A Liouville theorem for superlinear parabolic equations on the Heisenberg group

Juncheng Wei; Ke Wu

doi:10.1515/ans-2023-0119

Article Open Access

A Liouville theorem for superlinear parabolic equations on the Heisenberg group

Juncheng Wei and Ke Wu

Published/Copyright: March 1, 2024

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From the journal Advanced Nonlinear Studies Volume 24 Issue 1

Abstract

We consider a parabolic nonlinear equation on the Heisenberg group. Applying the Gidas–Spruck type estimates, we prove that under suitable conditions, the equation does not have positive solutions. As an application of the nonexistence result, we provide optimal universal estimates for positive solutions.

Keywords: Heisenberg group; Liouville theorem; blow up rate estimate; parabolic nonlinear equation; Gidas–Spruck type estimates

1991 Mathematics Subject Classification: Primary 35K58, 35B09; Secondary 35J60

1 Introduction

In this paper, we consider the parabolic equation

(1.1) ∂ s u = Δ H u + u p , in H n × ( − ∞ , + ∞ ) ,

where H n is the Heisenberg group. We recall that H n is the Lie group ( C n × R , ◦ ) equipped with the group action

( z , t ) ◦ ( ξ , t ′ ) = z + ξ , t + t ′ + 2 Im ∑ i = 1 n z i ξ ̄ i .

Denoting by Q = 2n + 2 the homogeneous dimension of H n and by | ( z , t ) | H the intrinsic metric, where

| ( z , t ) | H = ( | z | 4 + t 2 ) 1 4 .

The CR structure of H n is spanned by the left-invariant vector fields

Z i = ∂ ∂ z i + − 1 z ̄ i ∂ ∂ t , i = 1 , … , n .

For a smooth function u on H n , we denote

u i = Z i u , u i ̄ = Z i ̄ u , u 0 = ∂ u ∂ t ,

where

Z i ̄ = ∂ ∂ z ̄ i − − 1 z i ∂ ∂ t , i = 1 , … , n .

Then

| ∇ u | 2 = ∑ i = 1 n u i u i ̄

and Δ H is the subelliptic Laplacian defined by

Δ H = ∑ i = 1 n ( Z i Z i ̄ + Z i ̄ Z i ) .

In the Euclidean space, the analogous equation corresponding to (1.1) is

(1.2) ∂ s u = Δ u + u p , in R n × ( − ∞ , + ∞ ) ,

If u is a stationary solution of (1.2), then u satisfies the elliptic equation

(1.3) Δ u + u p = 0 , in R n .

For the Equation (1.3), Gidas and Spruck proved in their seminal paper [1] that if n > 2, 1 < p < (n + 2)/(n − 2) and if u is a non-negative C ² solution, then u ≡ 0. This result is obtained through the use of a special vector field constructed from the solution. The basic form of the vector field used in Ref. [1] appeared previously in a geometric result of Obata (see Ref. [2]) concerning conformal deformations of the usual metric on S ⁿ. In Ref. [3], Chen-Li give a new proof of the nonexistence result in Ref. [1]. The proof in Ref. [3] is based on the moving plane method. If p = (n + 2)/(n − 2), then (1.3) is a special case of the Yamabe problem in conformal geometry. In Ref. [4], using the asymptotic symmetry technique, Caffarelli, Gidas and Spruck classified all the positive solutions of (1.3) for n ≥ 3. They showed that any positive solutions of (1.3) can be written in the form

u a , λ ( x ) = λ n ( n − 2 ) λ 2 + | x − a | 2 n − 2 2 ,

where λ > 0 and a is some point in R n .

For the Equation (1.2), the exponent p _F = 1 + 2/n plays the role of a critical exponent for the Cauchy problem. It is known that if 1 < p ≤ p _F, then the Equation (1.2) does not admit any nontrivial non-negative distributional solution. However, in the range p _F < p < (n + 2)/(n − 2), it is more difficult to classify positive classical solutions. If we assume that u is radially symmetric, then it is proved in Ref. [5] that the nonexistence of positive bounded solutions holds for 1 < p < p _S, where

p S = + ∞ , if 1 ≤ n ≤ 2 , n + 2 n − 2 , if n ≥ 3 .

Without the symmetry assumption, it is proved in Ref. [6] that if 1 < p < p _B, where

p B = + ∞ , if n = 1 , n ( n + 2 ) ( n − 1 ) 2 , if n ≥ 2 ,

then (1.2) has no positive classical solution. The method of the proof in Ref. [6] is based on a modification of the technique of local, integral gradient estimates developed in Ref. [1] for elliptic problems. By using the monotonicity formula and the energy estimates for the rescaled problems, Quittner proved in Ref. [7] that the nonexistence of positive solutions holds when n > 2, 1 < p < n/(n − 2). In Ref. [8], Quittner proved the optimal Liouville theorem for the Equation (1.2) by further refining the method used in Ref. [7].

Because of these important contributions, the structure of positive solutions of (1.2) has been well understood. Compared with the Equation (1.2), the results concerning positive solutions of (1.1) are less known. For the Equation (1.1), it is proved in Ref. [9] that if 1 < p ≤ Q/(Q − 2) and if u is a nonnegative stationary solution, then u ≡ 0. By applying the moving plane method, Birindelli and Prajapat proved in Ref. [10] that if 1 < p < (Q + 2)/(Q − 2) and if u is a nonnegative stationary solution of (1.1) such that u(z, t) = u(|z|, t), then u ≡ 0. In Ref. [11] , Yu generalized the method in Ref. [10] to semilinear elliptic equations with general nonlinearities. In Ref. [12], Xu improved the Liouville type result in Ref. [9] to the range n > 1, 1 < p < (Q(Q + 2))/(Q − 1)². In an interesting paper [13], Ma and Ou gave a complete classification of nonnegative stationary solutions to the Equation (1.1) when 1 < p < (Q + 2)/(Q − 2). Their proof is based on a generalized formula of that found by Jerison and Lee [14]. In Ref. [15], by using a technique developed in Ref. [16], we constructed infinitely many axial symmetric singular positive solutions to the equation

Δ H u + u p = 0 , in H n \ { 0 }

when n > 1 and p is supercritical.

In this paper, we are interested in positive solutions of (1.1) which are not necessary to be stationary solutions. The main result is the following.

Theorem 1.1

If n > 1, 1 < p < 1 + (4Q − 10)/(Q ² − 4Q + 6), then the Equation (1.1) does not have positive solution.

Remark 1.2

If n > 1, 1 < p < 1 + 2/Q, then Theorem 1.1 is a special case of the results in Refs. [17], [18]. Indeed, some Fujita-type results for the Cauchy problem are obtained in these two papers.

Remark 1.3

It is easy to check that if n > 1, then

1 < 1 + 2 Q < 1 + 4 Q − 10 Q 2 − 4 Q + 6 .

Therefore, Theorem 1.1 is not covered by the existing non-existence results for the Cauchy-type problem.

Remark 1.4

Inspired by the Equation (1.2), it is natural to conjecture that if n ≥ 1, 1 < p < (Q + 2)/(Q − 2), then the Equation (1.1) does not possess positive classical solutions.

In 2007, Poláčik, Quittner and Souplet proved in their famous paper [19] that a Liouville theorem for scaling invariant superlinear parabolic problems would always imply optimal universal estimates for solutions of related initial value problems, including estimates of their singularities and decay. As a consequence of the main result in Ref. [19] and Theorem 1.1, we have the following result.

Proposition 1.5

Let n > 1, 1 < p < 1 + (4Q − 10)/(Q ² − 4Q + 6) and let u be a positive solution of the equation

(1.4) ∂ s u = Δ H u + u p , in H n × ( − ∞ , 0 ) ,

then there exists a positive constant c independent of n and p such that

(1.5) u ( z , t , s ) ≤ c ( − s ) − 1 p − 1 , in H n × ( − ∞ , 0 ) .

Remark 1.6

In a series of papers [20]–[22], Giga and Kohn studied the asymptotic behavior of positive classical solutions to the equation

(1.6) ∂ s u = Δ u + u p , in R n × ( − ∞ , 0 )

when 1 < p < p _S. By combining the results in Refs. [20]–[22], we know that if u is a positive classical solution of (1.6), then either

lim t → 0 ( − t ) 1 p − 1 u ( 0 , t ) = 1 p − 1 1 p − 1

or the origin is not a blow up point. It will be an interesting problem to know whether similar result holds for the Equation (1.4). The main difficulty is that we do not know how to establish a suitable monotonicity formula.

The content of this paper will be organized as follows. In Section 2, we derive a family of integral estimates relating any smooth function with its gradient and its sublaplacian. In Section 3, we give the proof of main results by using the estimates established in Section 2.

2 An integral estimate based on the Bochner formula

In this section, we provide a family of integral estimates for arbitrary smooth real functions. The proof relies on the Bochner identity.

Lemma 2.1

Let u be a real smooth function, then

(2.1) ∑ j = 1 n ( | ∇ u | 2 ) j j ̄ = ∑ i , j = 1 n ( u i j j ̄ u i ̄ + u i u i ̄ j j ̄ + u i j u i ̄ j ̄ + u i j ̄ u i ̄ j ) .

Proof

The proof is straightforward, so we will omit the details. □

Since u is real, we know from Lemma 2.1 that

(2.2) ∑ j = 1 n ( | ∇ u | 2 ) j ̄ j = ∑ i , j = 1 n ( u i ̄ j ̄ j u i + u i ̄ u i j ̄ j + u i j u i ̄ j ̄ + u i j ̄ u i ̄ j ) .

By combining (2.1) and (2.2), we have the following lemma.

Lemma 2.2

Let u be a real smooth function, then

(2.3) 1 2 Δ H ( | ∇ u | 2 ) = 1 2 ∑ i , j = 1 n ( u i j j ̄ u i ̄ + u i u i ̄ j j ̄ + u i ̄ j ̄ j u i + u i ̄ u i j ̄ j ) + ∑ i , j = 1 n ( u i j u i ̄ j ̄ + u i j ̄ u i ̄ j ) .

Remark 2.3

The Bochner formula is used by Greenleaf in Ref. [23] to derive an analogue of the Lichnerowicz estimate for the sublaplacian on a pseudo-hermitian manifold. It is also used in Ref. [24] to do the same thing.

In order to continue the proof, we define

(2.4) E i j ̄ u = u i j ̄ − 1 n ∑ α = 1 n u α α ̄ δ i j ̄ ,

then

(2.5) ∑ i , j = 1 n | E i j ̄ u | 2 = ∑ i , j = 1 n u i j ̄ − 1 n ∑ α = 1 n u α α ̄ δ i j ̄ u i ̄ j − 1 n ∑ α = 1 n u α ̄ α δ i ̄ j = ∑ i , j = 1 n u i j ̄ u i ̄ j − 1 n ∑ α = 1 n u α α ̄ ∑ α = 1 n u α ̄ α = ∑ i , j = 1 n u i j ̄ u i ̄ j − 1 n n − 1 u 0 + 1 2 Δ H u − n − 1 u 0 + 1 2 Δ H u = ∑ i , j = 1 n u i j ̄ u i ̄ j − 1 n n 2 u 0 2 + 1 4 ( Δ H u ) 2 .

By combing (2.3) and (2.5), we have the following lemma.

Lemma 2.4

Let u be a real smooth function, then

(2.6) 1 2 Δ H ( | ∇ u | 2 ) = 1 2 ∑ i , j = 1 n ( u i j j ̄ u i ̄ + u i u i ̄ j j ̄ + u i ̄ j ̄ j u i + u i ̄ u i j ̄ j ) + ∑ i , j = 1 n u i j u i ̄ j ̄ + | E i j ̄ u | 2 + 1 n n 2 u 0 2 + 1 4 ( Δ H u ) 2 .

Let ϕ ∈ C 0 ∞ ( H n ) , 0 ≤ ϕ ≤ 1 and let v = u 2 r + 2 , where r > 0 is a parameter which will be determined later. Since u = v r + 2 2 , then

(2.7) ∂ s u = r + 2 2 v r 2 ∂ s v ,

(2.8) | ∇ u | 2 = ( r + 2 ) 2 4 v r | ∇ v | 2 .

Multiplying the both sides of (2.6) by ϕ ^q with q be a parameter to be determined, then we have

0 = − I 1 + I 2 + I 3 + I 4 + I 5 ,

where

I 1 = 1 2 ∫ Δ H ( | ∇ u | 2 ) ϕ q , I 2 = 1 2 ∑ i , j = 1 n ∫ ( u i ̄ j j ̄ u i + u i j ̄ j u i ̄ ) ϕ q , I 3 = ∑ i , j = 1 n ∫ | E i j ̄ u | 2 ϕ q , I 4 = 1 n ∫ n 2 u 0 2 + 1 4 ( Δ H u ) 2 ϕ q , I 5 = 1 2 ∑ i , j = 1 n ∫ ( u i j u i ̄ j ̄ + u i u i ̄ j ̄ j + u i j u i ̄ j ̄ + u i ̄ u i j j ̄ ) ϕ q .

We will compute the integrals terms by terms.

(2.9) I 1 = 1 2 ∫ Δ H ( | ∇ u | 2 ) ϕ q = ( r + 2 ) 2 8 ∫ v r | ∇ v | 2 Δ H ( ϕ q ) . I 2 = 1 2 ∑ i , j = 1 n ∫ ( u i ̄ j j ̄ u i + u i j ̄ j u i ̄ ) ϕ q = 1 2 ∑ i , j = 1 n ∫ [ ( u j ̄ j i ̄ + 2 − 1 δ j j ̄ u 0 i ̄ − 2 − 1 δ j i ̄ u 0 j ̄ ) ] u i ϕ q + 1 2 ∑ i , j = 1 n ∫ [ ( u j j ̄ i − 2 − 1 δ j j ̄ u 0 i + 2 − 1 δ i j ̄ u 0 j ) ] u i ̄ ϕ q = ( n − 1 ) − 1 ∑ i = 1 n ∫ ( u 0 i ̄ u i − u 0 i u i ̄ ) ϕ q − 1 2 ∑ i , j = 1 n ∫ u j ̄ j ( u i i ̄ ϕ q + u i ( ϕ q ) i ̄ ) − 1 2 ∑ i , j = 1 n ∫ u j j ̄ ( u i ̄ i ϕ q + u i ̄ ( ϕ q ) i ) = 2 n ( n − 1 ) ∫ u 0 2 ϕ q − ( n − 1 ) − 1 ∑ i = 1 n ∫ u 0 u i ( ϕ q ) i ̄ − u i ̄ ( ϕ q ) i − ∫ − n − 1 u 0 + 1 2 Δ H u n − 1 u 0 + 1 2 Δ H u ϕ q − 1 2 ∑ i = 1 n ∫ − n − 1 u 0 + 1 2 Δ H u ( u i ( ϕ q ) i ̄ ) − 1 2 ∑ i = 1 n ∫ n − 1 u 0 + 1 2 Δ H u ( u i ̄ ( ϕ q ) i ) = n ( n − 2 ) ( r + 2 ) 2 4 ∫ v 0 2 ϕ q v r − 1 4 ∫ ( Δ H u ) 2 ϕ q − ( r + 2 ) 2 8 ( n − 2 ) − 1 ∑ i = 1 n ∫ v 0 ( v i ( ϕ q ) i ̄ − v i ̄ ( ϕ q ) i ) v r − r + 2 8 ∑ i = 1 n ∫ Δ H u ( v i ( ϕ q ) i ̄ + v i ̄ ( ϕ q ) i ) v r 2 .

(2.10) I 3 = ∑ i , j = 1 n ∫ | E i j ̄ u | 2 ϕ q = ∑ i , j = 1 n ∫ r ( r + 2 ) 4 v r − 2 2 v i v j ̄ − 1 n | ∇ v | 2 δ i j ̄ + r + 2 2 v r 2 E i j ̄ v × r ( r + 2 ) 4 v r − 2 2 v i ̄ v j − 1 n | ∇ v | 2 δ i ̄ j + r + 2 2 v r 2 E i ̄ j v ϕ q = ( r + 2 ) 2 4 ∑ i , j = 1 n ∫ | E i j ̄ v | 2 ϕ q v r + 1 − 1 n r 2 ( r + 2 ) 2 16 ∫ v r − 2 ϕ q | ∇ v | 4 + r ( r + 2 ) 2 8 ∑ i , j = 1 n ∫ ( v i ̄ j v i v j ̄ + v i j ̄ v i ̄ v j ) ϕ q v r − 1 − r ( r + 2 ) 2 8 n ∑ i , j = 1 n ∫ ∑ α = 1 n v α ̄ α δ i ̄ j v i v j ̄ + ∑ α = 1 n v α α ̄ δ i j ̄ v i ̄ v j ϕ q v r − 1 = ( r + 2 ) 2 4 ∑ i , j = 1 n ∫ | E i j ̄ v | 2 ϕ q v r + 1 − 1 n r 2 ( r + 2 ) 2 16 ∫ v r − 2 ϕ q | ∇ v | 4 + r ( r + 2 ) 2 8 ∑ i , j = 1 n ∫ [ ( v i v i ̄ ) j − v i ̄ v i j ] v j ̄ ϕ q v r − 1 + r ( r + 2 ) 2 8 ∑ i , j = 1 n ∫ [ ( v i v i ̄ ) j ̄ − v i v i ̄ j ̄ ] v j ϕ q v r − 1 − r ( r + 2 ) 2 8 n ∫ v r − 1 ϕ q | ∇ v | 2 Δ H v = ( r + 2 ) 2 4 ∑ i , j = 1 n ∫ | E i j ̄ v | 2 ϕ q v r + 1 − 1 n r 2 ( r + 2 ) 2 16 ∫ v r − 2 ϕ q | ∇ v | 4 − r ( r + 2 ) 2 8 ∑ j = 1 n ∫ | ∇ v | 2 ( v j ̄ j ϕ q v r − 1 + v j ̄ ( ϕ q ) j v r − 1 + v j ̄ ϕ q ( v r − 1 ) j ) − r ( r + 2 ) 2 8 ∑ j = 1 n ∫ | ∇ v | 2 ( v j j ̄ ϕ q v r − 1 + v j ( ϕ q ) j ̄ v r − 1 + v j ϕ q ( v r − 1 ) j ̄ ) − r ( r + 2 ) 2 8 n ∫ v r − 1 ϕ q | ∇ v | 2 Δ H v − r ( r + 2 ) 2 8 ∑ i , j = 1 n ∫ v r − 1 ϕ q ( v i v j v i ̄ j ̄ + v i ̄ v j ̄ v i j ) = ( r + 2 ) 2 4 ∑ i , j = 1 n ∫ | E i j ̄ v | 2 ϕ q v r − 1 + 1 n r ( r + 2 ) 4 ∫ v r − 2 2 Δ H u ϕ q | ∇ v | 2 − r ( r + 2 ) 2 8 − r 2 3 + 1 n + 2 ( r − 1 ) ∫ v r − 2 ϕ q | ∇ v | 4 − r ( r + 2 ) 2 8 ∑ j = 1 n ∫ v r − 1 | ∇ v | 2 ( ϕ q ) j v j ̄ + ( ϕ q ) j ̄ v j − r ( r + 2 ) 2 8 ∑ i , j = 1 n ∫ v r − 1 ϕ q ( v i ̄ v j ̄ v i j + v i v j v i ̄ j ̄ ) .

(2.11) I 4 = 1 n ∫ n 2 u 0 2 + 1 4 ( Δ H u ) 2 ϕ q = n ( r + 2 ) 2 4 ∫ v 0 2 v r ϕ q + 1 4 n ∫ ( Δ H u ) 2 ϕ q .

(2.12) I 5 = 1 2 ∑ i , j = 1 n ∫ ( u i j u i ̄ j ̄ + u i u i ̄ j ̄ j + u i j u i ̄ j ̄ + u i ̄ u i j j ̄ ) ϕ q = ∑ i , j = 1 n ∫ r + 2 2 v r 2 v i j + r ( r + 2 ) 4 v r − 2 2 v i v j × r + 2 2 v r 2 v i ̄ j ̄ + r ( r + 2 ) 4 v r − 2 2 v i ̄ v j ̄ ϕ q + 1 2 ∑ i , j = 1 n ∫ u i [ u j ̄ j i ̄ − 2 − 1 δ i ̄ j u j ̄ 0 ] ϕ q + 1 2 ∑ i , j = 1 n ∫ u i ̄ [ u j j ̄ i + 2 − 1 δ i j ̄ u j 0 ] ϕ q = ( r + 2 ) 2 4 ∑ i , j = 1 n ∫ v r ϕ q | v i j | 2 + r 2 ( r + 2 ) 2 16 ∫ v r − 2 ϕ q | ∇ v | 4 + r ( r + 2 ) 2 8 ∑ i , j = 1 n ∫ v r − 1 ϕ q ( v i j v i ̄ v j ̄ + v i ̄ j ̄ v i v j ) + 1 2 ∑ i = 1 n ∫ u i − n − 1 u 0 + 1 2 Δ H u i ̄ − 2 − 1 u i ̄ 0 ϕ q + 1 2 ∑ i = 1 n ∫ u i ̄ n − 1 u 0 + 1 2 Δ H u i + 2 − 1 u i 0 ϕ q .

Since

(2.13) ∑ i = 1 n ∫ u i − n − 1 u 0 + 1 2 Δ H u i ̄ − 2 − 1 u i ̄ 0 ϕ q = − ( n + 2 ) − 1 ∑ i = 1 n ∫ u i u 0 i ̄ ϕ q + 1 2 ∑ i = 1 n ∫ u i ( Δ H u ) i ̄ ϕ q = ( n + 2 ) − 1 ∫ n − 1 u 0 + 1 2 Δ H u u 0 ϕ q + ( n + 2 ) − 1 ∑ i = 1 n ∫ u i u 0 ( ϕ q ) i ̄ − 1 2 ∫ n − 1 u 0 + 1 2 Δ H u Δ H u ϕ q − 1 2 ∑ i = 1 n ∫ Δ H u u i ( ϕ q ) i ̄ ,

we have

(2.14) I 5 = ( r + 2 ) 2 4 ∑ i , j = 1 n ∫ v r ϕ q | v i j | 2 + r 2 ( r + 2 ) 2 16 ∫ v r − 2 ϕ q | ∇ v | 4 + r ( r + 2 ) 2 8 ∑ i , j = 1 n ∫ v r − 1 ϕ q ( v i j v i ̄ v j ̄ + v i ̄ j ̄ v i v j ) − n ( n + 2 ) ( r + 2 ) 2 4 ∫ v r ϕ q v 0 2 − 1 4 ∫ ( Δ H u ) 2 ϕ q − r + 2 8 ∑ i = 1 n ∫ Δ H u ( v i ( ϕ q ) i ̄ + v i ̄ ( ϕ q ) i ) v r 2 + ( r + 2 ) 2 8 ( n + 2 ) − 1 ∑ i = 1 n ∫ v 0 ( v i ( ϕ q ) i ̄ − v i ̄ ( ϕ q ) i ) v r .

On the other hand, using integration by parts, we have

(2.15) I 5 = 1 2 ∑ i , j = 1 n ∫ ( u i j u i ̄ j ̄ + u i u i ̄ j ̄ j + u i j u i ̄ j ̄ + u i ̄ u i j j ̄ ) ϕ q = − 1 2 ∑ i , j = 1 n ∫ u i u i ̄ j ̄ ( ϕ q ) j − 1 2 ∑ i , j = 1 n ∫ u i ̄ u i j ( ϕ q ) j ̄ = 1 2 ∑ i , j = 1 n ∫ u i i ̄ u j ̄ ( ϕ q ) j + 1 2 ∑ i , j = 1 n ∫ u i u j ̄ ( ϕ q ) j i ̄ + 1 2 ∑ i , j = 1 n ∫ u i ̄ i u j ( ϕ q ) j ̄ + 1 2 ∑ i , j = 1 n ∫ u i ̄ u j ( ϕ q ) j ̄ i = 1 2 ∑ j = 1 n ∫ r + 2 2 n − 1 v r 2 v 0 + 1 2 Δ H u r + 2 2 v r 2 v j ̄ ( ϕ q ) j + 1 2 ∑ j = 1 n ∫ − r + 2 2 n − 1 v r 2 v 0 + 1 2 Δ H u r + 2 2 v r 2 v j ( ϕ q ) j ̄ + ( r + 2 ) 2 8 ∑ i , j = 1 n ∫ ( v i v j ̄ ( ϕ q ) j i ̄ + v i ̄ v j ( ϕ q ) j ̄ i ) v r . = ( r + 2 ) 2 8 n − 1 ∑ i = 1 n ∫ v r ( ( ϕ q ) i v i ̄ − ( ϕ q ) i v i ̄ ) v 0 + r + 2 8 ∑ i = 1 n ∫ v r 2 Δ H u ( ( ϕ q ) i v i ̄ + ( ϕ q ) i ̄ v i ) + ( r + 2 ) 2 8 ∑ i , j = 1 n ∫ v r ( ( ϕ q ) j i ̄ v i v j ̄ + ( ϕ q ) j ̄ i v i ̄ v j ) .

We write I ₅ as η times (2.14) plus (1 − η) times (2.15), where η ∈ (0, 1) is a positive constant. From the above computations, we can get that

0 = − I 1 + I 2 + I 3 + I 4 + η I 5 + ( 1 − η ) I 5

is equivalent to

(2.16) ( r + 2 ) 2 4 ∑ i , j = 1 n ∫ | E i j ̄ v | 2 ϕ q v r + λ 1 ∫ v r − 2 ϕ q | ∇ v | 4 + λ 2 ∫ ( Δ H u ) 2 ϕ q + λ 3 ∫ v r ϕ q v 0 2 + λ 4 ∑ i , j = 1 n ∫ v r ϕ q | v i j | 2 + λ 5 ∫ ( Δ H u ) ϕ q | ∇ v | 2 v r − 2 2 = λ 6 ∑ i = 1 n ∫ v r − 1 | ∇ v | 2 ( ϕ q ) i v i ̄ + ( ϕ q ) i ̄ v i + λ 7 ∫ v r | ∇ v | 2 Δ H ( ϕ q ) + λ 8 ∑ i , j = 1 n ∫ v r ( ϕ q ) j i ̄ v i v j ̄ + ( ϕ q ) j ̄ i v i ̄ v j + λ 9 ∑ i = 1 n ∫ v r ( ϕ q ) i v i ̄ − ( ϕ q ) i ̄ v i v 0 + λ 10 ∑ i , j = 1 n ∫ v r − 1 ϕ q ( v i ̄ v j ̄ v i j + v i v j v i ̄ j ̄ ) + λ 11 ∑ i = 1 n ∫ v i ̄ ( ϕ q ) i + v i ( ϕ q ) i ̄ Δ H u v r 2 ,

where

(2.17) λ 1 = ( r + 2 ) 2 4 1 n − 1 − η r 2 2 + 1 2 ( 2 + η ) r 2 − r ( r − 1 ) , λ 2 = − n − 1 4 n − η 4 , λ 3 = [ n ( n − 1 ) − η n ( n + 2 ) ] ( r + 2 ) 2 4 , λ 4 = ( r + 2 ) 2 4 η , λ 5 = − 1 + 1 n r ( r + 2 ) 4 , λ 6 = r ( r + 2 ) 2 8 , λ 7 = ( r + 2 ) 2 8 , λ 8 = − ( r + 2 ) 2 8 ( 1 − η ) , λ 9 = ( r + 2 ) 2 4 − 1 [ ( n + 1 ) η − ( n − 1 ) ] , λ 10 = ( r + 2 ) 2 r 8 ( 1 − η ) , λ 11 = r + 2 4 η .

Remark 2.5

The reason why we split the term I ₅ into two forms is that in the next section, we need to use λ ₄ > 0. The idea that splitting the term I ₅ into two forms is borrowed from Refs. [12], [24].

Remark 2.6

In the next section, we also need to use λ ₃ > 0, n = 1 prevents λ ₃ from being positive.

3 The proof of the main results

In this section, we turn to the proof of the main results. The Proof of Theorem 1.1 is the analogue of the one used in Ref. [25] (Theorem 21.1) to prove the Liouville-type theorem for the Equation (1.2).

Proof of Theorem 1.1

Let u be a positive solution of the equation

(3.1) ∂ s u = Δ H u + u p , in H n × ( − ∞ , ∞ )

and let ϕ be a smooth function such that

ϕ ( z , t , s ) = 1 , in | ( z , t ) | H ≤ 1 2 × − 1 2 , 1 2 , 0 , in H n × ( − ∞ , ∞ ) \ { | ( z , t ) | H ≤ 1 } × ( − 1,1 ) , 0 ≤ ϕ ≤ 1 , in { | ( z , t ) | H ≤ 1 } × ( − 1,1 ) .

In the rest of this section, we will use the notation ∬ = ∫ ∞ ∞ ∫ H n for simplicity. By using (3.1) and integration by part, we have

(3.2) ∬ ( ∂ s u − u p ) 2 ϕ q = ( r + 2 ) 2 4 ∬ ( ∂ s v ) 2 ϕ q v r + ∬ ϕ q v p ( r + 2 ) − ( r + 2 ) ∬ ∂ s v ϕ q v − 1 + ( p + 1 ) ( r + 2 ) 2 = ( r + 2 ) 2 4 ∬ ( ∂ s v ) 2 ϕ q v r + ∬ ϕ q v p ( r + 2 ) + 2 q p + 1 ∬ ϕ q − 1 ∂ s ϕ v ( p + 1 ) ( r + 2 ) 2 .

Since

(3.3) ∬ ϕ q | ∇ v | 2 v r + ( p − 1 ) ( r + 2 ) 2 = 1 2 ( r + 1 ) + ( p − 1 ) ( r + 2 ) ∑ i = 1 n ∬ ϕ q v i v r + 1 + ( p − 1 ) ( r + 2 ) 2 i ̄ + 1 2 ( r + 1 ) + ( p − 1 ) ( r + 2 ) ∑ i = 1 n ∬ ϕ q v i ̄ v r + 1 + ( p − 1 ) ( r + 2 ) 2 i = − 1 2 ( r + 1 ) + ( p − 1 ) ( r + 2 ) ∬ ϕ q v r + 1 + ( p − 1 ) ( r + 2 ) 2 Δ H v − 1 2 ( r + 1 ) + ( p − 1 ) ( r + 2 ) ∑ i = 1 n ∬ v r + 1 + ( p − 1 ) ( r + 2 ) 2 ( v i ( ϕ q ) i ̄ + v i ̄ ( ϕ q ) i ) = − 1 2 ( r + 1 ) + ( p − 1 ) ( r + 2 ) ∑ i = 1 n ∬ v r + 1 + ( p − 1 ) ( r + 2 ) 2 ( v i ( ϕ q ) i ̄ + v i ̄ ( ϕ q ) i ) + 2 ( r + 2 ) [ 2 ( r + 1 ) + ( p − 1 ) ( r + 2 ) ] ∬ ϕ q v p ( r + 2 ) + r 2 ( r + 1 ) + ( p − 1 ) ( r + 2 ) ∬ ϕ q v r + ( p − 1 ) ( r + 2 ) 2 | ∇ v | 2 + 2 q [ 2 ( r + 1 ) + ( p − 1 ) ( r + 2 ) ] ( 1 + p ) ( r + 2 ) ∬ ϕ q − 1 ∂ s ϕ v ( p + 1 ) ( r + 2 ) 2 ,

we conclude that

(3.4) ∬ ϕ q | ∇ v | 2 v r + ( p − 1 ) ( r + 2 ) 2 = − 1 p ( r + 2 ) ∑ i = 1 n ∬ v r + 1 + ( p − 1 ) ( r + 2 ) 2 ( v i ( ϕ q ) i ̄ + v i ̄ ( ϕ q ) i ) + 2 p ( r + 2 ) 2 ∬ ϕ q v p ( r + 2 ) + 2 q p ( 1 + p ) ( r + 2 ) 2 ∬ ϕ q − 1 ∂ s ϕ v ( p + 1 ) ( r + 2 ) 2 .

Therefore,

(3.5) ∬ ( ∂ s u − u p ) ϕ q | ∇ v | 2 v r − 2 2 = r + 2 2 ∬ ∂ s v ϕ q | ∇ v | 2 v r − 1 − ∬ ϕ q | ∇ v | 2 v r + ( p − 1 ) ( r + 2 ) 2 = r + 2 2 ∬ ∂ s v ϕ q | ∇ v | 2 v r − 1 − 2 p ( r + 2 ) 2 ∬ ϕ q v p ( r + 2 ) + 1 p ( r + 2 ) ∑ i = 1 n ∬ v r + 1 + ( p − 1 ) ( r + 2 ) 2 ( v i ( ϕ q ) i ̄ + v i ̄ ( ϕ q ) i ) − 2 q p ( p + 1 ) ( r + 2 ) 2 ∬ ϕ q − 1 ∂ s ϕ v ( p + 1 ) ( r + 2 ) 2 .

By (2.16), (2.17), (3.2) and (3.4), we have

(3.6) r + 2 2 2 ∑ i , j = 1 n ∬ | E i j ̄ v | 2 ϕ q v r + λ 1 ∬ v r − 2 ϕ q | ∇ v | 4 + λ 2 ′ ∬ ϕ q v p ( r + 2 ) + λ 3 ∬ v r ϕ q v 0 2 + λ 4 ∑ i , j = 1 n ∬ v r ϕ q | v i j | 2 = λ 6 ∑ i = 1 n ∬ v r − 1 | ∇ v | 2 ( ϕ q ) i v i ̄ + ( ϕ q ) i ̄ v i + λ 7 ∬ v r | ∇ v | 2 Δ H ( ϕ q ) + λ 8 ∑ i , j = 1 n ∬ v r ( ϕ q ) j i ̄ v i v j ̄ + ( ϕ q ) j ̄ i v i ̄ v j + λ 9 ∑ i = 1 n ∬ v r ( ϕ q ) i v i ̄ − ( ϕ q ) i ̄ v i v 0 + λ 10 ∑ i , j = 1 n ∬ v r − 1 ϕ q ( v i ̄ v j ̄ v i j + v i v j v i ̄ j ̄ ) + r + 2 2 λ 11 ∑ i = 1 n ∬ v i ̄ ( ϕ q ) i + v i ( ϕ q ) i ̄ ∂ s v v r + λ 11 ′ ∑ i = 1 n ∬ v r + 1 + ( p − 1 ) ( r + 2 ) 2 ( ( ϕ q ) i ̄ v i + ( ϕ q ) i v i ̄ ) − λ 2 ( r + 2 ) 2 4 ∬ ( ∂ s v ) 2 ϕ q v r − r + 2 2 λ 5 ∬ ∂ s v ϕ q | ∇ v | 2 v r − 1 + λ 12 ∬ ϕ q − 1 ∂ s ϕ v ( p + 1 ) ( r + 2 ) 2 ,

where

(3.7) λ 2 ′ = λ 2 − 2 p ( r + 2 ) 2 λ 5 ,

(3.8) λ 11 ′ = − λ 11 − 1 p ( r + 2 ) λ 5 ,

(3.9) λ 12 = 2 ( p + 1 ) ( r + 2 ) − λ 2 ( r + 2 ) q + λ 5 q p ( r + 2 ) .

We estimate the right hand side of (3.6). By using Young’s inequality, we have

(3.10) λ 6 ∑ i = 1 n ∬ v r − 1 | ∇ v | 2 ( ϕ q ) i v i ̄ + ( ϕ q ) i ̄ v i = q λ 6 ∑ i = 1 n ∬ v r − 1 | ∇ v | 2 ϕ q − 1 ( ϕ i v i ̄ + ϕ i ̄ v i ) ≤ ϵ ∬ v r − 2 ϕ q | ∇ v | 4 + c ∬ v r + 2 ϕ q − 4 | ∇ ϕ | 4 .

Similarly, we have

(3.11) λ 7 ∬ v r | ∇ v | 2 Δ H ( ϕ q ) ≤ ϵ ∬ v r − 2 ϕ q | ∇ v | 4 + c ∬ v r + 2 ϕ q − 4 | Δ H ϕ | + | ∇ ϕ | 2 2 ,

(3.12) λ 8 ∑ i , j = 1 n ∬ v r ( ϕ q ) j i ̄ v i v j ̄ + ( ϕ q ) j ̄ i v i ̄ v j ≤ ϵ ∬ v r − 2 ϕ q | ∇ v | 4 + c ∬ v r + 2 ϕ q − 4 ( | ∇ 2 ϕ | + | ∇ ϕ | 2 ) 2 ,

(3.13) λ 9 ∑ i = 1 n ∬ v r ( ϕ q ) i v i ̄ − ( ϕ q ) i ̄ v i v 0 ≤ ϵ ∬ v r v 0 2 ϕ q + ϵ ∬ v r − 2 ϕ q | ∇ v | 4 + c ∬ v r + 2 ϕ q − 4 | ∇ ϕ | 4 ,

(3.14) λ 10 ∑ i , j = 1 n ∬ v r − 1 ϕ q ( v i ̄ v j ̄ v i j + v i v j v i ̄ j ̄ ) ≤ ( r + 2 ) 2 4 η ∑ i , j = 1 n ∬ v r ϕ q | v i j | 2 + ( r + 2 ) 2 r 2 ( 1 − η ) 2 16 η ∬ v r − 2 ϕ q | ∇ v | 4 .

Next, we have

(3.15) r + 2 2 λ 11 ∑ i = 1 n ∬ v i ̄ ( ϕ q ) i + v i ( ϕ q ) i ̄ ∂ s v v r ≤ ϵ ∬ v r − 2 ϕ q | ∇ v | 4 + c ∬ v r + 2 ϕ q − 4 | ∇ ϕ | 4 + ϵ ∬ ( ∂ s v ) 2 ϕ q v r ,

(3.16) − r + 2 2 λ 5 ∬ ∂ s v ϕ q | ∇ v | 2 v r − 1 ≤ ϵ ∬ v r − 2 ϕ q | ∇ v | 4 + c ∬ ( ∂ s v ) 2 ϕ q v r ,

(3.17) λ 12 ∬ ϕ q − 1 ∂ s ϕ v ( p + 1 ) ( r + 2 ) 2 ≤ ϵ ∬ ϕ q v p ( r + 2 ) + c ∬ ϕ q − 2 ( ∂ s ϕ ) 2 v r + 2 ,

(3.18) λ 11 ′ ∑ i = 1 n ∬ v r + 1 + ( p − 1 ) ( r + 2 ) 2 ( ( ϕ q ) i ̄ v i + ( ϕ q ) i v i ̄ ) ≤ ϵ ∬ ϕ q v p ( r + 2 ) + c ∬ v r + 2 ϕ q − 4 | ∇ ϕ | 4 + ϵ ∬ v r − 2 ϕ q | ∇ v | 4 .

Finally, we estimate ∬ ( ∂ s v ) 2 ϕ q v r . By combining (2.7) and (3.1) and integration by part, we have

(3.19) ∬ ( ∂ s v ) 2 ϕ q v r = 4 ( r + 2 ) 2 ∬ ( Δ H u + u p ) ϕ q ∂ s u = 4 ( r + 2 ) 2 ∬ ∑ i = 1 n ( u i i ̄ + u i ̄ i ) ϕ q ∂ s u + 4 ( r + 2 ) 2 ∬ ϕ q u p ∂ s u = 4 ( r + 2 ) 2 ∬ ∑ i = 1 n ( u i i ̄ + u i ̄ i ) ϕ q ∂ s u − 4 q ( r + 2 ) 2 ( p + 1 ) ∬ ϕ q − 1 ∂ s ϕ v ( p + 1 ) ( r + 2 ) 2 = − 4 ( r + 2 ) 2 ∬ ∑ i = 1 n u i ( ϕ q ) i ̄ + u i ̄ ( ϕ q ) i ∂ s u − 4 ( r + 2 ) 2 ∬ ∑ i = 1 n [ u i ( ∂ s u ) i ̄ + u i ̄ ( ∂ s u ) i ] ϕ q − 4 q ( r + 2 ) 2 ( p + 1 ) ∬ ϕ q − 1 ∂ s ϕ v ( p + 1 ) ( r + 2 ) 2 .

Since

∑ i = 1 n u i ( ∂ s u ) i ̄ + ∑ i = 1 n u i ̄ ( ∂ s u ) i = ∑ i = 1 n ( u i ∂ s u i ̄ + u i ̄ ∂ s u i ) = ∂ s ( | ∇ u | 2 ) ,

we have

(3.20) − 4 ( r + 2 ) 2 ∬ ∑ i = 1 n u i ( ϕ q ) i ̄ + u i ̄ ( ϕ q ) i ∂ s u − 4 ( r + 2 ) 2 ∬ ∑ i = 1 n [ u i ( ∂ s u ) i ̄ + u i ̄ ( ∂ s u ) i ] ϕ q − 4 q ( r + 2 ) 2 ( p + 1 ) ∬ ϕ q − 1 ∂ s ϕ v ( p + 1 ) ( r + 2 ) 2 = − ∬ ∑ i = 1 n v i ( ϕ q ) i ̄ + v i ̄ ( ϕ q ) i v r ∂ s v − 4 q ( r + 2 ) 2 ( p + 1 ) ∬ ϕ q − 1 ∂ s ϕ v ( p + 1 ) ( r + 2 ) 2 − 4 ( r + 2 ) 2 ∬ ∂ s ( | ∇ u | 2 ) ϕ q = − ∬ ∑ i = 1 n v i ( ϕ q ) i ̄ + v i ̄ ( ϕ q ) i v r ∂ s v − 4 q ( r + 2 ) 2 ( p + 1 ) ∬ ϕ q − 1 ∂ s ϕ v ( p + 1 ) ( r + 2 ) 2 + 4 q ( r + 2 ) 2 ∬ ϕ q − 1 | ∇ u | 2 ∂ s ϕ = − ∬ ∑ i = 1 n v i ( ϕ q ) i ̄ + v i ̄ ( ϕ q ) i v r ∂ s v − 4 q ( r + 2 ) 2 ( p + 1 ) ∬ ϕ q − 1 ∂ s ϕ v ( p + 1 ) ( r + 2 ) 2 + q ∬ ϕ q − 1 v r | ∇ v | 2 ∂ s ϕ .

We infer from (3.19) and (3.20) that

(3.21) ∬ ( ∂ s v ) 2 ϕ q v r ≤ ϵ ∬ ( ∂ s v ) 2 v r ϕ q + ϵ ∬ | ∇ v | 4 v r − 2 ϕ q + ϵ ∬ ϕ q v p ( r + 2 ) + c ∬ ϕ q − 2 ( ∂ s ϕ ) 2 v r + 2 + c ∬ ϕ q − 4 | ∇ ϕ | 4 v r + 2 .

We choose ϵ ≪ 1, then (3.21) implies

(3.22) ∬ ( ∂ s v ) 2 ϕ q v r ≤ ϵ ∬ | ∇ v | 4 v r − 2 ϕ q + ϵ ∬ ϕ q v p ( r + 2 ) + c ∬ ϕ q − 2 ( ∂ s ϕ ) 2 v r + 2 + c ∬ ϕ q − 4 | ∇ ϕ | 4 v r + 2 .

By the above estimates, we conclude that

(3.23) λ 1 − ( r + 2 ) 2 r 2 ( 1 − η ) 2 16 η − ϵ ∬ v r − 2 ϕ q | ∇ v | 4 + λ 2 ′ − ϵ ∬ ϕ q v p ( r + 2 ) + ( λ 3 − ϵ ) ∬ v r ϕ q v 0 2 ≤ c ∬ ϕ q − 4 v r + 2 | ∇ 2 ϕ | + | Δ H ϕ | + | ∇ ϕ | 2 + | ∂ s ϕ | 2 .

Since r > 0, we know from (3.23) that

(3.24) λ 1 − ( r + 2 ) 2 r 2 ( 1 − η ) 2 16 η − ϵ ∬ v r − 2 ϕ q | ∇ v | 4 + λ 2 ′ − ϵ ∬ ϕ q v p ( r + 2 ) + ( λ 3 − ϵ ) ∬ v r ϕ q v 0 2 ≤ c ∬ ϕ μ 2 | ∇ 2 ϕ | + | Δ H ϕ | + | ∇ ϕ | 2 + | ∂ s ϕ | 2 s 2 ,

where

s 2 = p p − 1 , μ 2 = q − 4 − q p s 2 .

It is easy to see that μ ₂ > 0 provided that q is large enough. For every R > 1, we take

ϕ R ( z , t , s ) = ϕ z R , t R 2 , s R 2

into (3.24), then

(3.25) λ 1 − ( r + 2 ) 2 r 2 ( 1 − η ) 2 16 η − ϵ ∬ v r − 2 ϕ R q | ∇ v | 4 + λ 2 ′ − ϵ ∬ ϕ R q v p ( r + 2 ) + ( λ 3 − ϵ ) ∬ v r ϕ R q v 0 2 ≤ c R 2 n + 4 − 4 s 2 .

Fix 0 < η < (n − 1)/(n + 2), then λ ₃ > 0. If ϵ is small enough, then λ ₃ − ϵ > 0. In order to finish the proof, it is enough to choose p > 1 and r > 0 such that

(3.26) 1 n + 3 + η − 4 ( r − 1 ) r − ( 1 − η ) 2 η > 0 ,

(3.27) − n − 1 4 n − η 4 + r 2 p ( r + 2 ) 1 + 1 n > 0 ,

(3.28) n + 2 − 2 s 2 < 0 .

Set δ = 2r/(r + 2), then (3.26)–(3.28) are equivalent to

(3.29) ( 1 + η ) 2 4 η − 1 + η + 1 n 4 δ − 1 < 0 ,

(3.30) − δ 1 + 1 n + 1 + η − 1 n p < 0 ,

(3.31) n + 2 − 2 p p − 1 < 0 .

In order that (3.29)–(3.31) hold, we need

(3.32) 0 < δ < 4 n η ( n − 1 ) η + n ,

(3.33) 1 < p < n + 1 n + n η − 1 δ

and

(3.34) 1 < p < 1 + 2 n .

It is easy to check that

f ( η ) = 4 n η ( n + 1 ) ( n + n η − 1 ) ( ( n − 1 ) η + n )

is monotone increasing in (0, (n − 1)/(n + 2)). If

δ = 4 n η ( n − 1 ) η + n , η = n − 1 n + 2 ,

then

(3.35) n + 1 n + n η − 1 δ = 2 n ( n + 2 ) ( 2 n 2 + 1 ) < 1 + 2 n .

By continuity, we know that for every

1 < p < 2 n ( n + 2 ) 2 n 2 + 1 = 1 + 4 Q − 10 Q 2 − 4 Q + 6 ,

there exist two small constants θ ₁ and θ ₂ such that (3.32) and (3.33) hold provided that we take

η = n − 1 n + 2 − θ 1 , δ = 4 n ( n − 1 ) 2 n 2 + 1 − θ 2 .

These parameters are exactly what we need, hence the Proof of Theorem 1.1 is completed. □

Proof of Proposition 1.5

Assume (1.5) is false, then there exist sequences {u _k} and {(z _k, t _k, s _k)} such that u _k solves (1.4) and the functions

M k = u k p − 1 2 + | ∇ H u k | p − 1 p + 1

satisfy

(3.36) M k ( z k , t k , s k ) > 2 k d p − 1 ( ( z k , t k , s k ) , ∂ ( H n × ( − ∞ , 0 ) ) ) = 2 k s k − 1 ,

where

d p ( ( z k , t k , s k ) , z k ′ , t k ′ , s k ′ ) = | z k ′ , t k ′ − 1 ◦ ( z k , t k ) | H + | s k − s k ′ | 1 2

and

d p ( ( z k , t k , s k ) , ∂ ( H n × ( − ∞ , 0 ) ) ) = inf z k ′ , t k ′ , s k ′ ∈ ∂ ( H n × ( − ∞ , 0 ) ) d p ( ( z k , t k , s k ) , z k ′ , t k ′ , s k ′ ) .

By applying the doubling lemma (Lemma 5.1 in Ref. [26], Lemma 5.1 in Ref. [19]), there exists { ( z k ̃ , t k ̃ , s k ̃ ) } such that

M k ( z k ̃ , t k ̃ , s k ̃ ) ≥ M k ( z k , t k , s k )

and

M k ( z , t , s ) ≤ 2 M k ( z k ̃ , t k ̃ , s k ̃ )

for all (z, t, s) with

d p ( ( z , t , s ) , ( z k ̃ , t k ̃ , s k ̃ ) ) ≤ k λ k ,

where λ k = M k − 1 ( z k ̃ , t k ̃ , s k ̃ ) . We rescale u _k by setting

v k ( z , t , s ) = λ k 2 p − 1 u k z k ̃ + λ k z , t k ̃ + λ k 2 t , s k ̃ + λ k 2 s ,

then v _k satisfies (1.4) with

v k p − 1 2 + | ∇ H v k | p − 1 p + 1 ( 0,0,0 ) = 1 .

By the regularity theories (see Ref. [27]), {v _k} converges to a nontrivial positive solution of (1.1), By Theorem 1.1, this is a contradiction. □

Corresponding author: Juncheng Wei, Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2, Canada, E-mail: jcwei@math.ubc.ca

Dedicated to Joel Spruck with admirations.

Acknowledgments

The authors are grateful to the referees for their careful reading of the manuscript and for their helpful remarks. The research of J. Wei is partially supported by NSERC of Canada. The research of K. Wu is supported by the China Postdoctoral Science Foundation (2023M732712).

Research ethics: Not applicable.
Author contributions: All authors contributed equally.
Competing interests: Authors state no conflict of interest.
Research funding: Not applicable.
Data availability: Data availability is not applicable to this article as no new data were created or analyzed in this study.

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Received: 2023-01-16

Accepted: 2023-12-18

Published Online: 2024-03-01

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Keywords for this article

Heisenberg group; Liouville theorem; blow up rate estimate; parabolic nonlinear equation; Gidas–Spruck type estimates

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