Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 110, pp. 1–9.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
NONEXISTENCE RESULTS FOR A PSEUDO-HYPERBOLIC
EQUATION IN THE HEISENBERG GROUP
MOKHTAR KIRANE, LAKHDAR RAGOUB
Abstract. Sufficient conditions are obtained for the nonexistence of solutions
to the nonlinear pseudo-hyperbolic equation
utt − ∆H utt − ∆H u = |u|p ,
(η, t) ∈ H × (0, ∞), p > 1,
where ∆H is the Kohn-Laplace operator on the (2N + 1)-dimensional Heisenberg group H. Then, this result is extended to the case of a 2 × 2-system of
the same type. Our technique of proof is based on judicious choices of the test
functions in the weak formulation of the sought solutions.
1. Introduction
In this article, we are concerned with the nonexistence of weak solutions to the
nonlinear pseudo-hyperbolic equation
utt − ∆H utt − ∆H u = |u|p ,
(η, t) ∈ H × (0, ∞), p > 1,
(1.1)
under the initial conditions
u(η, 0) = u0 (η),
ut (η, 0) = u1 (η),
η ∈ H,
(1.2)
where ∆H is the Kohn-Laplace operator on the (2N + 1)-dimensional Heisenberg
group H. In the Euclidean case, pseudo-hyperbolic equations served as models for
the unidirectional propagation of nonlinear dispersive long waves [2], creep buckling [5] for example. For further applications, one is referred to the valuable book [1]
where a sizeable number of pseudo-hyperbolic equations are studied. Our proofs
rely on the test function method [8, 12]. For the reader convenience, some background facts used in the sequel are recalled.
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.
2010 Mathematics Subject Classification. 47J35, 34A34, 35R03.
Key words and phrases. Nonexistence; nonlinear pseudo-hyperbolic equation;
systems of pseudo-hyperbolic equations; Heisenberg group.
c
2015
Texas State University - San Marcos.
Submitted January 5, 2015. Published April 22, 2015.
1
2
M. KIRANE, L. RAGOUB
EJDE-2015/110
On H it is natural to define a distance from η = (x, y, τ ) =: (z, τ ) to the origin
by
N
X
2 1/4
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 satisfies 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 a 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)
d2 v
dρ2
+
Q − 1 dv ,
ρ dρ
PN
−2
2
2
where ρ = |η|H , a(η) = ρ
i=1 (xi + yi ) and Q = 2N + 2 is the homogeneous dimension of H.
For more details on Heisenberg groups, we refer to [4, 7].
In this work, we first provide a sufficient condition for the nonexistence of weak
solutions to the nonlinear problem (1.1)-(1.2), then we extend the result to the case
of the 2 × 2 system
utt − ∆H utt − ∆H u = |v|q ,
p
vtt − ∆H vtt − ∆H v = |u| ,
(η, t) ∈ H × (0, ∞),
(η, t) ∈ H × (0, ∞),
u(η, 0) = u0 (η), u1 (η, 0) = u1 (η),
v(η, 0) = v0 (η), v(η, 0) = v1 (η),
η∈H
(1.3)
η ∈ H,
where p, q > 1 are real numbers, for which we provide a sufficient condition for the
nonexistence of weak solutions.
2. Results and proofs
Let HT = H × (0, T ), H = H × (0, ∞). For R > 0, let
UR = {(x, y, τ, t) ∈ H : 0 ≤ t4 + |x|4 + |y|4 + τ 2 ≤ 2R4 }.
EJDE-2015/110
NONEXISTENCE OF SOLUTIONS IN THE HEISENBERG GROUP
3
2.1. Case of a single equation. The definition of solutions we adopt for (1.1)(1.2) is:
We say that u is a local weak solution to (1.1)-(1.2) on H with initial data
u(0, ·) = u0 ∈ L1loc (H), if u ∈ Lploc (H) and satisfies
Z
Z
Z
|u|p ϕ dϑ dt +
u1 (ϑ)ϕ(ϑ, 0) dϑ +
u1 (ϑ)∆H ϕ(ϑ, 0) dϑ
H
H
H
Z
Z
Z
=
uϕtt dϑ dt +
u∆H ϕtt dt dϑ −
u∆H ϕ dt dϑ,
H
H
H
for any test function ϕ, ϕ(·, t) = 0, ϕt (·, t) = 0, t ≥ T . The solution u is said global
if it exists on (0, ∞).
Our first main result is given by the following theorem.
Theorem 2.1. Let u1 ∈ L1 (H). Suppose that
Z
u0 dϑ > 0.
(2.1)
H
If
1<p≤1+
2
,
Q−1
then any weak solution to (1.1)-(1.2) blows-up in a finite time.
Proof. Suppose that u is a weak solution to (1.1)-(1.2). Then for any regular test
function ϕ, we have
Z
Z
|u|p ϕ dϑ dt +
u1 (ϑ)ϕ(ϑ, 0) dϑ
H
H
Z
Z
≤
|u||ϕtt | dϑ dt +
|u||∆H ϕtt | dt dϑ
(2.2)
H
H
Z
Z
+
|u||∆H ϕ| dt dϑ +
|u1 (ϑ)||∆H ϕ(ϑ, 0)| dϑ.
H
H
Using the ε-Young inequality
0
ab ≤ εap + Cε bp ,
a, b, ε, Cε > 0, 1 < p, p0 , p + p0 = pp0 ,
with parameters p and p0 = p/(p − 1), we obtain
Z
Z
1
|u||ϕtt | dϑ dt =
|u|ϕ1/p ϕ− p |ϕtt | dϑ dt
H
H
Z
Z
p
1
p
≤ε
|u| ϕ dϑ dt + cε
ϕ− p−1 |ϕtt | p−1 dϑ dt,
H
H
for some positive constant cε .
Similarly, we have
Z
Z
Z
p
1
|u||∆H ϕtt | dt dϑ ≤ ε
|u|p ϕ dϑ dt + cε
ϕ− p−1 |∆H ϕtt | p−1 dt dϑ,
H
Z
ZH
ZH
1
P
|u||∆H ϕ| dt dϑ ≤ ε
|u|p ϕ dϑ dt + cε
ϕ− p−1 |∆H ϕ| P −1 dt dϑ.
H
H
(2.3)
H
(2.4)
(2.5)
4
M. KIRANE, L. RAGOUB
EJDE-2015/110
Using (2.2), (2.3), (2.4) and (2.5), for ε > 0 small enough, we obtain
Z
Z
|u|p ϕ dϑ dt +
u1 (ϑ)ϕ(ϑ, 0) dϑ
H
H
Z
≤ C Ap (ϕ) + Bp (ϕ) + Cp (ϕ) +
|u1 (ϑ)||∆H ϕ(ϑ, 0)| dϑ ,
(2.6)
H
where
Z
p
1
Ap (ϕ) =
ϕ− p−1 |ϕtt | p−1 dϑ dt,
Z H
p
1
Bp (ϕ) =
ϕ− p−1 |∆H ϕtt | p−1 dϑ dt,
ZH
p
1
Cp (ϕ) =
ϕ− p−1 |∆H ϕ| p−1 dϑ dt.
(2.7)
(2.8)
(2.9)
H
Now, let us consider the test function
t4 + |x|4 + |y|4 + τ 2 , R > 0, ω 1,
ϕR (t, ϑ) = φω
R4
where φ ∈ C0∞ (R+ ) is a decreasing function satisfying
(
1 if 0 ≤ r ≤ 1,
φ(r) =
0 if r ≥ 2.
(2.10)
Observe that supp(ϕR ) is a subset of UR , while supp(ϕR tt ), supp(∆H ϕR ) and
supp(∆H (ϕR )tt ) are subsets of
ΘR = {(t, x, y, τ ) ∈ H : R4 ≤ t4 + |x|4 + |y|4 + τ 2 ≤ 2R4 }.
Let
ρ=
t4 + |x|4 + |y|4 + τ 2
.
R4
Then we have
∆H ϕR (t, ϑ)
4ω(N + 4) 2
2
|x|
+
|y|
φ0 (ρ)φω−1 (ρ)
R4
16ω(ω − 1) +
(|x|6 + |y|6 ) + 2τ (|x|2 − |y|2 )x · y + τ 2 (|x|2 + |y|2 ) φ02 (ρ)φω−2 (ρ)
8
R
16ω + 8 (|x|6 + |y|6 ) + 2τ (|x|2 − |y|2 )x · y + τ 2 (|x|2 + |y|2 ) φ00 (ρ)φω−1 (ρ)
R
for example.
Observe first that (ϕR )t (ϑ, 0) = 0 as required in the definition. It follows that
there is a positive constant C > 0, independent of R, such that for all (t, ϑ) ∈ ×R ,
we have
|∆H ϕR (t, ϑ)| ≤ CR−2 φω−2 (ρ)χ(ρ),
(2.11)
where
χ(ρ) = |φ0 (ρ)|φ(ρ) + φ02 (ρ) + |φ00 (ρ)|φ(ρ),
and
=
|(∆H ϕR )t (t, ϑ)| ≤ CR−3 ,
|(ϕR )tt (t, ϑ)| ≤ CR
−4
.
(2.12)
(2.13)
EJDE-2015/110
NONEXISTENCE OF SOLUTIONS IN THE HEISENBERG GROUP
5
Using (2.11) and (2.12), we obtain
2p
Ap (ϕR ) ≤ CRQ+1− p−1 ,
Bp (ϕR ) ≤ CR
Cp (ϕR ) ≤ CR
4p
Q+1− p−1
2p
Q+1− p−1
(2.14)
,
(2.15)
.
(2.16)
Let us consider now the change of variables
(t, x, y, τ ) = (t, ϑ) 7→ (e
t, ve) = (e
t, x
e, ye, τe),
(2.17)
where
e
t = R−1 t,
τe = R−2 τ, x
e = R−1 x,
ye = R−1 y.
Let
ρe = e
t4 + |e
x|4 + |e
y |4 + τe2 ,
Cf
t, x
e, ye, τe) ∈ H : 1 ≤ ρe ≤ 2},
R = {(e
CR = {(x, y, τ ) ∈ H : R4 ≤ |x|4 + |y|4 + τ 2 ≤ 2R4 }.
Using (2.6), (2.15) and (2.16), we obtain
Z
Z
|u|p ϕR dϑ dt +
u1 (ϑ)ϕR (ϑ, 0) dϑ
H
H
Z
≤ C Rϑ1 + Rϑ2 +
|u1 (ϑ)||∆H ϕR (ϑ, 0)| dϑ ,
(2.18)
CR
where
ϑ1 = Q + 1 −
2p
p−1
and ϑ2 = Q + 1 −
4p
.
p−1
On the other hand, we have
Z
Z
|u|p ϕR dϑ dt +
u1 (ϑ)ϕR (ϑ, 0) dϑ
lim inf
R→∞ H
H
Z
Z
≥ lim inf
|u|p ϕR dϑ dt + lim inf
u1 (ϑ)ϕR (ϑ, 0) dϑ.
R→∞
R→∞
H
H
Using the monotone convergence theorem, we obtain
Z
Z
p
|u| ϕR dϑ dt =
|u|p dϑ dt.
lim inf
R→∞
H
H
1
Since u1 ∈ L (H), by the dominated convergence theorem, we have
Z
Z
lim inf
u1 (ϑ)ϕR (ϑ, 0) dϑ =
u1 (ϑ) dϑ.
R→∞
H
H
Now, we have
lim inf
Z
R→∞
p
Z
|u| ϕR dϑ dt +
Z
u1 (ϑ)ϕR (ϑ, 0) dϑ ≥
H
|u|p dϑ dt + `,
H
H
where from (2.1),
Z
`=
u1 (ϑ) dϑ > 0.
H
By the definition of the limit inferior, for every ε > 0, there exists R0 > 0 such that
Z
Z
p
|u| ϕR dϑ dt +
u1 (ϑ)ϕR (ϑ, 0) dϑ
H
H
6
M. KIRANE, L. RAGOUB
EJDE-2015/110
Z
Z
> lim inf
|u|p ϕR dϑ dt +
u1 (ϑ)ϕR (ϑ, 0) dϑ − ε
R→∞
H
H
Z
≥
|u|p dϑ dt + ` − ε,
H
for every R ≥ R0 . Taking ε = `/2, we obtain
Z
Z
Z
`
p
|u| ϕR dϑ dt +
u1 (ϑ)ϕR (ϑ, 0) dϑ ≥
|u|p dϑ dt + ,
2
H
H
H
for every R ≥ R0 . Then from (2.18), we have
Z
Z
`
|u0 (ϑ)||∆H ϕR (ϑ, 0)| dϑ ,
|u|p dϑ dt + ≤ C Rϑ1 + Rϑ2 +
2
CR
H
(2.19)
for R large enough.
Now, we require that ϑ1 = max{ϑ1 , ϑ2 } ≤ 0, which is equivalent to 1 < p ≤
2
1 + Q−1
. We distinguish two cases.
2
Case 1: 1 < p < 1 + Q−1
. In this case, letting R → ∞ in (2.19) and using the
dominated convergence theorem, we obtain
Z
`
|u|p dϑ dt + ≤ 0,
2
H
which is a contradiction as ` > 0.
2
Case 2: p = 1 + Q−1
. From (2.19), we obtain
Z
Z
p
|u| dϑ dt ≤ C < ∞ ⇒ lim
R→∞
H
|u|p ϕR dϑ dt = 0.
(2.20)
CR
Using the Hölder inequality with parameters p and p/(p − 1), from (2.2), we obtain
Z
1/p
Z
`
|u|p ϕR dϑ dt + ≤ C
|u|p ϕR dϑ dt
.
2
H
ΘR
Letting R → ∞ in the above inequality and using (2.20), we obtain
Z
`
|u|p dϑ dt + = 0.
2
H
A contradiction; the proof of the theorem is complete.
2.1.1. The case of system (1.3). The definition of solutions we adopt for (1.3) is:
We say that the pair (u, v) is a local weak solution to (1.3) on H with initial
data (u(0, ·), v(0, ·)) = (u0 , v0 ) ∈ L1loc (H) × L1loc (H), if (u, v) ∈ Lploc (H) × Lqloc (H)
and satisfies
Z
Z
|v|q ϕ dϑ dt +
u1 (ϑ)ϕ(ϑ, 0) dϑ
H
H
Z
Z
Z
Z
=
uϕtt dϑ dt +
u(∆H ϕ)tt dt dϑ −
u∆H ϕ dt dϑ +
u1 (ϑ)∆H ϕ(ϑ, 0) dϑ
H
H
H
H
and
Z
Z
|u|p ϕ dϑ dt +
v1 (ϑ)ϕ(ϑ, 0) dϑ
H
H
Z
Z
Z
Z
=
vϕtt dϑ dt +
v(∆H ϕ)tt dt dϑ −
v∆H ϕ dt dϑ +
v1 (ϑ)∆H ϕ(ϑ, 0) dϑ,
H
H
H
H
EJDE-2015/110
NONEXISTENCE OF SOLUTIONS IN THE HEISENBERG GROUP
7
for any test function ϕ, ϕ(·, t) = 0, ϕt (·, t) = 0, t ≥ T . The solution is said global
if it exists for T = +∞.
Our second main result is given by the following theorem.
Theorem 2.2. Let (u1 , v1 ) ∈ L1 (H) × L1 (H). Suppose that
Z
Z
u1 dϑ > 0 and
v1 dϑ > 0.
H
H
If 1 < pq ≤ (pq)∗ , where
2
max{p + 1, q + 1},
Q−1
then there exists no nontrivial weak solution to (1.3).
(pq)∗ = 1 +
Proof. Suppose that (u, v) is a nontrivial weak solution to (1.3). Then for any
regular test function ϕ, we have
Z
Z
|v|q ϕ dϑ dt +
u1 (ϑ)ϕ(ϑ, 0) dϑ
H
H
Z
Z
≤
|u||ϕtt | dϑ dt +
|u||(∆H ϕ)tt | dt dϑ
H
H
Z
Z
+
|u||∆H ϕ| dt dϑ +
|u1 (ϑ)||∆H ϕ(ϑ, 0)| dϑ
H
H
and
Z
p
Z
|u| ϕ dϑ dt +
v1 (ϑ)ϕ(ϑ, 0) dϑ
H
Z
Z
≤
|v||ϕtt | dϑ dt +
|v||(∆H ϕ)tt | dt dϑ
H
H
Z
Z
+
|v||∆H ϕ| dt dϑ +
|v1 (ϑ)||∆H ϕ(ϑ, 0)| dϑ.
H
H
H
Taking ϕ = ϕR , the test function given by (2.10), and using the Hölder inequality
with parameters p and p/(p − 1), we obtain
Z
Z
Z
q
|v| ϕR dϑ dt +
u1 (ϑ)ϕR (ϑ, 0) dϑ −
|u1 (ϑ)||∆H ϕR (ϑ, 0)| dϑ
H
H
H
Z
1/p
p−1
p−1
p−1
p
p
p
≤ Ap (ϕR )
+ Bp (ϕR )
+ Cp (ϕR )
|u|p ϕR dϑ dt
,
H
where Ap (ϕ), Bp (ϕ) and Cp (ϕ) are given respectively by (2.7), (2.8) and (2.9).
Similarly, by the Hölder inequality with parameters q and q/(q − 1), we get
Z
Z
Z
|u|p ϕR dϑ dt +
v1 (ϑ)ϕR (ϑ, 0) dϑ −
|v1 (ϑ)||∆H ϕR (ϑ, 0)| dϑ
H
H
H
Z
1/q
q−1
q−1
q−1
≤ Aq (ϕR ) q + Bq (ϕR ) q + Cq (ϕR ) q
|v|q ϕR dϑ dt
.
H
Without restriction of the generality, we may assume that for R large enough, we
have
Z
Z
u1 (ϑ)ϕR (ϑ, 0) dϑ −
|u1 (ϑ)||∆H ϕR (ϑ, 0)| dϑ ≥ 0,
ZH
ZH
v1 (ϑ)ϕR (ϑ, 0) dϑ −
|v1 (ϑ)||∆H ϕR (ϑ, 0)| dϑ ≥ 0.
H
H
8
M. KIRANE, L. RAGOUB
EJDE-2015/110
Slight modifications yield the proof in the general case (see the proof of Theorem
2.1). Then for R large enough, we have
Z
|v|q ϕR dϑ dt
H
(2.21)
Z
1/p
p−1
p−1
p−1
≤ Ap (ϕR ) p + Bp (ϕR ) p + Cp (ϕR ) p
|u|p ϕR dϑ dt
H
and
Z
|u|p ϕR dϑ dt
H
Z
1/q
q−1
q−1
q−1
|v|q ϕR dϑ dt
.
≤ Aq (ϕR ) q + Bq (ϕR ) q + Cq (ϕR ) q
(2.22)
H
Using the change of variables (2.17), from (2.21) and (2.22), we obtain
Z
1/p
Z
Q(p−1)−2
q
p
,
|v| ϕR dϑ dt ≤ CR
|u|p ϕR dϑ dt
H
ZH
Z
1/q
Q(q−1)−2
q
|u|p ϕR dϑ dt ≤ CR
|v|q ϕR dϑ dt
.
H
(2.23)
(2.24)
H
Combining (2.23) with (2.24), we obtain
1
1− pq
Z
|u|p ϕR dϑ dt
≤ CRυ1 ,
H
1
1− pq
Z
|v|q ϕR dϑ dt
≤ CRυ2 ,
(2.25)
(2.26)
H
where
υ1 =
Q(pq − 1) − 2(p + 1)
pq − 1
and υ2 =
Q(pq − 1) − 2(q + 1)
.
pq − 1
2
We require that υ1 ≤ 0 or υ2 ≤ 0 which is equivalent to 1 < pq ≤ 1 + Q
max{p +
1, q + 1}. We distinguish two cases.
2
Case 1: 1 < pq < 1 + Q
max{p + 1, q + 1}. Without loss of the generality, we may
suppose that 0 < q ≤ p. In this case, letting R → ∞ in (2.25), we obtain
Z
|u|p dϑ dt = 0,
H
which is a contradiction.
2
Case 2: pq = 1 + Q
max{p + 1, q + 1}. This case can be treated in the same way
as in the proof of Theorem 2.1.
Remark 2.3. If p = q and u = v in Theorem 2.2, we obtain the result for a single
equation given by Theorem 2.1.
References
[1] A. B. Al’shin, M. O. Korpusov, A. G. Sveshnikov; Blow-up in nonlinear Sobolev type, De
Gruyter, 2011.
[2] T. B. Benjamin, J. L. Bona, J. J. Mahony; Model equations for long waves in nonlinear
dispersive systems, Philos. Trans. R. Soc. Lond. Ser. A 272 (1220) (1972), 47–78.
[3] Y. Cao, J. Yin, C. Wang; Cauchy problems of semilinear pseudo-parabolic equations, J.
Differential Equations. 246 (2009), 4568–4590.
EJDE-2015/110
NONEXISTENCE OF SOLUTIONS IN THE HEISENBERG GROUP
9
[4] 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.
[5] N. J. Hoff; Creep buckling, Aeron. Quart. 7 (1956), No 1, 1–20.
[6] E. I. Kaikina, P. I. Naumkin, I. A. Shishmarev; The Cauchy problem for a Sobolev type
equation with power like nonlinearity, Izv Math. 69 (2005), 59–111.
[7] E. Lanconelli, F. Uguzzoni; Asymptotic behaviour and non existence theorems for semilinear
Dirichlet problems involving critical exponent on unbounded domains of the Heisenberg group,
Boll. Un. Mat. Ital. 1(1) (1998), 139–168.
[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) 3–383.
[9] S. I. Pokhozhaev; Nonexistence of Global Solutions of Nonlinear Evolution Equations, Differential Equations. 49, No. 5 (2013), 599-606.
[10] S.I. Pohozaev, L. Véron; Nonexistence results of solutions of semilinear differential i nequalities on the Heisenberg group, Manuscripta Math. 102 (2000) 85–99.
[11] S. L. Sobolev; On a new problem of mathematical physics, Izv. Akad. Nauk USSR Ser. Math.
18 (1954), 3–50.
[12] Qi S. Zhang; The critical exponent of a reaction diffusion equation on some Lie groups,
Math. Z. 228 (1) (1998), 51–72.
Mokhtar Kirane
Laboratoire de Mathématiques, Image et Applications, Pôle Sciences et Technologies,
Université de La Rochelle, Avenue M. Crépeau, 17042 La Rochelle, France.
NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
E-mail address: mkirane@univ-lr.fr
Lakhdar Ragoub
Al Yamamah University, College of Computers and Information Systems
P.O. Box 45180, Riyadh 11512, Saudi Arabia
E-mail address: l ragoub@yu.edu.sa