Available online at www.tjnsa.com
J. Nonlinear Sci. Appl. 9 (2016), 3279–3286
Research Article
Nonexistence of global weak solutions of a system of
wave equations on the Heisenberg group
Rabah Kellila,∗, Mokhtar Kiraneb
a
University of Majmaah, Faculty of Science Al Zulfi, Al Zulfi, KSA.
b
Laboratoire de Mathématiques, Image et Applications, Faculté des Sciences, Pôle Sciences et Technologies, Avenue M. Crépeau,
17000, La Rochelle, France.
Communicated by M. Jleli
Abstract
Sufficient conditions are obtained for the nonexistence of solutions to the nonlinear higher order pseudoparabolic equation
utt − ∆H u + (−∆H )δ/2 u = f (η, t)up ,
(η, t) ∈ H × (0, ∞), p > 1, u ≥ 0,
where ∆H is the Kohn–Laplace operator on the (2N + 1)-dimensional Heisenberg group H and f (η, t) is a
given function. Then, this result is extended to the case of a 2 × 2-system of the same type. Our technique
c
of proof is based on a duality argument. 2016
All rights reserved.
Keywords: Nonexistence, nonlinear hyperbolic equation, systems of hyperbolic equations, Heisenberg
group.
2010 MSC: 47J35, 34A34, 35R03.
1. Introduction and preliminaries
In this paper, we are first concerned with the nonexistence of global weak solutions to the nonlinear
hyperbolic equation
δ
utt − ∆H u + (−∆H ) 2 u = f (η, t)up ,
(η, t) ∈ H × (0, ∞), p > 1, u ≥ 0,
(1.1)
equipped with the initial conditions
u(η, 0) = u0 (η), ut (η, 0) = u1 (η), η ∈ H,
(1.2)
where (−∆H )δ/2 , 0 < δ < 2 is the fractional power of the Kohn–Laplace operator ∆H on the (2N + 1)∗
Corresponding author
Email addresses: kellilrabah@yahoo.fr, r.kellil@mu.edu.sa (Rabah Kellil), mokhtar.kirane@univ-lr.fr (Mokhtar
Kirane)
Received 2015-12-23
R. Kellil, M. Kirane, J. Nonlinear Sci. Appl. 9 (2016), 3279–3286
3280
dimensional Heisenberg group H. Then we extend our analysis to the 2 × 2 system

α
utt − ∆H u + (−∆H ) 2 u = f (η, t)v q , (η, t) ∈ H × (0, ∞),




 v − ∆ v + (−∆ ) β2 v = g(η, t)up , (η, t) ∈ H × (0, ∞),
tt
H
H


u(η, 0) = u0 (η), u1 (η, 0) = u1 (η),
η ∈ H,



v(η, 0) = v0 (η), v(η, 0) = v1 (η),
η ∈ H,
(1.3)
where u, v > 0 and p, q > 1, and f (η, t), g(η, t) are given functions.
Before we state and prove our results, let us dwell a while on the existing literature. In the Euclidean
case, hyperbolic equations and systems are well documented, one is referred to the valuable books [2],
[16]. However works on hyperbolic equations (systems) on the Heisenberg group are scarce. The study of
evolution equations on the Heisenberg group started with the paper of Nachman [11] and then followed by
Müller [10] , Zhang [17], Pascucci [12], Véron and Pokhozhaev [14], Elhamidi and Kirane [4], Elhamidi and
Obeid [5], D’Ambrosio [3], Goldstein and Kombe [7] and Mokrani [9].
For the reader convenience, we recall some background facts that will be used in the sequel.
The (2N + 1)-dimensional Heisenberg group H is the space R2N +1 equipped with the group operation
η ◦ η 0 = (x + x0 , y + y 0 , τ + τ 0 + 2(x · y 0 − x0 · y)),
for all η = (x, y, τ ), η 0 = (x0 , y 0 , τ 0 ) ∈ RN × RN × R, where · denotes the standard scalar product in RN . This
group operation endows H with the structure of a Lie group.
On H it is natural to define a distance from η = (x, y, τ ) =: (z, τ ) to the origin by

!2 1/4
N
X
1/4
|η|H = τ 2 +
(x2i + yi2 )  = τ 2 + |z|4
,
i=1
where x = (x1 , . . . , xN ) and y = (y1 , . . . , yN ).
The Laplacian ∆H over H can be defined from the vectors fields
and Yi = ∂yi − 2xi ∂τ
Xi = ∂xi + 2yi ∂τ
for i = 1, . . . , N , as follows
∆H =
N
X
(Xi2 + Yi2 ).
i=1
A simple computation gives the expression
∆H u =
N
X
∂x2i xi u + ∂y2i yi u + 4yi ∂x2i τ u − 4xi ∂y2i τ u + 4(x2i + yi )2 ∂τ2τ u .
i=1
The operator ∆H has the following properties:
• It is invariant with respect to the left multiplication in the group, i.e., for all η, η 0 ∈ H, we have
∆H (u(η ◦ η 0 )) = ∆H u(η ◦ η 0 );
• It is homogeneous with respect to dilatation. More precisely, for λ ∈ R and (x, y, τ ) ∈ H, we have
∆H (u(λx, λy, λ2 τ )) = λ2 (∆H u)(λx, λy, λ2 τ );
• If u(η) = v(|η|H ), then
∆H v(ρ) = a(η)
where ρ = |η|H , a(η) = ρ−2
PN
2
i=1 (xi
d2 v Q − 1 dv
+
dρ2
ρ dρ
,
+ yi2 ) and Q = 2N + 2 is the homogeneous dimension of H.
R. Kellil, M. Kirane, J. Nonlinear Sci. Appl. 9 (2016), 3279–3286
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For more details on Heisenberg groups, we refer to [6]. We also need the following lemmas.
Lemma 1.1 ([1]). Consider a convex function F ∈ C 2 (R) and assume that 0 ≤ ϕ ∈ C0∞ (R2N +1 ). Then
α
α
F 0 (ϕ)(−∆H ) 2 ϕ ≥ (−∆H ) 2 F (ϕ) f or 0 < α < 2.
(1.4)
Remark 1.2. When α = 2, the relation (1.4) is still valid as
σ
σ−1
∆H ϕ = σφ
σ−2
∆H ϕ + σ(σ − 1)ϕ
N
X
(|Xi ϕ|2 + |Yi ϕ|2 ) ≥ σϕσ−1 ∆H ϕ.
i=1
R
Lemma 1.3 ([13]). Let f ∈ L1 (R2N +1 ) be such that R2N +1 f dη ≥ 0. Then there exists a test function
0 ≤ ϕ ≤ 1 such that
Z
f ϕ dη ≥ 0.
R2N +1
2. Results and proofs
Let HT = H × (0, T ), H = H × (0, ∞). For R > 0, let
CR = {(x, y, τ, t) ∈ H : R4 ≤ t2 + |x|4 + |y|4 + τ 2 ≤ 2R4 }.
R
Define Lploc (HT , f (η, t) dηdt) = u : HT 7→ R | K |u|p f (η, t) dηdt < +∞ for any compact K ⊂ HT .
2.1. The case of a single equation.
Here we consider the problem (1.1)–(1.2). Let us start with the following definition.
+1
Definition 2.1. A local weak solution u of the equation (1.1) in R2N
, equipped with the initial data
+
u(., 0) = u0 ( . ) and ut (., 0) = u1 ( . ), is a function
1
u ∈ CB
[0, T ]; (L1loc (HB ) ∩ C((0, T ); Lploc (H)) ∩ L1 Lploc (H, f (η, t) dηdt)),
satisfying
Z
−
Z
δ/2
Z
u∆H ϕ dηdt +
(−∆H ) ϕu dηdt +
uϕtt dηdt
HT
HT
Z
Z
Z
p
=
f u ϕ dηdt +
u1 (η)ϕ(η, 0) dη −
u0 (η)ϕt (η, 0) dη
HT
R2N +1
HT
(2.1)
R2N +1
for any test function ϕ ∈ C02 (HT ). We say that the solution is global when T = +∞.
Theorem 2.2. Let p > 1 and
Z
lim inf
R→∞
u1 (η) dη > 0.
|η|<R
+1
Then the problem (1.1)–(1.2) does not admit a global weak solution defined on R2N
if
+
1<p≤
Q + δ − 1 + δ(2λ + 1)
= p∗ .
Q−1
Proof. Let u be a global weak solution and ϕ a regular nonnegative test function chosen so that ϕt (η, 0) = 0.
R. Kellil, M. Kirane, J. Nonlinear Sci. Appl. 9 (2016), 3279–3286
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From (2.1), we have
Z
Z
Z
Z
Z
u1 (η)ϕ(η, 0) dη +
f.up ϕ dηdt =
uϕtt dηdt −
u∆H ϕ dηdt +
R2N +1
H
H
H
HT
u(−∆H )δ/2 ϕ dηdt.
(2.2)
Let us take ϕσ , σ 1, rather than ϕ (this will be clear later) in (2.2); by using Young’s inequality, we
have
Z
Z
Z
0
1
0
0
−p
(σ−1− σp )p0
(σ−2− σp )p0
u(ϕσ )tt dηdt ≤
f p ϕ
f up ϕσ + C1
|ϕtt |p + ϕ
|ϕt |2p dηdt,
(2.3)
6 H
H
H
where p + p0 = pp0 . Now, using Lemma 1.3, we obtain the estimate
Z
Z
Z
0
1
0 −p
(σ−1− σp )p0
ϕ
f up ϕσ + C2
|∆H ϕ|p f p ,
−
u∆H ϕσ dηdt ≤
6 H
H
H
and similarly,
Z
H
1
ϕ dηdt ≤
6
δ/2 σ
u(−∆H )
Z
Z
p σ
(σ−1− σp )p0
ϕ
f u ϕ + C3
H
H
0
|(−∆H )δ/2 ϕ|p f
0
− pp
.
Collecting the above inequalities, we obtain
Z
Z
nZ
−p0
0
0
(σ−1− σp )p0
(σ−2− σp )p0
p
u1 (η)ϕ(η , 0) dη +
f u ϕ dη dt ≤ C
f p (ϕ
|ϕtt |p + ϕ
|ϕt |2p ) dη dt
R2N +1
H
ZH
0
0
(σ−1− σp )p0
f −p /p (ϕ
+
|∆H ϕ|p ) dη dt
Z H
o
δ
0
(σ−1− σp )p0 −p0 /p
+
ϕ
f
|(−∆) 2 ϕ|p dη dt .
H
At this stage, we take
τ 2 + |x|4 + |y|4 + tα 2
, α= ,
ϕ(η , t) = Φ
4
R
δ
2
+
where Φ ∈ C0 (R ) is nonincreasing, 0 ≤ Φ ≤ 1, and
0 if
r ≥ 2,
Φ(r) =
1 if 0 ≤ r ≤ 1.
We have
∆H ϕ(η , t) =
16
4(N + 2) 0 2
2
00
6
6
+
Φ
(%)
|x|
+
|y|
Φ
(%)
|x|
+
|y|
8
R4
R
+ τ 2 |x|2 + |y|2 + 2τ hx, yi |x|2 − |y|2 ,
where
%=
1 2
4
4
α
τ
+
|x|
+
|y|
+
t
.
R4
Passing now to the scaled variables
e
t=R
−4
α
t,
τe = R−2 τ,
x
e = R−1 x,
ye = R−1 y
and setting
α
%e = τe2 + |e
x|4 + |e
y |4 + te ,
α
Ω = {(e
τ , e
t) , τe2 + |e
x|4 + |e
y |4 + te ≤ 2},
α
T = {(e
τ , e
t) , 1 ≤ τe2 + |e
x|4 + |e
y |4 + te ≤ 2},
R. Kellil, M. Kirane, J. Nonlinear Sci. Appl. 9 (2016), 3279–3286
3283
we obtain
e
∆
ϕ(e
τ
,
t
)
τ, e
t )∈T,
H
≤ CR−2 , for (e
δ
+1
τ, e
t ) ≤ CR−δ , for (e
τ, e
t ) ∈ R2N
,
(−∆H ) 2 ϕ(e
+
4
τ, e
t )∈T,
τ, e
t ) ≤ CR− α , for (e
ϕt (e
2
τ, e
t ) ≤ CR− α , for (e
τ, e
t )∈T,
ϕtt (e
which, in turn, allow us to obtain the estimates
Z
0
0
2 0
4
0
0
−p
(σ−1− σp )p0
(σ−2− σp )p0
−(γ+ 4λ
)p −α
p +2N +1+ α
α p
|ϕtt |p + ϕ
|ϕt |2p dη dt ≤ CR
,
f p ϕ
H
Z
0
0
4
0
− p (σ−1− σp )p0
−(γ+ 4λ
) p −2p0 +2N +1+ α
α p
f pϕ
|∆H ϕ|p dηdt ≤ CR
,
H
Z
0
0
4
0
− p (σ−1− σp )p0
−(γ+ 4λ
) p −δp0 +2N +1+ α
α p
f pϕ
|(−∆H )δ/2 ϕ|p dηdt ≤ CR
.
H
For α = 2δ , we have the estimate
Z
Z
0
−(γ+2δλ) pp −δp0 +2N +2δ
0<
u1 (η)ϕ(η, 0)dη +
f up ϕ dηdt ≤ CR
.
R2N +1
(2.4)
H
Now, if p < p∗ , then passing to the limit in (2.4) when R → ∞, we obtain
Z
Z
0 < lim inf
u1 (η) dη +
f up dηdt = 0,
R→∞
|η|<R
H
which is a contradiction.
The case of p = p∗ may be treated as in [8].
2.2. The case of the system (1.3)
First, we introduce the definition of a weak solution of the system (1.3).
Definition 2.3. A local weak solution of the system (1.3) is a couple of functions
1
(u, v) ∈ CB
[0, T ]; (L1loc (HB
∩ C((0, T ); Lploc (H)) ∩ L1 Lploc (H, g(η, t) dηdt))×
1
CB
[0, T ]; (L1loc (HB ∩ C((0, T ); Lploc (H)) ∩ L1 Lploc (H, f (η, t) dηdt))
satisfying the equations
Z
−
Z
Z
u∆H ϕ +
u(−∆H )δ/2 ϕ +
uϕtt
HT
HT
HT
Z
Z
Z
q
=
gv +
u1 (η)ϕ(η, 0)dη −
HT
and
H
(2.5)
u0 (η)ϕt (η, 0) dη
R2N +1
Z
−
Z
Z
v∆H ϕ +
v(−∆H )%/2 ϕ +
vϕtt
HT
HT
HT
Z
Z
Z
p
=
fu +
v1 (η)ϕ(η, 0) dη −
HT
R2N +1
(2.6)
v0 (η)ϕt (η, 0) dη
R2N +1
for any test function ϕ ∈ C02 (HT ). We say the the solution is global when T = +∞.
R. Kellil, M. Kirane, J. Nonlinear Sci. Appl. 9 (2016), 3279–3286
Theorem 2.4. Let p, q > 1 be two real numbers. If
Z
u1 (η)dη > 0 and
lim inf
R→∞
3284
Z
v1 (η) dη > 0,
lim inf
R→∞
|η|
|η|
then the system (1.3) does not admit a global weak solution defined on H if
(Q + 1)(pq − 1) ≤ max{Λ1 , Λ2 },
where
Λ1 = (k + 1 + 2µ)p + min{1, %}pq + γ + 2λ,
and
Λ2 = (γ + 1 + 2λ)p + min{1, δ}pq + k + 2µ.
Proof. Let (u, v) be a global weak nontrivial solution and ϕ a regular nonnegative test function chosen so
that ϕt (η, 0) = 0. From (2.5), we have
Z
Z
Z
Z
q
u1 (η)ϕ(η, 0) dη +
gv dηdt =
uϕtt dηdt −
u∆H ϕ dηdt
HT
HT
HT
R2N +1
Z
u(−∆)δ/2 ϕ dηdt.
+
HT
Using ϕσ , σ 1 and Hölder’s inequality instead of Young’s one, we have
Z
1
Z
p
σ
u(ϕ )tt dηdt ≤ C
f up ϕσ A(f, p),
HT
HT
where
A(f, p) =
Z
f
−p0
p
ϕ
10
Z
p
|ϕtt | dηdt
+
(σ−1− σp )p0
p0
HT
0
(σ−2− σp )p0
f −p /p ϕ
10
0
p
|ϕt |2p dηdt .
HT
Similarly, by using the Hölder inequality, we obtain the estimate
Z
Z
1/p
σ
−
u∆H ϕ dηdt ≤ C
f up ϕσ dηdt
B(f, p),
HT
HT
where
B(f, p) =
Z
(σ−1− σp )p0
ϕ
10
0
0
p
|∆H ϕ|p f −p /p dηdt .
HT
Also, by Lemma 1.1 we have
Z
Z
δ/2 σ
u(−∆H ) ϕ dηdt ≤ C
HT
HT
where
T (f, p, δ) =
Z
ϕ
(σ−1− σp )p0
HT
So we get
Z
1/p
f up ϕσ dηdt
T (f, p, δ),
Z
gv q ϕσ dηdt ≤ C
u1 (η)ϕ(η, 0) dη +
10
0
0
p
|(−∆CR )−δ/2 ϕ|p f −p /p dηdt .
Z
f up ϕσ
1/p A(f, p) + B(f, p) + T (f, p, δ)
.
(2.7)
Proceeding in the same manner, we obtain the estimate
Z
Z
Z
1/q v1 (η)ϕ(η, 0) dη +
f up ϕσ dηdt ≤ C
gv q ϕσ
A(g, q) + B(g, q) + T (g, q, %) .
(2.8)
R2N +1
R2N +1
HT
HT
CR
CR
R. Kellil, M. Kirane, J. Nonlinear Sci. Appl. 9 (2016), 3279–3286
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To simplify, let us set
Z
Z
p σ
I=
f u ϕ dηdt
gv q ϕσ dηdt.
and J =
HT
HT
By using
Z
Z
u1 (η)ϕ(η, 0) dη > 0
and
v1 (η)ϕ(η, 0) dη > 0,
R2N +1
R2N +1
and Lemma 1.3 the inequalities (2.7) and (2.8) may be written as
1
I ≤ CJ q A(g, q) + B(g, q) + T (g, q, %) ,
and
1
J ≤ CI p A(f, p) + B(f, p) + T (f, p, δ) .
So
I
and
J
(2.9)
1
1− pq
1
1
1
≤ C A(g, q) + B(g, q) + T (g, q, %) × A q (f, p) + B q (f, p) + T q (f, p, δ) ,
(2.10)
1
1− pq
1
1
1
≤ C A(f, p) + B(f, p) + T (f, p, δ) × A p (g, q) + B p (g, q) + T p (g, q, δ) .
(2.11)
At this stage, we choose
ϕ(η, t) = Φ
τ 2 + |x|4 + |y|4 + t2
R4
,
(2.12)
where Φ is the same as in the proof of Theorem 2.2. Passing to the scaled variables
e
t = R−2 t, τe = R−2 τ, x
e = R−1 x and ye = R−1 y,
we obtain the estimates
−(γ+2λ) p1 −1+(2N +3) p10
≤ CRσ1 ,
(2.14)
−(γ+2λ) p1 −2+(2N +3) p10
≤ CRσ2 ,
(2.15)
A(f, p) ≤ CR
B(f, p) ≤ CR
(2.13)
T (f, p, δ) ≤ CR
≤ CRσ3 ,
(2.16)
−(k+2µ) 1q −1+(2N +3) q10
≤ CRθ1 ,
(2.17)
−(k+2µ) 1q −2+(2N +3) q10
≤ CRθ2 ,
(2.18)
A(g, q) ≤ CR
B(g, q) ≤ CR
−(γ+2λ) p1 −δ+(2N +3) p10
and
T (g, q, %) ≤ CR
−(k+2µ) 1q −%+(2N +3) q10
≤ CRθ3 .
(2.19)
Using these inequalities in (2.8) and in (2.10), we obtain
I
1
1− pq
≤C
9
X
Rωi ,
i=1
where
σ1
, for 1 ≤ i ≤ 3,
q
σ2
= θi + , for 1 ≤ i ≤ 3,
q
ωi = θ i +
ωi+3
and
ωi+6 = θi +
A similar inequality holds for J
1
1− pq
.
σ3
, for 1 ≤ i ≤ 3.
q
(2.20)
R. Kellil, M. Kirane, J. Nonlinear Sci. Appl. 9 (2016), 3279–3286
3286
Now, we require ωi ≤ 0 for 1 ≤ i ≤ 9. This is equivalent to
(Q + 1)(pq − 1) ≤ (k + 1 + 2µ)p + min{1, %}pq + γ + 2λ = Λ1 .
Considering J
1
1− pq
, we obtain
(Q + 1)(pq − 1) ≤ (γ + 1 + 2λ)p + min{1, δ}pq + k + 2µ = Λ2 .
Consequently, if
(Q + 1)(pq − 1) ≤ max{Λ1 , Λ2 },
then passing to the limit when R → ∞ in (2.11), we obtain
Z
f up dηdt = 0 =⇒ u = 0,
H
which in turn leads to v = 0 via (2.10). Returning to inequalities (2.7) and (2.8) we obtain a contradiction.
Acknowledgements
This project is funded by the Deanship of Scientific Research (DSR), Al-Majmaah University, Saudi
Arabia under no 15. The authors therefore aknowledge with thanks DSR For the financial support.
References
[1] B. Ahmad, A. Alsaedi, M. Kirane, Nonexistence of global solutions of some nonlinear space-nonlocal evolution
equations on the Heisenberg group, Electron. J. Differential Equations, 2015 (2015), 10 pages. 1.1
[2] A. B. Al’shin, M. O. Korpusov, A. G. Sveshnikov, Blow-up in Nonlinear Sobolev Type, Walter de Gruyter, Berlin,
(2011). 1
[3] L. D’Ambrosio, Critical degenerate inequalities on the Heisenberg group, Manuscripta Math., 106 (2001), 519–536.
1
[4] A. Elhamidi, M. Kirane, Nonexistence results of solutions to systems of semilinear differential inequalities on the
Heisenberg group, Abstr. Appl. Anal., 2004 (2004), 155–164. 1
[5] A. Elhamidi, A. Obeid, Systems of semilinear higher-order evolution inequalities on the Heisenberg group, J.
Math. Anal. Appl., 280 (2003), 77–90. 1
[6] G. B. Folland, E. M. Stein, Estimate for the ∂H complex and analysis on the Heisenberg group, Comm. Pure Appl.
Math., 27 (1974), 492–522. 1
[7] J. A. Goldstein, I. Kombe, Nonlinear degenerate parabolic equations on the Heisenberg group, Int. J. Evol. Equ.,
1 (2005), 122. 1
[8] E. Mitidieri, S. I. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential
equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), 1–384. 2.1
[9] H. Mokrani, F. Mint Aghrabatt, Semilinear parabolic equations on the Heisenberg group with a singular potential,
Abstr. Appl. Anal., 2012 (2012), 18 pages. 1
[10] D. Müller, E. M. Stein, Lp-estimates for the wave equation on the Heisenberg group, Rev. Mat. Iberoam., 15
(1999), 297–332. 1
[11] A. I. Nachman, The wave equation on the Heisenberg group, Comm. Partial Differential Equations, 7 (1982),
675–714. 1
[12] A. Pascucci, Semilinear equations on nilpotent Lie groups: global existence and blow-up of solutions, Matematiche,
53 (1998), 345–357. 1
[13] S. I. Pokhozhaev, Nonexistence of global solutions of nonlinear evolution equations, Differ. Equ., 49 (2013),
599–606. 1.3
[14] S. I. Pohozaev, L. Véron, Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg
group, Manuscripta Math., 102 (2000), 85–99. 1
[15] S. L. Sobolev, On a new problem of mathematical physics, Izv. Akad. Nauk SSSR Ser. Math., 18 (1954), 3–50.
[16] C. D. Sogge, Lectures on Non-Linear Wave Equations: Second edition, International Press, Boston, (2008). 1
[17] Q. S. Zhang, The critical exponent of a reaction diffusion equation on some Lie groups, Math. Z., 228 (1998),
51–72. 1