Int. J. Anal. Appl. (2023), 21:37

Asymptotic Behavior of Solution for Coupled Reaction Diffusion System by Order m

Mebarki Maroua1, Barrouk Nabila2,∗

1Faculty of Science and Technology, Departement of Mathematics and Computer Science, Amine

Elokkal El Hadj Moussa Ag Akhamouk University, P.O.Box 10034, Tamanrasset 11000, Algeria
2Faculty of Science and Technology, Department of Mathematics and Computer Science, Mohamed

Cherif Messaadia University, P.O.Box 1553, Souk Ahras 41000, Algeria

∗Corresponding author: n.barrouk@univ-soukahras.dz

Abstract. The aim of this paper is to prove that asymptotic behavior in the time of solutions for the

weakly coupled reaction diffusion system:


∂ui
∂t
−di ∆ui = fi (u1,u2, . . . ,um) in Ω ×R+,

∂ui
∂η

= 0 in ∂Ω ×R+,

ui (.,0) = u
0
i (.) in Ω,

(0.1)

where Ω is an open bounded domain of class C1 in Rn, ui (t,x), i = 1,m, t ≥ 0, x ∈ Ω are real valued
functions. We treat the system (0.1) as a dynamical system in C

(
Ω
)
×C

(
Ω
)
× ...×C

(
Ω
)
and apply

Lyapunov type stability techniques. A key ingredient in this analysis is a result which establishes that

the orbits of the dynamical system are precompact in C
(

Ω
)
×C

(
Ω
)
× ...×C

(
Ω
)
. As a consequence

of Arzela-Ascoli theorem, this will be satisfied if the orbits are, for example, uniformly bounded in

C1
(

Ω
)
×C1

(
Ω
)
× ...×C1

(
Ω
)
for t > 0.

1. Introduction

The existence, uniqueness, and asymptotic behavior of the solution of a balanced two component

reaction diffusion system have been investigated. It was shown that a global and unique solution

existed and it’s second component can be estimated using the Lyapunov Functional see [1, 14, 15].

Received: Mar. 8, 2023.

2020 Mathematics Subject Classification. 35K57, 35B40, 35B45.

Key words and phrases. reaction diffusion system; semigroup; global existence; asymptotic behavior.

https://doi.org/10.28924/2291-8639-21-2023-37
ISSN: 2291-8639

© 2023 the author(s).

https://doi.org/10.28924/2291-8639-21-2023-37


2 Int. J. Anal. Appl. (2023), 21:37

It was, also, demonstrated that each component of the solution converged, at infinity, to a constant

which can be used in terms of the reacting function and the initial data.

The results of the current research can be used in several areas of applied mathematics, especially

when the system equations originate from mathematical models of real systems such as in Biology,

Chemistry, Population Dynamics, and other disciplines. We know that the problem (0.1) has a unique

global solution see [5, 7, 17]. The main question we want to address is asymptotic behavior the

solutions for system (0.1). In fact the subject of the asymptotic behavior of reaction diffusion systems

has received a lot of attention in the last decades and several outstanding results have been proved

by some of the major experts in the field.

This question has been investigated by many authors by considering special forms of the nonlinear

terms fi.

In the case where i = 1, 2 : 


∂u1
∂t
−d1∆u1 = f1 (u1,u2) in Ω ×R+,

∂u2
∂t
−d2∆u2 = f2 (u1,u2) in Ω ×R+,

λi
∂ui
∂η

+ (1 −λi ) ui = 0, in ∂Ω ×R+,

(1.1)

when d1 6= d2, and nonnegative initial data arise, for example, as models for the diffusion of substances
which at the same time react with each other chemically (cf. [8, 16]). Also (1.1) is related to the

Rosenzweig-Mac Arthur equation in ecology (cf. [2]).

In the case where f1 (u1,u2) = −f2 (u1,u2) = −u1uσ2 , Note that, the behavior of non-negative total
solutions (1.1) is treated in the paper of Alikakos [2] obtained L∞-bounds of solutions global existence

when 1 < σ <
n + 2

n
, and Masuda [14] who showed that solutions exist globally for every σ ≥ 1 and,

in addition, showed that the solutions converge as t goes to +∞. Recently, Haraux and Youkana [13]
established a global existence result of a system (1.1) for a large class of the function f1 and f2. More

precisely, they showed that for

f1 (u1,u2) = f2 (u1,u2) = −u1Ψ (u2) ,

the problem (1.1) admits a global solution provided that the following condition holds:

lim
u2→+∞

[log (1 + Ψ (u2))]

u2
= 0.

In the case where d2 ≥ 0, systems of the type (1.1) occur in many applications (cf. [8]).
For triangular diffusion matrix, global bounds were proved by Kirane in [11] if u02 (x) ≥

d2
d1−d3 u

0
1 (x) ≥ 0,

x ∈ Ω,. The author proved also that the solution (u1,u2) converges to a constant vector k = (k1,k2)
as t →∞, uniformly in Ω̄. Furthermore, k1 ≥ 0, k2 ≥ 0 and k1Ψ (k2) = 0.



Int. J. Anal. Appl. (2023), 21:37 3

In this paper we shall generalize the results obtained in [11]. We prove the asymptotic behavior of

solutions of m-component reaction-diffusion systems with diagonal matrix and homogeneous Neumann

conditions. The reaction terms are assumed to be of polynomial growth.

We consider the following m-equations of reaction-diffusion system, with m ≥ 2:
∂U

∂t
−Am∆U = F (U) in Ω × (0, +∞), (1.2)

where Ω is an open bounded domain of class C1 in Rn, the vectors U and F and the matrix Am are
defined as:

U = (u1, ...,um)
T , F = (f1, ..., fm)

T ,

Am =




d1 0 0 · · · 0

0 d2 0
...

...

0 0 d3
... 0

...
...

...
... 0

0 · · · 0 0 dm



.

The constants (di )
m
i=1 , are supposed to be strictly positive which reflects the parabolicity of the system

and implies at the same time that the diffusion matrix Am is positive defnite. The boundary conditions

and initial data (respectively) for the proposed system are assumed to satisfy:

∂ηU = 0 on ∂Ω × (0, +∞),

and

U(0,x) = U0(x) = (u
0
1, ...,u

0
m)

T on Ω,

where ∂
∂η

denotes the outward normal derivative on the boundary ∂Ω, the vectors U0 are defined as:

U0 = (u
0
1, ...,u

0
m)

T .

2. Notations and Preliminary

In the following we denote by

‖.‖P the norm in L
P (Ω) for 1 ≤ P ≤ +∞,

‖.‖∞ the norm in C (Ω), and
‖.‖1,∞ the norm in C

1 (Ω).

For 1 < P < ∞, set 
 D (A) =

{
u : u ∈ W 2,p (Ω) : ∂u

∂η
= 0 on ∂Ω

}
,

Au = ∆u for u ∈ D (A) .

It is well known (cf. for example, [6]) that A is m-dissipative in LP (Ω) for 1 < P < ∞. Moreover,
the restriction of A to C

(
Ω̄
)
is m-dissipative.

Let us now recall an overview of the asymptotic behavior of the solution for coupled reaction diffusion

systems. This will pave the way to introduce our findings later on.



4 Int. J. Anal. Appl. (2023), 21:37

Consider the initial value problem {
ut (t) = Lu (t) + f (u (t))

u (0) = u0,
(IVP)

where L is the infinitesimal generator of a C0-semigroup S (t) on a real Banach space X with norm

‖.‖, f : X →R is a given function, and u0 ∈ X is a given initial datum.

Theorem 2.1. [19] Let T > 0. A function u : [0,T ] → X is a weak solution of (IVP) on [0,T ] if and
only if f (u (t)) ∈ L1 (0,T,X) and u satisfies the variation of constants formula

u (t) = S (t) u0 +

∫ t
0

S (t − s) f (u (s)) ds, for all s ∈ [0,T ] .

Definition 2.1. A function u : [0,T ] → X is called a strong solution of (IVP) if u (t) is strongly
continuously differentiable in the interval 0 < t < T, u (t) ∈ D (L) for 0 < t < T, that equation
(IVP) is satisfied for 0 < t < T and u (t) → u0 as t → 0.

Theorem 2.2. [18] Let f : X → X be locally Lipschitz continuous. Then for u0 ∈ X, (IVP)
has a unique weak solution defined in a maximal interval of existence [0,Tmax), Tmax > 0, u ∈
C ([0,Tmax) ,X). Moreover, if Tmax < ∞, then

lim
t→Tmax

‖u (t)‖ = +∞.

Now, let us recall the following definition.

Definition 2.2. Let {S (t)}t≥0 be a nonlinear semigroup on a compact metric space X. If(
u01,u

0
2, . . . ,u

0
m

)
∈ X, O

(
u01,u

0
2, . . . ,u

0
m

)
=

{
S (t)

(
u01,u

0
2, . . . ,u

0
m

)}
t≥0 is the orbit through(

u01,u
0
2, . . . ,u

0
m

)
, then the w-limite set for

(
u01,u

0
2, . . . ,u

0
m

)
is defined by

w
(
u01,u

0
2, . . . ,u

0
m

)
= {(u1,u2, . . . ,um) ∈ X : ∃tn →∞ :

S (tn)
(
u01,u

0
2, . . . ,u

0
m

)
→ (u1,u2, . . . ,um)}.

3. The Main Result

In this section, we state the main result.

Theorem 3.1. The solution w = (u1,u2, . . . ,um) of the system (1.2) converges a constant vector of

the form ξ = (ξ1,ξ2, . . . ,ξi ) as t →∞, uniformly in Ω i.e
(
ui →
t→∞

ξi

)
for i = 1,m.

Furthermore, we have:

ξi ≥ 0, i = 1,m, fi (ξ1,ξ2, . . . ,ξm) = 0,

and
m∑
i=1

ξi =
1

Ω

∫
Ω

m∑
i=1

u0i (x) dx.

The following lemma is a useful tool in the proof of the Theorem 3.1.



Int. J. Anal. Appl. (2023), 21:37 5

Lemma 3.1. Let (u1,u2, . . . ,um) be a solution of (1.2). We have∫
QT

|∇ui|
2
dxdt < ∞ for i = 1,m, here QT = Ω × [0,T ] and 0 < T < ∞.

Proof. We have For i = 1,m.

∂ui
∂t
−di ∆ui = fi (u1,u2, . . . ,um) . (3.1)

By integrating over (0,T ) is obtained∫ T
0

∂ui
∂t

(x,t) dt = di

∫ T
0

∆uidt +

∫ T
0

fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dt,

ui (x,T ) −ui (x, 0) = di
∫ T

0

∆uidt +

∫ T
0

fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dt,

and integrating a second time is collected over Ω∫
Ω

ui (x,T ) dx −
∫

Ω

ui (x, 0) dx = di

∫
Ω

∫ T
0

∆uidtdx

+

∫
Ω

∫ T
0

fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dtdx.

Green’s formula is applied to
∫

Ω
∆uidx, we gain∫

Ω

∆uidx =

∫
∂Ω

∂ui
∂η
dσ −

∫
Ω

∇ui∇1dx, therefore
∫

Ω

∆uidx = 0,

thus ∫
Ω

∫ T
0

fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dtdx =

∫
Ω

ui (x,T ) dx −
∫

Ω

u0i (x) dx < ∞,

as a result of ui (T ) ∈ C
(

Ω
)
we have∫

QT

fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dtdx < ∞, for i = 1,m.

Multiply now the ith equation of (1.2) by ui, for i = 1,m, and integrating over QT , we attain∫
Ω

∫ T
0

ui
∂ui
∂t

(x,t) dtdx = di

∫
Ω

∫ T
0

ui ∆uidtdx

+

∫
Ω

∫ T
0

uifi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dtdx,

by using the Green formula∫
Ω

ui ∆uidx =

∫
∂Ω

ui
∂ui
∂η
dσ −

∫
Ω

|∇ui|
2
dx, therefore

∫
Ω

ui ∆uidx = −
∫

Ω

|∇ui|
2
dx,

and a simple calculation, it becomes

1

2

∫
Ω

[
u2i (x,t)

]∣∣T
0
dx = −di

∫ T
0

∫
Ω

|∇ui|
2
dxdt

+

∫ T
0

∫
Ω

ui (x,t) fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dxdt,



6 Int. J. Anal. Appl. (2023), 21:37

then ∫
Ω

u2i (x,T ) + 2di

∫
QT

|∇ui|
2
dxdt =

∫
Ω

(
u0i (x)

)2
dx

+2

∫
QT

ui (x,t) fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dxdt,

consequently

2di

∫
QT

|∇ui|
2
dxdt ≤

∫
Ω

(
u0i (x)

)2
dx

+2

∫
QT

ui (x,t) fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dxdt, (3.2)

since ∫
Ω

(
u0i (x)

)2
dx < ∞, for i = 1,m.

and ∫
QT

ui (x,t) fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dxdt

≤‖ui‖L∞(QT )
∫
QT

fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dxdt < ∞, for i = 1,m.

hence

2di

∫
QT

|∇ui|
2
dxdt < ∞, for i = 1,m.

consequently ∫
QT

|∇ui|
2
dxdt < ∞, for i = 1,m. ∀T > 0.

�

4. Proof of the Main Result (Theorem 3.1)

We are now ready to prove the main result of this work:

Proof of Theorem 3.1. First, if we integrate the ith equation of (1.2) over Ω we have∫
Ω

∂ui
∂t

(x,t) dx = di

∫
Ω

∆uidx +

∫
Ω

fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dx,

use Green theorem to transform the terms ∆ui in the light of boundary conditions we observe that∫
Ω

∂ui
∂t

(x,t) dx =

∫
Ω

fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dx,

if we add this equations imliying∫
Ω

m∑
i=1

∂ui
∂t

(x,t) dx =

∫
Ω

m∑
i=1

fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dx, for i = 1,m,

if we assume that
m∑
i=1

∫
Ω

fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dx = 0,



Int. J. Anal. Appl. (2023), 21:37 7

we get ∫
Ω

m∑
i=1

∂ui
∂t

(x,t) dx = 0, for i = 1,m,

as ∫ t
0

∫
Ω

m∑
i=1

∂ui
∂t

(x,t) dxdt =

∫
Ω

∫ t
0

m∑
i=1

∂ui
∂t

(x,t) dtdx

=

∫
Ω

m∑
i=1

ui (x,t)

∣∣∣∣∣
t

0

dx

=

∫
Ω

m∑
i=1

ui (x,t) dx −
∫

Ω

m∑
i=1

u0i (x) dx = 0,

we deduce that ∫
Ω

m∑
i=1

ui (x,t) dx =

∫
Ω

m∑
i=1

u0i (x) dx. (4.1)

Integrating the ith equation of the system (1.2) in Ω, for i = 1,m, we have:∫
Ω

∂ui
∂t

(x,t) dx =

∫
Ω

fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dx > 0,

as a means that d
dt

∫
Ω
ui (x,t) dx > 0. then the fonction t →

∫
Ω
ui (x,t) dx is increasing and Ω

is bounded. Then t → 1|Ω|
∫

Ω
ui (x,t) dx is increasing and according to the positivity of ui was

1
|Ω|
∫

Ω
ui (x,t) dx ≥ 0. Therefore 1|Ω|

∫
Ω
ui (x,t) dx is bounded below and increasing.

Formerly ∃ lim
t→∞

1
|Ω|
∫

Ω
ui (x,t) dx = li, for i = 1,m.

On the other hand, since sets ∪
t≥0
{ui (t)}t≥0 , for i = 1,m are precompacts in C

(
Ω
)
. There exists a

sequence (tn)n≥0, tn →∞ such that

lim
n→∞

ui (tn) = u
s
i , for i = 1,m in C

(
Ω
)
,

Now, denote by w
(
u01,u

0
2, . . . ,u

0
m

)
the w-limite set for

(
u01,u

0
2, . . . ,u

0
m

)
and Φ the set of the solution

of the elliptic system{
−di ∆usi = fi

(
us1 (x,t) ,u

s
2 (x,t) , . . . ,u

s
m (x,t)

)
in Ω,

∂us
i

∂η
= 0 in ∂Ω,

(4.2)

and prove Φ = {(ξ1,ξ2, . . . ,ξm)} where ξ1,ξ2, . . . ,ξm are constants, in fact, multipliying the ith

equation of the problem (4.2) by usi for i = 1,m and integrating over Ω yields:

−di
∫

Ω

usi ∆u
s
i dx =

∫
Ω

usi fi (u
s
1 (x,t) ,u

s
2 (x,t) , . . . ,u

s
m (x,t)) dx.

Apply Green formular:

di

∫
Ω

|∇usi |
2
dx =

∫
Ω

usi fi (u
s
1 (x,t) ,u

s
2 (x,t) , . . . ,u

s
m (x,t)) dx.



8 Int. J. Anal. Appl. (2023), 21:37

Adding the ith equations yields

m∑
i=i

di

∫
Ω

|∇usi |
2
dx =

m∑
i=i

∫
Ω

usi fi (u
s
1 (x,t) ,u

s
2 (x,t) , . . . ,u

s
m (x,t)) dx.

Supposing
m∑
i=i

usi fi
(
us1 (x,t) ,u

s
2 (x,t) , . . . ,u

s
m (x,t)

)
≤ 0 for i = 1,m, then

0 ≤
m∑
i=i

di

∫
Ω

|∇usi |
2
dx ≤ 0,

therefore
m∑
i=i

di

∫
Ω

|∇usi |
2
dx = 0.

We deduce that ∫
Ω

|∇usi |
2
dx = 0 ⇒∇usi = 0 ⇒ u

s
i = ξi. (4.3)

Replacing usi = ξi, for i = 1,m in the i
th equation (4.2).

It is clear that fi (ξ1,ξ2, . . . ,ξm) = 0.

Hereafter, we are going to show that w
(
u01,u

0
2, . . . ,u

0
m

)
6= φ. Now, ∀x ∈ Ω, σ ∈ ]−1, 1[ and let

pni (x,σ) = ui (x,tn + σ) , for i = 1,m,

multiply the ith equation of the poblem (1.2) by ∂ui
∂t

∂ui
∂t

∂ui
∂t
−di

∂u

∂t
∆ui =

∂ui
∂t
fi (u1,u2, . . . ,um) ,

and integrate over Ω we get:

∫
Ω

(
∂ui
∂t

)2
dx −di

∫
Ω

∂ui
∂t

∆uidx =

∫
Ω

∂ui
∂t
fi (u1,u2, . . . ,um) dx,

as ∥∥∥∥∂ui∂t
∥∥∥∥2
L2(Ω)

= di

∫
Ω

∂ui
∂t

∆udx +

∫
Ω

∂ui
∂t
fi (u1,u2, . . . ,um) dx,

intégrating result over (t0, +∞) , we have:

∫ +∞
t0

∥∥∥∥∂ui∂t
∥∥∥∥2
L2(Ω)

dt = di

∫ +∞
t0

∫
Ω

∂ui
∂t

∆uidxdt +

∫ +∞
t0

∫
Ω

∂ui
∂t
fi (u1,u2, . . . ,um) dxdt < ∞,



Int. J. Anal. Appl. (2023), 21:37 9

thus ∂ui
∂t
∈ L2

(
t0, +∞,L2 (Ω)

)
, ∀σ ∈ ]−1, 1[ we get:

pni (x,σ) −ui (x,tn) = ui (x,tn + σ) −ui (x,tn)

=

∫ tn+σ
tn

∂ui
∂t

(x,t) dt

≤
∫ tn+1
tn−1

∂ui
∂t

(x,t) dt by reason of (tn − 1 < tn, σ < 1, tn + σ < tn + 1)

≤
(∫ tn+1

tn−1
(1)

2
dt

)1
2

(∫ tn+1
tn−1

(
∂ui
∂t

(x,t)

)2
dt

)1
2

,

≤
√

2

(∫ tn+1
tn−1

(
∂ui
∂t

(x,t)

)2
dt

)1
2

,

as follows

|pni (x,σ) −ui (x,tn)|
2

= 2

∫ tn+1
tn−1

(
∂ui
∂t

(x,t)

)2
dt,

integrating the latter inequality Ω yields∫
Ω

|pni (x,σ) −ui (x,tn)|
2
dx ≤ 2

∫
Ω

∫ tn+1
tn−1

(
∂ui
∂t

(x,t)

)2
dtdx,

we pass to the limit as n →∞, we have:

‖pni (x,σ) −u
s
i ‖

2
L2(Ω) ≤ 2 limn→∞

∫
Ω

∫ tn+1
tn−1

(
∂ui
∂t

(x,t)

)2
dtdx = 0,

so

‖pni (x,σ) −u
s
i ‖

2
L2(Ω) → 0n→∞ .

As a result, we will all σ ∈ ]−1, 1[ ,

‖pni (x,σ) −u
s
i ‖

2
L2(Ω) → 0n→∞, hence sup−1<σ<1

‖pni (x,σ) −u
s
i ‖

2
L2(Ω) → 0n→∞,

and by the same mode are obtained:

sup
−1<σ<1

‖pni (x,σ) −u
s
i ‖

2
L2(Ω) → 0n→∞, for i = 1,m.

Also, we can have:

sup
−1<σ<1

‖∇pni (x,σ) −∇u
s
i ‖

2
L2(Ω) →n→∞ 0 for i = 1,m,

through positivity and boundedness of the solution was:

0 ≤ ui (x,tn + σ) ≤ Mi,

remarkably fi ∈ C∞ (Rn), we can conclude, using Lebesgue’s theorem, that

fi (p1 (x,σ) ,p2 (x,σ) , . . . ,pm (x,σ)) → fi (us1,u
s
2, . . . ,u

s
m) in L

2 (Ω × (−1, 1)) weak.



10 Int. J. Anal. Appl. (2023), 21:37

Now, let %i ∈ C1
(

Ω
)
such that %i = 0 on ∂Ω where i = 1,m, and let γ ∈ C1

(
Ω
)
such that

suppγ ⊂ [−1, 1] , ∫ 1
−1
γ (s) ds = 1 and γ (−1) = γ (1) .

We multiply the ith equation of problem (1.2) by γ (t − tn) %i and integrate over Ω×(tn − 1,tn + 1) ,
we obtain ∫ tn+1

tn−1
∫

Ω
γ (t − tn) %i

∂ui
∂t
dxdt −di

∫ tn+1
tn−1

∫
Ω
γ (t − tn) %i ∆uidxdt

=
∫ tn+1
tn−1

∫
Ω
γ (t − tn) %ifi (u1,u2, . . . ,um) dxdt.

(4.4)

Forecast the integral
∫ tn+1
tn−1 γ (t − tn) %i

∂ui
∂t
dt by part, we find

∫ tn+1
tn−1

γ (t − tn) %i
∂ui
∂t
dt = −

∫ tn+1
tn−1

γ′ (t − tn) %iui (x,t) dt, (4.5)

to appraise
∫

Ω
γ (t − tn) %i ∆uidx applying Green’s formula∫

Ω

γ (t − tn) %i ∆uidx =
∫
∂Ω

γ (t − tn) %i
∂ui
∂η
dσ −

∫
Ω

∇γ (t − tn) %i∇uidx

= −
∫

Ω

∇γ (t − tn) %i∇uidx,

extremely ∫
Ω

γ (t − tn) %i ∆uidx = −
∫

Ω

∇γ (t − tn) %i∇uidx, (4.6)

injected (4.5) and (4.6) in (4.4) is accessed

−
∫ tn+1
tn−1

∫
Ω

γ′ (t − tn) %iui (x,t) dxdt + di
∫ tn+1
tn−1

∫
Ω

∇γ (t − tn) %i∇uidxdt (4.7)

+

∫ tn+1
tn−1

∫
Ω

γ (t − tn) %ifi (u1,u2, . . . ,um) dxdt = 0 for i = 1,m.

By making the following change of variable σ = t − tn → dσ = dt

if

{
t = tn − 1
t = tn + 1

→

{
σ = −1
σ = 1

accordingly the integral (4.7) becomes∫ +1
−1

∫
Ω

γ′ (σ) %ip
n
i (x,σ) dxdσ −di

∫ +1
−1

∫
Ω

∇γ (σ) %i∇pni (x,σ) dxdσ

∫ +1
−1

∫
Ω

γ (σ) %ifi (p1 (x,σ) ,p2 (x,σ) , . . . ,pm (x,σ)) dxdσ = 0, for i = 1,m. (4.8)



Int. J. Anal. Appl. (2023), 21:37 11

Applying Lesbegue’s theorem we gain: for i = 1,m

lim
n→∞

∫ +1
−1

∫
Ω

γ′ (σ) %ip
n
i (x,σ) dxdt =

∫ +1
−1

∫
Ω

γ′ (σ) %iu
s
i dxdσ

=

∫ +1
−1

γ′ (σ) dσ

∫
Ω

%iu
s
i dx

= γ (σ)|+1−1
∫

Ω

%iu
s
i dx = 0 by virtue of γ (1) = γ (−1) ,

from inequality (4.8), we make: for i = 1,m

−di
∫

Ω

∇%i∇usi +
∫

Ω

%ifi (u
s
1,u

s
2, . . . ,u

s
m) dx = 0,

Which is the variational formulation of (4.2), hence w = Φ.

Finally, combining (4.2) with (4.1) yields∫
Ω

m∑
i=1

ξidx =

∫
Ω

m∑
i=1

u0i dx,

|Ω|
m∑
i=1

ξi =

∫
Ω

m∑
i=1

u0i dx,

m∑
i=1

ξi =
1

|Ω|

∫
Ω

m∑
i=1

u0i dx,

the proof of the theorem is complete. �

Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi-

cation of this paper.

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	1. Introduction
	2. Notations and Preliminary
	3. The Main Result
	4. Proof of the Main Result (Theorem 3.1)
	References