Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group

Wei Shi

doi:10.1515/math-2023-0133

Open Access Published by De Gruyter Open Access November 13, 2023

Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group

Wei Shi

From the journal Open Mathematics

https://doi.org/10.1515/math-2023-0133

Abstract

In this article, the Liouville theorems for the Kirchhoff-type parabolic equations on the Heisenberg group were established. The main technique for proving the result relies on the method of test functions.

Keywords: Kirchhoff-type equation; Heisenberg group; heat equation

MSC 2010: 35K60; 35R03

1 Introduction

In this work, we focus our attention on the Kirchhoff-type heat equation in the Heisenberg group of the form:

(1) ∂ u m ∂ t − M t , ∫ H n ∣ ∇ H u ∣ 2 d η , ∫ H n ∣ Δ H u ∣ 2 d η Δ H u m = ∣ η ∣ H γ ∣ u ∣ p , u ( η , 0 ) = u 0 ( η ) ≥ 0 , η ∈ H n ,

where ( η , t ) ∈ H n × ( 0 , T ) , T > 0 , and p > m > 0 , γ > − 2 , ∣ η ∣ H is defined in (10) and Δ H is the sub-Laplacian on H n (see Section 2). M : ( 0 , T ) × R + × R + → R is satisfying the following condition:

(2) 0 < M ( t , ⋅ , ⋅ ) ≤ C ′ t β , ∀ t > 0 , 0 ≤ β < 1 .

We prove that any entire (weak) solutions are necessarily constant.

The second part of this article is devoted to study a Liouville-type result for the following system of Kirchhoff-type parabolic equations in Ω ≔ H n × ( 0 , T ) , T > 0 ,

(3) ∂ u m 1 ∂ t − M 1 Δ H u m 1 = ∣ η ∣ H γ 1 ∣ v ∣ p , ∂ v m 2 ∂ t − M 2 Δ H v m 2 = ∣ η ∣ H γ 2 ∣ u ∣ q , u ( η , 0 ) = u 0 ( η ) ≥ 0 , η ∈ H n , v ( η , 0 ) = v 0 ( η ) ≥ 0 , η ∈ H n ,

where p > m 2 > 0 , q > m 1 > 0 , and

(4) M i = M i t , ∫ H n ∣ ∇ H u ∣ 2 d η , ∫ H n ∣ ∇ H v ∣ 2 d η , ∫ H n ∣ Δ H u ∣ 2 d η , ∫ H n ∣ Δ H v ∣ 2 d η

are the bounded functions defined in Σ ≔ ( 0 , T ) × R + × R + × R + × R + , and there exist positive constants C i , such that

(5) 0 < M i ( t , ⋅ , ⋅ , ⋅ , ⋅ ) ≤ C i t β i , ∀ t > 0 , 0 ≤ β i < 1 ,

for i = 1 , 2 .

Kirchhoff’s model arose in the study of the changes in length of the string produced by transverse vibrations. Kirchhoff [1] proposed an equation of the form:

ρ u t t − p 0 h + E 2 L ∫ 0 L u x 2 d x u x x = 0 , t > 0 , x ∈ ( 0 , L ) ,

where ρ , p 0 , h , E , and L are all positive constants. In line with this observation, Li et al. [2] examined the existence of an positive solution to the problem:

a + λ ∫ R n ( ∣ ∇ u ∣ 2 + b u 2 ) d x ( − Δ u + b u ) = f ( u ) , x ∈ R n ,

with λ ∈ [ 0 , λ 0 ) , where a and b are the positive constants and λ ≥ 0 is a parameter.

On the other hand, the nonexistence of solutions is less investigated. Nonexistence results for positive solutions of nonlinear elliptic inequalities of the type

(6) Δ H u + ∣ η ∣ H γ u p ≤ 0 ,

were studied by Garofalo and Lanconelli [3] under restrictive assumptions on u and later on by Birindelli et al. [4] under a much less restrictive assumption.

To better describe the setting, we recall the seminal work of Fujita in his article [5], where he studied the following nonlinear heat equation:

(7) u t ( x , t ) − Δ u ( x , t ) = u 1 + p ( x , t ) , ( x , t ) ∈ R N × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) ≥ 0 , x ∈ R N .

He showed the following results. If 0 < p < 2 N , then a solution of Problem (7) blows up in finite time for N > 2 while being globally well posed for p > 2 N , causing to call this critical exponent “the critical exponent of Fujita” or just “Fujita’s exponent.”

To motivate our results, we recall that in the classical Riemannian setting, Jleli and Samet [6] considered the Cauchy problems for nonlinear Sobolev-type equations with potentials defined on complete noncompact Riemannian manifolds. Recently, Jleli and Samet [7] established new blow-up results for a higher-order evolution inequality involving a convection term in an exterior domain of R n . An important aspect most directly related to the present work is the fine analysis of blow-up solutions of the fully nonlinear elliptic equations [8–10].

In [11], Pohozaev and Véron found a critical exponent for the following non-linear diffusion equation with the Kohn-Laplacian on Heisenberg group:

∂ u ( x , t ) ∂ t − Δ H u ( x , t ) = ∣ u ( x , t ) ∣ p , ( x , t ) ∈ H n × ( 0 , + ∞ ) .

Also, the critical exponents for other equations with the Kohn-Laplacian on the Heisenberg group were derived in [12–15]. In particular, Ruzhansky and Yessirkegenov in [15] found the Fujita exponent on general unimodular Lie groups.

In recent years, increasing attention has been given to the analysis and partial differential equations on Heisenberg group [16–23]. There are many interesting results about Kirchhoff-type problems [24–29]. In particular, in the study [30], the authors established the Liouville theorems for the following system of elliptic differential inequalities:

(8) Δ H u m 1 + ∣ η ∣ H γ 1 ∣ v ∣ p ≤ 0 , Δ H v m 2 + ∣ η ∣ H γ 2 ∣ u ∣ q ≤ 0 ,

on different unbounded open domains of the Heisenberg group H n , including the whole space and halfspace of H n .

In this article, the Kirchhoff-type nonlinear systems of parabolic-type equations on the Heisenberg group were studied, the coefficient that expresses Kirchhoff-type nonlinearity denoted by M . The Liouville theorem of Problem (1) in halfspace of H n remains for further study.

Remark 1.1

As noted in [31], the following particular cases could be a few examples of the function M :

M = 1 .
M = 1 1 + a ∫ H n ∣ ∇ H u ∣ 2 d η + b ∫ H n ∣ Δ H u ∣ 2 d η , where a and b are some real constants.
M = ( 1 + c ∣ t ∣ γ ) − β , where c and γ are some real constants and β > 0 .
M = exp ( − ( a ∫ H n ∣ ∇ H u ∣ 2 d η + b ∫ H n ∣ Δ H u ∣ 2 d η ) ) , where a and b are the real constants.
M = log a 1 + b 1 ∫ H n ∣ ∇ H u ∣ 2 d η + c 1 ∫ H n ∣ Δ H u ∣ 2 d η a + b ∫ H n ∣ ∇ H u ∣ 2 d η + c ∫ H n ∣ Δ H u ∣ 2 d η , where a , b , c , a 1 , b 1 , and c 1 are some real constants.

Our main results of this study can be stated as follows. For our convenience, we denote

Λ 1 = m 1 m 2 γ 2 + 2 m 2 q − 2 m 2 β 1 q − 2 β 2 p q + m 1 γ 1 q + p q m 2 ( q − m 1 ) + q ( p − m 2 )

and

Λ 2 = m 1 m 2 γ 1 + 2 m 1 p − 2 m 1 β 2 p − 2 β 1 p q + m 2 γ 2 p + p q m 1 ( p − m 2 ) + p ( q − m 1 ) .

Theorem 1.2

Assume that the function M ( t , ⋅ , ⋅ ) satisfies (2): if γ > − 2 and p < ( Q + 2 + γ ) m Q + 2 + 2 β − 2 m , then Problem (1) does not have a global non-trivial weak solution, where Q = 2 n + 2 is called the homogeneous dimension of H n .

Theorem 1.3

Let p , q > 1 and M i , i = 1 , 2 satisfies (5): if γ 1 , γ 2 > − 2 and Q < 2 + max { Λ 1 , Λ 2 } , then System (3) does not admit a nontrivial weak solution.

The rest of this article is devoted to the proof of Liouville theorems. For completeness, in Section 2, we first collect some well-known results on the Heisenberg group. The proof of Liouville theorems is finally completed in Section 3.

2 Preliminaries

In this section, we recall some basic facts regarding the Heisenberg vector fields, which will be used in the sequel.

The Heisenberg group H n can be identified with ( R 2 n + 1 , ∘ ) , where 2 n + 1 stands for the topological dimension and the group multiplication “ ∘ ” is defined by:

ξ ˆ ∘ ξ ≔ x + x ˆ , y + y ˆ , t + t ˆ + 2 ∑ i = 1 n x i y ˆ i − y i x ˆ i ,

for any ξ = ( x , y , t ) , ξ ˆ = ( x ˆ , y ˆ , t ˆ ) in H n , with x = ( x 1 , … , x n ) , x ˆ = ( x ˆ 1 , … , x ˆ n ) , y = ( y 1 , … , y n ) , and y ˆ = ( y ˆ 1 , … , y ˆ n ) denoting the elements of R n . Moreover, they are homogeneous of degree one with respect to the dilations:

(9) δ λ ( ξ ) = ( λ x , λ y , λ 2 t ) ( λ > 0 ) .

We consider the norm on H n defined by:

(10) ∣ ξ ∣ H ≔ ρ ( ξ ) = ∑ i = 1 n x i 2 + y i 2 2 + t 2 1 4 ,

and the associated Heisenberg distance

d H ( ξ , ξ ˆ ) = ρ ( ξ ˆ − 1 ∘ ξ ) ,

where ξ ˆ − 1 is the inverse of ξ ˆ with respect to “ ∘ ,” i.e., ξ ˆ − 1 = − ξ ˆ . Let D R ( ξ ) denote the Koranyi ball with center at ξ and radius R associate the gauge distance d H ( ξ , ξ ˆ ) = ρ ( ξ ˆ − 1 ∘ ξ ) , and we will refer to it as Heisenberg ball; for every ξ ∈ H n and R > 0 , we will use the notation:

D R ( ξ ) ≔ { η ∈ H n ∣ d H ( ξ , η ) < R } .

It follows that

∣ D R ( ξ ) ∣ = ∣ D R ( 0 ) ∣ = ∣ D 1 ( 0 ) ∣ R Q ,

where ∣ D 1 ( 0 ) ∣ is the volume of the unit Heisenberg ball under the Haar measure, which is equivalent to 2 n + 1 -dimensional Lebesgue measure of R 2 n + 1 . The n -dimension Heisenberg algebra is the Lie algebra spanned by the left-invariant vector fields:

(11) X i ≔ ∂ ∂ x i + 2 y i ∂ ∂ t , i = 1 , … , n , Y i ≔ ∂ ∂ y i − 2 x i ∂ ∂ t , i = 1 , … , n , T ≔ ∂ ∂ t .

The Heisenberg gradient, or horizontal gradient of a regular function u , is then defined by:

∇ H u ≔ ( X 1 u , … , X n u , Y 1 u , … , Y n u ) .

However, its Heisenberg Hessian matrix is

∇ H 2 u = X 1 X 1 u ⋯ X n X 1 u Y 1 X 1 u ⋯ Y n X 1 u ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ X 1 X n u ⋯ X n X n u Y 1 X n u ⋯ Y n X n u X 1 Y 1 u ⋯ X n Y 1 u Y 1 Y 1 u ⋯ Y n Y 1 u ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ X 1 Y n u ⋯ X n Y n u Y 1 Y n u ⋯ Y n Y n u .

Consider the vector fields X j , Y j for j = 1 , … , n in (11), the sub-Laplacian on the Heisenberg group is the linear differential operator of the second order defined by:

(12) Δ H u ≔ ∑ j = 1 n X j 2 u + Y j 2 u = ∑ j = 1 n ∂ 2 u ∂ x j 2 + ∂ 2 u ∂ y j 2 + 4 y j ∂ 2 u ∂ x j ∂ t − 4 x j ∂ 2 u ∂ y j ∂ t + 4 ( x j 2 + y j 2 ) ∂ 2 u ∂ t 2 .

Define

S 2 2 ( H n ) = { u ∣ u ∈ L 2 ( H n ) , ∣ ∇ H u ∣ ∈ L 2 ( H n ) , ∣ Δ H u ∣ ∈ L 2 ( H n ) } .

3 Parabolic-type equations and system

Definition 3.1

A weak solution u of (1) in Ω = H n × ( 0 , T ) , with initial date 0 ≤ u 0 ∈ L loc 1 ( H n ) , is a locally integrable function such that u ∈ C 1 ( 0 , T ; S 2 2 ( H n ) ) ∩ L loc p ( Ω ) that satisfies

(13) ∫ Ω ∣ η ∣ H γ ∣ u ∣ p ψ d η d t + ∫ H n u 0 ( η ) ψ ( η , 0 ) d η = − ∫ Ω u m ψ t d η d t − ∫ Ω M t , ∫ H n ∣ ∇ H u ∣ 2 d η , ∫ H n ∣ Δ H u ∣ 2 d η u m Δ H ψ d η d t ,

for any test function 0 ≤ ψ ∈ C 2 ( ( 0 , T ) ; S 2 2 ( H n ) ) ∩ C 1 ( [ 0 , T ) ; S 2 2 ( H n ) ) .

Observe that all the integrals in (13) are well defined.

Now, we begin to prove Theorem 1.2.

Proof of Theorem 1.2

Let u be such a nontrivial weak solution of (1) and 0 ≤ ψ ≤ 1 be a smooth nonnegative test function such that

(14) ∫ Ω ∣ η ∣ H − γ m p − m ψ − m p − m ∣ Δ H ψ ( η , t ) ∣ p p − m t β p p − m d η d t + ∫ Ω ∣ η ∣ H − γ p − m ψ − 1 p − m ∣ ψ t ( η , t ) ∣ p p − m d η d t < ∞ .

From Identity (13), applying the Holder’s inequality, one has

∫ Ω ∣ η ∣ H γ ∣ u ∣ p ψ d η d t ≤ ∫ Ω ∣ η ∣ H γ ∣ u ∣ p ψ d η d t + ∫ H n u 0 ( η ) ψ ( η , 0 ) d η = − ∫ Ω u m ( η , t ) ψ t ( η , t ) d η d t − ∫ Ω M t , ∫ H n ∣ ∇ H u ∣ 2 d η , ∫ H n ∣ Δ H u ∣ 2 d η u m ( η , t ) Δ H ψ ( η , t ) d η d t ≤ C ′ ∫ Ω t β ∣ u ( η , t ) ∣ m ∣ Δ H ψ ( η , t ) ∣ d η d t + ∫ Ω ∣ u ( η , t ) ∣ m ∣ ψ t ( η , t ) ∣ d η d t ≤ C ′ ∫ Ω ∣ u ( η , t ) ∣ p ∣ η ∣ H γ ψ d η d t m p ∫ Ω ∣ η ∣ H − γ m p − m ψ − m p − m ∣ Δ H ψ ( η , t ) ∣ p p − m t β p p − m d η d t p − m p + ∫ Ω ∣ u ( η , t ) ∣ p ∣ η ∣ H γ ψ d η d t m p ∫ Ω ∣ η ∣ H − γ p − m ψ − m p − m ∣ ψ t ( η , t ) ∣ p p − m d η d t p − m p ≤ 1 2 ∫ Ω ∣ u ( η , t ) ∣ p ∣ η ∣ H γ ψ d η d t + C ∫ Ω ∣ η ∣ H − γ m p − m ψ − m p − m ∣ Δ H ψ ( η , t ) ∣ p p − m t β p p − m d η d t + ∫ Ω ∣ η ∣ H − γ p − m ψ − m p − m ∣ ψ t ( η , t ) ∣ p p − m d η d t .

In the sequel, C denotes a constant, which may vary from line to line but is independent of the terms that will take part in any limit processing. Therefore the inequality becomes

∫ Ω ∣ u ( η , t ) ∣ p ∣ η ∣ H γ ψ d η d t ≤ C ∫ Ω ∣ η ∣ H − γ m p − m ψ − m p − m ∣ Δ H ψ ( η , t ) ∣ p p − m t β p p − m d η d t + ∫ Ω ∣ η ∣ H − γ p − m ψ − m p − m ∣ ψ t ( η , t ) ∣ p p − m d η d t .

Taking

(15) ψ ( η , t ) = Ψ ∣ ξ ∣ 4 + ∣ ξ ˜ ∣ 4 + τ 2 + t 2 R 4 , η = ( ξ , ξ ˜ , τ ) ∈ H n , t > 0 , R > 0

with Ψ ∈ C c ∞ ( R + ) satisfies 0 ≤ Ψ ≤ 1 and

(16) Ψ ( r ) = 1 , if 0 ≤ r ≤ 1 , ↘ , if 1 < r ≤ 2 , 0 , if r > 2 .

We note that supp ( ψ ) and supp ( Δ H ψ ) are the subsets of

Σ R = { ( η , t ) = ( ξ , ξ ˜ , τ , t ) ∈ H n × R + : ∣ ξ ∣ 4 + ∣ ξ ˜ ∣ 4 + ∣ τ ∣ 2 + ∣ t ∣ 2 ≤ 2 R 4 } .

We note that from [11],

∣ Δ H ψ ∣ ≤ C R − 2 ,

and

∣ ψ t ∣ ≤ C R − 2 .

Let us denote

Ω 1 ≔ { ( η ˜ , t ˜ ) ∣ ( ξ ¯ . ξ ˆ , τ ˜ , t ˜ ) ∈ H n × R + : ∣ ξ ¯ ∣ 4 + ∣ ξ ˆ ∣ 4 + ∣ τ ˜ ∣ 2 + ∣ t ˜ ∣ 2 ≤ 2 }

and

μ = ∣ ξ ¯ ∣ 4 + ∣ ξ ˆ ∣ 4 + ∣ τ ˜ ∣ 2 + ∣ t ˜ ∣ 2 .

We perform the change of variables R ξ ¯ = ξ , R ξ ˆ = ξ ˜ , R 2 τ ˜ = τ , R 2 t ˜ = t , and we obtain

(17) ∫ Ω ∣ η ∣ H − γ p − m ψ − m p − m ∣ ψ t ( η , t ) ∣ p p − m d η d t ≤ R Q + 2 − γ p − m − 2 p p − m ∫ Ω 1 ( Ψ ∘ μ ) − p p − m ∂ ( Ψ ∘ μ ) ∂ t p p − m d η ˜ d t ˜ ≤ C R Q + 2 − 2 p + γ p − m

and

(18) ∫ Ω ∣ η ∣ H − γ m p − m ψ − m p − m ∣ Δ H ψ ( η , t ) ∣ p p − m t β p p − m d η d t ≤ C R 2 β p p − m − γ m p − m − 2 p p − m + Q + 2 ∫ Ω 1 t ˜ β p p − m ∣ Ψ ∘ μ ∣ − p p − m ∣ Δ H ( Ψ ∘ μ ) ∣ p p − m d η ˜ d t ˜ ≤ C R 2 β p p − m − γ m p − m − 2 p p − m + Q + 2 .

Combining (17) with (18), we obtain

(19) ∫ Ω ∣ u ( η , t ) ∣ p ∣ η ∣ H γ ψ d η d t ≤ C R 2 β p p − m − γ m p − m − 2 p p − m + Q + 2 .

Here, we choose

2 β p p − m − γ m p − m − 2 p p − m + Q + 2 < 0 ,

i.e.,

p < ( Q + 2 + γ ) m Q + 2 + 2 β − 2 m .

Letting R → ∞ , one has

∫ Ω ∣ u ( η , t ) ∣ p ∣ η ∣ H γ d η d t ≤ 0 ,

arriving at a contradiction.□

Now, we will show a Liouville-type result for the system of Kirchhoff-type parabolic equations in Ω ≔ H n × ( 0 , T ) , T > 0 .

Definition 3.2

A pair of solutions ( u , v ) ∈ C 1 ( 0 , T ; S 2 2 ( H n ) ) ∩ L q ( H n ) × C 1 ( 0 , T ; S 2 2 ( H n ) ) ∩ L p ( H n ) with p , q > 1 is a weak solution of System (3) on Ω = H n × ( 0 , T ) with the Cauchy date ( u 0 , v 0 ) ∈ L loc 1 ( H n ) × L loc 1 ( H n ) on the Heisenberg group, if the following identities

(20) ∫ Ω ∣ η ∣ H γ 1 ∣ v ∣ p ψ d η d t + ∫ H n u 0 ( η ) ψ ( η , 0 ) d η = − ∫ Ω u m 1 ψ t d η d t − ∫ Ω M 1 u m 1 Δ H ψ d η d t

and

(21) ∫ Ω ∣ η ∣ H γ 2 ∣ u ∣ q ψ d η d t + ∫ H n v 0 ( η ) ψ ( η , 0 ) d η = − ∫ Ω v m 2 ψ t d η d t − ∫ Ω M 2 v m 2 Δ H ψ d η d t ,

hold for any test functions 0 ≤ ψ ∈ C 2 ( ( 0 , T ) ; S 2 2 ( H n ) ) ∩ C 1 ( [ 0 , T ) ; S 2 2 ( H n ) ) , where M 1 and M 2 are defined in (4).

Proof of Theorem 1.3

Following the scheme of proof of Theorem 1.2, if ( u , v ) is a nontrivial weak solution of (3), from (20), we have

(22) ∫ Ω ∣ η ∣ H γ 1 ∣ v ∣ p ψ d η d t ≤ ∫ Ω ∣ η ∣ H γ 1 ∣ v ∣ p ψ d η d t + ∫ H n u 0 ( η ) ψ ( η , 0 ) d η = − ∫ Ω u m 1 ψ t d η d t − ∫ Ω M 1 u m 1 Δ H ψ d η d t ≤ C 1 ∫ Ω t β 1 ∣ u ∣ m 1 ∣ Δ H ψ ∣ d η d t + ∫ Ω ∣ u ∣ m 1 ∣ ψ t ∣ d η d t ≤ C 1 ∫ Ω ∣ u ∣ q ∣ η ∣ H γ 2 ψ d η d t m 1 q ∫ Ω t q β 1 q − m 1 ∣ η ∣ H − m 1 γ 2 q − m 1 ψ − m 1 q − m 1 ∣ Δ H ψ ∣ q q − m 1 d η d t q − m 1 q + ∫ Ω ∣ u ∣ q ∣ η ∣ H γ 2 ψ d η d t m 1 q ∫ Ω ψ − m 1 q − m 1 ∣ η ∣ H − m 1 γ 2 q − m 1 ∣ ψ t ∣ q q − m 1 d η d t q − m 1 q ≤ C 1 ∫ Ω ∣ u ∣ q ∣ η ∣ H γ 2 ψ d η d t m 1 q I 1 ,

where

(23) I 1 = ∫ Ω t q β 1 q − m 1 ∣ η ∣ H − m 1 γ 2 q − m 1 ψ − m 1 q − m 1 ∣ Δ H ψ ∣ q q − m 1 d η d t q − m 1 q + ∫ Ω ψ − m 1 q − m 1 ∣ η ∣ H − m 1 γ 2 q − m 1 ∣ ψ t ∣ q q − m 1 d η d t q − m 1 q < ∞ .

Similarly, we have

(24) ∫ Ω ∣ η ∣ H γ 2 ∣ u ∣ q ψ d η d t ≤ C 2 ∫ Ω ∣ v ∣ p ∣ η ∣ H γ 1 ψ d η d t m 2 p ∫ Ω t p β 2 p − m 2 ∣ η ∣ H − m 2 γ 1 p − m 2 ψ − m 2 p − m 2 ∣ Δ H ψ ∣ p p − m 2 d η d t p − m 2 p + ∫ Ω ∣ v ∣ p ∣ η ∣ H γ 1 ψ d η d t m 2 p ∫ Ω ψ − m 2 p − m 2 ∣ η ∣ H − m 2 γ 1 p − m 2 ∣ ψ t ∣ p p − m 2 d η d t p − m 2 p ≤ C 2 ∫ Ω ∣ v ∣ p ∣ η ∣ H γ 1 ψ d η d t m 2 p I 2 ,

where

(25) I 2 = ∫ Ω t p β 2 p − m 2 ∣ η ∣ H − m 2 γ 1 p − m 2 ψ − m 2 p − m 2 ∣ Δ H ψ ∣ p p − m 2 d η d t p − m 2 p + ∫ Ω ψ − m 2 p − m 2 ∣ η ∣ H − m 2 γ 1 p − m 2 ∣ ψ t ∣ p p − m 2 d η d t p − m 2 p < ∞ .

Finally, we obtain

(26) ∫ Ω ∣ v ∣ p ∣ η ∣ H γ 1 ψ d η d t 1 − m 1 m 2 p q ≤ C 3 I 2 m 1 q I 1 ,

where C 3 = C 2 m 1 q and 1 − m 1 m 2 p q > 0 .

Similarly, we have

(27) ∫ Ω ∣ u ∣ q ∣ η ∣ H γ 2 ψ d η d t 1 − m 1 m 2 p q ≤ C 4 I 1 m 2 p I 2 ,

where C 4 = C 1 m 2 p .

Using the homogeneous dimension Q of H n , one calculates

(28) I 1 = ∫ Ω t q β 1 q − m 1 ∣ η ∣ H − m 1 γ 2 q − m 1 ψ − m 1 q − m 1 ∣ Δ H ψ ∣ q q − m 1 d η d t q − m 1 q + ∫ Ω ψ − m 1 q − m 1 ∣ η ∣ H − m 1 γ 2 q − m 1 ∣ ψ t ∣ q q − m 1 d η d t q − m 1 q ≤ C R 2 q β 1 q − m 1 γ 2 q − 2 + ( Q + 2 ) ( q − m 1 ) q + R − m 1 γ 2 q − 2 + ( Q + 2 ) ( q − m 1 ) q ≤ C R 2 β 1 + ( Q + 2 ) ( q − m 1 ) − m 1 γ 2 q − 2 ≐ C R α 1 ,

where

α 1 = 2 β 1 + ( Q + 2 ) ( q − m 1 ) − m 1 γ 2 q − 2 .

Analogously, we obtain

(29) I 2 = ∫ Ω t p β 2 p − m 2 ∣ η ∣ H − m 2 γ 1 p − m 2 ψ − m 2 p − m 2 ∣ Δ H ψ ∣ p p − m 2 d η d t p − m 2 p + ∫ Ω ψ − m 2 p − m 2 ∣ η ∣ H − m 2 γ 1 p − m 2 ∣ ψ t ∣ p p − m 2 d η d t p − m 2 p ≤ C R 2 p β 2 p − m 2 γ 1 p − 2 + ( Q + 2 ) ( p − m 2 ) p + R − m 2 γ 1 p − 2 + ( Q + 2 ) ( p − m 2 ) p ≤ C R 2 β 2 + ( Q + 2 ) ( p − m 2 ) − m 2 γ 1 p − 2 ≐ C R α 2 ,

where

α 2 = 2 β 2 + ( Q + 2 ) ( p − m 2 ) − m 2 γ 1 p − 2 .

Consequently, inserting this estimation in the right-hand side of (27), we obtain

(30) ∫ Ω ∣ u ∣ q ∣ η ∣ H γ 2 ψ d η d t 1 − m 1 m 2 p q ≤ I 1 m 2 p I 2 ≤ C R m 2 α 1 p + α 2 .

If m 2 α 1 p + α 2 < 0 , by calculating, we have

(31) Q + 2 < m 1 m 2 γ 2 + 2 m 2 q − 2 m 2 β 1 q − 2 β 2 p q + m 1 γ 1 q + p q m 2 ( q − m 1 ) + q ( p − m 2 ) .

Repeating the aforementioned arguments, we obtain

(32) ∫ Ω ∣ v ∣ p ∣ η ∣ H γ 1 ψ d η d t 1 − m 1 m 2 p q ≤ I 2 m 1 q I 1 ≤ C R m 1 α 2 q + α 1 ,

and we can require

(33) Q + 2 < m 1 m 2 γ 1 + 2 m 1 p − 2 m 1 β 2 p − 2 β 1 p q + m 2 γ 2 p + p q m 1 ( p − m 2 ) + p ( q − m 1 ) .

Finally, letting R → ∞ , we arrive at a contradiction.□

Acknowledgment

The authors would like to thank the anonymous referees for their thorough reading and insightful comments.

Funding information: This research was supported by NNSF (11971061, 12271028), BNSF (1222017), and the Fundamental Research Funds for the Central Universities.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflicts of interest.
Data availability statement: No additional data are available.

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Received: 2022-07-16

Revised: 2023-09-18

Accepted: 2023-09-22

Published Online: 2023-11-13

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

Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group

Abstract

1 Introduction

Remark 1.1

Theorem 1.2

Theorem 1.3

2 Preliminaries

3 Parabolic-type equations and system

Definition 3.1

Proof of Theorem 1.2

Definition 3.2

Proof of Theorem 1.3

Acknowledgment

References

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