Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 02, pp. 1–10.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

LIFESPAN OF SOLUTIONS OF A FRACTIONAL EVOLUTION

EQUATION WITH HIGHER ORDER DIFFUSION ON THE

HEISENBERG GROUP

AHMED ALSAEDI, BASHIR AHMAD, MOKHTAR KIRANE, ABERRAZAK NABTI

Communicated by Jerome A. Goldstein

Abstract. We consider the higher order diffusion Schrödinger equation with

a time nonlocal nonlinearity

i∂tu− (−∆H)mu =
λ

Γ(α)

∫ t

0
(t− s)α−1|u(s)|p ds,

posed in (η, t) ∈ H×(0,+∞), supplemented with an initial data u(η, 0) = f(η),

where m > 1, p > 1, < α < 1, and ∆H is the Laplacian operator on the

(2N + 1)-dimensional Heisenberg group H. Then, we prove a blow up result
for its solutions. Furthermore, we give an upper bound estimate of the life

span of blow up solutions.

1. Introduction

In this article, we consider a nonlocal in time higher-order nonlinear Schrödinger
equation on the Heisenberg group

i∂tu− (−∆H)mu = λIα0|t|u(t)|p, η = (x, y, τ) ∈ H, t > 0, (1.1)

subject to the initial data

u(η, 0) = f(η), (1.2)

where u ≡ u(η, t) is a complex-valued unknown function, i2 = −1, λ = λ1 + iλ2 ∈
C\{0}, λi ∈ R (i = 1, 2), f = f(η) = f1(η) + if2(η), fi = fi(η) ∈ L1

loc(R2N+1)
(i = 1, 2) are real valued functions, and Iα0|tψ is the Riemann–Liouville fractional

integral of order (0 < α < 1) defined for a continuous function ψ(t), t > 0, by(
Iα0|tψ

)
(t) =

1

Γ(α)

∫ t

0

(t− s)α−1ψ(s) ds.

Here, Γ(·) stands for the gamma function.
First, for the sake of the reader, we give some known facts about the Heisenberg

group H and the operator ∆H. For their proof and more information, we refer
for example to [4, 5, 8, 9, 10]. The Heisenberg group H, whose elements are η =

2010 Mathematics Subject Classification. 35Q55, 35B44, 26A33, 35B30.
Key words and phrases. Schrödinger equation; Heisenberg group; life span;

Riemann-Liouville fractional integrals and derivatives.
c©2020 Texas State University.

Submitted June 8, 2019. Published January 7, 2020.

1



2 A. ALSAEDI, B. AHMAD, M. KIRANE, A. NABTI EJDE-2020/02

(x, y, τ) ≡ (z̃, τ) is the Lie group (R2N+1, ◦) with the group operation “◦” defined
by

η ◦ η̃ = (x+ x̃, y + ỹ, τ + τ̃ + 2(〈x, ỹ〉 − 〈x̃, y〉)),
where 〈·, ·〉 is the usual inner product in RN . The Laplacian ∆H over H is obtained
from the vector fields Xi = ∂xi + 2yi∂τ and Yi = ∂yi − 2xi∂τ , by

∆H =

N∑
i=1

(X2
i + Y 2

i );

explicitly, we have

∆H =

N∑
i=1

( ∂2
∂x2i

+
∂2

∂y2i
+ 4yi

∂2

∂xi∂τ
− 4xi

∂2

∂yi∂τ
+ 4(x2i + y2i )

∂2

∂τ2

)
.

A natural group of dilitations on H is given by

δγ(η) = (γx, γy, γ2τ), γ > 0,

whose Jacobian determinant is γQ, where

Q = 2N + 2

is the homogeneous dimension of H.
The operator ∆H is a degenerate elliptic operator. It is invariant with respect to

the left translation of H and homogeneous with respect to the dilatations δγ . More
precisely, we have

∆H(u(η ◦ η̃)) = (∆Hu)(η ◦ η̃), ∆H(u ◦ δγ) = γ2(∆Hu) ◦ δγ η, η̃ ∈ H.

The natural distance from η to the origin is

|η|H =
(
τ2 +

( N∑
i=1

x2i + y2i

)2)1/4
=
(
τ2 + |z̃|4

)1/4
.

Before we present our results, let us dwell a while on some existing literature. There
are many results about nonexistence of solutions of nonlinear Schrödinger equation
(see, e.g. [12, 18, 1, 6] and the references therein). Ikeda and Wakasugi [12] studied
the equation

i∂tu+ ∆u = λ|u|p, x ∈ RN , t > 0, (1.3)

with u(x, 0) = f(x), and showed that if 1 < p ≤ 1 + N/2, λ ∈ C\{0} and f ∈
L2(RN ), then the life span Tm must be finite and

lim
t→Tm

‖u(t)‖L2 = +∞.

Later, Kirane and Nabti [13] considered the equation

i∂tu+ ∆u =
λ

Γ(α)

∫ t

0

(t− s)α−1|u(s)|p ds, x ∈ RN , t > 0, (1.4)

with u(x, 0) = f(x), f ∈ L1(RN ) and proved that if 1 < p ≤ 1+2(α+1)/(N−2α)+,
λ ∈ C\{0}, λ1 > 0 and

∫
RN f2(x) dx < 0, then equation (1.4) has no global weak

solutions.
On the other hand, there are many papers concerning the life span of solutions

of various evolution equations (see [11, 14, 19, 13]); we mention in particular that
recently Ikeda [11] obtained the upper bound for the life span of solutions for



EJDE-2020/02 LIFESPAN OF SOLUTIONS OF AN EVOLUTION EQUATION 3

the nonlinear Schrödinger equations (1.3) supplemented with the initial condition
u(x, 0) = εf(x), of the form Tε ≤ Cε1/ρ, C > 0, ρ := k/2− 1/(p− 1) < 0.

Our present work is motivated by [16, 2]. Pohozaev and Véron [16] gave some
results about nonexistence of weak solutions of the differential inequality

∂tu−∆H(au) ≥ |η|γH|u|
p, a ∈ L∞, η ∈ H, t > 0, (1.5)

subjected to the initial condition u(x, 0) = u0(x), for γ > −2, 1 < p ≤ (Q+2+γ)/Q
and

∫
R2N+1 u0(x) dx ≥ 0. Recently Cazenave and al. [2] studied the global solutions,

and blow up solutions for the parabolic equation with nonlocal in time nonlinearity

∂tu−∆u =

∫ t

0

(t− s)−γ |u|p−1u(s) ds, x ∈ RN , t > 0, (1.6)

with 0 ≤ γ < 1, p > 1, u0 ∈ C0(RN ), and proved some results concerning the
nonexistence of global weak solutions.

Using the test function method, we study the blow up of weak solutions of
problem (1.1)–(1.2). Then we obtain an upper bound of the life span of blow up
solutions of equation (1.1) with initial data of the form u(η, 0) = εf(η), ε > 0.

2. Blow up solutions

In this section, we prove a blow up result for problem (1.1)–(1.2). At first, let us
recall some definitions and properties concerning fractional integrals and derivatives
(see [17] for more on fractional integrals and derivatives).

We denote by Dα
0|tψ(t) and Dα

t|Tψ(t) the left-handed and right-handed Riemann-

Liouville fractional derivatives of order (0 < α < 1) of a continuous function ψ(t),
t > 0 defined by (

Dα
0|tψ

)
(t) =

1

Γ(1− α)

d

dt

∫ t

0

(t− s)−αψ(s) ds,

(
Dα
t|Tψ

)
(t) = − 1

Γ(1− α)

d

dt

∫ T

t

(s− t)−αψ(s) ds.

Let AC([0, T ]) be the space of absolutely continuous on [0, T ] with T finite. We
introduce the following lemmas that will be use hereafter.

Lemma 2.1. Let ψ,ϕ,Dα
0|tψ,D

α
t|Tϕ ∈ C([0, T ]), we have the formula of integration

by parts (see [17, (2.64) p. 46])∫ T

0

(
Dα

0|tψ
)
(t)ϕ(t) dt =

∫ T

0

ψ(t)
(
Dα
t|Tϕ

)
(t) dt. (2.1)

Lemma 2.2. Let ψ ∈ AC2([0, T ]) := {ψ : [0, T ]→ R such that Dψ ∈ AC([0, T ])}.
Then, we have

−D ·Dα
t|Tψ(t) = Dα+1

t|T ψ(t), (2.2)

where D := d/dt is the usual derivative. Moreover, for all 1 ≤ q ≤ ∞, the equality

Dα
0|tI

α
0|t = IdLq (0, T ) (2.3)

holds almost everywhere on [0, T ].



4 A. ALSAEDI, B. AHMAD, M. KIRANE, A. NABTI EJDE-2020/02

Lemma 2.3 ((See [3])). Let

ψ(t) =
(

1− t

T

)σ
+

with t ≥ 0, T > 0 and σ � 1, then for all α ∈ (0, 1), we have

Dα
t|Tψ(t) = C1T

−α
(

1− t

T

)σ−α
+

, (2.4)

Dα+1
t|T ψ(t) = C2T

−α−1
(

1− t

T

)σ−α−1
+

, (2.5)(
Dα
t|Tψ

)
(T ) = 0,

(
Dα
t|Tψ

)
(0) = C1T

−α, (2.6)

where

C1 =
(1− α+ σ)Γ(σ + 1)

Γ(2− α+ σ)
, C2 =

(1− α+ σ)(σ − α)Γ(σ + 1)

Γ(2− α+ σ)
.

Lemma 2.4 (see [15, Lemma 3.1]). Let χ ∈ L1(R2N+1) and
∫
R2N+1 χ(η) dη < 0.

Then there exists a test function 0 ≤ ω ≤ 1 such that∫
R2N+1

χ(η)ω(η) dη < 0. (2.7)

Definition 2.5. Let T > 0. A function u is called a local weak solution of (1.1)–
(1.2), if u ∈ C([0, T );Lploc(R2N+1)) and satisfies

λ

∫ T

0

∫
R2N+1

Iα0|t|u|
pφ(η, t) dη dt+ i

∫
R2N+1

f(η)φ(η, 0) dη

= −
∫ T

0

∫
R2N+1

u (−∆H)mφ(η, t) dηdt− i
∫ T

0

∫
R2N+1

u ∂tφ(η, t) dη dt

(2.8)

for any φ ∈ C∞,10 (R2N+1 × (0, T )), φ ≥ 0, φ(·, T ) = 0. If T = +∞, we say that u is
a global weak solution of problem (1.1)–(1.2).

Let f = f1 + if2 satisfy one the the following set of assumptions

f1 ∈ L1(R2N+1), λ2

∫
R2N+1

f1(η) dη > 0,

or

f2 ∈ L1(R2N+1), λ1

∫
R2N+1

f2(η) dη < 0.

(2.9)

Now, we are in a position to announce our results.

Theorem 2.6. Suppose that p > 1 and

p ≤ p∗ =
Q+ 2m

Q− 2αm
, (2.10)

where if the equality holds, we assume p > Q/(Q− 2m) with Q > 2mmax{1, 1/α}.
If the initial data f satisfies (2.9), then problem (1.1)–(1.2) does not admit a global
weak solution.



EJDE-2020/02 LIFESPAN OF SOLUTIONS OF AN EVOLUTION EQUATION 5

Proof. The proof is done by contradiction. Suppose that u is a global bounded
weak solution. First we choose the test function. For this aim, we shall use a
non-negative smooth function φ1 which was constructed in [7].

φ1(x) = φ1(|x|), φ1(0) = 1, 0 < φ1(r) ≤ 1, for r ≥ 0, (2.11)

where φ1(r) is decreasing and φ1(r) → 0 as r → ∞ sufficiently fast. Moreover,
there exists a constant km such that

|∆m
H φ1| ≤ kmφ1, η ∈ R2N+1, (2.12)

and ‖φ1‖L1 = 1. Let

φ2(t) =
(
1− t

T

)σ
, T > 0, σ � 1,

φ(η, t) := φ1
( η
R

)
φ2
( t

R2m

)
, R > 0.

Let Q := R2N+1× [0, TR2m). We consider the case
∫
R2N+1 f2(η) dη < 0 and λ1 > 0

only, since the other cases can be treated similarly (see Remark 2.7).
Using (2.8), we have

λ

∫
Q
Iα0|t|u|

pφ(η, t) dηdt+ i

∫
R2N+1

f(η)φ(η, 0) dη

= −
∫
Q
u(−∆H)mφ(η, t)dηdt− i

∫
Q
u∂tφ(η, t)dηdt.

(2.13)

Replacing φ(η, t) by Dα
t|TR2mφ(η, t), we arrive at

λ

∫
Q
Iα0|t|u|

pDα
t|TR2mφ(η, t) dη dt+ i

∫
R2N+1

f(η)Dα
t|TR2mφ(η, 0) dη

= −
∫
Q
u(−∆H)mDα

t|TR2mφ(η, t) dηdt− i
∫
Q
uDDα

t|TR2mφ(η, t) dη dt.

(2.14)

Furthermore, by taking the real parts, using (2.1) and (2.3) in the left-hand side of
(2.14), and (2.2) in the right-hand side, we obtain

λ1

∫
Q
|u|pφ(η, t) dη dt−Dα

t|TR2mφ2(0)

∫
R2N+1

f2(η)φ1(η/R)dη

= −
∫
Q

(Re u)(−∆H)mφ1(η/R)Dα
t|TR2mφ2

(
t/R2m

)
dη dt

−
∫
Q

(Imu)φ1(η/R)Dα+1
t|TR2mφ2

(
t/R2m

)
dη dt.

By the assumption on f2 and using the Lemma 2.4, we have

Dα
t|TR2mφ2(0)

∫
R2N+1

f2(η)φ1(η/R) dη = CT−αR−2αm
∫
R2N+1

f2(η)φ1(η/R) dη ≤ 0.

Setting

IR :=

∫
Q
|u|pφ(η, t) dη dt,



6 A. ALSAEDI, B. AHMAD, M. KIRANE, A. NABTI EJDE-2020/02

we may write the estimate

λ1IR ≤ −
∫
Q

(Re u)(−∆H)mφ1(η/R)Dα
t|TR2mφ2

(
t/R2m

)
dηdt

−
∫
Q

(Imu)φ1(η/R)Dα+1
t|TR2mφ2

(
t/R2m

)
dηdt

≤
∫
Q
|u| |∆m

H φ1(η/R)||Dα
t|TR2mφ2

(
t/R2m

)
|dηdt

+

∫
Q
|u|φ1(η/R)|Dα+1

t|TR2mφ2
(
t/R2m

)
|dηdt ≡ A1 +A2.

(2.15)

Now, applying ε-Young’s inequality,

XY ≤ εXp + C(ε)Y q, X ≥ 0, Y ≥ 0, p+ q = pq,

with 0 < ε� 1, C(ε) = (1/q)(pε)−q/p) in

A1 with X = |u|φ(η, t)1/p, Y = φ(η, t)−1/p|∆m
H φ1(η/R)| |Dα

t|TR2mφ2
(
t/R2m

)
|,

A2 with X = |u|φ(η, t)1/p, Y = φ(η, t)−1/pφ1(η/R)|Dα+1
t|TR2mφ2

(
t/R2m

)
|,

we obtain

(λ1 − 2ε)IR

≤ C(ε)

∫
Q
φ1(η/R)−

1
p−1 |∆m

H φ1(η/R)|
p
p−1φ2

(
t/R2m

)− 1
p−1

× |Dα
t|TR2mφ2

(
t/R2m

)
|
p
p−1 dηdt

+ C(ε)

∫
Q
φ1(η/R)φ2

(
t/R2m

)− 1
p−1 |Dα+1

t|TR2mφ2
(
t/R2m

)
|
p
p−1 dη dt

≡ A3 +A4.

(2.16)

At this stage, we pass to the scaled variables s = t/R2m, η̃ = (x̃, ỹ, τ̃) such that
τ̃ = τ/R2, x̃ = x/R, ỹ = y/R, we obtain

A3 ≤ CRβ
∫ T

0

∫
R2N+1

φ1(η̃)φα1
2 (s) dη̃ds,

A4 ≤ CRβ
∫ T

0

∫
R2N+1

φ1(η̃)φα2
2 (s) dη̃ds,

where

α1 =
p(σ − α)− σ
σ(p− 1)

, α2 =
p(σ − α− 1)− σ

σ(p− 1)
, β = Q+ 2m− 2mp(α+ 1)

p− 1
.

Finally, we arrive at

(λ1 − 2ε)IR ≤ CRβ . (2.17)

Note that inequality (2.10) is equivalent to β ≤ 0. So, we have to consider two
cases:
• Case β < 0: we pass to the limit in (2.17) as R goes to +∞; we obtain∫ ∞

0

∫
R2N+1

|u|p dηdt = 0 =⇒ u ≡ 0,

this is a contradiction.



EJDE-2020/02 LIFESPAN OF SOLUTIONS OF AN EVOLUTION EQUATION 7

• Case β = 0: using inequality (2.17) with R → +∞, and taking into account
the fact that p = p∗, we obtain

u ∈ Lp((0,+∞)× R2N+1).

On the other hand, repeating the same calculations as above, with

φ(x, t) = φ1
( η

RL−1
)
φ2
( t

R2m

)
,

where 1 ≤ L < R is large enough such that when R→ +∞ we do not have L→ +∞
at the same time, we arrive at

λIR ≤ CL−Q + CL
2pm
p−1−Q, (2.18)

thanks to the change of variables τ̃ = τ/(RL−1)2, x̃ = x/RL−1, ỹ = y/RL−1 and
s = t/R2m. Thus, using p > Q/(Q− 2m) and passing to the limit when R→ +∞,
and then when L→ +∞ in (2.18), we obtain∫ ∞

0

∫
R2N+1

|u|p dηdt = 0 =⇒ u ≡ 0,

which is also a contradiction. �

Remark 2.7. For the other cases, setting

IR ≡


−
∫
Q λ1|u|

p
φ(η, t) dηdt if λ1 < 0, λ1

∫
R2N+1 f2(η) dη < 0,∫

Q λ2|u|
pφ(η, t) dηdt if λ2 > 0, λ2

∫
R2N+1 f1(η) dη > 0,

−
∫
Q λ2|u|

pφ(η, t) dηdt if λ2 < 0, λ2
∫
R2N+1 f1(η) dη > 0,

we can prove the same conclusion in the same manner as above.

3. Life span of blow up solutions

To estimate the life span of blow up solutions, we assume that f satisfies one of
the two sets of conditions

f1 ∈ L1
loc(R2N+1), λ2f1(η) ≥ |η|−kH , |η|H > 1,

or

f2 ∈ L1
loc(R2N+1), −λ1f2(η) ≥ |η|−kH , |η|H > 1,

(3.1)

where

Q− 2αm < k <
2m(α+ 1)

p− 1
. (3.2)

We also consider the case when λ1 > 0 only; the other cases can be treated in a
similar manner.

Theorem 3.1. Suppose that conditions (3.1), (2.10) and (3.2) are satisfied, and
let u be the solution of (1.1) with the initial data u(η, 0) = εf(η), where ε > 0.
Denote by [0, Tε) the life span of u. Then there exists a positive constant C such
that

Tε ≤ Cε1/ρ,
where ρ = k

2m −
α+1
p−1 < 0.

Remark 3.2. When p = Q+2m
Q−2αm , we have ρ = k−Q+2αm

2m .



8 A. ALSAEDI, B. AHMAD, M. KIRANE, A. NABTI EJDE-2020/02

Proof of Theorem 3.1. First, repeating the same calculations as in Theorem 2.6,
we obtain

λ1IR − CT−αR−2αm
∫
R2N+1

εf2(η)φ1(η/R) dη

≤
∫
Q
|u||∆m

H φ1(η/R)||Dα
t|TR2mφ2

(
t/R2m

)
|dηdt

+

∫
Q
|u|φ1(η/R)|Dα+1

t|TR2mφ2
(
t/R2m

)
|dηdt ≡ A1 +A2.

(3.3)

By Hölder’s inequality applied to A1 and A2, we have

λ1IR − CT−αR−2αm
∫
R2N+1

εf2(η)φ1(η/R) dη

≤ I1/pR

(∫
Q
φ1(η/R)φ2

(
t/R2m

)− 1
p−1 |Dα+1

t|TR2mφ2
(
t/R2m

)
|
p
p−1 dηdt

) p−1
p

+ I
1/p
R

(∫
Q
φ1(η/R)−

1
p−1 |∆m

H φ1(η/R)|
p
p−1φ2

(
t/R2m

)− 1
p−1

× |Dα
t|TR2mφ2

(
t/R2m

)
|
p
p−1 dηdt

) p−1
p

.

(3.4)

Using (2.4), (2.5), and passing to the scaled variables s = t/TR2m, η̃ = (x̃, ỹ, τ̃)
such that τ̃ = τ/R2, x̃ = x/R, ỹ = y/R, we arrive at

λ1IR + CT−αVR ≤ R
β
q I

1/p
R (A(T ) +B(T )), (3.5)

where

VR := εR−2αm
∫
R2N+1

−f2(η)φ1(η/R) dη,

A(T ) := CT−α
(∫ T

0

∫
R2N+1

φ1(η̃)φ2(s)α1 dη̃ds
) p−1

p

,

B(T ) := CT−(α+1)
(∫ T

0

∫
R2N+1

φ1(η̃)φ2(s)α2 dη̃ds
) p−1

p

.

Thus

VR ≤ Cλ1Tα
(R β

q

λ1
(A(T ) + B(T )) I

1/p
R − IR

)
.

We clearly have

A(T ) =
C

(σ + 1− qα)1/q
T
p−1
p −α = apT

p−1
p −α, (3.6)

B(T ) =
C

(σ + 1− q(α+ 1))1/q
T
p−1
p −(α+1) = bpT

p−1
p −(α+1). (3.7)

Note that

max
x>0

(γxw − x) = (1− w)ww/(1−w)γ1/(1−w),

for γ > 0 and 0 < w < 1. Whereupon

VR ≤ CTαRβE(T )q, (3.8)

for any T > 0 and R > 0, where

C = λ
−1/(p−1)
1 (p− 1)(1/p)q,



EJDE-2020/02 LIFESPAN OF SOLUTIONS OF AN EVOLUTION EQUATION 9

E(T ) = A(T ) + B(T ) = apT
1− pα+1

p + bpT
− pα+1

p .

On the other hand, by the definition of VR and the assumption on the initial data
f , we have

VR = εR−2αm
∫
R2N+1

−f2(η)φ1(η/R) dη

≥ εR−2αm
∫
|η|H≥1

−f2(η)φ1(η/R) dη

≥ ελ−11 R−2αm
∫
|η|H≥1

|η|−kH φ1(η/R) dη;

passing to the scaled variables η̃ = (x̃, ỹ, τ̃) such that x̃ = x/R, ỹ = y/R, τ̃ = τ/R2,
we obtain

VR ≥ εRQ−k−2αmλ−11

∫
|η̃|H≥ 1

R

|η̃|−kH φ1(η̃) dη̃

≥ εRQ−k−2αmλ−11

∫
|η̃|H≥ 1

R0

|η̃|−kH φ1(η̃) dη̃

= CkεR
Q−k−2αm,

for any R > R0, where R0 is a constant independent of R and ε.
Now, let t0 ∈ (0, Tε) and R > R0. By using (3.8) with T = t0R

−2m, we obtain

ε ≤ CR2αm+k−Q
(
T
α
q R

β
q E(t0R

−2m)
)q
≡ CH(t0, R). (3.9)

Furthermore,

H(t0, R) =
(
ap t

1− pα+1
p

0 R
k(p−1)
p −2m + bp t

− pα+1
p

0 R
k(p−1)
p

) p
p−1

= t
−α+1
p−1

0

(
ap t0R

k(p−1)
p −2m + bpR

k(p−1)
p

) p
p−1

.

(3.10)

Substituting R = t
1/2m
0 in (3.10), we can restate inequality (3.9) as

ε ≤ CH(t0, t
1/2m
0 ) ≤ Ct

k
2m−

α+1
p−1

0 ,

with some C > 0. Consequently, the inequality

t0 ≤ Cε1/ρ

holds for any t0 ∈ (0, Tε). This completes the proof of the theorem. �

Acknowledgements. This project was funded by the Deanship of Scientific Re-
search (DSR) at King Abdulaziz University, Jeddah, Saudi Arabia, under grant no.
(RG-36-130-40). The authors, therefore, acknowledge with thanks DSR technical
and financial support.

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Ahmed Alsaedi

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Sci-

ences, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Email address: aalsaedi@hotmail.com

Bashir Ahmad
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Sci-
ences, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Email address: bashirahmad qau@yahoo.com

Mokhtar Kirane

LASIE, Faculté des Sciences et Technologies, Université de La Rochelle, Avenue M.
Crépeau, 17000, La Rochelle, France.

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Sci-
ences, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Email address: mkirane@univ-lr.fr

Abderrazak Nabti
Laboratoire de Mathématiques, Informatiques et Systèmes (LAMIS), Université Larbi
Tebessi, 12002 Tebessa, Algeria

Email address: abderrazaknabti@gmail.com


	1. Introduction
	2. Blow up solutions
	3. Life span of blow up solutions
	Acknowledgements

	References