R}$, such that

$$u(z^k,t_k)\ge {\epsilon }_0>0$$

and

$$w_{{\Lambda }_k}(z^k,t_k)=-m_k+\sigma {\displaystyle m_k^2}<0,$$

where $\sigma \ge 0$. Combining the fact that $u(z,t)=0\mathrm{in}({\mathbb{R}}^n\backslash {\displaystyle {\mathbb{R}}_{+}^n})\times \mathbb{R}$ with the fact that $u(z,t)$ are continuous, there exists $r_0>0$, which is irrespective of k, such that

$$u(z,t)\ge \frac{{\epsilon }_0}{2}>0\mathrm{for}(z,t)\in B_{r_0}(z^0)\times (t_k-{\displaystyle r_0^{\frac{2s}{\alpha }}},t_k+{\displaystyle r_0^{\frac{2s}{\alpha }}}]\subset {\displaystyle {\mathbb{R}}_{+}^n}\times \mathbb{R}.$$

(33)

Next, we show that ${\delta }_k:=dist(z^k,T_{{\Lambda }_k})={\Lambda }_k-{\displaystyle z_1^k}$ is bounded and ${\delta }_k\ne 0$ for a large enough k. If not, it follows that ${\delta }_k\to 0$ as $k\to \infty$. Let

$$v_k(z,t):=w_{{\Lambda }_k}(z,t)-\sigma {\displaystyle m_k^2}{\eta }_k(z,t),$$

where ${\eta }_k(z,t)$ is a series of smooth cut-off functions,

$${\eta }_k(z,t)=\eta (\frac{z-z_k}{{\delta }_k}-\frac{t-t_k}{{\displaystyle {\delta }_k^{\frac{2s}{\alpha }}}})\in {\displaystyle C_0^{\infty }}(B_{{\delta }_k}(z^k)\times (-{\displaystyle {\delta }_k^{\frac{2s}{\alpha }}}+t_k,{\displaystyle {\delta }_k^{\frac{2s}{\alpha }}}+t_k)),$$

satisfying

$${\eta }_k(z,t)\left\{\begin{array}{ccc}=1,\hfill & \hfill & (z,t)\in B_{\frac{{\delta }_k}{2}}(z^k)\times (-\frac{{\displaystyle {\delta }_k^{\frac{2s}{\alpha }}}}{2}+t_k,\frac{{\displaystyle {\delta }_k^{\frac{2s}{\alpha }}}}{2}+t_k),\hfill \\ \in [0,1],\hfill & \hfill & (z,t)\in [B_{{\delta }_k}(z^k)\backslash B_{\frac{{\delta }_k}{2}}(z^k)]\times [(-{\displaystyle {\delta }_k^{\frac{2s}{\alpha }}}+t_k,-\frac{{\displaystyle {\delta }_k^{\frac{2s}{\alpha }}}}{2}+t_k)\cup (\frac{{\displaystyle {\delta }_k^{\frac{2s}{\alpha }}}}{2}+t_k,r^{\frac{2s}{\alpha }}+t_k)],\hfill \\ =0,\hfill & \hfill & (z,t)\notin B_{{\delta }_k}(z^k)\times (t_k-{\displaystyle {\delta }_k^{\frac{2s}{\alpha }}},t_k+{\displaystyle {\delta }_k^{\frac{2s}{\alpha }}}).\hfill \end{array}\right.$$

Denote

$$Q_{{\delta }_k}(z^k,t_k):=B_{{\delta }_k}(z^k)\times (t_k-{\displaystyle {\delta }_k^{\frac{2s}{\alpha }}},t_k+{\displaystyle {\delta }_k^{\frac{2s}{\alpha }}}),$$

therefore, direct calculation can be obtained.

$$v_k(z^k,t_k)=w_{{\Lambda }_k}(z^k,t_k)-{\displaystyle m_k^2}{\eta }_k(z^k,t_k)=-m_k+\sigma {\displaystyle m_k^2}-\sigma {\displaystyle m_k^2}=-m_k,$$

and

$$v_k(z,t)=w_{{\Lambda }_k}(z,t)\ge -m_k\mathrm{in}{\displaystyle Q_{{\delta }_k}^c}(z^k,t_k)\cap ({\Sigma }_{{\Lambda }_k}\times \mathbb{R}).$$

Then, there exists a point $({\overline{z}}^k,{\overline{t}}^k)$ and $({\overline{z}}^k,{\overline{t}}^k)\in Q_{{\delta }_k}(z^k,t_k)$ such that

$$-m_k-\sigma {\displaystyle m_k^2}\le v_k({\overline{z}}^k,{\overline{t}}^k)={\displaystyle \underset{{\Sigma }_{{\Lambda }_k}\times \mathbb{R}}{inf}}{v}_k(z,t)\le -m_k.$$

By definition of $v_k$, we have

$$-m_k\le w_{{\Lambda }_k}({\overline{z}}^k,{\overline{t}}^k)\le -m_k+\sigma {\displaystyle m_k^2}<0.$$

For the minimum point $({\overline{z}}^k,{\overline{t}}^k)$ of $v_k(z,t)$, we have

$${\displaystyle {\partial }_t^{\alpha }}{v}_k({\overline{z}}^k,{\overline{t}}^k)=C_{\alpha }{\displaystyle {\int }_{-\infty }^{{\overline{t}}^k}}\frac{v_k({\overline{z}}^k,{\overline{t}}^k)-v_k({\overline{z}}^k,\tau )}{{({\overline{t}}^k-\tau )}^{1+\alpha }}d\tau \le 0,$$

and

$$\begin{array}{cc}\hfill & {(-{\Delta }_g)}^s{v}_k({\overline{z}}^k,{\overline{t}}^k)\hfill \\ \hfill =& \mathrm{P}.\mathrm{V}.\left({\displaystyle {\int }_{{\Sigma }_{{\Lambda }_k}}^{ }}+{\displaystyle {\int }_{{\displaystyle {\Sigma }_{{\Lambda }_k}^c}}^{ }}\right)g\left(\frac{v_k({\overline{z}}^k,{\overline{t}}^k)-v_k(\vartheta ,{\overline{t}}^k)}{|{\overline{z}}^k{-\vartheta |}^s}\right)\frac{1}{|{\overline{z}}^k{-\vartheta |}^{n+s}}d\vartheta \hfill \\ \hfill \le & C\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\Sigma }_{{\Lambda }_k}}^{ }}g\left(\frac{2v_k({\overline{z}}^k,{\overline{t}}^k)+\sigma {\displaystyle m_k^2}{\eta }_k(\vartheta ,{\overline{t}}^k)}{|{\overline{z}}^k-{\vartheta }^{{\Lambda }_k}{|}^s}\right)\frac{1}{|{\overline{z}}^k-{\vartheta }^{{\Lambda }_k}{|}^{n+s}}d\vartheta \hfill \\ \hfill \le & C\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\Sigma }_{{\Lambda }_k}}^{ }}g\left(\frac{2v_k({\overline{z}}^k,{\overline{t}}^k)}{|{\overline{z}}^k-{\vartheta }^{{\Lambda }_k}{|}^s}\right)\frac{1}{|{\overline{z}}^k-{\vartheta }^{{\Lambda }_k}{|}^{n+s}}d\vartheta +C\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\Sigma }_{{\Lambda }_k}}^{ }}g\left(\frac{\sigma {\displaystyle m_k^2}{\eta }_k(\vartheta ,{\overline{t}}^k)}{|{\overline{z}}^k-{\vartheta }^{{\Lambda }_k}{|}^s}\right)\frac{1}{|{\overline{z}}^k-{\vartheta }^{{\Lambda }_k}{|}^{n+s}}d\vartheta \hfill \\ \hfill \le & -\frac{C}{{\displaystyle {\delta }_k^{2s}}}+\frac{C{\displaystyle m_k^2}}{{\displaystyle {\delta }_k^{2s}}}.\hfill \end{array}$$

By Equation (27), Corollary 5.2 in [29], and properties of g, it follows that

$$\begin{array}{cc}\hfill & {\displaystyle {\partial }_t^{\alpha }}{v}_k({\overline{z}}^k,{\overline{t}}^k)+{(-{\Delta }_g)}^s{v}_k({\overline{z}}^k,{\overline{t}}^k)\hfill \\ \hfill =& {\displaystyle {\partial }_t^{\alpha }}{w}_{{\Lambda }_k}({\overline{z}}^k,{\overline{t}}^k)+{(-{\Delta }_g)}^s{u}_{{\Lambda }_k}({\overline{z}}^k,{\overline{t}}^k)-{(-{\Delta }_g)}^{s}u({\overline{z}}^k,{\overline{t}}^k)-\sigma {\displaystyle m_k^2}{\displaystyle {\partial }_t^{\alpha }}{\eta }_k({\overline{z}}^k,{\overline{t}}^k)\hfill \\ \hfill & +{(-{\Delta }_g)}^s{v}_k({\overline{z}}^k,{\overline{t}}^k)-[{(-{\Delta }_g)}^s{u}_{{\Lambda }_k}({\overline{z}}^k,{\overline{t}}^k)-{(-{\Delta }_g)}^{s}u({\overline{z}}^k,{\overline{t}}^k)]\hfill \\ \hfill \ge & \left(\frac{q{\displaystyle {\xi }_1^{q-1}}\left({\overline{z}}^k,{\overline{t}}^k\right)}{{\displaystyle {\xi }_1^q}\left({\overline{z}}^k,{\overline{t}}^k\right)+1}+\frac{p{\displaystyle {\xi }_2^{p-1}}\left({\overline{z}}^k,{\overline{t}}^k\right)}{{\displaystyle {\xi }_2^p}\left({\overline{z}}^k,{\overline{t}}^k\right)+1}\right)w_{{\Lambda }_k}({\overline{z}}^k,{\overline{t}}^k)-\frac{C{\displaystyle m_k^2}}{{\displaystyle {\delta }_k^{2s}}}\hfill \\ \hfill -& C\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}\frac{g^{\prime }(\zeta (\vartheta ,{\overline{t}}^k))(w_{{\Lambda }_k}({\overline{z}}^k,{\overline{t}}^k)-w_{{\Lambda }_k}(\vartheta ,{\overline{t}}^k))}{|{\overline{z}}^k{-\vartheta |}^{n+2s}}d\vartheta \hfill \\ \hfill \ge & \left(\frac{q{\displaystyle {\xi }_1^{q-1}}\left({\overline{z}}^k,{\overline{t}}^k\right)}{{\displaystyle {\xi }_1^q}\left({\overline{z}}^k,{\overline{t}}^k\right)+1}+\frac{p{\displaystyle {\xi }_2^{p-1}}\left({\overline{z}}^k,{\overline{t}}^k\right)}{{\displaystyle {\xi }_2^p}\left({\overline{z}}^k,{\overline{t}}^k\right)+1}\right)w_{{\Lambda }_k}({\overline{z}}^k,{\overline{t}}^k)-\frac{C{\displaystyle m_k^2}}{{\displaystyle {\delta }_k^{2s}}}\hfill \\ \hfill -& C{\displaystyle {\int }_{{\Sigma }_{{\Lambda }_k}}^{ }}\frac{g^{\prime }(\zeta (\vartheta ,{\overline{t}}^k))(w_{{\Lambda }_k}({\overline{z}}^k,{\overline{t}}^k)-w_{{\Lambda }_k}(\vartheta ,{\overline{t}}^k))}{|{\overline{z}}^k-{\vartheta }^{{\Lambda }_k}{|}^{n+2s}}d\vartheta -C{\displaystyle {\int }_{{\Sigma }_{{\Lambda }_k}}^{ }}\frac{g^{\prime }(\zeta (\vartheta ,{\overline{t}}^k))(w_{{\Lambda }_k}({\overline{z}}^k,{\overline{t}}^k)+w_{{\Lambda }_k}(\vartheta ,{\overline{t}}^k))}{|{\overline{z}}^k-{\vartheta }^{{\Lambda }_k}{|}^{n+2s}}d\vartheta \hfill \\ \hfill \ge & -C{w}_{{\Lambda }_k}({\overline{z}}^k,{\overline{t}}^k)-\frac{C{\displaystyle m_k^2}}{{\displaystyle {\delta }_k^{2s}}}-C{\displaystyle {\int }_{{\Sigma }_{{\Lambda }_k}}^{ }}\frac{g^{\prime }(\zeta (\vartheta ,{\overline{t}}^k))2w_{{\Lambda }_k}({\overline{z}}^k,{\overline{t}}^k)}{|{\overline{z}}^k-{\vartheta }^{{\Lambda }_k}{|}^{n+2s}}d\vartheta \hfill \\ \hfill \ge & -C{m}_k-\frac{C{\displaystyle m_k^2}}{{\displaystyle {\delta }_k^{2s}}},\hfill \end{array}$$

where

$$g^{\prime }(\zeta (\vartheta ,{\overline{t}}^k))\mathrm{between}\frac{u_{{\Lambda }_k}({\overline{z}}^k,{\overline{t}}^k)-u_{{\Lambda }_k}(\vartheta ,{\overline{t}}^k)}{{\left|{\overline{z}}^k-\vartheta \right|}^s}\mathrm{and}\frac{u({\overline{z}}^k,{\overline{t}}^k)-u(\vartheta ,{\overline{t}}^k)}{{\left|{\overline{z}}^k-\vartheta \right|}^s},$$

and

$${\xi }_1({\overline{z}}^k,{\overline{t}}^k)\mathrm{and}{\xi }_2({\overline{z}}^k,{\overline{t}}^k)\mathrm{fall}\mathrm{in}\mathrm{between}{u}_{{\Lambda }_k}({\overline{z}}^k,{\overline{t}}^k)\mathrm{and}u({\overline{z}}^k,{\overline{t}}^k).$$

In summary, we obtain

$$-\frac{C}{{\displaystyle {\delta }_k^{2s}}}\ge -C{m}_k-\frac{C{\displaystyle m_k^2}}{{\displaystyle {\delta }_k^{2s}}}.$$

The inequality is multiplied by $-{\displaystyle {\delta }_k^{2s}}$ on both sides; then, by Equation (32) and the assumption ${\displaystyle \underset{k\to \infty }{lim}}{\delta }_k=0$, we have

$$0<C\le C{m}_k{\displaystyle {\delta }_k^{2s}}+C{\displaystyle m_k^2}\to 0\mathrm{as}k\to \infty .$$

This is a contradiction. So, ${\delta }_k$ is bounded and ${\displaystyle \underset{k\to \infty }{lim}}{\delta }_k\ne 0$ for sufficiently large k. In addition, since ${\Lambda }_k\to {\Lambda }_0$ when $k\to \infty$, we can infer that there is a subsequence of $\{z^k,t_k\}$ (which we will also denote as $\{z^k,t_k\}$), satisfying $\{z^k,t_k\}\subset {\Sigma }_{{\Lambda }_0}\times \mathbb{R}$ and dist $\{z^k,T_{{\Lambda }_0}\}\ge {\delta }_0>0$. According to Equation (33), we further choose a radius, $r_1:=min\{r_0,{\delta }_0\}$, such that

$$u(z,t)\ge \frac{{\epsilon }_0}{2}>0\mathrm{in}{B}_{r_1}(z^k)\times (t_k-{\displaystyle r_1^{\frac{2s}{\alpha }}},t_k+{\displaystyle r_1^{\frac{2s}{\alpha }}}]\subset {\Omega }_{{\Lambda }_0}\times \mathbb{R}.$$

(34)

Before continuing, let ${\overline{z}}^k=(2{\Lambda }_0,{(z^k)}^{\prime })$, since $\mathrm{dist}(T_{{\Lambda }_0},T_{2{\Lambda }_0})={\Lambda }_0>2r_1$; then, we have $\overline{B_{r_1}(z^k)}\cap B_{2r_1}({\overline{z}}^k)=⌀$. Next, we demonstrate that there exists a positive constant ${\epsilon }_1$ such that

$$u(z,t)\ge {\epsilon }_1>0\mathrm{in}{B}_{r_1}({\overline{z}}^k)\times (t_k-{\displaystyle r_1^{\frac{2s}{\alpha }}},t_k+{\displaystyle r_1^{\frac{2s}{\alpha }}}].$$

(35)

Otherwise, we have

$$u(z,t)<\epsilon \mathrm{in}{B}_{r_1}({\overline{z}}^k)\times (t_k-{\displaystyle r_1^{\frac{2s}{\alpha }}},t_k+{\displaystyle r_1^{\frac{2s}{\alpha }}}]\mathrm{for}\mathrm{any}\epsilon >0,$$

(36)

then, we derive

$$|(ln\left|u^q\left(z,t\right)+1\right|+ln\left|u^p\left(z,t\right)+1\right|)-(ln\left|0^q+1\right|+ln\left|0^p+1\right|)|\le C\left|(u(z,t))-0\right|<C\epsilon .$$

Furthermore, it follows that

$$ln\left|u^q\left(z,t\right)+1\right|+ln\left|u^p\left(z,t\right)+1\right|\ge -C\epsilon \mathrm{in}{B}_{r_1}({\overline{z}}^k)\times (t_k-{\displaystyle r_1^{\frac{2s}{\alpha }}},t_k+{\displaystyle r_1^{\frac{2s}{\alpha }}}]\mathrm{for}\mathrm{any}\epsilon >0.$$

Then, by combining Theorem 5, Equations (26) and (34) and $u(z,t)$ are continuous, and it follows that

$$u(z,t)\ge {\epsilon }_1>0,(z,t)\in B_{\frac{r_1}{2}}({\overline{z}}^k)\times \left(t_k-{(\frac{r_1}{2})}^{\frac{2s}{\alpha }},t_k+{(\frac{r_1}{2})}^{\frac{2s}{\alpha }}\right],$$

which contradicts with Equation (36). So, we obtain Equation (35).

Let ${\widehat{z}}^k=(0,{(z^k)}^{\prime })$. Due to the fact that $u(z,t)=0$ in $({\mathbb{R}}^n\backslash {\displaystyle {\mathbb{R}}_{+}^n})\times \mathbb{R}$ and $u(z,t)$ are continuous, there exists $r_2<\frac{r_1}{2}$, which is irrespective of k, such that

$$u(z,t)\le \frac{{\epsilon }_1}{2},(z,t)\in (B_{r_2}({\widehat{z}}^k)\cap {\displaystyle {\mathbb{R}}_{+}^n})\times (t_k-{\displaystyle r_1^{\frac{2s}{\alpha }}},t_k+{\displaystyle r_1^{\frac{2s}{\alpha }}}].$$

(37)

For any point $z\in B_{r_2}({\widehat{z}}^k)\cap {\displaystyle {\mathbb{R}}_{+}^n}$, $z^{{\Lambda }_0}$ denotes the reflection point of z about $T_{{\Lambda }_0}$, and $z^{{\Lambda }_0}\in B_{r_2}({\overline{z}}^k)\cap {\Sigma }_{2{\Lambda }_0}\subset B_{\frac{r_1}{2}}({\overline{z}}^k)$. Combining Equations (35) and (37), we obtain

$$w_{{\Lambda }_0}(z,t)=u_{{\Lambda }_0}(z,t)-u(z,t)\ge {\epsilon }_1-\frac{{\epsilon }_1}{2}=\frac{{\epsilon }_1}{2}>0,(z,t)\in (B_{r_2}({\widehat{z}}^k)\cap {\displaystyle {\mathbb{R}}_{+}^n})\times (t_k-{\displaystyle r_1^{\frac{2s}{\alpha }}},t_k+{\displaystyle r_1^{\frac{2s}{\alpha }}}].$$

(38)

Now, we mainly show that there is a positive constant ${\epsilon }_2$, such that

$$w_{{\Lambda }_0}(z,t)\ge {\epsilon }_2>0\mathrm{for}(z,t)\in B_{\frac{r_1}{2}}(z^k)\times \left(t_k-{(\frac{r_1}{2})}^{\frac{2s}{\alpha }},t_k+{(\frac{r_1}{2})}^{\frac{2s}{\alpha }}\right].$$

(39)

If not, we obtain

$$w_{{\Lambda }_0}(z,t)<\epsilon \mathrm{for}(z,t)\in B_{\frac{r_1}{2}}(z^k)\times \left(t_k-{(\frac{r_1}{2})}^{\frac{2s}{\alpha }},t_k+{(\frac{r_1}{2})}^{\frac{2s}{\alpha }}\right]\mathrm{for}\mathrm{any}\epsilon >0.$$

(40)

Combining Equation (27) and the definition of ${\Lambda }_0$, it follows that

$$\left\{\begin{array}{c}{\displaystyle {\partial }_t^{\alpha }}{w}_{{\Lambda }_0}(z,t)+{(-{\Delta }_g)}^s{u}_{{\Lambda }_0}(z,t)-{(-{\Delta }_g)}^{s}u(z,t)=\left(\frac{q{\displaystyle {\xi }_1^{q-1}}\left(z,t\right)}{{\displaystyle {\xi }_1^q}\left(z,t\right)+1}+\frac{p{\displaystyle {\xi }_2^{p-1}}\left(z,t\right)}{{\displaystyle {\xi }_2^p}\left(z,t\right)+1}\right)w_{{\Lambda }_0}(z,t),\hfill \\ \begin{array}{cc}& \mathrm{in}{B}_{\frac{r_1}{2}}(z^k)\times \left(t_k-{(\frac{r_1}{2})}^{\frac{2s}{\alpha }},t_k+{(\frac{r_1}{2})}^{\frac{2s}{\alpha }}\right],\hfill \\ w_{{\Lambda }_0}(z,t)\ge 0,\hfill & \mathrm{in}\left({\Sigma }_{{\Lambda }_0}\backslash B_{\frac{r_1}{2}}(z^k)\right)\times \left(t_k-{(\frac{r_1}{2})}^{\frac{2s}{\alpha }},t_k+{(\frac{r_1}{2})}^{\frac{2s}{\alpha }}\right],\hfill \\ w_{{\Lambda }_0}(z,t)\ge 0,\hfill & \mathrm{in}{B}_{\frac{r_1}{2}}(z^k)\times \left(-\infty ,t_k+{(\frac{r_1}{2})}^{\frac{2s}{\alpha }}\right].\hfill \end{array}\end{array}\right.$$

where ${\xi }_1(z,t)$ and ${\xi }_2(z,t)$ fall in-between $u_{{\Lambda }_0}(z,t)$ and $u(z,t)$. Then, by Equation (40) and the boundedness of $\left(\frac{q{\displaystyle {\xi }_1^{q-1}}\left(z,t\right)}{{\displaystyle {\xi }_1^q}\left(z,t\right)+1}+\frac{p{\displaystyle {\xi }_2^{p-1}}\left(z,t\right)}{{\displaystyle {\xi }_2^p}\left(z,t\right)+1}\right)$, we have

$$\left(\frac{q{\displaystyle {\xi }_1^{q-1}}\left(z,t\right)}{{\displaystyle {\xi }_1^q}\left(z,t\right)+1}+\frac{p{\displaystyle {\xi }_2^{p-1}}\left(z,t\right)}{{\displaystyle {\xi }_2^p}\left(z,t\right)+1}\right)w_{{\Lambda }_0}(z,t)>-C\epsilon ,(z,t)\in B_{\frac{r_1}{2}}(z^k)\times \left(t_k-{(\frac{r_1}{2})}^{\frac{2s}{\alpha }},t_k+{(\frac{r_1}{2})}^{\frac{2s}{\alpha }}\right]$$

for any $\epsilon >0$. Hence, by Equation (38), Theorem 6 and $u(z,t)$ are continuous, and we obtain

$$w_{{\Lambda }_0}(z,t)\ge {\epsilon }_2>0,(z,t)\in B_{\frac{r_1}{4}}(z^k)\times \left(t_k-{(\frac{r_1}{4})}^{\frac{2s}{\alpha }},t_k+{(\frac{r_1}{4})}^{\frac{2s}{\alpha }}\right]$$

for some positive constant ${\epsilon }_2$, which contradicts with Equation (40). So, we obtain Equation (39). Moreover, by utilizing the continuity of $w_{\Lambda }(z,t)$, ${\Lambda }_k\to {\Lambda }_0$ when $k\to \infty$ and Equation (39), we can ultimately derive

$$w_{{\Lambda }_k}(z,t)\ge \frac{{\epsilon }_2}{2}>0\mathrm{in}{B}_{\frac{r_1}{2}}(z^k)\times \left(t_k-{(\frac{r_1}{2})}^{\frac{2s}{\alpha }},t_k+{(\frac{r_1}{2})}^{\frac{2s}{\alpha }}\right],$$

which means that $w_{{\Lambda }_k}(z^k,t_k)\ge \frac{{\epsilon }_2}{2}>0$ for a large enough k. Therefore, this contradicts the assumption that the sequence $\{(z^k,t_k)\}\subset {\displaystyle {\Sigma }_{{\Lambda }_k}^{-}}\times \mathbb{R}$; then, we must have ${\Lambda }_0=+\infty$.

Step 3: Our most critical step is to prove that the positive solution $u(z,t)$ of Equation (26) is strictly increasing in half space along the $z_1$-direction for any $t\in \mathbb{R}$.

Based on the previous two steps, for any $\Lambda >0$, we derive

$$w_{\Lambda }(z,t)\ge 0\mathrm{in}{\Sigma }_{\Lambda }\times \mathbb{R}.$$

In fact, we just have to prove that

$$w_{\Lambda }(z,t)>0\mathrm{in}{\Omega }_{\Lambda }\times \mathbb{R}\mathrm{for}\mathrm{any}\Lambda >0.$$

(41)

Assume that Equation (41) is invalid; there must be a point, $(z^0,t_0)\in {\Omega }_{{\Lambda }_0}\times \mathbb{R}$, and ${\Lambda }_0>0$ such that

$$w_{{\Lambda }_0}(z^0,t_0)={\displaystyle \underset{{\Sigma }_{{\Lambda }_0}\times \mathbb{R}}{min}}{w}_{{\Lambda }_0}(z,t)=0.$$

By $w_{{\Lambda }_0}(z,t_0)\ne 0$ in ${\Sigma }_{{\Lambda }_0}$, $u(z,t)=0$ in $(\mathbb{R}\backslash {\displaystyle {\mathbb{R}}_{+}^n})\times \mathbb{R}$, $\frac{1}{|z^0-{\vartheta }^{{\Lambda }_0}{|}^{n+2s}}<\frac{1}{|z^0{-\vartheta |}^{n+2s}}$ in ${\Sigma }_{{\Lambda }_0}$, and the properties of g, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle {\partial }_t^{\alpha }}{w}_{{\Lambda }_0}(z^0,t_0)+{(-{\Delta }_g)}^s{u}_{{\Lambda }_0}(z^0,t_0)-{(-{\Delta }_g)}^{s}u(z^0,t_0)\hfill \\ \hfill =& C_{\alpha }{\displaystyle {\int }_{-\infty }^{t_0}}\frac{w_{{\Lambda }_0}(z^0,t_0)-w_{{\Lambda }_0}(z^0,\tau )}{{(t_0-\tau )}^{1+\alpha }}d\tau \hfill \\ \hfill & +\mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}\left[g\left(\frac{u_{{\Lambda }_0}(z^0,t_0)-u_{{\Lambda }_0}(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)-g\left(\frac{u(z^0,t_0)-u(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\right)\right]\frac{1}{|z^0{-\vartheta |}^{n+s}}d\vartheta \hfill \\ \hfill \le & \mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\mathbb{R}}^n}^{ }}\frac{g^{\prime }(\vartheta ,t_0)(w_{{\Lambda }_0}(z^0,t_0)-w_{{\Lambda }_0}(\vartheta ,t_0))}{{\left|z^0-\vartheta \right|}^{n+2s}}d\vartheta \hfill \\ \hfill \le & \mathrm{P}.\mathrm{V}.{\displaystyle {\int }_{{\Sigma }_{{\Lambda }_0}}^{ }}{w}_{{\Lambda }_0}(\vartheta ,t_0)g^{\prime }(\vartheta ,t_0)\left(\frac{1}{|z^0-{\vartheta }^{{\Lambda }_0}{|}^{n+2s}}-\frac{1}{|z^0{-\vartheta |}^{n+2s}}\right)d\vartheta \hfill \\ \hfill <& 0,\hfill \end{array}$$

where

$$g^{\prime }(\vartheta ,t_0)\mathrm{between}\frac{u_{{\Lambda }_0}(z^0,t_0)-u_{{\Lambda }_0}(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}\mathrm{and}\frac{u(z^0,t_0)-u(\vartheta ,t_0)}{|z^0{-\vartheta |}^s}.$$

This contradicts

$${\displaystyle {\partial }_t^{\alpha }}{w}_{{\Lambda }_0}(z^0,t_0)+{(-{\Delta }_g)}^s{u}_{{\Lambda }_0}(z^0,t_0)-{(-{\Delta }_g)}^{s}u(z^0,t_0)=0.$$

Then, Equation (41) is valid.

Finally, for every fixed $t\in \mathbb{R}$, utilizing Equation (41), for any $\overline{z}=({\overline{z}}_1,z^{\prime })$ and $\widehat{z}=({\widehat{z}}_1,z^{\prime })$ in ${\displaystyle {\mathbb{R}}_{+}^n}$, and satisfying ${\overline{z}}_1<{\widehat{z}}_1$, when we select $\Lambda =\frac{{\overline{z}}_1+{\widehat{z}}_1}{2}$, we can conclude that

$$0<w_{\Lambda }(\overline{z},t)=u_{\Lambda }(\overline{z},t)-u(\overline{z},t)=u(\widehat{z},t)-u(\overline{z},t).$$

Therefore, we prove that the positive solution $u(z,t)$ of Equation (26) is strictly increasing in half space along the $z_1$-direction for any $t\in \mathbb{R}$.  □

Example 1.

Consider the following equation:

$$\left\{\begin{array}{cc}{\displaystyle {\partial }_t^{\frac{1}{4}}}u(z,t)+{(-{\Delta }_g)}^{\frac{1}{2}}u(z,t)=ln\left|u^2\left(z,t\right)+1\right|+ln\left|u^3\left(z,t\right)+1\right|,\hfill & \mathrm{in}{\displaystyle {\mathbb{R}}_{+}^2}\times \mathbb{R},\hfill \\ u(z,t)=0,\hfill & \mathrm{in}\left({\mathbb{R}}^2\backslash {\displaystyle {\mathbb{R}}_{+}^2}\right)\times \mathbb{R}.\hfill \end{array}\right.$$

(42)

If $u(z,t)\le C(1+|z{|}^{\frac{1}{2}})$ is a positive solution to Equation (42), then according to Theorem 7, the positive solution $u(z,t)$ of Equation (42) is strictly increasing in ${\displaystyle {\mathbb{R}}_{+}^2}$ along the $z_1$-direction for any $t\in \mathbb{R}$.

5. Conclusions

In this paper, we study the double index logarithmic nonlinear fractional $g\text{-}$Laplacian parabolic equations with the Marchaud fractional time derivatives ${\displaystyle {\partial }_t^{\alpha }}$ by using the direct moving plane method. We successfully overcome the difficulties caused by the double non-locality of space-time and the nonlinearity of the fractional $g\text{-}$Laplacian. The results of this paper provide an important tool and method for the study of qualitative properties of solutions, especially for the unbounded solutions of fractional elliptic and parabolic problems. In the future work, we will continue to deeply study other properties and numerical simulations of this class of equations and explore its real-world applications.