Non-existence, radial symmetry, monotonicity, and Liouville theorem of master equations with fractional p-Laplacian

1 Introduction

The primary aim of this article is to study

(1) The radial symmetry and strict monotonicity of the solution of the master equation with fractional p -Laplacian in the unit ball

(2) The Liouville theorem of the following homogeneous master equation with fractional p -Laplacian

(3) The non-existence of the solution for the master equation with fractional p -Laplacian

Here, the operator ( ∂ t − Δ p ) s is defined later.

The new operator proposed in this article originates from the famous fully fractional heat operator ( ∂ t − Δ ) s introduced by Riesz [25], which is defined as follows:

Because the value of ( ∂ t − Δ ) s at a point ( z , t ) is related to the value of u in the entire R n and all past times up to t , this operator is non-local in both space and time. Interestingly, when u depends only on z , then it can be expressed as

where ( − Δ ) s represents the famous fractional Laplacian. When u depends only on t , then

where ∂ t s represents the Marchaud fractional derivative [20], commonly denoted as D left s , which is the Marchaud left fractional derivative, defined by

The Marchaud fractional derivative has application in describing numerous physical phenomena, including plasma turbulence [4], wave propagation [8] and magneto-thermoelastic heat conduction [9]. Furthermore, when s → 1 , the operator ( ∂ t − Δ ) s becomes the local operator ∂ t − Δ .

The master equation has been applied in physical phenomena to anomalous diffusion [3] and chaotic dynamics [14]. In the biological field, it is used to study biological invasions [1]. In the financial field, it can simulate the relationship between waiting time and the increase in trading prices [24]. From a probabilistic perspective, the master equation plays a crucial role in the theory of continuous-time random walks, where u represents the distribution of particles that perform random jumps while experiencing stochastic time delays [21]. Unlike the nonlocal parabolic equation

and the dual fractional parabolic equation

the master equation accounts for the strong correlation between waiting times and particle jumps, which is absent in the nonlocal parabolic equation (1.4) and the dual fractional parabolic equation (1.5). Obviously, the master equation is of significant value in different fields. Therefore, further study of the master equation will enhance our understanding of complex phenomena. At present, the radial symmetry, existence, regularity, uniqueness, Liouville theorem and Gibbons’ conjecture of solutions for the master equation have obtained a series of outstanding results [2,5,7,17,19,26].

The fractional p -Laplacian appears in various fields such as engineering, chemistry, physics, image processing, and biological modeling. In recent years, many scholars have shown great interest in problems with the fractional p -Laplacian. For instance, in 2020, Wang and Yang investigated the non-existence of positive solutions of the fractional p -Laplacian equation with indefinite nonlinearities [31]. In 2021, Mi et al. studied a class of fractional p -Laplacian equations involving asymptotically periodic conditions and established the existence results of two new ground state solutions by variational methods [22]. In 2023, Hamdani et al. studied a novel tri-nonlocal Kirchhoff problem, which involves the p ( x ) -fractional Laplacian equations of variable order [11]. In 2024, Liao established the local Hölder regularity of weak solutions of a fractional p -Laplacian parabolic equation involving only measurable kernels [16]. The other results regarding the fractional p -Laplacian equation can be referred in [6,23,28,29,34,36]. In the particular case when p = 2 , ( − Δ ) p s transforms into the well-known fractional Laplacian. For the fractional Laplacian, Zhao et al. combined the minimax method, bifurcation theory, and Morse theory to study the multiplicity of nontrivial solutions for a class of nonlinear fractional Laplace problems with a parameter and superlinear nonlinearity [35]. Kossowski and Przeradzki established the existence theory of solutions for the heat equations with generalized fractional Laplacian [12]. Another special case is that when s = 1 , the fractional p -Laplacian degenerates into the classical p -Laplacian. For the p -Laplacian, Wang and Sun studied the normalized solutions of the p -Laplacian equation with a trapping potential and L r -supercritical growth, where r = p or r = 2 [30]. Wu et al. studied the local solvability of a singular parabolic p -Laplacian equation with logarithmic nonlinearity by the technique of cut-off combining with the method of Faedo-Galerkin approximation. Based on the potential well method and Hardy-Sobolev inequality, they derived the global existence of solutions [33]. Many fruitful results can be found in [10,18,37].

Inspired by the above articles, we introduce a new operator that combines the fractional p -Laplacian and the fully fractional heat operator, which is defined as follows:

where 0 < s < 1 , 2 ≤ p < ∞ , C n , s p = s p 2 ( 1 − s ) 2 2 s − 1 − n − s p π n − 1 2 Γ ( p + 1 2 ) Γ ( 2 − s ) is the normalized positive constant and the integral with respect to ζ is considered in the sense of Cauchy principal values. To make sure that the integral in (1.6) is well defined, let

where L ( R n × R ) is a class of slowly increasing functions defined as

and the local parabolic Hölder space C z , t , loc 2 s + ε , s + ε ( R n × R ) is defined in [19]. In the following, we recall the definition of the parabolic Hölder space C z , t 2 α , α ( R n × R ) (cf. [13]).

(i) When 0 < α ≤ 1 2 , if u ( z , t ) ∈ C z , t 2 α , α ( R n × R ) , then there exists a constant C > 0 such that

for any z , ζ ∈ R n and t , ϱ ∈ R .

(ii) When 1 2 < α ≤ 1 , we say that

if u is α -Hölder continuous in t uniformly with respect to z and its gradient ∇ z u is ( 2 α − 1 ) -Hölder continuous in z uniformly with respect to t and ( α − 1 2 ) -Hölder continuous in t uniformly with respect to z .

(iii) When α > 1 , if u ( z , t ) ∈ C z , t 2 α , α ( R n × R ) , then it means that

Then, we can analogously define the local parabolic Hölder space C z , t , loc 2 α , α ( R n × R ) . Interestingly, when u depends only on z , then

where ( − Δ ) p s represents the famous fractional p -Laplacian. When u depends only on t , then

where ∂ t s represents the Marchaud fractional derivative [20]. Moreover, when p = 2 , the operator ( ∂ t − Δ p ) s transforms into the famous fully fractional heat operator ( ∂ t − Δ ) s .

Due to the non-local and strong correlation exhibited by ( ∂ t − Δ p ) s in the context of the master equation with fractional p -Laplacian, many traditional methods for solving local operators are no longer applicable in non-local environments. However, for the exploration of explosion rate, prior estimation and optimal global estimation for non-local parabolic problems with initial boundary values, the study of the qualitative properties of the solution plays a vital role. Therefore, this article mainly overcomes the challenge of non-local and strong correlation of operator ( ∂ t − Δ p ) s and establishes the direct method of moving planes suitable for this operator ( ∂ t − Δ p ) s . Consequently, we obtain the Liouville theorem, radial symmetry, monotonicity, and non-existence of the solution of the master equation with fractional p -Laplacian.

The primary conclusions of this article are as follows.

Radial symmetry and monotonicity: Assume that

is a positive bounded solution of equation (1.1). If f ∈ C 1 ( [ 0 , + ∞ ) ) with f ( 0 ) ≥ 0 , f ′ ( 0 ) ≤ 0 , the solution u of equation (1.1) is radially symmetric and strictly decreasing with respect to the origin in z ∈ B 1 ( 0 ) for any t ∈ R .

Liouville theorem: Assume that

is a bounded solution of equation (1.2). Then, u ( z , t ) must be a constant.

Non-existence: Assume that

is a positive bounded solution of equation (1.3). Suppose f satisfies

1. f ( u ) = z 1 ψ ( u ) ,

2. ψ is monotonically increasing with respect to u ,

3. ψ > C , where C is a positive constant.

The structure of this article is as follows. In Section 2, we establish the narrow region principle. On this basis, using the direct method of moving planes, we obtain equation (1.1), which is radially symmetric and strictly decreasing from the origin within the unit ball for any t ∈ R . In Section 3, we introduce the maximum principle in unbounded domains. By applying the Fourier transform, the Liouville theorem of the homogeneous equation (1.2) is obtained, that is, all bounded solutions must be constants. In Section 4, we prove that equation (1.3) has no bounded positive solution by utilizing the maximum principle in bounded domains.

Notations. Let z 1 be any given direction in R n . For any λ ∈ R , let us take

as the moving planes, and it’s perpendicular to z 1 -axis. Let

and

represent the left region of T λ . Let

be reflection point of z with respect to T λ . Denote

and

Then, w λ is an antisymmetric function satisfying

2 Radial symmetry and monotonicity

In this section, we first establish the narrow region principle (i.e., Theorem 2.1). This principle is essential for proving the radial symmetry and monotonicity of the solution of the master equation with fractional p -Laplacian. Second, Theorem 2.1 and the direct method of moving planes are applied to prove the radial symmetry and strict monotonicity of the solution of the master equation with fractional p -Laplacian (i.e., Theorem 2.2). In the entire article, C represents a positive constant, which value may vary from line to line.