Hopf's lemma, asymptotic radial symmetry, and monotonicity of solutions to the logarithmic Laplacian parabolic system

Guotao Wang; Jing Wang

doi:10.1515/anona-2025-0134

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Hopf's lemma, asymptotic radial symmetry, and monotonicity of solutions to the logarithmic Laplacian parabolic system

Guotao Wang and Jing Wang

Published/Copyright: November 26, 2025

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From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

In this article, we extend the asymptotic method of moving planes to the following logarithmic Laplacian parabolic system:

∂ z ∂ t ( η , t ) + ( − △ ) ℒ z ( η , t ) = f ( t , v ( η , t ) ) , ( η , t ) ∈ B 1 ( 0 ) × [ 0 , ∞ ) , ∂ v ∂ t ( η , t ) + ( − △ ) ℒ v ( η , t ) = g ( t , z ( η , t ) ) , ( η , t ) ∈ B 1 ( 0 ) × [ 0 , ∞ ) , z ( η , t ) = 0 , v ( η , t ) = 0 , ( η , t ) ∈ B 1 c ( 0 ) × [ 0 , ∞ ) ,

where B 1 ( 0 ) denotes the unit ball in R N . We first establish a maximum principle, an asymptotic narrow region principle, an asymptotic maximum principle, and an asymptotic strong maximum principle for this system. By applying these principles together with the asymptotic method of moving planes, we then prove the asymptotic radial symmetry and monotonicity of positive solutions to the logarithmic Laplacian parabolic system. Finally, we prove a Hopf’s lemma for the parabolic system involving the logarithmic Laplacian ( − Δ ) ℒ , as well as an asymptotic Hopf’s lemma applicable to antisymmetric functions.

Keywords: logarithmic Laplacian; parabolic system; asymptotic radial symmetry; moving planes; Hopf’s lemma

MSC 2020: 35K55; 35B06; 35B50

1 Introduction

This work is devoted to study the Hopf’s lemma, asymptotic radial symmetry and monotonicity of the positive solutions to the following logarithmic Laplacian parabolic system within the unit ball B 1 ( 0 ) :

(1.1) ∂ z ∂ t ( η , t ) + ( − △ ) ℒ z ( η , t ) = f ( t , v ( η , t ) ) , ( η , t ) ∈ B 1 ( 0 ) × [ 0 , ∞ ) , ∂ v ∂ t ( η , t ) + ( − △ ) ℒ v ( η , t ) = g ( t , z ( η , t ) ) , ( η , t ) ∈ B 1 ( 0 ) × [ 0 , ∞ ) , z ( η , t ) = 0 , v ( η , t ) = 0 , ( η , t ) ∈ B 1 c ( 0 ) × [ 0 , ∞ ) ,

where ( − △ ) ℒ , which refers to the logarithmic Laplacian, is given by the following expression for each t > 0 .

Definition 1.1

If z ( ⋅ , t ) ∈ L 0 1 ( R N ) is Dini continuous with respect to η ∈ R N , Ω ⊂ R N is an open subset. For η ∈ Ω , the logarithmic Laplacian ( − △ ) ℒ has the following integral representation:

( − Δ ) ℒ z ( η , t ) = C N ∫ Ω z ( η , t ) − z ( ξ , t ) ∣ η − ξ ∣ N d ξ − C N ∫ R N \ Ω z ( ξ , t ) ∣ η − ξ ∣ N d ξ + [ h Ω ( η ) + μ N ] z ( η , t ) ,

where

h Ω ( η ) = C N ∫ B 1 ( η ) \ Ω 1 ∣ η − ξ ∣ N d ξ − C N ∫ Ω \ B 1 ( η ) 1 ∣ η − ξ ∣ N d ξ ,

C N = π − N 2 Γ N 2 , μ N = 2 log 2 + κ N 2 − β , Γ is the Gamma function, κ = Γ ′ Γ is the Digamma function, and β = − Γ ′ ( 1 ) is the Euler-Mascheroni constant.

As a novel class of nonlocal operators, the logarithmic Laplacian was first introduced by Chen and Weth [14] in 2019. They systematically studied its origin, derived its fundamental properties, and established a functional analytic framework for the associated Dirichlet problem in bounded domains. In particular, they characterized the asymptotic behavior of the principal Dirichlet eigenvalue and eigenfunctions of the fractional Laplacian ( − Δ ) s as s → 0 . Notably, the logarithmic Laplacian ( − Δ ) ℒ can be formally defined by

( − Δ ) ℒ ≔ d d s s = 0 ( − Δ ) s ,

that is, it corresponds to the formal derivative of the fractional Laplacian with respect to the order s at s = 0 . Thus, it may be regarded as a limiting case of the fractional Laplacian and belongs to the broader class of nonlocal pseudo-differential operators. The fractional Laplacian, as a prototypical nonlocal operator, has found extensive applications in modeling diverse physical phenomena, including anomalous diffusion, spectral graph theory, quasi-geostrophic flows, minimal surfaces, water waves, image processing, and the relativistic quantum mechanics of astrophysical objects (refer [1,11,19,20,34] and their cited references). The mathematical theory concerning equations and systems involving the fractional Laplacian has been intensively developed over the past decades (refer [6,7,12,13,22,23,26,29,30,36,37] and references therein). The nonlocal characteristic of the logarithmic Laplacian significantly increases the analytical difficulty in studying equations and systems governed by this operator. To overcome the challenges arising from nonlocality, Caffarelli and Silvestre [13] proposed the celebrated extension method, which transforms nonlocal problems into local ones in higher dimensions (refer [2,17,18] for details). In addition, Chen et al. developed an integral-form moving plane method (refer [4,9,27,38] for details). However, both methods generally require additional assumptions or rely on the corresponding equivalent integral equation, which may limit their applicability in more general contexts. As shown in [8], these techniques fail to apply to the fully nonlinear nonlocal operator of the form

H β ( z ( η ) ) = C N , β PV ∫ R N G ( z ( η ) − z ( ξ ) ) ∣ η − ξ ∣ N + β d ξ ,

highlighting the necessity of developing methods capable of directly tackling nonlocal operators. In this direction, Chen et al. [7] put forward the direct method of moving planes, which does not rely on auxiliary conditions and has proven to be an effective tool for analyzing nonlocal equations. This method has since been widely applied to establish many qualitative characteristics of solutions to a range of fractional equations and systems, including radial symmetry, monotonicity, among others (refer [5,16,21,31,32,40,44]). For example, Liu and Ma [28] applied the direct method of moving planes to a coupled nonlinear fractional Laplacian system and obtained two radial symmetry results for decaying solutions of the systems. Zhang et al. [42] generalized the direct technique of moving planes to a fractional p -Laplacian system and further demonstrated that its positive solutions exhibit symmetry and monotonicity within the unit ball and the whole space. Xu et al. [41] proved radial symmetry for nonnegative solutions to a fractional Laplacian system involving different negative powers. Subsequently, Qie et al. [35] established key decay principles at infinity and a boundary estimate within the direct technique of moving planes. Moreover, they obtained the radial symmetry and monotonicity results for the fractional p -Laplacian system with negative powers in the whole space. In [3], Chen centered his attention on a Schrödinger-type system that incorporates tempered fractional p -Laplacians, showing that the positive solutions of this system exhibit radial symmetry and monotonicity within the unit ball and the whole space. Luo and Zhang [33] further investigated the radial symmetry of positive solutions for a nonlinear system governed by fully nonlinear nonlocal operators.

It is particularly noteworthy that, for the logarithmic Laplacian addressed in this work, Zhang and Nie [45] developed a version of the direct technique of moving planes specifically adapted to this operator and established radial symmetry and monotonicity results for two classes of elliptic equations. Subsequently, they proved a Hopf’s lemma for the logarithmic Laplacian in [46]. More recently, Zhang et al. [43] extended the direct technique of moving planes to Logarithmic Laplacian elliptic system and then analyzed the symmetry characteristics of their solutions. However, as far as our knowledge goes, no outcomes have been presented concerning the solution properties of the logarithmic Laplacian parabolic system to date. To fill this gap, the present study overcomes the challenges posed by the coupling nature of the logarithmic Laplacian parabolic system (1.1). This work establishes a series of critical maximum principles and a narrow region principle, while developing an asymptotic method of moving planes specifically tailored to this system. Through these efforts, the asymptotic radial symmetry and monotonicity of positive solutions to system (1.1) are derived, with Hopf’s lemma further established; this aims to advance theoretical insights into this emerging research area.

For the convenience, we make the following notations.

Notations. Let η 1 be any given direction in R N . We define the moving plane T α perpendicular to the η 1 -axis as

T α ≔ { η = ( η 1 , η ′ ) ∈ R N ∣ η 1 = α for α ∈ R } ,

where

η ′ ≔ ( η 2 , … , η n ) ∈ R N − 1 .

We denote by

Σ α ≔ { η ∈ R N ∣ η 1 < α }

the region to the left of T α . Let η α be the reflection of η with respect to T α , meaning

η α ≔ ( 2 α − η 1 , η 2 , … , η n ) .

Let z ( η , t ) , v ( η , t ) be a pair of solutions to system (1.1) and

z α ( η , t ) ≔ z ( η α , t ) , v α ( η , t ) ≔ v ( η α , t ) .

Define

Z α ( η , t ) ≔ z α ( η , t ) − z ( η , t ) , V α ( η , t ) ≔ v α ( η , t ) − v ( η , t ) .

Then, they are anti-symmetric about the hyperplane T α , meaning

Z α ( η 1 , η 2 , … , η n , t ) = − Z α ( 2 α − η 1 , η 2 , … , η n , t ) , V α ( η 1 , η 2 , … , η n , t ) = − V α ( 2 α − η 1 , η 2 , … , η n , t ) .

We define the ϖ -limit sets of z and v as follows:

ϖ ( z ) ≔ { φ ∣ φ = lim z ( ⋅ , t i ) for some t i → ∞ } , ϖ ( v ) ≔ { ς ∣ ς = lim v ( ⋅ , t i ) for some t i → ∞ } .

For any φ ( η ) ∈ ϖ ( z ) and ς ( η ) ∈ ϖ ( v ) , denote

ϕ α ( η ) = φ ( η α ) − φ ( η ) = φ α ( η ) − φ ( η ) , ζ α ( η ) = ς ( η α ) − ς ( η ) = ς α ( η ) − ς ( η ) ,

where ϕ α ( η ) and ζ α ( η ) are the ϖ -limit sets of Z α ( η , t ) and V α ( η , t ) , respectively.

2 Preliminaries

This section is devoted to establishing several essential maximum principles, including maximum principle (Theorem 2.1), asymptotic narrow region principle for antisymmetric functions (Theorem 2.2), asymptotic maximum principle for antisymmetric functions (Theorem 2.3), and asymptotic strong maximum principle for antisymmetric functions (Theorem 2.4). These theorems serve as pivotal tools in Section 3.

For the subsequent discussion, we present the following assumptions:

Ω ⊂ R N is a bounded Lipschitz domain.
h Ω + μ N ≥ 0 .
z ( η , t ) , v ( η , t ) ∈ L 0 1 ( R N ) × C 1 [ 0 , ∞ ) are continuous with respect to η on Ω ¯ × [ 0 , + ∞ ) as well as Dini continuous with respect to η on Ω × [ 0 , + ∞ ) .
Z α ( η , t ) , V α ( η , t ) ∈ L 0 1 ( R N ) × C 1 [ 0 , ∞ ) are continuous with respect to η on Ω ¯ × [ 0 , ∞ ) , Dini continuous with respect to η on Ω × [ 0 , ∞ ) and h Σ α + μ N ≥ 0 .

Theorem 2.1

Suppose ( A 1 ) , ( A 2 ) , ( A 3 ) hold and z ( η , t ) , v ( η , t ) satisfy

(2.1) ∂ z ∂ t ( η , t ) + ( − Δ ) ℒ z ( η , t ) ≥ c 1 ( η , t ) v ( η , t ) , ( η , t ) ∈ Ω × [ 0 , + ∞ ) , ∂ v ∂ t ( η , t ) + ( − Δ ) ℒ v ( η , t ) ≥ c 2 ( η , t ) z ( η , t ) , ( η , t ) ∈ Ω × [ 0 , + ∞ ) , z ( η , t ) ≥ 0 , v ( η , t ) ≥ 0 , ( η , t ) ∈ ( R N \ Ω ) × ( 0 , + ∞ ) , z ( η , 0 ) ≥ 0 , v ( η , 0 ) ≥ 0 , η ∈ Ω ,

where 0 < c k ( η , t ) < h Ω ( η , t ) + μ N ( k = 1 , 2 ) . Then,

z ( η , t ) ≥ 0 , v ( η , t ) ≥ 0 , ∀ T > 0 , ( η , t ) ∈ Ω × [ 0 , T ] .

Proof

For some T > 0 , the following inequality must hold

z ( η , t ) ≥ inf Ω z ( η , 0 ) ≥ 0 , v ( η , t ) ≥ inf Ω v ( η , 0 ) ≥ 0 , ( η , t ) ∈ Ω × [ 0 , T ] .

Otherwise, assuming no loss of generality, one can find a point ( η 0 , t 0 ) ∈ Ω × ( 0 , T ] such that

z ( η 0 , t 0 ) = min Ω × [ 0 , T ] z ( η , t ) < 0 ,

then

∂ z ∂ t ( η 0 , t 0 ) ≤ 0 .

According to the first inequality of system (2.1), we can obtain

(2.2) 0 ≤ ∂ z ∂ t ( η 0 , t 0 ) + ( − Δ ) ℒ z ( η 0 , t 0 ) − c 1 ( η 0 , t 0 ) v ( η 0 , t 0 ) = ∂ z ∂ t ( η 0 , t 0 ) + C N ∫ Ω z ( η 0 , t 0 ) − z ( ξ , t 0 ) ∣ η 0 − ξ ∣ N d ξ − C N ∫ R N \ Ω z ( ξ , t 0 ) ∣ η 0 − ξ ∣ N d ξ + [ h Ω ( η 0 , t 0 ) + μ N ] z ( η 0 , t 0 ) − c 1 ( η 0 , t 0 ) v ( η 0 , t 0 ) < [ h Ω ( η 0 , t 0 ) + μ N ] z ( η 0 , t 0 ) − c 1 ( η 0 , t 0 ) v ( η 0 , t 0 ) = K z ( η 0 , t 0 ) − c 1 ( η 0 , t 0 ) v ( η 0 , t 0 ) ,

where [ h Ω ( η 0 , t 0 ) + μ N ] is denoted as K . This inequality requires that v ( η 0 , t 0 ) < 0 , which implies that there exists a point ( η 1 , t 1 ) ∈ Ω × [ 0 , T ] such that

v ( η 1 , t 1 ) = min Ω × [ 0 , T ] v ( η , t ) < 0

and

∂ v ∂ t ( η 1 , t 1 ) ≤ 0 .

Combining the second inequality of systems (2.1) and (2.2), we obtain

0 ≤ ∂ v ∂ t ( η 1 , t 1 ) + ( − Δ ) ℒ v ( η 1 , t 1 ) − c 2 ( η 1 , t 1 ) z ( η 1 , t 1 ) = ∂ v ∂ t ( η 1 , t 1 ) + C N ∫ Ω v ( η 1 , t 1 ) − v ( ξ , t 1 ) ∣ η 1 − ξ ∣ N d ξ − C N ∫ R N \ Ω v ( ξ , t 1 ) ∣ η 1 − ξ ∣ N d ξ + [ h Ω ( η 1 , t 1 ) + μ N ] v ( η 1 , t 1 ) − c 2 ( η 1 , t 1 ) z ( η 1 , t 1 ) < [ h Ω ( η 1 , t 1 ) + μ N ] v ( η 1 , t 1 ) − c 2 ( η 1 , t 1 ) z ( η 1 , t 1 ) = L v ( η 1 , t 1 ) − c 2 ( η 1 , t 1 ) z ( η 1 , t 1 ) ≤ L v ( η 1 , t 1 ) − c 2 ( η 1 , t 1 ) z ( η 0 , t 0 ) < L v ( η 1 , t 1 ) − 1 K c 1 ( η 0 , t 0 ) c 2 ( η 1 , t 1 ) v ( η 0 , t 0 ) ≤ L v ( η 1 , t 1 ) − 1 K c 1 ( η 0 , t 0 ) c 2 ( η 1 , t 1 ) v ( η 1 , t 1 ) = L 1 − 1 L K c 1 ( η 0 , t 0 ) c 2 ( η 1 , t 1 ) v ( η 1 , t 1 ) < 0 ,

where [ h Ω ( η 1 , t 1 ) + μ N ] is denoted as L . Consequently, we derive a contradiction, thereby Theorem 2.1 is proven.□

Theorem 2.2

Suppose ( A 1 ) , ( A 4 ) hold and Ω ⊂ { η ∣ α − ι < η 1 < α } ⊂ Σ α is a narrow region with small ι . For large enough t ¯ , Z α ( η , t ) , V α ( η , t ) satisfy

(2.3) ∂ Z α ∂ t ( η , t ) + ( − △ ) ℒ Z α ( η , t ) = c 1 α ( η , t ) V α ( η , t ) , ( η , t ) ∈ Ω × [ t ¯ , ∞ ) , ∂ V α ∂ t ( η , t ) + ( − △ ) ℒ V α ( η , t ) = c 2 α ( η , t ) Z α ( η , t ) , ( η , t ) ∈ Ω × [ t ¯ , ∞ ) , Z α ( η , t ) ≥ 0 , V α ( η , t ) ≥ 0 , ( η , t ) ∈ ( Σ α \ Ω ) × [ t ¯ , ∞ ) , Z α ( η , t ) = − Z α ( η α , t ) , V α ( η , t ) = − V α ( η α , t ) , ( η , t ) ∈ Ω × [ t ¯ , ∞ ) ,

where c k α ( η , t ) ( k = 1 , 2 ) are bounded from above. Then, for small enough ι ,

lim t → ∞ ¯ Z α ( η , t ) ≥ 0 , lim t → ∞ ¯ V α ( η , t ) ≥ 0 , ∀ η ∈ Ω .

Proof

For any T > t ¯ , we claim that the following inequality must hold:

(2.4) Z α ( η , t ) ≥ min { 0 , inf Ω Z α ( η , t ¯ ) } , ( η , t ) ∈ Ω × [ t ¯ , T ] , V α ( η , t ) ≥ min { 0 , inf Ω V α ( η , t ¯ ) } , ( η , t ) ∈ Ω × [ t ¯ , T ] .

If (2.4) does not hold, assuming no loss of generality, one can find a point ( η 0 , t 0 ) ∈ Ω × ( t ¯ , T ] such that

Z α ( η 0 , t 0 ) = min Σ α × ( t ¯ , T ] Z α ( η , t ) < min { 0 , inf Ω Z α ( η , t ¯ ) } .

Then,

(2.5) ∂ Z α ∂ t ( η 0 , t 0 ) ≤ 0

and

( − Δ ) ℒ Z α ( η 0 , t 0 ) = C N ∫ Σ α Z α ( η 0 , t 0 ) − Z α ( ξ , t 0 ) ∣ η 0 − ξ ∣ N d ξ − C N ∫ R N \ Σ α Z α ( ξ , t 0 ) ∣ η 0 − ξ ∣ N d ξ + [ h Σ α ( η 0 , t 0 ) + μ N ] Z α ( η 0 , t 0 ) = C N ∫ Σ α Z α ( η 0 , t 0 ) − Z α ( ξ , t 0 ) ∣ η 0 − ξ ∣ N d ξ − C N ∫ Σ α Z α ( ξ α , t 0 ) ∣ η 0 − ξ α ∣ N d ξ + [ h Σ α ( η 0 , t 0 ) + μ N ] Z α ( η 0 , t 0 ) ≤ C N ∫ Σ α Z α ( η 0 , t 0 ) − Z α ( ξ , t 0 ) ∣ η 0 − ξ α ∣ N d ξ + C N ∫ Σ α Z α ( ξ , t 0 ) ∣ η 0 − ξ α ∣ N d ξ + [ h Σ α ( η 0 , t 0 ) + μ N ] Z α ( η 0 , t 0 ) = C N ∫ Σ α Z α ( η 0 , t 0 ) ∣ η 0 − ξ α ∣ N d ξ + [ h Σ α ( η 0 , t 0 ) + μ N ] Z α ( η 0 , t 0 ) ≤ C N ∫ Σ α Z α ( η 0 , t 0 ) ∣ η 0 − ξ α ∣ N d ξ .

Let

H = ξ = ( ξ 1 , ξ ′ ) ∈ Σ α ∣ ι < ξ 1 − ( η 0 ) 1 < 2 ι , ∣ ξ ′ − η ′ ∣ < ι ,

then

∫ Σ α 1 ∣ η 0 − ξ α ∣ N d ξ ≥ ∫ H 1 ∣ η 0 − ξ α ∣ N d ξ ≥ c ( ln 1 − ln ι ) → + ∞ ( ι → 0 ) .

Therefore,

( − Δ ) ℒ Z α ( η 0 , t 0 ) → − ∞ .

Combining the boundedness of c 1 α ( η , t ) , by choosing ι sufficiently small, we have

∂ Z α ∂ t ( η 0 , t 0 ) = − ( − △ ) ℒ Z α ( η 0 , t 0 ) + c 1 λ ( η 0 , t 0 ) V α ( η 0 , t 0 ) → + ∞ ,

which is contradictory to (2.5). Therefore, (2.4) holds. Let t → ∞ , we derive that

lim t → ∞ ¯ Z α ( η , t ) ≥ 0 , lim t → ∞ ¯ V α ( η , t ) ≥ 0 , ∀ η ∈ Ω .

Theorem 2.2 is now fully demonstrated.□

Theorem 2.3

Suppose ( A 1 ) , ( A 4 ) hold and Z α ( η , t ) , V α ( η , t ) satisfy the following system:

(2.6) ∂ Z α ∂ t ( η , t ) + ( − Δ ) ℒ Z α ( η , t ) ≥ c 1 α ( η , t ) V α ( η , t ) , ( η , t ) ∈ Ω × [ 0 , + ∞ ) , ∂ V α ∂ t ( η , t ) + ( − Δ ) ℒ V α ( η , t ) ≥ c 2 α ( η , t ) Z α ( η , t ) , ( η , t ) ∈ Ω × [ 0 , + ∞ ) , Z α ( η , t ) = − Z α ( η α , t ) , V α ( η , t ) = − V α ( η α , t ) , ( η , t ) ∈ Σ α × [ 0 , + ∞ ) , Z α ( η , t ) ≥ 0 , V α ( η , t ) ≥ 0 , ( η , t ) ∈ ( Σ α \ Ω ) × [ 0 , + ∞ ) , Z α ( η , 0 ) ≥ 0 , V α ( η , 0 ) ≥ 0 , η ∈ Ω ,

where 0 < c k α ( η , t ) < h Σ α + μ N ( k = 1 , 2 ) . Then,

Z α ( η , t ) ≥ 0 , V α ( η , t ) ≥ 0 , ∀ T > 0 , ( η , t ) ∈ Ω × [ 0 , T ] .

Proof

For some T > 0 , the following inequality must hold:

Z α ( η , t ) ≥ inf Ω Z α ( η , 0 ) ≥ 0 , V α ( η , t ) ≥ inf Ω V α ( η , 0 ) ≥ 0 , ( η , t ) ∈ Ω × [ 0 , T ] .

Otherwise, assuming no loss of generality, one can find a point ( η 0 , t 0 ) ∈ Ω × ( 0 , T ] such that

Z α ( η 0 , t 0 ) = min Σ α × [ 0 , T ] Z α ( η , t ) < 0 ,

then

∂ Z α ∂ t ( η 0 , t 0 ) ≤ 0 .

According to the first equation of system (2.6), we obtain

(2.7) 0 ≤ ∂ Z α ∂ t ( η 0 , t 0 ) + ( − Δ ) ℒ Z α ( η 0 , t 0 ) − c 1 α ( η 0 , t 0 ) V α ( η 0 , t 0 ) = ∂ Z α ∂ t ( η 0 , t 0 ) + C N ∫ Σ α Z α ( η 0 , t 0 ) − Z α ( ξ , t 0 ) ∣ η 0 − ξ ∣ N d ξ − C N ∫ R N \ Σ α Z α ( ξ , t 0 ) ∣ η 0 − ξ ∣ N d ξ + [ h Σ α ( η 0 , t 0 ) + μ N ] Z α ( η 0 , t 0 ) − c 1 α ( η 0 , t 0 ) V α ( η 0 , t 0 ) < [ h Σ α ( η 0 , t 0 ) + μ N ] Z α ( η 0 , t 0 ) − c 1 α ( η 0 , t 0 ) V α ( η 0 , t 0 ) = K ˜ Z α ( η 0 , t 0 ) − c 1 α ( η 0 , t 0 ) V α ( η 0 , t 0 ) ,

where [ h Σ α ( η 0 , t 0 ) + μ N ] is denoted as K ˜ .

This inequality requires that V α ( η 0 , t 0 ) < 0 , which implies that one can find a point ( η 1 , t 1 ) ∈ Ω × [ 0 , T ] such that

V α ( η 1 , t 1 ) = min Σ α × [ 0 , T ] V α ( η , t ) < 0

and

∂ V α ∂ t ( η 1 , t 1 ) ≤ 0 .

Combining the second inequality of systems (2.6) and (2.7), we have

0 ≤ ∂ V α ∂ t ( η 1 , t 1 ) + ( − Δ ) ℒ V α ( η 1 , t 1 ) − c 2 α ( η 1 , t 1 ) Z α ( η 1 , t 1 ) = ∂ V α ∂ t ( η 1 , t 1 ) + C N ∫ Σ α V α ( η 1 , t 1 ) − V α ( ξ , t 1 ) ∣ η 1 − ξ ∣ N d ξ − C N ∫ R N \ Σ α V α ( ξ , t 1 ) ∣ η 1 − ξ ∣ N d ξ + [ h Σ α ( η 1 , t 1 ) + μ N ] V α ( η 1 , t 1 ) − c 2 α ( η 1 , t 1 ) Z α ( η 1 , t 1 ) < [ h Σ α ( η 1 , t 1 ) + μ N ] V α ( η 1 , t 1 ) − c 2 α ( η 1 , t 1 ) Z α ( η 1 , t 1 ) = L ˜ V α ( η 1 , t 1 ) − c 2 α ( η 1 , t 1 ) Z α ( η 1 , t 1 ) ≤ L ˜ V α ( η 1 , t 1 ) − c 2 α ( η 1 , t 1 ) Z α ( η 0 , t 0 ) < L ˜ V α ( η 1 , t 1 ) − 1 K ˜ c 1 α ( η 0 , t 0 ) c 2 α ( η 1 , t 1 ) V α ( η 0 , t 0 ) ≤ L ˜ V α ( η 1 , t 1 ) − 1 K ˜ c 1 α ( η 0 , t 0 ) c 2 α ( η 1 , t 1 ) V α ( η 1 , t 1 ) = L ˜ 1 − 1 L ˜ K ˜ c 1 α ( η 0 , t 0 ) c 2 α ( η 1 , t 1 ) V α ( η 1 , t 1 ) < 0 ,

where [ h Σ α ( η 1 , t 1 ) + μ N ] is denoted as L ˜ . Consequently, we derive a contradiction, thereby Theorem 2.3 is proven.□

Theorem 2.4

For sufficiently large t ¯ , let Z α ( η , t ) , V α ( η , t ) ∈ L 0 1 ( R N ) × C 1 [ 0 , ∞ ) be bounded and satisfy

(2.8) ∂ Z α ∂ t ( η , t ) + ( − △ ) ℒ Z α ( η , t ) = c 1 α ( η , t ) V α ( η , t ) , ( η , t ) ∈ Σ α × [ t ¯ , ∞ ) , ∂ V α ∂ t ( η , t ) + ( − △ ) ℒ V α ( η , t ) = c 2 α ( η , t ) Z α ( η , t ) , ( η , t ) ∈ Σ α × [ t ¯ , ∞ ) , Z α ( η , t ) = − Z α ( η α , t ) , V α ( η , t ) = − V α ( η α , t ) , ( η , t ) ∈ Σ α × [ t ¯ , ∞ ) , lim t → ∞ ¯ Z α ( η , t ) ≥ 0 , lim t → ∞ ¯ V α ( η , t ) ≥ 0 , η ∈ Σ α ,

where 0 < c k α ( η , t ) < h Σ α + μ N ( k = 1 , 2 ) . Assume ϕ α ( η ) > 0 , ζ α ( η ) > 0 somewhere in Σ α , then ϕ α ( η ) > 0 , ζ α ( η ) > 0 in Σ α .

Proof

Based on the definition of ϖ ( z ) , ϖ ( v ) , for any φ ∈ ϖ ( z ) , ς ∈ ϖ ( v ) , a sequence { t i } can be found such that

Z α ( η , t i ) → ϕ α ( η ) as t i → ∞ , V α ( η , t i ) → ζ α ( η ) as t i → ∞ .

Define

Z i ( η , t ) = Z α ( η , t + t i − 1 ) , V i ( η , t ) = V α ( η , t + t i − 1 ) .

It is apparent that

∂ Z i ∂ t ( η , t ) + ( − △ ) ℒ Z i ( η , t ) = c 1 i ( η , t ) V i ( η , t ) , ( η , t ) ∈ Σ α × [ t ¯ , ∞ ) , ∂ V i ∂ t ( η , t ) + ( − △ ) ℒ V i ( η , t ) = c 2 i ( η , t ) Z i ( η , t ) , ( η , t ) ∈ Σ α × [ t ¯ , ∞ ) ,

where c 1 i ( η , t ) = c 1 α ( η , t + t i − 1 ) , c 2 i ( η , t ) = c 2 α ( η , t + t i − 1 ) .

Then, we can find sequences Z i ( η , t ) (retained with the same notation Z i ( η , t ) ) and V i ( η , t ) (likewise retained) such that Z i → Z ∞ and V i → V ∞ uniformly on Σ α × [ 1 − ε 0 , 1 + ε 0 ] , where Z ∞ and V ∞ are the respective limit functions. They satisfy

∂ Z i ∂ t ( η , t ) + ( − Δ ) ℒ Z i ( η , t ) → ∂ Z ∞ ∂ t ( η , t ) + ( − Δ ) ℒ Z ∞ ( η , t ) , c 1 i ( η , t ) → c 1 ∞ ( η , t ) , as i → ∞ , ∂ V i ∂ t ( η , t ) + ( − Δ ) ℒ V i ( η , t ) → ∂ V ∞ ∂ t ( η , t ) + ( − Δ ) ℒ V ∞ ( η , t ) , c 2 i ( η , t ) → c 2 ∞ ( η , t ) , as i → ∞ .

Owing to the boundedness of c k α ( k = 1 , 2 ) , the limit functions c k ∞ ( k = 1 , 2 ) are also bounded. Additionally, Z ∞ , V ∞ are continuous functions with respect to η as well as t .

For t = 1 , it follows that

Z α ( η , t i ) = Z i ( η , 1 ) → Z ∞ ( η , 1 ) = ϕ α ( η ) as i → ∞ , V α ( η , t i ) = V i ( η , 1 ) → V ∞ ( η , 1 ) = ζ α ( η ) as i → ∞ .

By (2.8), we can find 0 < ε 0 < 1 such that

(2.9) ∂ Z ∞ ∂ t ( η , t ) + ( − △ ) ℒ Z ∞ ( η , t ) = c 1 ∞ V ∞ ≥ 0 , ( η , t ) ∈ Σ α × [ 1 − ε 0 , 1 + ε 0 ] , ∂ V ∞ ∂ t ( η , t ) + ( − △ ) ℒ V ∞ ( η , t ) = c 2 ∞ Z ∞ ≥ 0 , ( η , t ) ∈ Σ α × [ 1 − ε 0 , 1 + ε 0 ] .

Noting that ϕ α , ζ α are continuous and positive at some point in Σ α , we can find a subset Q ⋐ Σ α such that

(2.10) ϕ α ( η ) > c > 0 , ζ α ( η ) > c > 0 , η ∈ Q .

Moreover, since Z ∞ ( η , t ) is continuous with respect to t , we obtain

(2.11) Z ∞ ( η , t ) > c ⁄ 2 , ( η , t ) ∈ Q × [ 1 − ε 0 , 1 + ε 0 ] .

Given any η ¯ ∈ Σ α \ Q , take ℓ = min { dist ( η ¯ , Q ) , dist ( η ¯ , T α ) } > 0 . Then, the ball B ℓ ( η ¯ ) ⊂ Σ α \ Q , allowing us to construct a subsolution in B ℓ ( η ¯ ) × [ 1 − ε 0 , 1 + ε 0 ] .

Set

Z ̲ ( η , t ) = χ Q ∪ Q α ( η ) Z ∞ ( η , t ) + ε θ ( t ) h ( η ) ,

where Q α denotes the reflection of Q with respect to the hyperplane T α , and

χ Q ( η ) = 1 , η ∈ Q , 0 , η ∈ Q c .

θ ( t ) ∈ C 0 ∞ ( 1 − ε 0 , 1 + ε 0 ) satisfies

θ ( t ) = 1 , t ∈ 1 − ε 0 2 , 1 + ε 0 2 , 0 , t ∉ ( 1 − ε 0 , 1 + ε 0 ) .

And

h ( η ) = ( ℓ 2 − ∣ η − η ¯ ∣ 2 ) + − ( ℓ 2 − ∣ η − η ¯ α ∣ 2 ) + .

Through simple calculation, we can obtain h ( η ¯ ) = ℓ 2 and h ( η α ) = − h ( η ) .

Define ℘ ( η ) = c ( 1 − ∣ η ∣ 2 ) + . By selecting an appropriate c , it can be deduced that

( − Δ ) ℒ ℘ ( η ) = 1 , η ∈ B 1 ( 0 ) , γ ( η ) = 0 , η ∈ B 1 c ( 0 ) .

Define ℘ ℓ ( η ) = c ( ℓ 2 − ∣ η − η ¯ ∣ 2 ) + and ℘ ℓ α ( η ) = c ( ℓ 2 − ∣ x − η ¯ α ∣ 2 ) + .

We can also obtain

(2.12) ( − Δ ) ℒ ℘ ℓ ( η ) = 1 , η ∈ B ℓ ( η ¯ ) , ℘ ℓ ( η ) = 0 , η ∈ B ℓ c ( η ¯ ) ,

and

(2.13) ( − Δ ) ℒ ℘ ℓ α ( η ) = 1 , η ∈ B ℓ ( η ¯ α ) , ℘ ℓ α ( η ) = 0 , η ∈ B ℓ c ( η ¯ α ) .

Combining (2.12) with (2.13), we obtain for η ∈ B ℓ ( η ¯ ) ,

(2.14) ∣ ( − Δ ) ℒ h ( η ) ∣ = 1 c ∣ ( − Δ ) ℒ ℘ ℓ ( η ) − ( − Δ ) ℒ ℘ ℓ α ( η ) ∣ = 1 c 1 − C N ∫ B ℓ ( η ¯ ) ℘ ℓ α ( η ) − ℘ ℓ α ( ξ ) ∣ η − ξ ∣ N d ξ − C N ∫ R N \ B ℓ ( η ¯ ) ℘ ℓ α ( ξ ) ∣ η − ξ ∣ N d ξ + [ h B ℓ ( η ¯ ) ( η ) + μ N ] ℘ ℓ α ( η ) = 1 c 1 − C N ∫ B ℓ ( η ¯ ) c ( ℓ 2 − ∣ η − η ¯ α ∣ 2 ) + − c ( ℓ 2 − ∣ ξ − η ¯ α ∣ 2 ) + ∣ η − ξ ∣ N d ξ − C N ∫ B ℓ ( η ¯ α ) c ( ℓ 2 − ∣ ξ − η ¯ α ∣ 2 ) + ∣ η − ξ ∣ N d ξ + [ h B ℓ ( η ¯ ) ( η ) + μ N ] c ( ℓ 2 − ∣ η − η ¯ α ∣ 2 ) + ≤ C 0 ,

where C 0 > 0 .

From the definition of ( − Δ ) ℒ along with (2.11), for each t ∈ [ 1 − ε 0 , 1 + ε 0 ] fixed and any η ∈ B ℓ ( η ¯ ) , we obtain

(2.15) ( − Δ ) ℒ ( χ Q ∪ Q α ( η ) Z ∞ ( η , t ) ) = C N ∫ B ℓ ( η ¯ ) χ Q ∪ Q α ( η ) Z ∞ ( η , t ) − χ Q ∪ Q α ( ξ ) Z ∞ ( ξ , t ) ∣ η − ξ ∣ N d ξ − C N ∫ R N \ B ℓ ( η ¯ ) χ Q ∪ Q α ( ξ ) Z ∞ ( ξ , t ) ∣ η − ξ ∣ N d ξ + [ h B ℓ ( η ¯ ) ( η ) + μ N ] χ Q ∪ Q α ( η ) Z ∞ ( η , t ) = 0 − C N ∫ Q ∪ Q α Z ∞ ( ξ , t ) ∣ η − ξ ∣ N d ξ + 0 = C N ∫ Q − Z ∞ ( ξ , t ) ∣ η − ξ ∣ N d ξ + C N ∫ Q α − Z ∞ ( ξ α , t ) ∣ η − ξ α ∣ N d ξ = C N ∫ Q − Z ∞ ( ξ , t ) ∣ η − ξ ∣ N d ξ + C N ∫ Q Z ∞ ( ξ , t ) ∣ η − ξ α ∣ N d ξ = C N ∫ Q 1 ∣ η − ξ α ∣ N d ξ − 1 ∣ η − ξ ∣ N Z ∞ ( ξ , t ) d ξ ≤ − C 1 ,

where C 1 > 0 . By (2.14) and (2.15), we obtain

∂ Z ̲ ( η , t ) ∂ t + ( − Δ ) ℒ Z ̲ ( η , t ) = ε θ ′ ( t ) h ( η ) + ( − Δ ) ℒ ( χ Q ∪ Q α Z ∞ ( η , t ) ) + ε θ ( t ) ( − Δ ) ℒ h ( η ) ≤ ε θ ′ ( t ) h ( η ) − C 1 + ε θ ( t ) C 0 , ( η , t ) ∈ B ℓ ( η ¯ ) × [ 1 − ε 0 , 1 + ε 0 ] .

Choosing ε > 0 small enough, we have

(2.16) ∂ Z ̲ ∂ t ( η , t ) + ( − Δ ) ℒ Z ̲ ( η , t ) ≤ 0 , ( η , t ) ∈ B ℓ ( η ¯ ) × [ 1 − ε 0 , 1 + ε 0 ] .

Set Z ( η , t ) = Z ∞ ( η , t ) − Z ̲ ( η , t ) . Obviously, Z ( η , t ) is an antisymmetric function that satisfies Z ( η , t ) = − Z ( η α , t ) . Combining (2.9) and (2.16), we obtain

(2.17) ∂ Z ∂ t ( η , t ) + ( − Δ ) ℒ Z ( η , t ) ≥ 0 , ( η , t ) ∈ B ℓ ( η ¯ ) × [ 1 − ε 0 , 1 + ε 0 ] .

Based on the definition of Z ̲ ( η , t ) , it follows that

(2.18) Z ( η , t ) ≥ 0 in ( Σ α \ B ℓ ( η ¯ ) ) × [ 1 − ε 0 , 1 + ε 0 ]

and

(2.19) Z ( η , 1 − ε 0 ) ≥ 0 in B ℓ ( η ¯ ) .

Similarly, set

V ̲ ( η , t ) = χ Q ∪ Q α ( η ) V ∞ ( η , t ) + ε θ ( t ) h ( η ) , V ( η , t ) = V ∞ ( η , t ) − V ̲ ( η , t ) .

We can also obtain

(2.20) ∂ V ∂ t ( η , t ) + ( − △ ) ℒ V ( η , t ) ≥ 0 , ( η , t ) ∈ B ℓ ( η ¯ ) × [ 1 − ε 0 , 1 + ε 0 ] , V ( η , t ) ≥ 0 , ( η , t ) ∈ ( Σ α \ B ℓ ( η ¯ ) ) × [ 1 − ε 0 , 1 + ε 0 ] , V ( η , 1 − ε 0 ) ≥ 0 , ( η , t ) ∈ B ℓ ( η ¯ ) .

Now, combining (2.17), (2.18), (2.19), (2.20), and applying Theorem 2.3, we have

Z ( η , t ) ≥ 0 , V ( η , t ) ≥ 0 , ( η , t ) ∈ B ℓ ( η ¯ ) × [ 1 − ε 0 , 1 + ε 0 ] .

It implies that

Z ( η , t ) = Z ∞ ( η , t ) − ε h ( η ) θ ( t ) ≥ 0 , ( η , t ) ∈ B ℓ ( η ¯ ) × [ 1 − ε 0 , 1 + ε 0 ] , V ( η , t ) = V ∞ ( η , t ) − ε h ( η ) θ ( t ) ≥ 0 , ( η , t ) ∈ B ℓ ( η ¯ ) × [ 1 − ε 0 , 1 + ε 0 ] .

Particularly,

Z ∞ ( η , 1 ) ≥ ε h ( η ) , η ∈ B ℓ ( η ¯ ) , V ∞ ( η , 1 ) ≥ ε h ( η ) , η ∈ B ℓ ( η ¯ ) .

From h ( η ¯ ) = ℓ 2 , we obtain

(2.21) ϕ α ( η ¯ ) = Z ∞ ( η ¯ , 1 ) ≥ ε ℓ 2 > 0 , ζ α ( η ¯ ) = V ∞ ( η ¯ , 1 ) ≥ ε ℓ 2 > 0 .

Using the arbitrary nature of η ¯ in Σ α \ Q , we can combine (2.10) and (2.21) to deduce that

ϕ α ( η ) > 0 , ζ α ( η ) > 0 , η ∈ Σ α .

This completes the proof of Theorem 2.4.□

3 Asymptotic radial symmetry of solutions in B 1 ( 0 )

This section is devoted to establish the asymptotic radial symmetry of solutions to the following system:

(3.1) ∂ z ∂ t ( η , t ) + ( − △ ) ℒ z ( η , t ) = f ( t , v ( η , t ) ) , ( η , t ) ∈ B 1 ( 0 ) × [ 0 , ∞ ) , ∂ v ∂ t ( η , t ) + ( − △ ) ℒ v ( η , t ) = g ( t , z ( η , t ) ) , ( η , t ) ∈ B 1 ( 0 ) × [ 0 , ∞ ) , z ( η , t ) = 0 , v ( η , t ) = 0 , ( η , t ) ∈ B 1 c ( 0 ) × [ 0 , ∞ ) .

Theorem 3.1

Let z ( η , t ) and v ( η , t ) be a pair of positive, uniformly bounded solution to system (3.1) satisfying ( A 3 ). Suppose that f ( t , v ) , g ( t , z ) ∈ L ∞ ( R + × R ) are uniformly Lipschitz continuous and equicontinuous functions and satisfy

( F ) : f ( t , 0 ) , g ( t , 0 ) ≥ 0 , t ≥ 0 , and 0 < ∂ f ∂ v , ∂ g ∂ z < h Ω + μ N .

Then, for any φ ∈ ϖ ( z ) , ς ∈ ϖ ( v ) , it follows that either φ ( η ) ≡ 0 , ς ( η ) ≡ 0 or they are radially symmetric and strictly decreasing about the origin.

Proof

We start at the origin and take an arbitrary direction as the positive direction of the η 1 -axis. Set

Ω α = Σ α ∩ B 1 ( 0 ) = { η ∈ B 1 ( 0 ) ∣ η 1 < α } ,

then Z α ( η , t ) , V α ( η , t ) satisfy

(3.2) ∂ Z α ∂ t ( η , t ) + ( − △ ) ℒ Z α ( η , t ) = c 1 α ( η , t ) V α ( η , t ) , ( η , t ) ∈ Ω α × [ 0 , ∞ ) , ∂ V α ∂ t ( η , t ) + ( − △ ) ℒ V α ( η , t ) = c 2 α ( η , t ) Z α ( η , t ) , ( η , t ) ∈ Ω α × [ 0 , ∞ ) , Z α ( η , t ) = − Z α ( η α , t ) , V α ( η , t ) = − V α ( η α , t ) , ( η , t ) ∈ Ω α × [ 0 , ∞ ) ,

where 0 < c 1 α ( η , t ) = f ′ v ( t , ξ ( η , t ) ) < h Ω + μ N and 0 < c 2 α ( η , t ) = g ′ z ( t , θ ( η , t ) ) < h Ω + μ N , with ξ ( η , t ) between v ( η , t ) and v α ( η , t ) , θ ( η , t ) between z ( η , t ) and z α ( η , t ) .

We organized the proof in two steps for clarity.

Step 1. We first determine a suitable starting point from which the planes will begin to move. When α > − 1 and sufficiently close to − 1 , the domain Ω α becomes a narrow strip. From the non-negativity of z and v , together with

z ( η , t ) = 0 , v ( η , t ) = 0 , ( η , t ) ∈ B 1 c ( 0 ) × [ 0 , ∞ ) ,

we obtain

Z α ( η , t ) ≥ 0 , V α ( η , t ) ≥ 0 , ( η , t ) ∈ ( Σ α \ Ω α ) × [ 0 , ∞ ) .

In view of (3.2) together with Theorem 2.2, we infer that

(3.3) ϕ α ( η ) ≥ 0 , ζ α ( η ) ≥ 0 , ∀ η ∈ Ω α .

This provides the starting point for the moving planes.

Step 2. Next we proceed to move the hyperplane T α rightward to its limiting position, under the condition that (3.3) holds.

Define

(3.4) α 0 = sup { α ≤ 0 ∣ ϕ μ ( η ) ≥ 0 , ζ μ ( η ) ≥ 0 , ∀ φ ∈ ϖ ( z ) , ς ∈ ϖ ( v ) , η ∈ Ω μ , μ ≤ α } .

Our goal is to prove that α 0 (as defined in (3.4)) must be 0.

Case 1:

First, note that φ ( η ) ≡ 0 if and only if ς ( η ) ≡ 0 , for any φ ∈ ϖ ( z ) , ς ∈ ϖ ( v ) . Suppose φ ( η ) ≡ 0 , then

∂ ϕ α 0 ∂ t ( η , t ) + ( − Δ ) ℒ ϕ α 0 ( η , t ) = ∂ ϕ α 0 ∂ t ( η , t ) + C N ∫ Σ α 0 ϕ α 0 ( η , t ) − ϕ α 0 ( ξ , t ) ∣ η − ξ ∣ N d ξ − C N ∫ R N \ Σ α 0 ϕ α 0 ( ξ , t ) ∣ η − ξ ∣ N d ξ + [ h Σ α 0 ( η , t ) + μ N ] ϕ α 0 ( η , t ) ≡ 0 = c 1 α 0 ( η , t ) ζ α 0 ( η , t ) .

Since c 1 α 0 ( η , t ) > 0 , it follows that ζ α 0 ( η , t ) ≡ 0 , which implies ς ( η ) ≡ 0 . Similarly, we can obtain if ς ( η ) ≡ 0 , then ζ α 0 ( η , t ) ≡ 0 .

Case 2:

Afterwards, we consider the case where φ ( η ) ≢ 0 , ς ( η ) ≢ 0 , for any φ ∈ ϖ ( z ) , ς ∈ ϖ ( v ) . Given α 0 < 0 , the definition implies

ϕ α 0 ( η ) ≥ 0 , ζ α 0 ( η ) ≥ 0 , ∀ η ∈ Ω α 0 .

First, we claim that for any φ ∈ ϖ ( z ) , ς ∈ ϖ ( v ) , one can find a point η φ ∈ Σ α 0 such that

ϕ α 0 ( η φ ) > 0 , ζ α 0 ( η φ ) > 0 .

Assume the contrary. Then, for some φ ¯ ∈ ϖ ( z ) , we must have

ϕ ¯ α 0 ( η ) = φ ¯ α 0 ( η ) − φ ¯ ( η ) ≡ 0 in Σ α 0 ,

or likewise, for some ς ¯ ∈ ϖ ( v ) ,

ζ ¯ α 0 ( η ) = ς ¯ α 0 ( η ) − ς ¯ ( η ) ≡ 0 in Σ α 0 .

Assuming no loss of generality, suppose φ ¯ ∈ ϖ ( z ) satisfies

ϕ ¯ α 0 ( η ) = φ ¯ α 0 ( η ) − φ ¯ ( η ) ≡ 0 in Σ α 0 .

Since z ( η , t ) = 0 , ( η , t ) ∈ B 1 c ( 0 ) × ( 0 , ∞ ) , we obtain φ ¯ ( η ) ≡ 0 in B 1 c ( 0 ) ∩ Σ α 0 . Then, there exists η 0 ∈ B 1 ( 0 ) such that φ ¯ ( η 0 ) = 0 .

For the above φ ¯ ( η ) , the definition of ϖ ( z ) ensures the existence of a sequence { t i } such that z ( η , t i ) → φ ¯ ( η ) as t i → ∞ .

Set

z i ( η , t ) = z ( η , t + t i − 1 ) .

Then, clearly, z ∞ ( η , 1 ) = φ ¯ ( η ) , z ∞ ( η 0 , 1 ) = φ ¯ ( η 0 ) ≡ 0 . Since z ∞ ( η , t ) ≥ 0 , this gives

∂ z ∞ ∂ t ( η 0 , 1 ) ≤ 0

and

( − Δ ) ℒ z ∞ ( η 0 , 1 ) = C N ∫ B 1 ( 0 ) z ∞ ( η 0 , 1 ) − z ∞ ( ξ , 1 ) ∣ η 0 − ξ ∣ N d ξ − C N ∫ R N \ B 1 ( 0 ) z ∞ ( ξ , 1 ) ∣ η 0 − ξ ∣ N d ξ + [ h B 1 ( 0 ) ( η 0 , 1 ) + μ N ] z ∞ ( η 0 , 1 ) < 0 .

The above inequality depends on z ∞ ( ξ , 1 ) ≢ 0 in B 1 ( 0 ) (because φ ¯ ≢ 0 in B 1 ( 0 ) ). According to the above analysis, we obtain

∂ z ∞ ∂ t ( η 0 , 1 ) + ( − △ ) ℒ z ∞ ( η 0 , 1 ) < 0 .

Set

v i ( η , t ) = v ( η , t + t i − 1 ) , f i ( t , v ) = f ( t + t i − 1 , v ) .

Similarly, the definition of ϖ ( v ) also ensures the existence of a sequence { t i } such that v ( η , t i ) → ς ¯ ( η ) as t i → ∞ . From condition ( F ) , we know that there exist some functions f ˜ such that f i → f ˜ . In particular, setting t = 1 leads to a contradiction

0 > ∂ Z ∞ ∂ t ( η 0 , 1 ) + ( − Δ ) ℒ Z ∞ ( η 0 , 1 ) = f ˜ ( 1 , v ∞ ( η 0 , 1 ) ) = f ˜ ( 1 , ς ( η 0 ) ) ( ς ( η 0 ) ≥ 0 ) ≥ f ˜ ( 1 , 0 ) ≥ 0 .

This contradiction implies that for any φ ∈ ϖ ( z ) , ς ∈ ϖ ( v ) , a point η φ ∈ Σ α 0 exists such that

ϕ α 0 ( η φ ) > 0 , ζ α 0 ( η φ ) > 0 .

Using Theorem 2.4, we deduce that for any φ ∈ ϖ ( z ) , ς ∈ ϖ ( v )

(3.5) ϕ α 0 ( η ) > 0 , ζ α 0 ( η ) > 0 , η ∈ Ω α 0 .

Furthermore, we can find a universal constant C 0 such that for all φ ∈ ϖ ( z ) , ς ∈ ϖ ( v ) ,

(3.6) ϕ α 0 ( η ) ≥ C 0 > 0 , ζ α 0 ( η ) ≥ C 0 > 0 , η ∈ Ω α 0 − δ ¯ .

There also exists a universal ε > 0 such that

(3.7) ϕ α ( η ) ≥ C 0 2 > 0 , ζ α ( η ) ≥ C 0 2 > 0 , η ∈ Ω α 0 − δ ¯ , ∀ α ∈ ( α 0 , α 0 + ε ) .

Thus, for large enough t , this implies

Z α ( η , t ) ≥ 0 , V α ( η , t ) ≥ 0 , η ∈ Ω α 0 − δ ¯ , ∀ α ∈ ( α 0 , α 0 + ε ) .

For small δ > 0 , (3.7) ensures that a small ε > 0 can be chosen so that the region Ω α \ Ω α 0 − δ is narrow for α ∈ ( α 0 , α 0 + ε ) . By Theorem 2.2, it holds that

(3.8) ϕ α ( η ) ≥ 0 , ζ α ( η ) ≥ 0 , ∀ η ∈ Ω α \ Ω α 0 − δ .

Together with (3.7), we obtain

ϕ α ( η ) ≥ 0 , ζ α ( η ) ≥ 0 , ∀ η ∈ Ω α .

This contradicts the definition of α 0 . Hence α 0 = 0 . Therefore, for all φ ∈ ϖ ( z ) , ς ∈ ϖ ( v ) ,

ζ 0 ( η ) ≥ 0 , ϕ 0 ( η ) ≥ 0 , ∀ η ∈ Ω 0 .

Or equivalently, for all φ ∈ ϖ ( z ) , ς ∈ ϖ ( v ) ,

(3.9) φ ( − η 1 , η ′ ) ≤ φ ( η 1 , η ′ ) , ς ( − η 1 , η ′ ) ≤ ς ( η 1 , η ′ ) , 0 < η 1 < 1 .

Due to the arbitrariness of the direction of η 1 , φ ( η ) and ς ( η ) must exhibit radial symmetry centered at the origin. Furthermore, following a similar argument as in (3.5) and combining

ϕ α ( η ) > 0 , ζ α ( η ) > 0 , η ∈ Ω α , ∀ − 1 < α < 0 ,

we conclude that ϕ α ( η ) and ζ α ( η ) exhibit radial symmetry and are strictly decreasing with respect to the origin. This completes the proof of Theorem 3.1.□

4 Hopf’s lemma

Hopf’s lemma has long been a cornerstone in the analysis of elliptic and parabolic PDEs. It provides crucial support for the development of various maximum and strong maximum principles. It also serves as a pivotal tool for applying the moving plane technique to investigate qualitative characteristics of solutions to PDEs.

Against the backdrop of advancing research on nonlocal operators such as ( − Δ ) s and ( − Δ ) p s among others, fractional Hopf’s lemmas have increasingly emerged in recent years. Significant progress has been made in the development of Hopf’s lemma for elliptic equations (refer [10,25,46] and references cited therein); however, research on Hopf’s lemma for fractional parabolic equations remains relatively scarce. Jin and Xiong [24] demonstrated a strong maximum principle along with a Hopf’s lemma for odd solutions of a specific linear fractional parabolic equation over finite time periods. Chen and Wu [15] established a Hopf’s lemma for antisymmetric functions and obtained the monotonicity of solutions to fractional parabolic equations in a half space. Wang and Chen [39] first established a Hopf’s lemma for parabolic equation involving the fractional p -Laplacian (with p ≥ 2 ), and further obtained an asymptotic variant for antisymmetric solutions. Additionally, Zhang and Nie [46] obtained Hopf’s lemma for an elliptic equation with the logarithmic Laplacian.

To the best of our knowledge, a Hopf’s lemma for parabolic systems involving the logarithmic Laplacian has not been previously established in the literature. In this section, building on the aforementioned literature, particularly [39] and [46], we rigorously derive Theorem 4.1 (Hopf’s lemma for logarithmic Laplacian parabolic systems) and Theorem 4.2 (asymptotic Hopf’s lemma for antisymmetric functions). These results provide critical tools for analyzing qualitative properties of solutions in this emerging research area.

Theorem 4.1

Suppose that Ω has a smooth boundary and satisfies ( A 1 ), ( A 2 ). Let z ( η , t ) , v ( η , t ) be a pair of positive solutions to the following system:

(4.1) ∂ z ∂ t ( η , t ) + ( − △ ) ℒ z ( η , t ) = f ( t , v ( η , t ) ) , ( η , t ) ∈ Ω × [ 0 , ∞ ) , ∂ v ∂ t ( η , t ) + ( − △ ) ℒ v ( η , t ) = g ( t , z ( η , t ) ) , ( η , t ) ∈ Ω × [ 0 , ∞ ) , z ( η , t ) = 0 , v ( η , t ) = 0 , ( η , t ) ∈ Ω c × [ 0 , ∞ ) ,

and satisfy ( A 3 ). Meanwhile, f ( t , v ) , g ( t , z ) ∈ L ∞ ( R + × R ) are uniformly Lipschitz continuous and equicontinuous functions, satisfying f ( t , 0 ) = 0 , g ( t , 0 ) = 0 .

If there exists an interior sphere tangent to ∂ Ω at η 0 ∈ ∂ Ω , then there exists c 0 > 0 such that for any t 0 ∈ ( 0 , T ) and all η sufficiently close to ∂ Ω ,

z ( η , t 0 ) ≥ c 0 d ( η ) , v ( η , t 0 ) ≥ c 0 d ( η ) , d ( η ) = dist ( η , ∂ Ω ) .

Consequently, their outward normal derivatives satisfy

∂ z ( η , t 0 ) ∂ ν < 0 , ∂ v ( η , t 0 ) ∂ ν < 0 , η ∈ ∂ Ω , t 0 ∈ ( 0 , T ) ,

with ν defined as the outward normal vector at the tangency point η 0 on ∂ Ω .

Proof

Fix η 0 ∈ ∂ Ω . Suppose there exists an interior sphere B δ ( η ¯ ) ⊂ Ω tangent to ∂ Ω at η 0 . In view of this, define the small parabolic cylindrical neighborhood B ⊂ Ω × ( 0 , T ) as

B ≔ B δ ( η ¯ ) × [ t 0 − ε 0 , t 0 + ε 0 ] ,

where t 0 ∈ ( 0 , T ) , and ε 0 > 0 is a small positive constant. By coordinate translation and scaling, the problem can be transformed to the small parabolic cylinder

B ≔ B 1 ( 0 ) × [ t 0 − ε 0 , t 0 + ε 0 ] .

Take Q ⊂ Ω where Q ∩ B 1 ( 0 ) = ∅ and then construct a compact cylinder

B ˜ = Q × [ t 0 − ε 0 , t 0 + ε 0 ] .

It is clear that dist ( B ˜ , B ) > 0 . Owing to the positivity and continuity of z , there exists a constant c 1 > 0 such that

z ( η , t ) ≥ c 1 , ∀ ( η , t ) ∈ B ˜ .

Next we construct the subsolution

z ̲ ( η , t ) = χ Q ( η ) z ( η , t ) + ε θ ( t ) h ( η ) , ( η , t ) ∈ B 1 ( 0 ) × [ t 0 − ε 0 , t 0 + ε 0 ]

with ε > 0 to be determined afterwards. Here

χ Q ( η ) = 1 , η ∈ Q , 0 , η ∈ Q c , and h ( η ) = ( 1 − ∣ η ∣ 2 ) + .

The function θ ( t ) ∈ C ∞ ( t 0 − ε 0 , t 0 + ε 0 ) is defined by

θ ( t ) = 1 , t ∈ t 0 − ε 0 2 , t 0 + ε 0 2 , 0 , t ∉ ( t 0 − ε 0 , t 0 + ε 0 ) .

Then, for any ( η , t ) ∈ B 1 ( 0 ) × [ t 0 − ε 0 , t 0 + ε 0 ] , we obtain

( − △ ) ℒ ( χ Q ( η ) z ( η , t ) ) = C N ∫ B 1 ( 0 ) χ Q ( η ) z ( η , t ) − χ Q ( ξ ) z ( ξ , t ) ∣ η − ξ ∣ N d ξ − C N ∫ R N \ B 1 ( 0 ) χ Q ( ξ ) z ( ξ , t ) ∣ η − ξ ∣ N d ξ + [ h B 1 ( 0 ) ( η ) + μ N ] χ Q ( η ) z ( η , t ) = 0 − C N ∫ Q z ( ξ , t ) ∣ η − ξ ∣ N d ξ + 0 ≤ − C 1

and

( − △ ) ℒ h ( η ) = C N ∫ B 1 ( 0 ) ( 1 − ∣ η ∣ 2 ) + − ( 1 − ∣ ξ ∣ 2 ) + ∣ η − ξ ∣ N d ξ − C N ∫ R N \ B 1 ( 0 ) ( 1 − ∣ ξ ∣ 2 ) + ∣ η − ξ ∣ N d ξ + [ h B 1 ( 0 ) ( η ) + μ N ] ( 1 − ∣ η ∣ 2 ) + ≤ C 2 ,

where C 1 > 0 , C 2 > 0 . Consequently, we derive

∂ z ̲ ∂ t ( η , t ) + ( − △ ) ℒ z ̲ ( η , t ) = ε θ ′ ( t ) ( 1 − ∣ η ∣ 2 ) + + ( − △ ) ℒ ( χ Q ( η ) z ( η , t ) ) + ε θ ( t ) ( − △ ) ℒ h ( η ) ≤ ε θ ′ ( t ) ( 1 − ∣ η ∣ 2 ) + − C 1 + C 2 ε η ( t ) , ( η , t ) ∈ B 1 ( 0 ) × ( t 0 − ε 0 , t 0 + ε 0 ) .

Define Z ( η , t ) = z ( η , t ) − z ̲ ( η , t ) . Then, Z ( η , t ) satisfies:

∂ Z ∂ t ( η , t ) + ( − △ ) ℒ Z ( η , t ) = ∂ z ∂ t ( η , t ) + ( − △ ) ℒ z ( η , t ) − ∂ z ̲ ∂ t ( η , t ) + ( − △ ) ℒ z ̲ ( η , t ) ≥ f ( t , v ( η , t ) ) − ε θ ′ ( t ) ( 1 − ∣ η ∣ 2 ) + + C 1 − C 2 ε θ ( t ) = C ( η , t ) v ( η , t ) − ε θ ′ ( t ) ( 1 − ∣ η ∣ 2 ) + + C 1 − C 2 ε θ ( t ) ,

where

C ( η , t ) = ∂ f ( t , ψ ( η , t ) ) ∂ v , for some ψ ( η , t ) between 0 and v ( η , t ) .

Then, C ( η , t ) is bounded.

Thus, taking sufficiently small ε , for any ( η , t ) ∈ B 1 ( 0 ) × [ t 0 − ε 0 , t 0 + ε 0 ] , it follows that:

(4.2) ∂ Z ∂ t ( η , t ) + ( − △ ) ℒ Z ( η , t ) = C ( η , t ) v ( η , t ) − ε θ ′ ( t ) ( 1 − ∣ η ∣ 2 ) + + C 1 − C 2 ε θ ( t ) ≥ 0 .

Based on the definition of z ̲ ( η , t ) , we can directly derive that

z ( η , t ) ≥ χ Q ( η ) z ( η , t ) = z ̲ ( η , t ) , ( η , t ) ∈ B 1 c ( 0 ) × [ t 0 − ε 0 , t 0 + ε 0 ] .

This directly gives

(4.3) Z ( η , t ) ≥ 0 , ( η , t ) ∈ B 1 c ( 0 ) × [ t 0 − ε 0 , t 0 + ε 0 ] .

We can also have

(4.4) Z ( η , t 0 − ε 0 ) = z ( η , t 0 − ε 0 ) − z ̲ ( η , t 0 − ε 0 ) = z ( η , t 0 − ε 0 ) − [ χ Q ( η ) z ( η , t 0 − ε 0 ) + ε θ ( t 0 − ε 0 ) ( 1 − ∣ η ∣ 2 ) + ] ≥ 0 , η ∈ B 1 ( 0 ) .

Similarly, set

v ̲ ( η , t ) = χ Q ( η ) v ( η , t ) + ε θ ( t ) h ( η ) , V ( η , t ) = v ( η , t ) − v ̲ ( η , t ) .

Then,

(4.5) ∂ V ∂ t ( η , t ) + ( − △ ) ℒ V ( η , t ) ≥ 0 , ( η , t ) ∈ B 1 ( 0 ) × [ t 0 − ε 0 , t 0 + ε 0 ] , V ( η , t ) ≥ 0 , ( η , t ) ∈ B 1 c ( 0 ) × [ t 0 − ε 0 , t 0 + ε 0 ] , V ( η , t 0 − ε 0 ) ≥ 0 , η ∈ B 1 ( 0 ) .

Therefore, (4.2), (4.3), (4.4), (4.5), and Theorem 2.1 imply that

Z ( η , t ) ≥ 0 , V ( η , t ) ≥ 0 , ( η , t ) ∈ B 1 ( 0 ) × [ t 0 − ε 0 , t 0 + ε 0 ] .

Hence, the inequality

z ( η , t ) ≥ z ̲ ( η , t ) = ε θ ( t ) ( 1 − ∣ η ∣ 2 ) +

holds in the cylindrical region B 1 ( 0 ) × [ t 0 − ε 0 , t 0 + ε 0 ] . Particularly, for fixed ε and η ∈ B 1 ( 0 ) , according to the definition of θ ( t ) , we have

z ( η , t 0 ) ≥ ε ( 1 − ∣ η ∣ 2 ) + = ε ( 1 − ∣ η ∣ ) ( 1 + ∣ η ∣ ) + = c 0 d ( η ) , c 0 > 0 .

Consequently,

lim η → ∂ Ω z ( η , t 0 ) d ( η ) ≥ c 0 > 0 , ∀ t 0 ∈ ( 0 , T ) .

With ν defined as the outward normal vector to ∂ Ω at the tangency η 0 , the fact that z ( η 0 , t 0 ) = 0 implies

∂ z ( η , t 0 ) ∂ ν < 0

for any η ∈ ∂ Ω , t 0 ∈ ( 0 , T ) . Similarly, the same conclusion holds for v , namely,

∂ v ( η , t 0 ) ∂ ν < 0

for any η ∈ ∂ Ω , t 0 ∈ ( 0 , T ) . □

Theorem 4.2

Assume that Σ ˜ α lies on the right-hand side of the hyperplane T α , let Ω ⊂ Σ ˜ α be a bounded Lipschitz domain with smooth boundary and h Σ ˜ α + μ N ≥ 0 . Moreover, assume ( A 4 ) holds and Z α ( η , t ) , V α ( η , t ) satisfy the following system:

(4.6) ∂ Z α ∂ t ( η , t ) + ( − △ ) ℒ Z α ( η , t ) = c 1 α ( η , t ) V α ( η , t ) , ( η , t ) ∈ Σ ˜ α × [ 0 , ∞ ) , ∂ V α ∂ t ( η , t ) + ( − △ ) ℒ V α ( η , t ) = c 2 α ( η , t ) Z α ( η , t ) , ( η , t ) ∈ Σ ˜ α × [ 0 , ∞ ) , Z α ( η , t ) = − Z α ( η α , t ) , V α ( η , t ) = − V α ( η α , t ) , ( η , t ) ∈ Σ ˜ α × [ 0 , ∞ ) , lim t → ∞ ¯ Z α ( η , t ) ≥ 0 , lim t → ∞ ¯ V α ( η , t ) ≥ 0 , η ∈ Σ ˜ α ,

where 0 < c k α ( η , t ) < h Σ ˜ α + μ N ( k = 1 , 2 ) . If ϕ α ( η ) , ζ α ( η ) are nonnegative functions and not identically zero in Σ ˜ α , then

∂ ϕ α ∂ ν ( η ) < 0 , ∂ ζ α ∂ ν ( η ) < 0 , η ∈ ∂ Σ ˜ α ,

in which ν refers to the outward normal vector on ∂ Σ ˜ α .

Proof

For simplicity and with no loss of generality, take α = 0 . We shall prove that ∂ ϕ α ∂ η 1 ( 0 ) > 0 , ∂ ζ α ∂ η 1 ( 0 ) > 0 . Based on the definition of ϖ ( z ) , ϖ ( v ) , for any φ ∈ ϖ ( z ) , ς ∈ ϖ ( v ) , a sequence { t i } can be found such that

Z α ( η , t i ) → ϕ α ( η ) as t i → ∞ , V α ( η , t i ) → ζ α ( η ) as t i → ∞ .

Define

Z i ( η , t ) = Z α ( η , t + t i − 1 ) , V i ( η , t ) = V α ( η , t + t i − 1 ) .

It is straightforward to verify that

∂ Z i ∂ t ( η , t ) + ( − △ ) ℒ Z i ( η , t ) = c 1 i ( η , t ) V i ( η , t ) , ( η , t ) ∈ Σ ˜ α × [ 0 , ∞ ) , ∂ V i ∂ t ( η , t ) + ( − △ ) ℒ V i ( η , t ) = c 2 i ( η , t ) Z i ( η , t ) , ( η , t ) ∈ Σ ˜ α × [ 0 , ∞ ) ,

where c 1 i ( η , t ) = c 1 α ( η , t + t i − 1 ) , c 2 i ( η , t ) = c 2 α ( η , t + t i − 1 ) .

Then, we can find sequences Z i ( η , t ) (retained with the same notation Z i ( η , t ) ) and V i ( η , t ) (likewise retained) such that Z i → Z ∞ and V i → V ∞ uniformly on Σ ˜ α × [ 1 − ε 0 , 1 + ε 0 ] , where Z ∞ and V ∞ are the respective limit functions. They satisfy

Owing to the boundedness of c k α ( k = 1 , 2 ) , the limit functions c k ∞ ( k = 1 , 2 ) are also bounded. Additionally, Z ∞ , V ∞ are continuous with respect to η as well as t .

For t = 1 , it follows that

Z α ( η , t i ) = Z i ( η , 1 ) → Z ∞ ( η , 1 ) = ϕ α ( η ) as i → ∞ , V α ( η , t i ) = V i ( η , 1 ) → V ∞ ( η , 1 ) = ζ α ( η ) as i → ∞ .

By (4.6), we have

(4.7) ∂ Z ∞ ∂ t ( η , t ) + ( − △ ) ℒ Z ∞ ( η , t ) = c 1 ∞ V ∞ ≥ 0 , ( η , t ) ∈ Σ ˜ α × [ 1 − ε 0 , 1 + ε 0 ] , ∂ V ∞ ∂ t ( η , t ) + ( − △ ) ℒ V ∞ ( η , t ) = c 2 ∞ Z ∞ ≥ 0 , ( η , t ) ∈ Σ ˜ α × [ 1 − ε 0 , 1 + ε 0 ] .

Noting that ϕ α , ζ α are continuous and positive at some point in Σ ˜ α , we can find a subset Q ⊂ ⊂ Σ ˜ α such that

ϕ α ( η ) > c > 0 , ζ α ( η ) > c > 0 , η ∈ Q ,

where c > 0 . Moreover, in view of the continuity of Z ∞ ( η , t ) with respect to t , there exists 0 < ε 0 < 1 such that

(4.8) Z ∞ ( η , t ) > c ⁄ 2 , ( η , t ) ∈ Q × [ 1 − ε 0 , 1 + ε 0 ] .

Set

Z ̲ ( η , t ) = χ Q ∪ Q α ( x ) Z ∞ ( η , t ) + ε θ ( t ) h ( η ) .

Here

χ Q ∪ Q α ( η ) = 1 , η ∈ Q ∪ Q α , 0 , η ∉ Q ∪ Q α ,

where Q α denotes the reflection of Q with respect to the hyperplane T α . θ ( t ) ∈ C 0 ∞ [ 1 − ε 0 , 1 + ε 0 ] satisfies

θ ( t ) = 1 , t ∈ 1 − ε 0 2 , 1 + ε 0 2 , 0 , t ∉ ( 1 − ε 0 , 1 + ε 0 ) .

Define h ( η ) = η 1 ρ ( η ) , where ρ ( η ) ∈ C 0 ∞ ( B 2 ℓ ( 0 ) ) satisfies

ρ ( η ) = ρ ( ∣ η ∣ ) = 1 , ∣ η ∣ < ℓ , ∈ [ 0 , 1 ] , ℓ ≤ ∣ η ∣ ≤ 2 ℓ , 0 , ∣ η ∣ > 2 ℓ ,

with dist ( T 0 , ∂ Q ) > 2 ℓ for ℓ > 0 . Here B ℓ ( 0 ) denotes the ball of radius ℓ centered at the origin, and B 2 ℓ ( 0 ) is similarly defined as the ball of radius 2 ℓ centered at the origin. It is clear that h ( η ) is antisymmetric about the hyperplane T 0 , meaning

h ( − η 1 , η ′ ) = − h ( η 1 , η ′ ) .

From h ( η ) ∈ C 0 ∞ ( B 2 ℓ ( 0 ) ) , we have

∣ ( − Δ ) ℒ h ( η ) ∣ = ∣ C N ∫ B 2 ℓ ( 0 ) η 1 ρ ( η ) − η 1 ρ ( ξ ) ∣ η − ξ ∣ N d ξ − C N ∫ R N \ B 2 ℓ ( 0 ) η 1 ρ ( ξ ) ∣ η − ξ ∣ N d ξ + [ h B 2 ℓ ( 0 ) ( η ) + μ N ] η 1 ρ ( η ) ∣ ≤ C 0 ,

where C 0 > 0 . From the definition of ( − Δ ) ℒ along with (4.8), for each t ∈ [ 1 − ε 0 , 1 + ε 0 ] fixed and any η ∈ B 2 ℓ ( 0 ) ∩ Σ ˜ α , we obtain

( − Δ ) ℒ ( χ Q ∪ Q α ( η ) Z ∞ ( η , t ) ) = C N ∫ B 2 ℓ ( 0 ) χ Q ∪ Q α ( η ) Z ∞ ( η , t ) − χ Q ∪ Q α ( ξ ) Z ∞ ( ξ , t ) ∣ η − ξ ∣ N d ξ − C N ∫ R N \ B 2 ℓ ( 0 ) χ Q ∪ Q α ( ξ ) Z ∞ ( ξ , t ) ∣ η − ξ ∣ N d ξ + [ h B 2 ℓ ( 0 ) ( η ) + μ N ] χ Q ∪ Q α ( η ) Z ∞ ( η , t ) = 0 − C N ∫ Q ∪ Q α Z ∞ ( ξ , t ) ∣ η − ξ ∣ N d ξ + 0 = C N ∫ Q − Z ∞ ( ξ , t ) ∣ η − ξ ∣ N d ξ + C N ∫ Q α − Z ∞ ( ξ α , t ) ∣ η − ξ α ∣ N d ξ = C N ∫ Q − Z ∞ ( ξ , t ) ∣ η − ξ ∣ N d ξ + C N ∫ Q Z ∞ ( ξ , t ) ∣ η − ξ α ∣ N d ξ = C N ∫ Q 1 ∣ η − ξ α ∣ N d ξ − 1 ∣ η − ξ ∣ N Z ∞ ( ξ , t ) d ξ ≤ − C 1 ,

where C 1 > 0 . Therefore, based on the above analysis, we have

∂ Z ̲ ( η , t ) ∂ t + ( − Δ ) ℒ Z ̲ ( η , t ) = ε θ ′ ( t ) h ( η ) + ( − Δ ) ℒ ( χ Q ∪ Q α ( η ) Z ∞ ( η , t ) ) + ε θ ( t ) ( − Δ ) ℒ h ( η ) ≤ ε θ ′ ( t ) h ( η ) − C 1 + ε θ ( t ) C 0 , ( η , t ) ∈ ( B 2 ℓ ( 0 ) ∩ Σ ˜ α ) × [ 1 − ε 0 , 1 + ε 0 ] .

Choosing small enough ε > 0 , we have

(4.9) ∂ Z ̲ ∂ t ( η , t ) + ( − Δ ) ℒ Z ̲ ( η , t ) ≤ 0 , ( η , t ) ∈ ( B 2 ℓ ( 0 ) ∩ Σ ˜ α ) × [ 1 − ε 0 , 1 + ε 0 ] .

Set Z ˜ ( η , t ) = Z ∞ ( η , t ) − Z ̲ ( η , t ) . Obviously, Z ˜ ( η , t ) is an antisymmetric function that satisfies Z ˜ ( η , t ) = − Z ˜ ( η α , t ) . According to (4.7) and (4.9), we obtain Z ˜ ( η , t ) satisfying

(4.10) ∂ Z ˜ ∂ t ( η , t ) + ( − Δ ) ℒ Z ˜ ( η , t ) ≥ 0 , ( η , t ) ∈ ( B 2 ℓ ( 0 ) ∩ Σ ˜ α ) × [ 1 − ε 0 , 1 + ε 0 ] .

From the definition of Z ̲ ( η , t ) , we can also know

(4.11) Z ˜ ( η , t ) ≥ 0 , ( η , t ) ∈ ( Σ ˜ α \ B 2 ℓ ( 0 ) ∩ Σ ˜ α ) × [ 1 − ε 0 , 1 + ε 0 ]

and

(4.12) Z ˜ ( η , 1 − ε 0 ) ≥ 0 , η ∈ Σ ˜ α .

Similarly, set

V ̲ ( η , t ) = χ Q ∪ Q α ( η ) V ∞ ( η , t ) + ε θ ( t ) h ( η ) , V ˜ ( η , t ) = V ∞ ( η , t ) − V ̲ ( η , t ) .

We can also obtain

(4.13) ∂ V ˜ ∂ t ( η , t ) + ( − △ ) ℒ V ˜ ( η , t ) ≥ 0 , ( η , t ) ∈ ( B 2 ℓ ( 0 ) ∩ Σ ˜ α ) × [ 1 − ε 0 , 1 + ε 0 ] , V ˜ ( η , t ) ≥ 0 , ( η , t ) ∈ ( Σ ˜ α \ B 2 ℓ ( 0 ) ∩ Σ ˜ α ) × [ 1 − ε 0 , 1 + ε 0 ] , V ˜ ( η , 1 − ε 0 ) ≥ 0 , η ∈ Σ ˜ α .

Now, combining (4.10), (4.11), (4.12), (4.13), and applying Theorem 2.3, we obtain

Z ˜ ( η , t ) ≥ 0 , V ˜ ( η , t ) ≥ 0 , ( η , t ) ∈ ( B 2 ℓ ( 0 ) ∩ Σ ˜ α ) × [ 1 − ε 0 , 1 + ε 0 ] .

This gives

Z ˜ ( η , t ) = Z ∞ ( η , t ) − ε h ( η ) θ ( t ) ≥ 0 , ( η , t ) ∈ ( B 2 ℓ ( 0 ) ∩ Σ ˜ α ) × [ 1 − ε 0 , 1 + ε 0 ] , V ˜ ( η , t ) = V ∞ ( η , t ) − ε h ( η ) θ ( t ) ≥ 0 , ( η , t ) ∈ ( B 2 ℓ ( 0 ) ∩ Σ ˜ α ) × [ 1 − ε 0 , 1 + ε 0 ] .

In particular, when t = 1 , we have

Z ∞ ( η , 1 ) ≥ ε h ( η ) , η ∈ B 2 ℓ ( 0 ) ∩ Σ ˜ α , V ∞ ( η , 1 ) ≥ ε h ( η ) , η ∈ B 2 ℓ ( 0 ) ∩ Σ ˜ α ,

and

Z ∞ ( η , 1 ) ≥ ε η 1 , η ∈ B ℓ ( 0 ) ∩ Σ ˜ α , V ∞ ( η , 1 ) ≥ ε η 1 , η ∈ B ℓ ( 0 ) ∩ Σ ˜ α .

Since

Z ∞ ( η , 1 ) = ϕ α ( η ) = ϕ 0 ( η ) ≡ 0 , η ∈ T 0 , V ∞ ( η , 1 ) = ζ α ( η ) = ζ 0 ( η ) ≡ 0 , η ∈ T 0 ,

particularly, Z ∞ ( 0 , 1 ) = 0 , V ∞ ( 0 , 1 ) = 0 .

Hence,

Z ∞ ( η , 1 ) − 0 η 1 − 0 ≥ ε > 0 , η ∈ B ℓ ( 0 ) ∩ Σ ˜ α , V ∞ ( η , 1 ) − 0 η 1 − 0 ≥ ε > 0 , η ∈ B ℓ ( 0 ) ∩ Σ ˜ α .

Clearly, regardless of how small ℓ is, we obtain

∂ ϕ α ∂ η 1 ( 0 ) > 0 , ∂ ζ α ∂ η 1 ( 0 ) > 0 .

Therefore,

□ ∂ ϕ α ∂ ν ( η ) < 0 , ∂ ζ α ∂ ν ( η ) < 0 , η ∈ ∂ Σ ˜ α .

Funding information: The work was supported by Fundamental Research Program of Shanxi Province, China (No. 202403021221163), the Graduate Education Innovation Program of Shanxi, China (No. 2024JG103), and Postgraduate Education Innovation Program of Shanxi Normal University, China (No. 2024YJSKCSZSFK-06).
Author contributions: All authors contributed equally to the writing of this article.
Conflict of interest: This work does not have any conflicts of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2025-09-07

Revised: 2025-10-17

Accepted: 2025-11-04

Published Online: 2025-11-26

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/anona-2025-0134

Keywords for this article

logarithmic Laplacian; parabolic system; asymptotic radial symmetry; moving planes; Hopf’s lemma

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BY 4.0