Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 71, pp. 1–12.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.71

GLOBAL EXISTENCE AND BLOW-UP FOR THE VISCOELASTIC DAMPED

WAVE EQUATION ON THE HEISENBERG GROUP

XINGQUAN LI, YULI FENG, HAN YANG

Abstract. The purpose of this article is to study the Cauchy problem for the viscoelastic

damped wave equation on the Heisenberg group. We first prove the global existence of small
data solutions for p ∈ [2, Q/(Q− 4)] if n = 2, 3, p > 2 if n = 1 using the contraction principle.

Then, a blow-up result is obtained by using the test function method under certain integral

sign assumptions for the Cauchy data when 1 < p ≤ 1 + 2/(Q − 1), where Q = 2n + 2 is the
homogeneous dimension of the Heisenberg group. Moreover, we obtain the upper bound for the

lifespan of the solution by employing a revisited test function method.

1. Introduction

In this article, we consider the Cauchy problem for the viscoelastic damped wave equation on
the Heisenberg group

utt −∆Hu−∆Hut = |u|p, t > 0, η ∈ Hn,

u(0, η) = ϵu0, ut(0, η) = ϵu1, η ∈ Hn,
(1.1)

where p > 1 and ϵ is a positive small parameter. The Heisenberg group is the Lie groupHn = R2n+1

equipped with the law

η ◦ η′ =
(
x+ x′, y + y′, τ + τ ′ +

1

2
(x · y′ − x′ · y)

)
,

for η = (x, y, τ), η′ = (x′, y′, τ ′) ∈ Rn ×Rn ×R, where · denotes the scalar product in Rn. The Lie
algebra of left-invariant vector fields is spanned by

Xj = ∂xj −
yj
2
∂τ , Yj = ∂yj +

xj
2
∂τ , T = ∂τ ,

for each j = 1, . . . , n, satisfying the commutation relations

[Xj , Yk] = T, [Xj , Xk] = [Yj , Yk] = [Xj , T ] = [Yj , T ] = 0.

The Kohn-Laplacian on Heisenberg is defined by

∆H =

n∑
j=1

(X2
j + Y 2

j ) = ∆x +∆y +
1

4
(|x|2 + |y|2)∂2τ +

1

4

n∑
j=1

(xj∂
2
yjτ − yj∂

2
xjτ ), (1.2)

where ∆x and ∆y stand for the Laplacian operators on Rn.
We first recall classical results for the semilinear damped wave equation in the Euclidean setting

utt −∆u+ ut = |u|p, t > 0, x ∈ Rn,

u(0, x) = u0, ut(0, x) = u1, x ∈ Rn,
(1.3)

where ut corresponds to the friction damping. The foundational work by Matsumura [13] es-
tablished basic decay estimates for the solutions to the linear equation associated to (1.3). Sub-
sequently, there is some work concerned with the global well-posedness results for (1.3). For

2020 Mathematics Subject Classification. 35A01, 35B44, 35R03.
Key words and phrases. Viscoelastic damped wave equation; Heisenberg group; test function method;
global existence; blow up; lifespan estimates.
©2025. This work is licensed under a CC BY 4.0 license.

Submitted March 24, 2025. Published July 14, 2025.
1



2 X. LI, Y. FENG, H. YANG EJDE-2025/71

further details, we refer to [14, 8, 18] and the references therein. Among these results, Todorova-
Yordanov [4] investigated the global existence of solutions by assuming small, compactly supported
data for p > pcrit = 1 + 2/n, and the blow-up of the solution is obtained for the initial data∫
Rn uidx > 0, i = 0, 1 when p ∈ (1, pcrit). Through the construction of an energy functional and
application of convexity techniques, Zhang [19] obtained the blow-up result for the critical case
p = pcrit. Later, Ikehata-Tanizawa [11] obtained a similar conclusion for p > pcrit = 1+2/n by no
longer requiring a compact support for the initial data.

Additionally, some authors have also considered the following viscoelastic damped wave equa-
tion

utt −∆u−∆ut = |u|p, t > 0, x ∈ Rn,

u(0, x) = u0, ut(0, x) = u1, x ∈ Rn,
(1.4)

where −∆ut corresponds to the viscoelastic damping. Several L2(Rn)− L2(Rn) decay estimates,
augmented by L1(Rn) constraints on the initial data, were established in [9, 1, 2]. Subsequently,
Shibata [17] established Lp(Rn)−Lq(Rn) estimates for solutions to the linear problem (1.4) using
Fourier analysis and energy methods. D’Abbicco and Reissig [4] proved the global existence of
small data solutions to the problem (1.4) by Banach’s fixed point theorem under p ∈ [2, n

n−4 ] if

5 ≤ n ≤ 8, p > 1 + 3/(n− 1) if 2 ≤ n ≤ 4, and used the test function method to get the blow-up
of solutions when 1 < p ≤ 1 + 2/(n− 1). Unfortunately, there exists a gap between the exponent
of global existence and the exponent of blow-up. Consequently, the critical exponent remains an
open problem.

The study of the semilinear damped wave equation has also been extended to the non-Euclidean
framework. The authors [6, 15] considered the problem (1.3) to the Heisenberg group and studied
the following equations

utt −∆Hu+ ut = |u|p, t > 0, η ∈ Hn,

u(0, η) = u0, ut(0, η) = u1, η ∈ Hn.
(1.5)

Palmieri [15] established L2(Hn)-L
2(Hn) decay estimates for solutions to the linear equation (1.5)

on the Heisenberg group. Then, Georgiev and Palmieri [6] proved the global existence of small
initial data solutions in an exponentially weighted energy space when p > 1 + 2/Q. On the other
hand, a blow-up result for 1 < p ≤ 1+2/Q under certain integral sign assumptions for the Cauchy
data is obtained by using the test function method.

Liu and Li [12] investigated the linear wave equation associated with the problem (1.1) on the
Heisenberg group. By using the group Fourier transform on Hn and the properties of the Hermite
functions, they derived some L2(Hn) − L2(Hn) estimates with additional L1(Hn) regularity on
initial data for the solution and its higher-order horizontal gradients. In this paper, we study the
global existence of small data solutions for p ∈ [2, Q/(Q− 4)] when n = 2, 3, and p > 2 for n = 1
via the contraction principle. On the other hand, the blow-up result is obtained by using the test
function method when 1 < p ≤ 1+2/(Q−1), with

∫
Hn

[u1(η)+(−∆Hu0)(η)] dη > 0. Furthermore,

we derive the upper bound for the lifespan of the form

T (ϵ) ≤

{
Cϵ−(

2p
p−1−Q−1)

−1

, if 1 < p < 1 + 2
Q−1 ,

exp(Cϵ−(p−1)), if p = 1 + 2
Q−1 ,

through employing a technique which has been developed recently by Ikeda-Sobajima in [10].
This article is organized as follows. The decay estimates for the solution of the linear problem

will be given in Section 2. In Section 3, we prove the global existence of solutions to (1.1). Then,
the blow-up of solutions at a finite time will be shown in Section 4. Finally, we obtain the upper
bound for the lifespan in Section 5.

1.1. Notion. In this paper, f ≲ g means that there exists a positive constant C such that f ≤ Cg.
Moreover, f ≃ g implies that f ≲ g and g ≲ f . We define

Dk = (Hk(Hn) ∩ L1(Hn))× (L2(Hn) ∩ L1(Hn)),

∥(u0, u1)∥Dk = ∥u0∥L1(Hn) + ∥u0∥Hk(Hn) + ∥u1∥L2(Hn) + ∥u1∥L1(Hn),



EJDE-2025/71 GLOBAL EXISTENCE AND BLOW-UP 3

for k ≥ 0, k ∈ N. The Sobolev space Hk(Hn) is defined as follows

Hk+1(Hn) = {f ∈ Hk(Hn) : ∇Hf ∈ Hk(Hn)},
and equipped with the norm

∥f∥2Hk+1(Hn)
= ∥f∥2Hk(Hn)

+

n∑
i=1

(
∥Xif∥2Hk(Hn)

+ ∥Yif∥2Hk(Hn)

)
for all k ∈ N, where the horizontal gradient of a function f is given by

∇Hf = (X1f, · · · , Xnf, Y1f, · · · , Ynf).
For the sake of clarity, we denote H0(Hn) = L2(Hn). For further information regarding the
Heisenberg group, we refer to reference [5, Chapters 6].

1.2. Main results.

Theorem 1.1. Let p ∈ [2, Q/(Q − 4)] if n = 2, 3, p > 2 if n = 1. There exists a constant
C > 0 such that for any (u0, u1) ∈ D2 with ∥(u0, u1)∥D2 < C, then, there is a unique solution
u ∈ C([0,∞), H2(Hn) ∩ C1([0,∞), L2(Hn)) to (1.1). Moreover, the following estimates hold

∥u∥L2(Hn) ≲ (1 + t)−
Q−2

4 ∥(u0, u1)∥L2(Hn)∩L1(Hn),

∥ut∥L2(Hn) ≲ (1 + t)−
Q
4 ∥(u0, u1)∥L2(Hn)∩L1(Hn),

∥∇Hu∥L2(Hn) ≲ (1 + t)−
Q
4 ∥(u0, u1)∥D1 ,

∥∇2
Hu∥L2(Hn) ≲ (1 + t)−

Q+2
4 ∥(u0, u1)∥D2 .

Remark 1.2. The natural dilations {δr}r≥0 on the Heisenberg group are given by

δr(x, y, τ) = (rx, ry, r2τ),

which leads to

d(δr(η)) = r2n+2 dη,

so the homogeneous dimension of the Heisenberg group is Q = 2n+ 2.

Remark 1.3. The restriction on the number of dimensions arises from the application of the
Gagliardo-Nirenberg-type inequality on the Heisenberg group, which is employed to handle the
nonlinear term |u|p.

Before giving the blow-up result, we recall the definition of weak solution to (1.1).

Definition 1.4. A weak solution of the Cauchy problem (1.1) in [0, T ) × Hn is a function u ∈
Lp
loc(Hn) that satisfies∫ T

0

∫
Hn

|u(t, η)|pφ (t, η) dη dt+ ϵ

∫
Hn

(u1(η)−∆Hu0(η))φ(0, η) dη

= ϵ

∫
Hn

φt(0, η)u0(η) dη +

∫ T

0

∫
Hn

u(t, η)(φtt(t, η)−∆Hφ(t, η) + ∆Hφt(t, η)) dη dt,

(1.6)

for any φ ∈ C∞
0 ([0, T )×Hn). If T = ∞, we call u a global in time weak solution to (1.1), else we

call u a local in time weak solution.

For the weak solution to (1.1), we introduce the lifespan of the solution as follows.

Definition 1.5. Let u be a weak solution to (1.1) with a fixed initial parameter ϵ > 0. We define
the lifespan of the solution as

T (ϵ) = sup
T>0

{u is a weak solution to (1.1) in[0, T )×Hn}.

Now we state the blow-up result in this paper, which gives an upper bound of T (ϵ) for suitable
given initial data (u0, u1).



4 X. LI, Y. FENG, H. YANG EJDE-2025/71

Theorem 1.6. Assuming that the data u1, u0 ∈ C∞
0 ([0, T )×Hn) satisfy∫

Hn

(u1 −∆Hu0) dη > 0, (1.7)

and the exponent p satisfies

1 < p ≤ 1 +
2

Q− 1
. (1.8)

Then the weak solution (1.6) to (1.1) blows up in finite time. If the initial data is compactly
supported with suppu0, suppu1 ⊂ {(x, y, τ) ∈ Hn : |x|2 + |y|2 + |τ | < R2

0} for some R0 > 0, then
there exists ϵ0 > 0 such that for any ϵ ∈ (0, ϵ0] it holds

T (ϵ) ≤

{
Cϵ−(

2p
p−1−Q−1)

−1

, if 1 < p < 1 + 2
Q−1 ,

exp(Cϵ−(p−1)), if p = 1 + 2
Q−1 ,

(1.9)

where C is a positive constant independent of ϵ.

Remark 1.7. The conclusion of 1.6 is consistent with the blow-up result in [4]. Unfortunately,
there is still a gap between the exponent of global existence and the exponent of blow-up.

2. Decay estimates for solutions of the linear problem

First, we consider a linear version of problem (1.1)

utt −∆Hu−∆Hut = 0, t > 0, η ∈ Hn,

u(0, η) = ϵu0, ut(0, η) = ϵu1, η ∈ Hn.
(2.1)

The fundamental solutions to the Cauchy problem (2.1) are denoted by E0(t, η), E1(t, η) i.e., the
distributional solutions with data (u0, u1) = (δ0, 0) and (u0, u1) = (0, δ0), where δ0 is the Dirac
distribution in the η variable. If we denote by ∗(η) the group convolution with respect to the
variable η, then the solutions to the Cauchy problem (2.1) can be expressed as

u(t, η) = ϵu0 ∗(η) E0(t, η) + ϵu1 ∗(η) E1(t, η).

Decay estimates for the solution of the linear problem can be obtained from the following propo-
sition.

Proposition 2.1. Let (u0, u1) ∈ D2. The solution u = u(t, η) to (2.1) fulfills the (L2 ∩ L1)-L2

estimates

∥u∥L2(Hn) ≲ (1 + t)−
Q−2

4 ∥(u0, u1)∥L2(Hn)∩L1(Hn), (2.2)

∥ut∥L2(Hn) ≲ (1 + t)−
Q
4 ∥(u0, u1)∥L2(Hn)∩L1(Hn), (2.3)

∥∇Hu∥L2(Hn) ≲ (1 + t)−
Q
4 ∥(u0, u1)∥D1 , (2.4)

∥∇2
Hu∥L2(Hn) ≲ (1 + t)−

Q+2
4 ∥(u0, u1)∥D2 , (2.5)

and the L2 − L2 estimates

∥∇Hu∥L2(Hn) ≲ ∥(u0, u1)∥H1×L2 , (2.6)

∥∇2
Hu∥L2(Hn) ≲ (1 + t)−

1
2 ∥(u0, u1)∥H2×L2 . (2.7)

Proof. The proof of (2.3) is analogous to that of (2.2), so we omit it here. The remaining part
has been done in [12]. □



EJDE-2025/71 GLOBAL EXISTENCE AND BLOW-UP 5

3. Proof of Theorem 1.1

First, let us define the space

X(T ) = C([0, T ], H2(Hn)) ∩ C1([0, T ], L2(Hn)),

with the norm

∥u∥X(T ) = sup
0≤t≤T

{ 2∑
k=0

(1 + t)
Q−2+2k

4 ∥∇k
Hu∥L2(Hn) + (1 + t)Q/4∥ut∥L2(Hn)

}
.

For all T > 0, we define the operator N for u ∈ X(T ) as

Nu = ulin + unlin = ϵu0 ∗(η) E0(t, η) + ϵu1 ∗(η) E1(t, η) +

∫ t

0

|u|p ∗(η) E1(t− s, η)ds

Theorem 1.1 will be proved by showing that

∥Nu∥X(T ) ≲ ϵ∥(u0, u1)∥D2 + ∥u∥pX(T ), (3.1)

∥Nu−Nv∥X(T ) ≲ ∥u− v∥X(T )(∥u∥p−1
X(T ) + ∥v∥p−1

X(T )). (3.2)

In fact, let u(j) = N(u(j−1)), u(0) = 0, for j = 1, 2, 3, . . . . The combination of ∥(u0, u1)∥D2 < C
and (3.1) allows for the derivation

∥u(j)∥X(T ) ≲ ϵ. (3.3)

Once the uniform estimate (3.3) is established, we use (3.2) to obtain

∥u(j+1) − u(j)∥X(T ) ≤ Cϵp−1, (3.4)

∥u(j+1) − u(j)∥X(T ) ≤
1

2
∥u(j) − u(j−1)∥X(T ), (3.5)

for ϵ sufficiently small. {u(j)} is a Cauchy sequence in the Banach space X(T ) converging to the
unique solution of Nu = u. As all of the constants are independent of t, taking j → ∞ allows us
obtain the existence of the global solution.

From Proposition 2.1, It is easy to obtain that

∥ulin∥X(T ) ≲ ϵ∥(u0, u1)∥D2 ,

Due to Nu = ulin + unlin, then it is remain to demonstrate that

∥unlin∥X(T ) ≲ ∥u∥pX(T ),

under some conditions for the exponent p. To do this, we now need to estimate the nonlinearity
|u|p in the L1 and L2 norms, respectively. Applying the Gagliardo-Nirenberg type inequality [16]
on the Heisenberg group

∥v∥Lq(Hn) ≤ C∥∇k
Hv∥θL2(Hn)

∥v∥1−θ
L2(Hn)

,

where k = 1, 2, C is a nonnegative constant and θ = Q
k (

1
2 − 1

q ) ∈ [0, 1], we have

∥|u|p∥L1(Hn) = ∥u∥pLp(Hn)
≲ ∥u∥(1−θ1)p

L2(Hn)
∥∇2

Hu∥
θ1p
L2(Hn)

≲ (1 + t)−
Qp−p−Q

2 ∥u∥pX(T ), (3.6)

with θ1 = Q
2 (

1
2 − 1

p ) ∈ [0, 1], that is 2 ≤ p ≤ 2Q/(Q − 4) with n ≥ 2 or 2 ≤ p with n = 1.

Analogously,

∥|u|p∥L2(Hn) = ∥u∥pL2p(Hn)
≲ ∥u∥(1−θ2)p

L2(Hn)
∥∇2

Hu∥
θ2p
L2(Hn)

≲ (1 + t)−
2Qp−2p−Q

4 ∥u∥pX(T ), (3.7)

with θ2 = Q
2 (

1
2 −

1
2p ) ∈ [0, 1], that is 1 ≤ p ≤ Q/(Q− 4) with n ≥ 2 or 1 ≤ p with n = 1. It follows

from (3.6) and (3.7) that

∥|u|p∥L1(Hn)∩L2(Hn) ≲ (1 + t)−
Qp−p−Q

2 ∥u∥pX(T ),

where we restricted
2 ≤ p ≤ Q/(Q− 4), n = 2, 3

2 ≤ p, n = 1.
(3.8)



6 X. LI, Y. FENG, H. YANG EJDE-2025/71

We apply the derived L2 ∩ L1 − L2 estimates in [0, t] to obtain

∥unlin∥L2(Hn) ≲
∫ t

0

(1 + s)−
Qp−p−Q

2 (1 + t− s)−
Q−2

4 ds∥u∥pX(T )

≲ (1 + t)−
Q−2

4

∫ t
2

0

(1 + s)−
Qp−p−Q

2 ds∥u∥pX(T )

+ (1 + t)−
Qp−p−Q

2

∫ t

t
2

(1 + t− s)−
Q−2

4 ds∥u∥pX(T ).

It is easy to show that

(1 + t)−
Q−2

4

∫ t
2

0

(1 + s)−
Qp−p−Q

2 ds∥u∥pX(T ) ≲ (1 + t)−
Q−2

4 ∥u∥pX(T ).

This holds under the condition −Qp−p−Q
2 + 1 < 0, i.e.,

p > 1 +
3

Q− 1
. (3.9)

Combining (3.8) and (3.9), we deduce that p ∈ [2, Q/(Q− 4)] if n = 2, 3, and p > 2 if n = 1. Now

let n = 3, (1 + t− s)−
Q−2

4 is integrable over [ t2 , t]. Therefore,

(1 + t)−
Qp−p−Q

2

∫ t

t
2

(1 + t− s)−
Q−2

4 ds ≲ (1 + t)−
Qp−p−Q

2 ≤ (1 + t)−
Q−2

4 .

If n ≤ 2, it is easy to see that

(1 + t)−
Qp−p−Q

2

∫ t

t/2

(1 + t− s)−
Q−2

4 ds ≲

{
(1 + t)−

Qp−p−Q
2 +1−Q−2

4 , n = 1,

(1 + t)−
Qp−p−Q

2 ln(1 + t), n = 2.

In both cases, the decay is controlled by (1+t)−
Q−2

4 , due to p > 1+3/(Q−1). The same reasoning
leads to

∥∂tunlin∥L2(Hn) ≲ (1 + t)−
Q
4 ∥u∥pX(T ).

Consequently,

∥∂jt unlin∥L2(Hn) ≲ (1 + t)−
Q−2+2j

4 ∥u∥pX(T ),

where j = 0, 1. Next, we apply the derived (L2 ∩ L1 − L2) estimates (2.4), (2.5) in [0, t
2 ], and

L2 − L2 estimates (2.6), (2.7) in [ t2 , t] to obtain

∥∇k
Hu

nlin∥L2(Hn) ≲ (1 + t)−
Q+2(k−1)

4

∫ t
2

0

(1 + s)−
Qp−p−Q

2 ds∥u∥pX(T )

+ (1 + t)−
2Qp−2p−Q

4

∫ t

t
2

(1 + t− s)−
k−1
2 ds∥u∥pX(T )

≲ (1 + t)−
Q+2(k−1)

4 ∥u∥pX(T ) + (1 + t)−
2Qp−2p−Q

4 +1− k−1
2 ∥u∥pX(T )

≲ (1 + t)−
Q+2(k−1)

4 ∥u∥pX(T ),

for k = 1, 2. The last inequality in the above equation needs to satisfy the condition Q+2(k−1)
4 >

− 2Qp−2p−Q
4 + 1− k−1

2 for it to hold true, i.e., p > 1 + 3
Q−1 .

To prove (3.2), we notice that

∥Nu−Nv∥X(T ) = ∥
∫ t

0

(|u(s, ·)|p − |v(s, ·)|p) ∗(η) E1(t− s, ·)ds∥X(T ).

By Hölders inequality, one obtains

∥|u(s, ·)|p − |v(s, ·)|p∥Lm ≲ ∥u(s, ·)− v(s, ·)∥Lmp(∥u(s, ·)∥p−1
Lmp + ∥v(s, ·)∥p−1

Lmp),



EJDE-2025/71 GLOBAL EXISTENCE AND BLOW-UP 7

with m = 1, 2. Ultimately, by employing the Gagliardo-Nirenberg type inequality to estimate the
three terms on the right-hand side of the aforementioned inequality, we derive the desired estimate
(3.2). The proof of Theorem 1.1 is complete.

4. Proof of Theorem 1.6 (Blow-up)

The proof is divided into two parts. In this section, we first prove the blow-up result.
Proof. We apply the so-called test function method. We prove it by contradiction, and now
assume that there exists a global in time weak solution u to (1.1). For R > 1, the test function
φR ∈ C∞

0 ([0, T )×Hn with separate variables is defined as follows

φR(t, x, y, τ) = Φ
( t
R

)
Φ
( |x|
R

)
Φ
( |y|
R

)
Φ
( |τ |
R2

)
, (4.1)

where Φ is a smooth nonnegative non-increasing function such that Φ ∈ C∞
0 ([0,+∞)), Φ(r) = 1,

for any r ∈ [0, 12 ] and Φ(r) = 0, for any r ≥ 1 and satisfies

Φ ≲ Φ1/p, |Φ′| ≲ Φ1/p, |Φ′′| ≲ Φ1/p. (4.2)

For the existence of Φ, we refer to reference [7, 3]. Taking the test function in the definition of the
weak solution (1.6) as φR(t, x, y, τ). Hence,∫ R

0

∫
B
|u(t, η)|pφR(t, η) dη dt+ ϵ

∫
D
(u1(η)−∆Hu0(η))φR(0, η) dη

≤
∫ R

0

∫
D
|u(t, η)|(|∂2t φR(t, η)|+ |∆HφR(t, η)|+ |∆H∂tφR(t, η))| dη dt,

= I1 + I2 + I3,

(4.3)

where

B = {η = (x, y, τ) ∈ Hn; |x|2, |y|2, |τ | ≤ R2},

D = {η = (x, y, τ) ∈ Hn;
R

2
≤ |x|, |y| ≤ R,

R2

2
≤ |τ | ≤ R2},

and

I1 =

∫ R

0

∫
D
|u(t, η)||∂2t φR(t, η)| dη dt,

I2 =

∫ R

0

∫
D
|u(t, η)||∆HφR(t, η)| dη dt,

I3 =

∫ R

0

∫
D
|u(t, η)||∆H∂tφR(t, η)| dη dt.

Let us estimate I1. From the relation

∂2t φR = R−2Φ′′
( t
R

)
Φ
( |x|
R

)
Φ
( |y|
R

)
Φ
( τ

R2

)
and (4.2), it follows that

I1 ≲ R−2

∫ R

0

∫
D
|u(t, η)|φR(t, η)

1/p dη dt.

To estimate I2, using (1.2), we have

|∆HφR(t, η)| ≤Φ
( t
R

) ∣∣∣∣∆xΦ
( |x|
R

)∣∣∣∣Φ( |y|R )
Φ
( τ

R2

)
+Φ

( t
R

)
Φ
( |x|
R

) ∣∣∣∣∆yΦ
( |y|
R

)∣∣∣∣Φ( τ

R2

)
+

n∑
j=1

|xj |Φ
( t
R

)
Φ
( |x|
R

) ∣∣∣∣∂yjΦ
( |y|
R

)∣∣∣∣ ∣∣∣∂τΦ( τ

R2

)∣∣∣



8 X. LI, Y. FENG, H. YANG EJDE-2025/71

+

n∑
j=1

|yj |Φ
( t
R

) ∣∣∣∣∂xj
Φ
( |x|
R

)∣∣∣∣Φ( |y|R ) ∣∣∣∂τΦ( τ

R2

)∣∣∣
+

1

4
(|x|2 + |y|2)Φ

( t
R

)
Φ
( |x|
R

)
Φ
( |y|
R

) ∣∣∣∂2τΦ( τ

R2

)∣∣∣ .
By letting

x̃ =
x

R
, ỹ =

y

R
, τ̃ =

τ

R2
,

we conclude that

|∆HφR(t, η)| ≤ R−2Φ
( t
R

)
|∆x̃Φ(|x̃|)|Φ(|ỹ|)Φ(|τ̃ |)

+R−2Φ
( t
R

)
Φ(|x̃|)|∆ỹΦ(|ỹ|)|Φ(|τ̃ |)

+R−2
n∑

j=1

|x̃j |Φ
( t
R

)
Φ(|x̃|)|∂ỹj

Φ(|ỹ|)||∂τ̃Φ(|τ̃ |)|

+R−2
n∑

j=1

|ỹj |Φ
( t
R

)
|∂x̃j

Φ(|x̃|)|Φ(|ỹ|)|∂τ̃Φ(|τ̃ |)

+R−2 1

4
(|x̃|2 + |ỹ|2)Φ

( t
R

)
Φ(|x̃|)Φ(|ỹ|)|∂2τ̃Φ(|τ̃ |)|.

A direct calculation yields

I2 ≲ R−2

∫ R

0

∫
D
|u(t, η)|φR(t, η)

1/p dη dt.

In the same way, we find the estimate

|∆H∂tφR(t, η)| ≲ R−3|φR(t, η)|1/p ≲ R−2|φR(t, η)|1/p.

By Hölder inequality and (4.3), we have∫ R

0

∫
B
|u(t, η)|pφR(t, η) dη dt+ ϵ

∫
D
(u1(η)−∆Hu0(η))φR(0, η) dη

≤ R−2

∫ R

0

∫
D
|u(t, η)|φ1/p

R (t, η) dη dt

≤ R−2
(∫ R

0

∫
D
|u(t, η)|pφR(t, η) dη dt

)1/p(∫ R

0

∫
D
dη dt

) p−1
p

≤ R−2+
(p−1)(Q+1)

p

(∫ R

0

∫
B
|u(t, η)|pφR(t, η) dη dt

)1/p

.

(4.4)

If p < 1 + 2
Q−1 , it follows immediately that −2 + (p−1)(Q+1)

p < 0, and then, combining (1.7), as

R→ ∞, we conclude that

0 ≤ lim
R→∞

(∫ R

0

∫
B
|u(t, η)|pφR(t, η) dη dt

) p−1
p ≤ 0.

Thus,

lim
R→∞

∫ R

0

∫
B
|u(t, η)|pφR(t, η) dη dt = 0.

However, this is not possible because of (4.4) and (1.7).



EJDE-2025/71 GLOBAL EXISTENCE AND BLOW-UP 9

For the critical case p = 1 + 2
Q−1 , using again (4.4) and letting R → ∞, the result is u ∈

Lp((0,∞)×Hn), which implies that

lim
R→∞

∫ R

0

∫
D
|u(t, η)|pφR(t, η) dη dt

= lim
R→∞

∫ R

0

∫
B
|u(t, η)|pφR(t, η) dη dt− lim

R→∞

∫ R

0

∫
D0

|u(t, η)|pφR(t, η) dη dt

=

∫ ∞

0

∫
Hn

|u(t, η)|p dη dt−
∫ ∞

0

∫
Hn

|u(t, η)|p dη dt = 0,

(4.5)

where

D0 = {η = (x, y, τ) ∈ Hn; |x|, |y| ≤
R

2
, |τ | ≤ R2

2
}.

In conclusion, the application of (4.4), (4.5), the dominated convergence and the fact that φR(0, η) →
1 as R→ ∞, we conclude that

0 <

∫
Hn

(u1 −∆Hu0) dη ≤ 0.

It is a contradiction. This proves the first conclusion of Theorem 1.6.

5. Proof of Theorem 1.6 (Lifespan estimate)

In this section, we will derive an upper bound for the lifespan. The following Lemma will be
used in the proof.

Lemma 5.1 ([7]). If g(s) is a measurable function that satisfies the following properties: g(s) is
a decreasing function for s > 1

2 , and g(s) = 0 for s ∈ [0, 12 ] ∪ [1,∞), then it holds that∫ R

0

g( A
r2 )

r
dr ≤ log 2

2
g

(
A

R2

)
,

for any R > 0, A > 0.

Now we give the proof process for the upper bound of the lifespan.
Proof. Without loss of generality, it is assumed that

√
2R0 < T (ϵ). Indeed, if T (ϵ) ≤

√
2R0, then

T (ϵ) ≤

{√
2R0ϵ

−( 2p
p−1−Q−1)−1

, if 1 < p < 1 + 2
Q−1 ,

exp(log(
√
2R0)ϵ

−(p−1)), if p = 1 + 2
Q−1 ,

since ϵ is sufficiently small, so (1.9) is trivially fulfilled. Let us consider a new test function ψR(t, η)
defined as follows

ψR(t, η) =
[
Φ
( t2 + |x|2 + |y|2 + |τ |

R2

)]3p′

,

for any t ≥ 0, R > 1, η = (x, y, τ) ∈ Hn, where Φ(r) is defined in Section 4 and p′ is the conjugate

index of p. Note that the requirement R >
√
2R0 implies that ψR(0, · ) ≡ 1, ∂tψR(0, · ) ≡ 0. Thus

by applying (1.6) with test function ψR, we obtain∫ T

0

∫
Hn

|u(t, η)|pψR(t, η) dη dt+ ϵ

∫
Hn

(u1(η)−∆Hu0(η)) dη

≤
∫ T

0

∫
Hn

|u(t, η)|(|∂2t ψR(t, η)|+ |∆HψR(t, η)|+ |∆H∂tψR(t, η))| dη dt.
(5.1)

A direct calculation yields the following result

∂2t ψR(t, η) = 12p′(3p′ − 1)t2R−4Φ3p′−2
R (Φ′

R)
2 + 12p′t2R−4Φ3p′−1

R Φ′′
R + 6p′R−2Φ3p′−1

R Φ′
R.

Moreover, we define

Φ∗(r) =

{
0, if r ∈ [0, 12 ),

Φ(r), if r ∈ [ 12 ,∞),



10 X. LI, Y. FENG, H. YANG EJDE-2025/71

which implies immediately

|∂2t ψR(t, η)| ≲ R−4t2(Φ∗
R)

3p′
p ΦR|Φ′

R|2 +R−4t2(Φ∗
R)

3p′
p Φ′′

R +R−2(Φ∗
R)

3p′
p Φ′

R

≲ R−2(Φ∗
R)

3p′
p ,

(5.2)

where

ΦR = Φ
( t2 + |x|2 + |y|2 + |τ |

R2

)
, Φ∗

R = Φ∗
( t2 + |x|2 + |y|2 + |τ |

R2

)
.

Similarly, plugging the relations

∂2xj
ψR(t, η) = 12p′(3p′ − 1)x2jR

−4Φ3p′−2
R (Φ′

R)
2

+ 12p′x2jR
−4Φ3p′−1

R Φ′′
R + 6p′R−2Φ3p′−1

R Φ′
R,

∂2xjτψR(t, η) = 6p′(3p′ − 1)xjR
−4Φ3p′−2

R (Φ′
R)

2 + 6p′xjR
−4Φ3p′−1

R Φ′′
R,

∂2τψR(t, η) = 3p′(3p′ − 1)R−4Φ3p′−2
R (Φ′

R)
2 + 3p′R−4Φ3p′−1

R Φ′′
R,

and analogous relations for ∂2yj
ψR(t, η) and ∂

2
yjτψR(t, η) in the definition of Kohn-Laplacian from

(1.2), we find the estimate

|∆HψR(t, η)| ≲ R−2(Φ∗
R)

3p′
p . (5.3)

The same calculations yield the estimate

|∆H∂tψR(t, η)| ≲ R−2(Φ∗
R)

3p′
p . (5.4)

So, it follows from (5.1), (5.2), (5.3), (5.4) and Hölder inequality that∫ T

0

∫
Hn

|u(t, η)|pψR(t, η) dη dt+ ϵ

∫
Hn

(u1(η)−∆Hu0(η)) dη

≤
∫ T

0

∫
Hn

|u(t, η)|(|∂2t ψR(t, η)|+ |∆HψR(t, η)|+ |∆H∂tψR(t, η))| dη dt

≲ R−2

∫ T

0

∫
Hn

|u(t, η)|(Φ∗
R)

3p′
p dη dt

≲ R
−2+Q+1

p′
(∫ T

0

∫
Hn

|u(t, η)|(Φ∗
R)

3p′
dη dt

)1/p

,

(5.5)

for any R ∈ (
√
2R0, T (ϵ)). Using Lemma (5.1) with g = (Φ∗

R)
3p′
, A = t2 + |x|2 + |y|2 + |τ |, we

easily obtain
2

log 2

∫ R

0

∫ T

0

∫
Hn

|u(t, η)|p(Φ∗
r)

3p′
r−1 dη dt dr

=

∫ T

0

∫
Hn

|u(t, η)|p
∫ R

0

2

log 2
(Φ∗

r)
3p′
r−1dr dη dt

≤
∫ T

0

∫
Hn

|u(t, η)|p(Φ∗
R)

3p′
dη dt

≤
∫ T

0

∫
Hn

|u(t, η)|p(ΦR)
3p′
dη dt

=

∫ T

0

∫
Hn

|u(t, η)|pψR(t, η) dη dt.

(5.6)

We define

J(R) =

∫ R

0

∫ T

0

∫
Hn

|u(t, η)|p(Φ∗
r)

3p′
r−1 dη dt dr,

combining the estimates (5.5) with (5.6), we have

2J(R)

log 2
+ ϵI(u0, u1) ≲ R

−2+Q+1
p′ (RJ ′(R))1/p,



EJDE-2025/71 GLOBAL EXISTENCE AND BLOW-UP 11

which gives

R
2p− p(Q+1)

p′ −1 ≲ J ′(R)
(2J(R)

log 2
+ ϵI(u0, u1)

)−p

, (5.7)

where

I(u0, u1) =

∫
Hn

(u1(η)−∆Hu0(η)) dη.

Integrating R over [
√
2R0, T (ϵ)] on both sides of (5.7), we obtain∫ T

√
2R0

R
2p− p(Q+1)

p′ −1
dR ≲

∫ T

√
2R0

J ′(R)
(2J(R)

log 2
+ ϵI(u0, u1)

)−p

dR.

A straightforward calculations yield∫ T

√
2R0

R
2p− p(Q+1)

p′ −1
dR ≃

{
T

2p− p(Q+1)

p′ − (
√
2R0)

2p− p(Q+1)

p′ , if p ∈ (1, 1 + 2
Q−1 ),

log T − log(
√
2R0), if p = 1 + 2

Q−1 ,

and ∫ T

√
2R0

J ′(R)
(2J(R)

log 2
+ ϵI(u0, u1)

)−p
dR

=
log 2

2(p− 1)

[(2J(√2R0)

log 2
+ ϵI(u0, u1)

)1−p

−
(2J(T )

log 2
+ ϵI(u0, u1)

)1−p]
≤ log 2

2(p− 1)

(2J(√2R0)

log 2
+ ϵI(u0, u1)

)1−p

≲ ϵ−(p−1).

Here, we derive

T
2p− p(Q+1)

p′ ≲ ϵ−(p−1), if p ∈ (1, 1 +
2

Q− 1
),

log T ≲ ϵ−(p−1), if p = 1 +
2

Q− 1
,

which implies that (1.9) holds. So, the proof of the Theorem 1.6 is completed.

Acknowledgments. The authors would like to thank the handling editor and the referee for
their detailed comments.

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Xingquan Li
School of Mathematics, Southwest Jiaotong University, Chengdu 611756, China

Email address: lixingquan@my.swjtu.edu.cn

Yuli Feng

School of Mathematics, Southwest Jiaotong University, Chengdu 611756, China

Email address: fyl@my.swjtu.edu.cn

Han Yang

School of Mathematics, Southwest Jiaotong University University, Chengdu 611756, China
Email address: hanyang95@263.net


	1. Introduction
	1.1. Notion
	1.2. Main results

	2. Decay estimates for solutions of the linear problem
	3. Proof of Theorem 1.1
	4. Proof of Theorem 1.6 (Blow-up)
	5. Proof of Theorem 1.6 (Lifespan estimate)
	Acknowledgments

	References