International Journal of Analysis and Applications

Volume 18, Number 5 (2020), 748-773

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-18-2020-748

THE EXISTENCE RESULT OF RENORMALIZED SOLUTION FOR NONLINEAR

PARABOLIC SYSTEM WITH VARIABLE EXPONENT AND L1 DATA

FAIROUZ SOUILAH1, MESSAOUD MAOUNI2,∗ AND KAMEL SLIMANI2

1University 20th August 1955, Skikda, Algeria

2Laboratory of Applied Mathematics and History and Didactics of Maths ”LAMAHIS”, Algeria

∗Corresponding author: m.maouni@univ-skikda.dz; maouni21@gmail.com

Abstract. In this paper, we prove the existence result of a renormalized solution to a class of nonlinear

parabolic systems, which has a variable exponent Laplacian term and a Leary lions operator with data

belong to L1.

1. Introduction

Let Ω is bounded open domain of RN, (N ≥ 2) with lipschiz boundary ∂Ω, T is a positive number

oure aime is to study the existence of renormalized solution for a class of nonlinear parabolic systeme with

variable exponent and L1 data. More precisely, we study the asymptotic behavrior of the problem


(b1(u))t − divA(x,t,∇u) + γ(u) = f1(x,t,u,v) in Q = Ω×]0,T[,

(b2(v))t − ∆v+ = f2(x,t,u,v) in Q = Ω×]0,T[,

u = v = 0 on Σ = ∂Ω×]0,T[,

b1(u)(t = 0) = b1(u0) in Ω,

b2(v)(t = 0) = b2(v0) in Ω,

(1.1)

where divA(x,t,∇u) = div(|∇u|p(x)−2 ∇u) is a Leary lions operator (see assumptions (3.1)-(3.3)) with

p : Ω −→ [1, +∞) be a continuous real-valued function and let p− = minx∈Ω p(x) and p
+ = maxx∈Ω p(x)

Received January 27th, 2020; accepted February 25th, 2020; published June 25th, 2020.

2010 Mathematics Subject Classification. 35J70, 35D05.

Key words and phrases. nonlinear parabolic systems; variable exponent; renormalized solutions; L1 data.

©2020 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

748

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-748


Int. J. Anal. Appl. 18 (5) (2020) 749

with 1 < p− ≤ p+ < N. Let γ : R → R with γ(s) = λ |s|p(x)−2 s is a continuous increasing function for

λ > 0 and γ(0) = 0 such that γ is assumed to belong to L1(Q). The function fi : Q× R × R → R for i =

1, 2 be a Carathéodory function (see assumptions (3.5)-(3.7)).

Finally the function b : R → R is a strictly increasing C1-function lipchizienne with bi(0) = 0 (see (3.4) ),

the data fi and (b1(u0),b2(v0)) is in (L
1)2, for i = 1, 2.

The study of differential equations and variational problems with nonstandard growth conditions arouses

much interest with the development of elastic mechanics, electro-rheological fluid dynamics and image pro-

cessing, etc ( see [9], [19] ) .

Problems of this type have been studied by serval a authors. In 2007 H. Redwane, studied the existence of

solutions for a class of nonlinear parabolic systems see [18], in 2013 Youssef. B and all studied the existence

of a renormalized solution for the nonlinear parabolic systems with unbounded nonlinearities see [2] agin

in 2016 B . El Hamdaoui and all in [11] studied the renormalized solutions for nonlinear parabolic systems

in the Lebesgue Sobolev Space with variable exponent and L1 data. In 2016 [17] authors proved the exis-

tence and uniqueness of renormalized solution of a reaction diffusion systems which has a variable exponent

Laplacian term and could be applied to image denoising for the case of parabolic equations. In 2010 T. M.

Bendahmane, P. Wittbold and A.Zimmermann [7] have proved the existence and uniqueness of renormalized

solution to nonlinear parabolic equations with variable exponent and L1 data. C. Zhang and S. Zhou studied

the renormalized and entropy solution for nonlinear parabolic equation with variable exponent and L1 data.

Moreover, they obtain the equivalence of renormalized solution and entropy solution(see [23]).

In the present paper we prove the existence of renormalized solution for nonlinear parabolic systems with

variable exponent and L1 data of (1.1). The notion of renormalized solution was introduced by Diperna and

Lions [10] in their study of the Boltzmann equation, and this result can be seen as a generalization of the

results obtained by F. Souilah and all in [12].

The paper is organized as follows: Section 2, to recall some basic notations and properties of variable expo-

nent Lebesgue Sobolev space. Section 3, is devoted to specify the assumptions on, A(x,t,ξ), γ, b1, b2, f1,

f2, b1(u0) and b2(v0) needed in the present study. Section 4, to give the definition of a renormalized solution

of (1.1), and we establish (Theorem (4.1) ) the existence of such a solution.

2. The Mathematical Preliminaries on Variable Exponent Sobolev Spaces

In this section, we first recall some results on generalized Lebesgue-Sobolev spaces Lp(.)(Ω), W 1,p(.)(Ω)

and W
1,p(.)
0 (Ω) where Ω is an open subset of R

N . We refer to [13] for further properties of Lebesgue

Sobolev spaces with variable exponents. Let p : Ω −→ [1, +∞) be a continuous real-valued function and let

p− = minx∈Ω p(x), p
+ = maxx∈Ω p(x) with 1 < p(.) < N.



Int. J. Anal. Appl. 18 (5) (2020) 750

2.1. Generalized Lebesgue-Sobolev Spaces. First, denote the variable exponent Lebesgue space

Lp(.)(Ω) by

Lp(.)(Ω) = {u measurable function in Ω : ρp(.)(u) =
∫
Ω

|u|p(x) dx}

,

which is equipped with the Luxemburg norm

‖u‖Lp(.)(Ω) = inf


µ > 0,

∫
Ω

∣∣∣∣u(x)µ
∣∣∣∣p(x) dx ≤ 1


 . (2.1)

The space Lp(x)(Ω) is also called a generalized Lebesgue space.

The space (Lp(.)(Ω);‖.‖p(.)) is a separable Banach space. Moreover, if 1 < p
− ≤ p+ < +∞, then Lp(.)(Ω)

is uniformly convex, hence reflexive and its dual space is isomorphic to Lp′(.)(Ω), where 1
p(x)

+ 1
p′(x)

= 1, for

x ∈ Ω .

The following inequality will be used later:

min
{
‖u‖p

−

Lp(.)(Ω)
,‖u‖p

+

Lp(.)(Ω)

}
≤
∫
Ω

|u(x)|p(x) dx ≤ max
{
‖u‖p

−

Lp(.)(Ω)
,‖u‖p

+

Lp(.)(Ω)

}
. (2.2)

Finally, the Hölder type inequality∣∣∣∣∣∣
∫
Ω

uvdx

∣∣∣∣∣∣ ≤
(

1

p−
+

1

p+

)
‖u‖

p(.)
‖v‖

p′(.)
, (2.3)

for all u∈ Lp(.)(Ω) and v∈ Lp
′(.)(Ω).

Next, define the variable exponent Sobolev space W 1,p(.)(Ω) by

W 1,p(.)(Ω) =
{
u ∈ Lp(.)(Ω), |∇u| ∈ Lp(.)(Ω)

}
, (2.4)

which is Banach space equiped with the following norm

‖u‖
1,p(.)

= ‖u‖
p(.)

+ ‖∇u‖
p(.)

. (2.5)

The space (W 1,p(.)(Ω);‖.‖1,p(.)) is a separable and reflexive Banach space. An important role in manipulating

the generalized Lebesgue and Sobolev spaces is played by the modular ρp(.) of the space L
p(.)(Ω). To have

the following result:

Proposition 2.1. If un,u ∈ Lp(.)(Ω) and p+ < +∞, the following properties hold true.

(i) ‖u‖
p(.)

> 1 =⇒‖u‖p
+

p(.)
< ρp(.)(u) < ‖u‖

p−

p(.)
,

(ii) ‖u‖
p(.)

< 1 =⇒‖u‖p
−

p(.)
< ρp(.)(u) < ‖u‖

p+

p(.)
,

(iii) ‖u‖
p(.)

< 1 (respectively = 1; > 1)⇐⇒ ρp(.)(u) < 1 (respectively = 1; > 1),

(iv) ‖un‖
p(.)
−→ 0 (respectively −→ +∞)⇐⇒ ρp(.)(un) < 1(respectively −→ +∞),



Int. J. Anal. Appl. 18 (5) (2020) 751

(v) ρp(.)

(
u

‖u‖
p(.)

)
= 1.

For a measurable function u : Ω −→ R, we introduce the following notation

ρ1,p(.) =

∫
Ω

|u|p(x) dx +
∫
Ω

|∇u|p(x) dx.

Proposition 2.2. If u ∈ W 1,p(.)(Ω) and p+ < +∞, the following properties hold true.

(i)‖u‖
1,p(.)

> 1 =⇒‖u‖p
+

1,p(.)
< ρ1,p(.)(u) < ‖u‖

p−

1,p(.)
,

(ii)‖u‖
1,p(.)

< 1 =⇒‖u‖p
−

1,p(.)
< ρ1,p(.)(u) < ‖u‖

p+

1,p(.)
,

(iii)‖u‖
1,p(.)

< 1 (respectively = 1; > 1)⇐⇒ ρ1,p(.)(u) < 1(respectively = 1; > 1).

Extending a variable exponent p : Ω −→ [1, +∞) to Q = [0,T]×Ω by setting p(x,t) = p(x) for all (x,t) ∈ Q.

We may also consider the generalized Lebesgue space

Lp(.)(Q) =


u : Q −→ R mesurable such that

∫
Q

|u(x,t)|p(x) d(x,t) < ∞


 ,

endowed with the norm

‖u‖Lp(.)(Q) = inf


µ > 0,

∫
Q

∣∣∣∣u(x,t)µ
∣∣∣∣p(x) d(x,t) ≤ 1


 ,

which share the same properties as Lp(.)(Ω).

3. The Assumptions on The Data

This paper, we assume that the following assumptions hold true:

Let Ω be a bounded open set of RN (N ≥ 2), T > 0 is given and we set Q = Ω×]0,T[, and A : Q×RN → RN

be a Carathéodory function such that for all ξ,η ∈ RN,ξ 6= η

A(x,t,ξ).ξ > α |ξ|p(x) , (3.1)

|A(x,t,ξ)|6 β
[
L(x,t) + |ξ|p(x)−1

]
, (3.2)

(A(x,t,ξ) −A(x,t,η)).(ξ −η) > 0, (3.3)

where 1 < p− ≤ p+ < +∞, α,β are positives constants and L is a nonnegative function in Lp
′(.)(Q),

γ : R → R is a continuous increasing function with γ(0) = 0.

Let bi : R → R is a strictly increasing C1−function lipchizienne with bi(0) = 0 and for any ρ,τ

are positives constants and for i = 1, 2 such that

ρ ≤ b′i(s) ≤ τ, ∀s ∈ R, (3.4)



Int. J. Anal. Appl. 18 (5) (2020) 752

fi : Q× R × R → R be a Carathéodory function such that for any k > 0, there exists σk > 0, ck ∈ L1(Q)

such that

|f1(x,t,s1,s2)| ≤ ck(x,t) + σk|s2|2, (3.5)

for almost every (x,t) ∈ (Q), for every s1 such that |s1| ≤ k, and for every s2 ∈ R.

For any k > 0, there exists ζk > 0 and Gk ∈ Lp
′(.)(Q) such that

|f2(x,t,s1,s2)| ≤ Gk(x,t) + ζk|s1|p(x)−1, (3.6)

for almost every (x,t) ∈ (Q), for every s2 such that |s2| ≤ k, and for every s1 ∈ R.

f1(x,t,s1,s2)s1 ≥ 0 and f2(x,t,s1,s2)s2 ≥ 0, (3.7)

(b1(u0),b2(v0)) ∈ (L1(Ω))2. (3.8)

4. The Main Results

In this section, we study the existence of renormalized solutions to problem (1.1).

Definition 4.1. Let 2 −
1

N + 1
< p− ≤ p+ < N and (b1(u0),b2(v0)) ∈ (L1 (Ω))2. A measurable functions

(u,v) ∈ (C(]0,T[; L1(Ω)))2 is a renormalized solution of the problem (1.1) if ,

Tk(u) ∈ Lp
−

(]0,T[; W
1,p(.)
0 (Ω)),Tk(v) ∈ L

2(]0,T[; H10 (Ω)) for any k > 0 , (4.1)

γ(u) ∈ L1 (Q) and fi(x,t,u,v) ∈ (L1 (Q))2, ∀i = 1, 2,

b1(u) ∈ L∞
(
]0,T[; L1 (Ω)

)
∩Lq

−
(]0,T[; W

1,q(.)
0 (Ω)) (4.2)

and b2(v) ∈ L∞
(
]0,T[; L1 (Ω)

)
∩L2(]0,T[; H10 (Ω)),

for all continuous functions q(x) on Ω satisfying q(x) ∈
[
1,p(x) − N

N+1

)
for all x ∈ Ω,

lim
n→∞

∫
{n≤|u|≤n+1}

A(x,t,∇u)∇udxdt + lim
n→∞

∫
{n≤|v|≤n+1}

|∇v|2dxdt = 0, (4.3)

and if, for every function S ∈ W 2,∞(R) which is piecewise C1 and such that S′ has compact support on R,

to have,

(B1S(u))t −div(A(x,t,∇u)S
′(u)) + S′′(u)A(x,t,∇u)∇u + γ(u)S′(u) (4.4)

= f1(x,t,u,v)S
′(u) in D′(Q),

(B2S(v))t −div(∇vS
′(v)) + S′′(v)∇v = f2(x,t,u,v)S′(v) in D′(Q), (4.5)



Int. J. Anal. Appl. 18 (5) (2020) 753

B1S(u)(t = 0) = S(b1(u0)) in Ω, (4.6)

B2S(v)(t = 0) = S(b2(v0)) in Ω, (4.7)

where BiS(z) =
∫ z

0
b′i(r)S

′(r)dr, for i = 1, 2.

The following remarks are concerned with a few comments on definition (4.1).

Remark 4.1. Note that, all terms in (4.4) are well defined. Indeed, let k > 0 such that supp(S′) ⊂ [K,K],

we have BiS(u) belongs to L
∞(Q) for all i = 1, 2 because

|B1S(u)| ≤
∫ u

0

|b′1(r)S
′(r)|dr ≤ τ‖S′‖L∞(R),

and

|B2S(v)| ≤
∫ v

0

|b′2(r)S
′(r)|dr ≤ τ‖S′‖L∞(R),

and

S(u) = S(Tk(u)) ∈ Lp−(]0,T[; W
1,p(.)
0 (Ω)),S(v) = S(Tk(v)) ∈ L

2(]0,T[; H10 (Ω))

and
∂BiS(u)
∂t

∈ (D′(Q))2 for i = 1, 2. The term S′(u)A(x,t,∇Tk(u)) identifes with S′(Tk(u))A(x,t,∇(Tk(u)))

a.e. in Q, where u = Tk(u) in {|u| ≤ k}, assumptions (3.2) imply that

|S′(Tk(u))A(x,t,∇Tk(u))|

≤ β‖S′‖L∞(R)
[
L(x,t) + |∇(Tk(u))|

p(x)−1
]

a.e in Q.

(4.8)

Using (3.2) and (4.1), it follows that S′(u)A(x,t,∇u) ∈ (Lp
′(.)(Q))N . The term S′′(u)A(x,t,∇u)∇(u)

identifes with S′′(u)A(t,x,∇(Tk(u)))∇Tk(u) and in view of (3.2), (4.1) and (4.8), to obtain

S′′(u)A(x,t,∇u)∇(u) ∈ L1(Q) and S′(u)γ(u) ∈ L1(Q). Finally f1(x,t,u,v) S′(u) = f1(x,t,Tk(u),v)S′(u)

a.e in Q . Since |Tk(u)| ≤ k and S′(u) ∈ L∞(Q), ck(x,t) ∈ L1(Q), to obtain from (3.5) that

f1(x,t,Tk(u),v)S
′(u) ∈ L1(Q), and f2(x,t,u,v) S′(v) = f2(x,t,u,Tk(v))S′(v) a.e in Q. Since |Tk(v)| ≤ k

and S′(v) ∈ L∞(Q), Gk(x,t) ∈ Lp
′(.)(Q) , to obtain from (3.6) that f2(x,t,u,Tk(v))S

′(v) ∈ L1(Q). Also
∂B1S(u)
∂t

∈ L(p
−)′(]0,T[; W−1,p

′(.)(Ω)) + L1(Q) and B1S(u) ∈ L
p−(]0,T[; W

1,p(.)
0 (Ω)) ∩ L

∞(Q), and
∂B2S(v)
∂t

∈

L2(]0,T[; H−1(Ω)) + L1(Q) and B2S(v) ∈ L
2(]0,T[; H10 (Ω)) ∩ L∞(Q), which implies that (B1S(u),B

2
S(v))

∈ (C(]0,T[; L1(Ω)))2.



Int. J. Anal. Appl. 18 (5) (2020) 754

4.1. The Existence Theorem.

Theorem 4.1. Let (b1(u0),b2(v0)) ∈ (L1(Ω))2, assume that (3.1)-(3.8) hold true, then there exists at least

one renormalized solution (u,v) ∈ (C(]0,T[,L1(Ω)))2 of Problem (1.1) ( in the sens of Definition (4.1) ).

Proof. of Theorem (4.1) The above theorem is to be proved in five steps.

• Step 1: Approximate problem and a priori estimates. Let us define the following approxi-

mation of b and f for ε > 0 fixed and for i = 1, 2

biε(r) = T1
ε
(bi(r)) a.e in Ω for ε > 0, ∀r ∈ R, (4.9)

biε(u
ε
0) are a sequence of (C

∞
c (Ω))

2 functions such that (4.10)

(b1ε(u
ε
0),b

2
ε(v

ε
0)) → (b1(u0),b2(v0)) in (L

1(Ω))2 as ε tends to 0.

fε1 (x,t,r1,r2) = f1(x,t,T1
ε
(r1),r2), (4.11)

fε2 (x,t,r1,r2) = f2(x,t,r1,T1
ε
(r2)),

in view of (3.5), (3.6) and (3.7), there exist Gεk ∈ L
p′(.)(Q), cεk ∈ L

1(Q) and σεk,ζ
ε
k > 0 such that

|fε1 (x,t,s1,s2)| ≤ c
ε
k(x,t) + σ

ε
k|s2|

2, (4.12)

|fε2 (x,t,s1,s2)| ≤ G
ε
k(x,t) + ζ

ε
k|s1|

p(x)−1, (4.13)

for almost every (x,t) ∈ (Q), s1,s2 ∈ R,

fε1 (x,t,s1,s2)s1 ≥ 0 and f
ε
2 (x,t,s1,s2)s2 ≥ 0. (4.14)

Let us now consider the approximate problem:

(
b1ε(u

ε)
)
t
−divA(x,t,∇uε) + γ (uε) = fε1 (x,t,u

ε,vε) in Q, (4.15)

(
b2ε(v

ε)
)
t
− ∆vε = fε2 (x,t,u

ε,vε) in Q, (4.16)

uε = vε = 0 on ]0,T[ ×∂Ω, (4.17)

b1ε(u
ε) (t = 0) = b1ε(u

ε
0) in Ω, (4.18)

b2ε(v
ε) (t = 0) = b2ε(v

ε
0) in Ω. (4.19)



Int. J. Anal. Appl. 18 (5) (2020) 755

As a consequence, proving existence of a weak solution uε ∈ Lp
−

(]0,T[; W
1,p(.)
0 (Ω)) and v

ε ∈

L2(]0,T[; H10 (Ω)) of (4.15)-(4.18) is an easy task (see [15]).

we choose Tk(u
ε)χ(0,t) as a test function in (4.15), to get

∫
Ω

B
1,ε
k (u

ε)(t)dx +

t∫
0

∫
Ω

A(x,t,∇uε)∇Tk(uε) +
t∫
0

∫
Ω

γ (uε) Tk(u
ε)dxds

=

t∫
0

∫
Ω

fε1 (x,t,u
ε,vε)Tk(u

ε)dxds +

∫
Ω

B
1,ε
k (u

ε
0)dx, (4.20)

for almost every t in (0,T), and where

B
i,ε
k (r) =

∫ r
0

Tk(s)
∂biε(s)

∂s
ds.∀i = 1, 2.

Under the definition of B
i,ε
k (r) the inequality

0 ≤
∫

Ω

B
1,ε
k (u

ε
0)(t)dx ≤ k

∫
Ω

|b1ε(u
ε
0)|dx, k > 0.

Using (3.1), fε1 (x,t,u
ε,vε)Tk(u

ε) ≥ 0, and we have γ(uε) = λ|uε|p(x)−1uε ≥ 0 because 1 < p− ≤

p(x) ≤ +∞ and the definition of Bεk(r) in (4.20), to obtain∫
Ω

Bεk(u
1,ε)(t)dx + α

∫
Ek

|∇uε|p(x) dxds ≤ k
∥∥b1ε(uε0)∥∥L1(Q) , (4.21)

where Ek = {(x,t) ∈ Q : |uε| ≤ k}, using B
ε

k(u
ε)(t) ≥ 0 and inequality (2.2) in (4.21), to get

T

α

∫
0

min
{
‖∇Tk(uε)‖

p−

Lp(x)(Ω)
,‖∇Tk(uε)‖

p+

Lp(x)(Ω)

}
≤ α

∫
{(x,t)∈Q: |uε|≤k}

|∇uε|p(x) dxdt

≤ C, (4.22)

then is Tk(u
ε) is bounded in Lp−(]0,T[ ; W

1,p(x)
0 (Ω)).

Similarly, we choose Tk(v
ε)χ(0,t) as a test function in (4.16), to get∫

Ω

B
2,ε
k (v

ε)(t)dx + α

∫
Fk

|∇vε|2 dxds ≤ k
∥∥b2ε(vε0)∥∥L1(Q) , (4.23)

where Fk = {(x,t) ∈ Q : |vε| ≤ k}, then is Tk(vε) is bounded in L2(]0,T[ ; H10 (Ω)). Adding (4.21)

and (4.23), one gets∫
Ω

B
1,ε
k (u

ε)(t)dx +

∫
Ω

B
2,ε
k (v

ε)(t)dx ≤ k
∥∥(b1ε(uε0),b2ε(vε0))∥∥L1(Q)×L1(Q) . (4.24)

Also, to obtain

k

∫
{(t,x)∈Q:|uε|>k}

|γ(uε)|dxdt ≤ k‖bε(uε0)‖L1(Q) . (4.25)



Int. J. Anal. Appl. 18 (5) (2020) 756

Hence

k

∫
{(x,t)∈Q:|uε|>k}

|fε1 (x,t,u
ε,vε)|dxdt + k

∫
{(x,t)∈Q:|vε|>k}

|fε2 (x,t,u
ε,vε)|dxdt

≤ k
∥∥(b1ε(uε0),b2ε(vε0))∥∥L1(Q)×L1(Q) . (4.26)

Now, let T1(s − Tk(s)) = Tk,1(s) and take Tk,1(b1ε(uε)) as test function in (4.15). Reasoning as

above, by ∇Tk,1(s) = ∇sχ{k≤|s|≤k+1} and the young’s inequality, to obtain

α

∫
{k≤|b1ε(uε)|≤k+1}

b′1,ε(u
ε) |∇(uε)|p(x) dxdt ≤ k

∫
{|b1ε(uε0)|>k}

∣∣b1ε(uε0)∣∣dx
+ Ck

∫
{|b1ε(uε)|>k}

|γ(uε)|dxdt

+ Ck

∫
{|b1ε(uε)|>k}

|fε1 (x,t,u
ε,vε)|dxdt

≤ C1,

inequality (2.2) implies that

T∫
0

αχ{k≤|b1ε(uε)|≤k+1} min
{∥∥∇(b1ε(uε))∥∥p−Lp(x)(Ω) ,∥∥∇(b1ε(uε))∥∥p+Lp(x)(Ω)}

≤ α
∫

{k≤|b1ε(uε)|≤k+1}

b′1,ε(u
ε) |∇(uε)|p(x) dxdt ≤ C1. (4.27)

Similarly, we choose Tk(b
2
ε(v

ε)) as test function in (4.16), to have∫
{|b2ε(vε)|≤k}

b′2,ε(v
ε) |∇(vε)|2 dxdt ≤ k

∫
{|b2ε(vε0 )|>k}

∣∣b2ε(vε0)∣∣dx
+Ck

∫
{|b2ε(vε)|>k}

|fε2 (x,t,u
ε,vε)|dxdt ≤ C2,

we know that properties of B
i,ε
k (u

ε), (B
i,ε
k (r

ε) ≥ 0, Bi,εk (r
ε)) ≥ ρ(|r|−1), for all i = 1, 2, to obtain∫

Ω

∣∣∣B1,εk (uε)(t)∣∣∣dx + ∫
Ω

∣∣∣B2,εk (vε)(t)∣∣∣dx ≤ k∫
Ω

∣∣b1ε(uε)(t)∣∣dx + k∫
Ω

∣∣b2ε(vε)(t)∣∣dx
≤ ρ

(
2meas(Ω) + k

∥∥(b1ε(uε0),b2ε(vε0))∥∥L1(Q)×L1(Q)) . (4.28)
From the estimation (4.22), (4.23), (4.27) , (4.28) and the properites of B

i,ε
k and b

1
ε(u

ε
0), b

2
ε(v

ε
0), we

deduce that

b1ε(u
ε) and b2ε(v

ε) is bounded in L∞
(
]0,T[; L1 (Ω)

)
, (4.29)

uε and vε is bounded in L∞
(
]0,T[; L1 (Ω)

)
, (4.30)



Int. J. Anal. Appl. 18 (5) (2020) 757

and

b1ε(u
ε) is bounded in Lp−(]0,T[ ; W

1,p(x)
0 (Ω)), (4.31)

and

b2ε(v
ε) is bounded in L2(]0,T[ ; H10 (Ω)), (4.32)

by (4.27), (4.28) and Lemma 2.1 in [7] by and if

2 −
1

N + 1
< p(.) < N,

to obtain

b1ε(u
ε) is bounded in Lq−(]0,T[ ; W

1,q(x)
0 (Ω)), (4.33)

for all continuous variable exponents q ∈ C(Ω) satisfying

1 ≤ q(x) <
N(p(x) − 1) + p(x)

N + 1
,

for all x ∈ Ω.

And

Tk (u
ε) is bounded in Lp

−
(

]0,T[; W
1,p(.)
0 (Ω)

)
, (4.34)

and

Tk (v
ε) is bounded in L2

(
]0,T[; H10 (Ω)

)
. (4.35)

By (4.25) and (4.26), we may conclude that

γ(uε) is bounded in L1
(
]0,T[; L1 (Ω)

)
, (4.36)

and

fε1 (x,t,u
ε,vε) and fε2 (x,t,u

ε,vε) is bounded in L1
(
]0,T[; L1 (Ω)

)
, (4.37)

independently of ε.

Proceeding as in [3], [4] that for any S ∈ W 2,∞(R) such that S′ is compact (supp S′ ⊂ [−k,k]),

S (uε) is bounded in Lp−
(

]0,T[; W
1,p(.)
0 (Ω)

)
, (4.38)

and

S (vε) is bounded in L2
(
]0,T[; H10 (Ω)

)
, (4.39)

and

(S (uε))t is bounded in L
1 (Q) + L(p−)

′
(

]0,T[; W−1,p
′(.) (Ω)

)
, (4.40)

and

(S (vε))t is bounded in L
1 (Q) + L2

(
]0,T[; H−1 (Ω)

)
. (4.41)



Int. J. Anal. Appl. 18 (5) (2020) 758

In fact, as a consequence of (4.34), by Stampacchia’s Theorem, we obtain (4.38). To show that (4.40)

holds true, we multiply the equation (4.15) by S′(uε) and the equation (4.16) by S′(vε), to obtain

(
B1S (u

ε)
)
t

= div(S′ (uε)A(x,t,∇uε)) −A(x,t,∇uε)∇(S′ (uε)) (4.42)

−γ (uε) S′ (uε) + fε1 (x,t,u
ε,vε)S′ (uε) in D′ (Q) .

And

(
B2S (v

ε)
)
t

= div(S′ (vε)∇vε) −∇(S′ (vε)) (4.43)

+fε2 (x,t,u
ε,vε)S′ (vε) in D′ (Q) .

Since supp(S′) and supp(S′′) are both included in [−k; k]; uε may be replaced by Tk(uε) in

{|uε| ≤ k}. To have

|S′ (uε)A(x,t,∇uε)| (4.44)

≤ β‖S′‖L∞
[
L(x,t) + |∇Tk(uε)|

p(x)−1
]
,

as a consequence, each term in the right hand side of (4.42) is bounded either in

L(p−)
′
(

]0,T[; W−1,p
′(.) (Ω)

)
or in L1(Q), and obtain (4.40).

Now we look for an estimate on a sort of energy at infinity of the approximating solutions. For

any integer n ≥ 1, consider the Lipschitz continuous function θn defined through

θn (s) = Tn+1 (s) −Tn (s) =




0 if |s| ≤ n,

(|s|−n) sign(s) if n ≤ |s| ≤ n + 1,

sign(s) if |s| ≥ n.

Remark that ||θn||L∞ ≤ 1 for any n ≥ 1 and that θn (s) → 0, for any s when n tends to infinity.

Using the admissible test function θn(u
ε) in (4.15) leads to

∫
Ω

θ̃n (u
ε) (t) dx +

∫
Q

A(x,t,∇uε)∇(θn(uε)) dxdt +
∫
Q

γ (uε) θn(u
ε)dxdt

=

∫
Q

fε(x,t,uε)θn(u
ε)dxdt +

∫
Ω

θ̃n (u
ε
0) dx, (4.45)

where θ̃n (r) (t) =
∫ r

0
θn(s)

∂biε(s)

∂s
ds, for all i = 1, 2,

for almost any t in ]0,T[ and where θ̃n(r) =
r∫
0

θn(s)ds ≥ 0. Hence, dropping a nonnegative term

∫
{n≤|uε|≤n+1}

A(x,t,∇uε)∇uεdxdt (4.46)



Int. J. Anal. Appl. 18 (5) (2020) 759

≤
∫
Q

γ (uε) θn(u
ε)dxdt +

∫
Q

fε1 (x,t,u
ε,vε)θn(u

ε)dxdt +

∫
Ω

θ̃n (u
ε
0) dx

≤
∫

{|uε|≥n}

|γ (uε)|dxdt +
∫

{|uε|≥n}

|fε1 (x,t,u
ε,vε)|dxdt +

∫
{|b1ε(uε0)|≥n}

∣∣b1ε(uε0)∣∣dx.

Similarly, we take test function θn(v
ε) in (4.16) leads to

∫
{n≤|vε|≤n+1}

|∇vε|2dxdt (4.47)

≤
∫
Q

fε2 (x,t,u
ε,vε)θn(v

ε)dxdt +

∫
Ω

θ̃n (v
ε
0) dx ≤

∫
{|vε|≥n}

|fε2 (x,t,u
ε,vε)|dxdt

+

∫
{|b2ε(vε0 )|≥n}

∣∣b2ε(vε0)∣∣dx.

Next, we study the convergence of (un)n∈N and (vn)n∈N in C(]0,T[; L
1(Ω)).

Lemma 4.1. Both (uεn)n∈N and (v
εn)n∈N are Cauchy sequences in C(]0,T[; L

1(Ω)).

Proof. Let εn and εm two positive integers. It follows frome (4.15) and (4.16) that

∫
Ω

∂b1εn(u
εn −uεm)
∂t

ϕdx +

t∫
0

∫
Ω

(A(x,t,∇uεn) −A(x,t,∇uεm))∇ϕdxdt

+

t∫
0

∫
Ω

λ
[
|uεn|p(x)−2 uεn −|uεm|p(x)−2 uεm

]
φdxds

=

t∫
0

∫
Ω

[fεn1 (x,t,u
εn,vεn) −fεn1 (x,t,u

εm,vεm)] ϕdxds, (4.48)

and

∫
Ω

∂b2εn(v
εn −vεm)
∂t

φdx +

t∫
0

∫
Ω

(∇vεn −∇vεm)∇φdxdt (4.49)

=

t∫
0

∫
Ω

[fεn2 (x,t,u
εn,vεn) −fεn2 (x,t,u

εm,vεm)] φdxds,



Int. J. Anal. Appl. 18 (5) (2020) 760

where ϕ ∈ L∞(]0,T[; W 1,p(.)(Ω)) and φ ∈ L2(]0,T[; H10 (Ω)). To do this fix τ ∈ [0,T]. Taking

ϕ = 1
k
Tk(u

εn −uεm)1{[0,τ[} in (4.48) and φ = 1kTk(v
εn −vεm)1{[0,τ[} in (4.49), one gets

1

k

∫
Ω

B
1,εn
k (u

εn(τ) −uεn(τ))dx−
1

k

∫
Ω

B
1,εn
k (u

εn(0) −uεm(0))dx

+

τ∫
0

∫
Ω

1

k
(A(x,t,∇uεn) −A(x,t,∇uεm))∇Tk(uεn −uεm)dxdt (4.50)

+

τ∫
0

∫
Ω

λ

k

[
|uεn|p(x)−2 uεn −|uεm|p(x)−2 uεm

]
Tk(u

εn −uεm)dxds

=

t∫
0

∫
1

k
Ω

[fεn1 (x,t,u
εn,vεn) −fεn1 (x,t,u

εm,vεm)] Tk(u
εn −uεm)dxds,

and

1

k

∫
Ω

B
2,εn
k (v

εn(τ) −vεm(τ))dx−
1

k

∫
Ω

B
2,εn
k (v

εn(0) −vεm(0))dx

+
1

k

t∫
0

∫
Ω

∇(vεn −vεm)∇Tk(vεn −vεm)dxdt (4.51)

=

t∫
0

∫
Ω

1

k
[fεn2 (x,t,u

εn,vεn) −fεn2 (x,t,u
εm,vεm)] Tk(v

εn −vεm)dxds,

where

B
i,εn
k (r) =

∫ r
0

Tk(s)
∂biεn(s)

∂s
ds. ∀i = 1, 2,

adding (4.50) and (4.51), we get

1

k

∫
Ω

B
1,εn
k (u

εn(τ) −uεm(τ))dx +
1

k

∫
Ω

B
2,εn
k (v

εn(τ) −vεm(τ))dx

≤
τ∫
0

∫
Ω

λ
[
|uεn|p(x)−2 uεn −|uεm|p(x)−2 uεm

]
dxdt +

τ∫
0

∫
Ω

[fεn1 (x,t,u
εn,vεn) −fεn1 (x,t,u

εm,vεm)] dxdt +

τ∫
0

∫
Ω

[fεn2 (x,t,u
εn,vεn) −fεn2 (x,t,u

εm,vεm)] dxdt +

∫
Ω

∣∣b1εn(uεn0 −uεm0 )∣∣dx +
∫
Ω

∣∣b2εn(vεn0 −vεm0 )∣∣dx,



Int. J. Anal. Appl. 18 (5) (2020) 761

since B
i,εn
k (r) ≥ ρ

∫ r
0
Tk(s)ds ≥ ρ (|s|− 1) .∀i = 1, 2∫

Ω

|uεn(τ) −uεm(τ)|dx +
∫
Ω

|vεn(τ) −vεm(τ)|dx

≤ 2k meas(Ω) +
τ∫
0

∫
Ω

kλ
[
|uεn|p(x)−2 uεn −|uεm|p(x)−2 uεm

]
dxdt

+k

τ∫
0

∫
Ω

[fεn1 (x,t,u
εn,vεn) −fεn1 (x,t,u

εm,vεm)] dxdt

+k

τ∫
0

∫
Ω

[fεn2 (x,t,u
εn,vεn) −fεn2 (x,t,u

εm,vεm)] dxdt

+k

∫
Ω

∣∣b1εn(uεn0 −uεm0 )∣∣dx + k
∫
Ω

∣∣b2εn(vεn0 −vεm0 )∣∣dx,
letting εn, εm →∞ and them k → 0, to obtain

sup
τ∈[0,T]

∫
Ω

|uεn(τ) −uεm(τ)|dx + sup
τ∈[0,T]

∫
Ω

|vεn(τ) −vεm(τ)|dx

≤
τ∫
0

∫
Ω

kλ
[
|uεn|p(x)−2 uεn −|uεm|p(x)−2 uεm

]
dxdt

+k

τ∫
0

∫
Ω

[fεn1 (x,t,u
εn,vεn) −fεn1 (x,t,u

εm,vεm)] dxdt

+k

τ∫
0

∫
Ω

[fεn2 (x,t,u
εn,vεn) −fεn2 (x,t,u

εm,vεm)] dxdt

+k

∫
Ω

∣∣b1εn(uεn0 −uεm0 )∣∣dx + k
∫
Ω

∣∣b2εn(vεn0 −vεm0 )∣∣dx.
�

• Step 2: The limit of the solution of the approximated problem. Arguing again as in

[ [3], [4], [5]] estimates (4.38), (4.40), (4.39) and (4.41) imply that, for a subsequence still indexed by

ε,

(uε,vε) converge almost every where to (u,v), (4.52)

using (4.15), (4.34), (4.35) and (4.44), to get

Tk(u
ε) converge weakly to Tk(u) in L

p−
(

]0,T[ ; W
1,p(.)
0 (Ω)

)
, (4.53)

and

Tk(v
ε) converge weakly to Tk(v) in L

2
(
]0,T[ ; H10 (Ω)

)
, (4.54)



Int. J. Anal. Appl. 18 (5) (2020) 762

χ{|uε|≤k}A(x,t,∇uε) ⇀ ηk weakly in
(
Lp
′(.) (Q)

)N
, (4.55)

as ε tends to 0 for any k > 0 and any n ≥ 1 and where for any k > 0, ηk belongs to
(
Lp
′(.) (Q)

)N
.

Since γ(uε) is a continuous incrassing function, from the monotone convergence theorem and (4.25)

and by (4.52), to obtain that

γ(uε) converge weakly to γ(u) in L1(Q). (4.56)

We now establish that (b1(u),b2(v)) belongs to (L
∞
(
]0,T[ ; L1 (Ω)

)
)2. Indeed using (4.20) and∣∣∣Bi,εk (s)∣∣∣ ≥ ρ(|s|− 1), ∀i = 1, 2, leads to∫

Ω

∣∣b1ε(uε)∣∣ (t)dx + ∫
Ω

∣∣b2ε(vε)∣∣ (t)dx ≤ ρ(2meas(Ω)
+ ‖(fε1 (x,t,u

ε,vε),fε2 (x,t,u
ε,vε))‖(L1(Q))2

+ k‖γ (uε)‖L1(Q)

+ k
∥∥(b1ε(uε0),b2ε(vε0))∥∥(L1(Ω))2 ).

By lemma (4.1) and (4.46), (4.47), we conclude that there exist two subsequences of uεn and vεn, still

denoted by themselves for convenience, such that uεn converges to a function u in C(]0,T[; L1(Ω)),

vεnconverges to a function v in C(]0,T[; L1(Ω)). Using (4.25) and (4.10),(4.26), we have (b1(u),b2(v))

belongs to (L∞
(
]0,T[ ; L1 (Ω)

)
)2. We are now in a position to exploit (4.46) and (4.47). Since (uε,vε)

is bounded in (L∞
(
]0,T[ ; L1 (Ω)

)
)2, to get

lim
n→+∞

(
sup
ε
meas{|uε| ≥ n}

)
= 0. (4.57)

and

lim
n→+∞

(
sup
ε
meas{|vε| ≥ n}

)
= 0. (4.58)

The equi-integrability of the sequence fεi (x,t,u
ε,vε) in (L1(Q))2. We shall now prove that

fεi (x,t,u
ε,vε) converges to fi(x,t,u,v) strongly in (L

1(Q))2, for all i = 1, 2 by using Vitali’s theorem.

Since fεi (x,t,u
ε,vε) → fi(x,t,u,v) a.e in Q it suffices to prove that fεi (x,t,u

ε,vε) are equi-integrable

in Q. Let δ1 > 0 and A be a measurable subset belonging to Ω×]0,T[, we define the following sets

Gδ1 = {(x,t) ∈ Q : |un| ≤ δ1}, (4.59)

Fδ1 = {(x,t) ∈ Q : |un| > δ1}. (4.60)

Using the generalized Hölder’s inequality and Poincaré inequality, to have∫
A

|fε1 (x,t,u
ε,vε)|dxdt =

∫
A∩Gδ1

|fε1 (x,t,u
ε,vε)|dxdt +

∫
A∩Fδ1

|fε1 (x,t,u
ε,vε)|dxdt,



Int. J. Anal. Appl. 18 (5) (2020) 763

therfore

∫
A

|fε1 (x,t,u
ε,vε)|dxdt ≤

∫
A∩Gδ1

(
ck,ε(x,t) + σk,ε |vε|

2
)
dxdt

+

∫
A∩Fδ1

|fε1 (x,t,u
ε,vε)|dxdt

≤
∫
A

ck,ε(x,t)dxdt + σk,ε

∫
Q

|∇Tδ1 (v
ε)|2 dxdt

+

∫
A∩Fδ1

|fε1 (x,t,u
ε,vε)|dxdt

≤
∫
A

ck,ε(x,t)dxdt + σk,ε (meas(Q) + 1)
1
2


∫
QT

|∇Tδ1 (v
ε)|2 dxdt




1
2

+

∫
A∩Fδ1

|fε1 (x,t,u
ε,vε)|dxdt

≤ K1 + C2
(
k

α

∥∥b2ε(vε0)∥∥L1(Ω)
)1

2

+

∫
A∩Fδ1

1

|uε|
|uεfε1 (x,t,u

ε,vε)|dxdt

≤ K2 +
∫

A∩Fδ1

1

δ1
|uεfε1 (x,t,u

ε,vε)|dxdt

≤ K2 +
1

δ1

(
1

p−
+

1

p′−

) ∫
A∩Fδ1

|uε|p(x) dxdt




1

p−


 ∫
A∩Fδ1

|fε1 (x,t,u
ε,vε)|p

′(x)(p(x)−1)
dxdt




1

p′−

→ 0 when meas(A) → 0.

Which shows that fε1 (x,t,u
ε,vε) is equi-integrable. By using Vitali’s theorem, to get

fε1 (x,t,u
ε,vε) → f1(x,t,u,v) strongly in L1(Q). (4.61)

Now we prove that

fε2 (x,t,u
ε,vε) → f2(x,t,u,v) strongly in L1(Q). (4.62)



Int. J. Anal. Appl. 18 (5) (2020) 764

Let δ2 > 0 and A be a measurable subset belonging to Ω×]0,T[, we define the following sets

Gδ2 = {(x,t) ∈ Q : |vn| ≤ δ2}, (4.63)

Fδ2 = {(x,t) ∈ Q : |vn| > δ2}. (4.64)

Using the generalized Hölder’s inequality and Poincaré inequality, to get∫
A

|fε2 (x,t,u
ε,vε)|dxdt =

∫
A∩Gδ2

|fε2 (x,t,u
ε,vε)|dxdt +

∫
A∩Fδ2

|fε2 (x,t,u
ε,vε)|dxdt,

therfore∫
A

|fε2 (x,t,u
ε,vε)|dxdt ≤

∫
A∩Gδ2

(
Gεk(x,t) + ξ

ε
k |u

ε|p(x)−1
)
dxdt

+

∫
A∩Fδ2

|fε2 (x,t,u
ε,vε)|dxdt

≤
∫
A

Gεk(x,t)dxdt + ξ
ε
k

∫
Q

|∇Tδ2 (u
ε)|p(x)−1 dxdt

+

∫
A∩Fδ2

|fε2 (x,t,u
ε,vε)|dxdt

≤
∫
A

Gεk(x,t)dxdt + ξ
ε
k

(
1

p−
+

1

p′−

)
(meas(Q) + 1)

1

p−


∫
QT

|∇Tδ2 (u
ε)|(p(x)−1)p

′(x)
dxdt




1

p′−

+

∫
A∩Fδ2

|fε2 (x,t,u
ε,vε)|dxdt

≤ K3 + C4
(
k

α

∥∥b1ε(uε0)∥∥L1(Ω)
)1

2

+

∫
A∩Fδ2

1

|vε|
|vεfε2 (x,t,u

ε,vε)|dxdt

≤ K4 +
∫

A∩Fδ2

1

δ2
|vεfε2 (x,t,u

ε,vε)|dxdt

≤ K4 +
1

δ2


 ∫
A∩Fδ2

|vε|2 dxdt




1
2

 ∫
A∩Fδ2

|fε2 (x,t,u
ε,vε)|2 dxdt




1
2

→ 0 when meas(A) → 0.

Which shows that fε2 (x,t,u
ε,vε) is equi-integrable. By using Vitali’s theorem, to get

fε2 (x,t,u
ε,vε) → f2(x,t,u,v) strongly in L1(Q). (4.65)



Int. J. Anal. Appl. 18 (5) (2020) 765

Using (4.56), (4.61) and the equi-integrability of the sequence |b1ε(uε0)| in L1(Ω) and |b2ε(vε0)| in

L1(Ω), we deduce that

lim
n→+∞


sup

ε


 ∫
{n≤|uε|≤n+1}

A(x,t,∇uε)∇uεdxdt +
∫

{n≤|vε|≤n+1}

|∇vε|2dxdt




 = 0. (4.66)

• Step 4: Strong convergence. The specifie time regularization of Tk(u) (for fixed k ≥ 0) is defined

as follows. Let (v
µ
0 )µ be a sequaence in L

∞ (Ω)∩W 1,p(.)0 (Ω) such that ‖v
µ
0‖L∞(Ω) ≤ k, ∀µ > 0, and

v
µ
0 → Tk(u0) a.e in Ω with

1
µ
‖vµ0‖Lp(.)(Ω) → 0 as µ → +∞.

For fixed k ≥ 0 and µ > 0, let us consider the unique solution Tk(u)µ ∈ L∞ (Ω) ∩

Lp−
(

]0,T[; W
1,p(.)
0 (Ω)

)
of the monotone problem

∂Tk(u)µ
∂t

+ µ (Tk(u)µ −Tk(u)) = 0 in D′ (Q) , (4.67)

Tk(u)µ(t = 0) = v
µ
0 . (4.68)

The behavior of Tk(u)µ as µ → +∞ is investigated in [9] and we just recall here that (4.67)-(4.68)

imply that

Tk(u)µ → Tk(u) strongly in Lp−
(

]0,T[; W
1,p(.)
0 (Ω)

)
a.e in Q, as µ → +∞, (4.69)

with ‖Tk(u)µ‖L∞(Ω) ≤ k, for any µ, and
∂Tk(u)µ

∂t
∈ L(p−)

′
(

]0,T[; W−1,p
′(.) (Ω)

)
.

The main estimate is the following

Lemma 4.2. Let S be an increasing C∞ (R)− function such that S(r) = r for r ≤ k, and suppS′

is compact. Then

lim inf
µ→+∞

lim
ε→0

T∫
0

〈
∂B1S(u

ε)

∂t
, (Tk(u

ε)µ −Tk(u))
〉
dt ≥ 0,

where here 〈., .〉 denotes the duality pairing between L1(Ω) + W−1,p
′(.) (Ω) and L∞ (Ω) ∩W 1,p(.)0 (Ω),

and where B1S(z) =
∫ z

0
b′1(r)S

′(r)dr.

Proof. See [5], Lemma 1. �

Now we are to prove that the weak limit ηk and we prove the weak L
1 convergence of the ”truncted”

energy A(x,t,∇Tk(uε)) as ε tends to 0. In order to show this result we recall the lemma below.

Lemma 4.3. The subsequence of uε defined in step 3 satisfies

lim sup
ε→0

∫
Q

A(x,t,∇uε)∇Tk(uε)dxdt ≤
∫
Q

ηk∇Tk(u)dxdt, (4.70)



Int. J. Anal. Appl. 18 (5) (2020) 766

lim
ε→0

∫
Q

[
A
(
x,t,∇uεχ{|uε|≤k}

)
−A

(
x,t,∇uχ{|u|≤k}

)]

×
[
∇uεχ{|uε|≤k} −∇uχ{||≤k}

]
dxdt = 0 (4.71)

ηk = A
(
x,t,∇uχ{|u|≤k}

)
a.e in Q, for any k ≥ 0, as ε tends to 0.

A(x,t,∇uε)∇Tk(uε) →A(x,t,∇u)∇Tk(u) weakly in L1 (Q) . (4.72)

Proof. Let us introduce a sequence of increasing C∞(R)-functions Sn such that, for any n ≥ 1


Sn(r) = r if |r| ≤ n,

supp (S′n) ⊂ [−(n + 1), (n + 1)] ,

‖S′′n‖L∞(R) ≤ 1.

(4.73)

For fixed k ≥ 0, we consider the test function S′n(uε)
(
Tk(uε) − (Tk(u))µ

)
in (4.15), we use the

definition (4.73) of S′n and we definie W
ε
µ = Tk(uε) − (Tk(u))µ, to get

T∫
0

〈(
B1S(u

ε)
)
t
,Wεµ

〉
dt +

∫
Q

S′n(u
ε)A(x,t,∇uε)∇Wεµdxdt (4.74)

+

∫
Q

S′′n(u
ε)A(x,t,∇uε)∇uεWεµdxdt +

∫
Q

γ(uε)S′n(u
ε)Wεµdxdt

=

∫
Q

fε1 (x,t,u
ε,vε)S′n(u

ε)Wεµdxdt.

Now we pass to the limit in (4.74) as ε → 0, µ → +∞, n → +∞ for k real number fixed. In order

to perform this task, we prove below the following results for any k ≥ 0 :

lim inf
µ→+∞

lim
ε→0

T∫
0

〈(
B1S(u

ε)
)
t
,Wεµ

〉
dt ≥ 0 for any n ≥ k, (4.75)

lim
n→+∞

lim
µ→+∞

lim
ε→0

∫
Q

S′′n(u
ε)A(x,t,∇uε)∇uεWεµdxdt = 0, (4.76)

lim
µ→+∞

lim
ε→0

∫
Q

γ(uε)S′n(u
ε)Wεµdxdt = 0, for any n ≥ 1, (4.77)

lim
µ→+∞

lim
ε→0

∫
Q

fε1 (x,t,u
ε,vε)S′n(u

ε)Wεµdxdt = 0, for any n ≥ 1. (4.78)

Proof of (4.75). In view of the definition Wεµ, we apply lemma (4.2) with S = Sn for fixed n ≥ k.

As a consequence, (4.75) hold true. �



Int. J. Anal. Appl. 18 (5) (2020) 767

Proof of (4.76). For any n ≥ 1 fixed, we have supp(S′′n) ⊂ [−(n + 1),−n]∪ [n,n + 1] ,
∥∥Wεµ∥∥L∞(Q) ≤

2k and ‖S′′n‖L∞(R) ≤ 1, to get ∣∣∣∣∣∣
∫
Q

S′′n(u
ε)A(x,t,∇uε)∇uεWεµdxdt

∣∣∣∣∣∣ (4.79)
≤ 2k

∫
{n≤|uε|≤n+1}

A(x,t,∇uε)∇uεdxdt,

for any n ≥ 1, by (4.66) it possible to etablish (4.76) �

Proof of (4.77). For fixed n ≥ 1 and in view (4.56) . Lebesgue’s convergence theorem implies that

for any µ > 0 and any n ≥ 1

lim
ε→0

∫
Q

γ(uε)S′n(u
ε)Wεµ dxdt =

∫
Q

γ(u)S′n(u)(Tk(u) −Tk (u)µ)dxdt. (4.80)

Appealing now to (4.69) and passing to the limit as µ → +∞ in (4.80) allows to conclude that (4.77)

holds true. �

Proof of (4.78). By (4.11), (4.61) and Lebesgue’s convergence theorem implies that for any µ > 0

and any n ≥ 1, it is possible to pass to the limit for ε → 0

lim
ε→0

∫
Q

fε1 (x,t,u
ε,vε)S′n(u

ε)Wεµ dxdt =

∫
Q

f1(x,t,u,v)S
′
n(u)(Tk(u) −Tk (u)µ)dxdt,

using (4.69) permits to the limit as µ tends to +∞ in the above equality to obtain (4.78). �

Now turn back to the proof of Lemma (4.3), due to (4.75)-(4.78), we are in a position to pass to

the limit-sup when ε → 0, then to the limit-sup when µ → +∞ and then to the limit as n → +∞

in (4.74). Using the definition of Wεµ, we deduce that for any k ≥ 0,

lim
n→+∞

lim sup
µ→+∞

lim sup
ε→0

∫
Q

A(x,t,∇uε)S′n(u
ε)∇(Tk(uε) −Tk(u)µ) dxdt ≤ 0.

Since A(x,t,∇uε)S′n(uε)∇Tk(uε) = A(x,t,∇uε)∇Tk(uε) fo k ≤ n, the above inequality implies that

for k ≤ n,

lim sup
ε→0

∫
Q

A(x,t,∇uε)∇Tk(uε)dxdt (4.81)

≤ lim
n→+∞

lim sup
µ→+∞

lim sup
ε→0

∫
Q

A(t,x,∇uε)S′n(u
ε)∇Tk(u)µdxdt.

Due to (4.55), to have

A(x,t,∇uε)S′n(u
ε) → ηn+1S′n(u) weakly in

(
Lp
′(.) (Q)

)N
as ε → 0,



Int. J. Anal. Appl. 18 (5) (2020) 768

and the strong convergence of Tk(u)µ to Tk(u) in L
p−(]0,T[; W

1,p
0 (Ω)) as µ → +∞, to get

lim
µ→+∞

lim
ε→0

∫
Q

A(x,t,∇uε)S′n(u
ε)∇Tk(u)µdxdt (4.82)

=

∫
Q

S′n(u)ηn+1∇Tk(u)dxdt =
∫
Q

ηn+1∇Tk(u)dxdt,

as soon as k ≤ n, since S′n(s) = 1 for |s| ≤ n. Now, for k ≤ n, to have

S′n(u
ε)A(x,t,∇uε)χ{|uε|≤k} = A(x,t,∇u

ε)χ{|uε|≤k} a.e in Q.

Letting ε → 0, to obtain

ηn+1χ{|u|≤k} = ηkχ{|u|≤k} a.e in Q−{|u| = k} for k ≤ n.

Recalling (4.81) and (4.82) allows to conclude that (4.70) holds true. �

Proof of (4.71). Let k ≥ 0 be fixed. We use the monotone character (3.3) of A(x,t,ξ) with respest

to ξ, to obtain

Iε =

∫
Q

(
A(x,t,∇uεχ{|uε|≤k}) −A(x,t,∇uχ{|u|≤k})

)(
∇uεχ{|uε|≤k} −∇uχ{|u|≤k}

)
dxdt ≥ 0. (4.83)

Inequality (4.83) is split into Iε = Iε1 + I
ε
2 + I

ε
3 where

Iε1 =

∫
Q

A(x,t,∇uεχ{|uε|≤k})∇uεχ{|uε|≤k}dxdt,

Iε2 = −
∫
Q

A(x,t,∇uεχ{|uε|≤k})∇uχ{|u|≤k}dxdt,

Iε3 = −
∫
Q

A(x,t,∇uχ{|u|≤k})
(
∇uεχ{|uε|≤k} −∇uχ{|u|≤k}

)
dxdt.

We pass to the limit-sup as ε → 0 in Iε1, Iε2 and Iε3. Let us remark that we have uε = Tk(uε) and

∇uεχ{|uε|≤k} = ∇Tk(uε) a.e in Q, and we can assume that k is such that χ{|uε|≤k} almost everywhere

converges to χ{|u|≤k}(in fact this is true for almost every k, see Lemma 3.2 in [6]). Using (4.70), to

obtain

lim
ε→0

Iε1 = lim
ε→0

∫
Q

A(x,t,∇uε)∇Tk(uε)dxdt (4.84)

≤
∫
Q

ηk∇Tk(u)dxdt.



Int. J. Anal. Appl. 18 (5) (2020) 769

In view of (4.53) and (4.55), to have

lim
ε→0

Iε2 = −lim
ε→0

∫
Q

A(x,t,∇uεχ{|uε|≤k}) (∇Tk(u)) dxdt (4.85)

= −
∫
Q

ηk (∇Tk(u)) dxdt.

As a consequence of (4.53), we have for all k > 0

lim
ε→0

Iε3 = −
∫
Q

A(x,t,∇uχ{|u|≤k}) (∇Tk(uε) −∇Tk(u)) dxdt = 0. (4.86)

Taking the limit-sup as ε → 0 in (4.83) and using (4.84), (4.85) and (4.86) show that (4.71) holds

true. �

Proof of (4.72). Using (4.71) and the usual Minty argument applies it follows that (4.72) holds true.

Lemma 4.4. ∇Tk(vε) converges to ∇Tk(v) in (L2(Q))N .

Proof. Denote V εµ = Tk(vε) − (Tk(v))µ and choose S
′
n(v

ε)
(
Tk(vε) − (Tk(v))µ

)
the test function in

(4.16). One can get that

T∫
0

〈(
B2S(v

ε)
)
t
,V εµ

〉
dt +

∫
Q

S′n(v
ε)∇vε∇V εµdxdt (4.87)

+

∫
Q

S′′n(v
ε)|∇vε|2V εµdxdt =

∫
Q

fε2 (x,t,u
ε,vε)S′n(v

ε)V εµdxdt.

By a similar discussion, one has

lim inf
µ→+∞

lim
ε→0

T∫
0

〈(
B2S(v

ε)
)
t
,V εµ

〉
dt ≥ 0 for any n ≥ k, (4.88)

lim
n→+∞

lim
µ→+∞

lim
ε→0

∫
Q

S′′n(v
ε)|∇vε|2V εµdxdt = 0, (4.89)

and

lim
µ→+∞

lim
ε→0

∫
Q

fε2 (x,t,u
ε,vε)S′n(v

ε)V εµdxdt = 0, for any n ≥ 1. (4.90)

Hence

lim
n→+∞

lim
µ→+∞

lim
ε→0

∫
Q

S′n(v
ε)∇vε∇V εµdxdt ≤ 0. (4.91)

�

Similarly, one gets that ∇Tk(vε) converges to ∇Tk(v) in (L2(Q))N . �



Int. J. Anal. Appl. 18 (5) (2020) 770

• Step 5: In this step we prove that (u,v) satisfies (4.3), (4.4)-(4.7) . For any fixed n ≥ 0 one has∫
{n≤|uε|≤n+1}

A(x,t,∇uε)∇uεdxdt

=

∫
Q

A(x,t,∇uε)∇Tn+1(uε)dxdt−
∫
Q

A(x,t,∇uε)∇Tn(uε)dxdt.

According to (4.55) and (4.72) one is at liberty to pass to the limit as ε tends to 0 for fixed n ≥ 1

and to obtain

lim
ε→0

∫
{n≤|uε|≤n+1}

A(x,t,∇uε)∇uεdxdt (4.92)

=

∫
Q

A(x,t,∇u)∇Tn+1(u)dxdt−
∫
Q

A(x,t,∇u)∇Tn(u)dxdt

=

∫
{n≤|uε|≤n+1}

A(x,t,∇u)∇udxdt.

Letting n tends to +∞ in (4.92), it follows from estimate (4.66), that

lim
ε→0

lim

∫
{n≤|uε|≤n+1}

A(x,t,∇uε)∇uεdxdt = 0.

Similarly, one can prove

lim
ε→0

lim

∫
{n≤|vε|≤n+1}

|∇vε|2dxdt = 0.

Let S be a function in W 2,∞(R) such that S′ has a compact. Let k be a positive real number such that

supp(S′) ⊂ [−k,k]. Pontwise multiplication of that approximate equation (4.15) by (S′(uε),S′(vε))

leads to

(
B1S(u

ε)
)
t
−div(S′(uε)A(x,t,∇uε)) (4.93)

+S′′(uε)A(x,t,∇uε)∇(uε) + γ(uε)S′(uε) = fε1 (x,t,u
ε,vε)S′(uε) in D′(Q),

and

(
B2S(v

ε)
)
t
−div(S′(vε)∇vε) (4.94)

+S′′(vε)|∇(vε)|2 = fε2 (x,t,u
ε,vε)S′(vε) in D′(Q).

In what follows to pass to the limit as ε tends to 0 in each term of (4.93). Since S is bounded, and

(S(uε),S(vε)) converges to (S(u),S(v)) a.e in Q and in (L∞(Q))2 *-weak, then



Int. J. Anal. Appl. 18 (5) (2020) 771

(
(
B1S(u

ε)
)
t
,
(
B2S(v

ε)
)
t
) converges to (

(
B1S(u)

)
t
,
(
B1S(v)

)
t
) in D′(Q) as ε tends to 0. Since supp(S′) ⊂

[−k,k],

S′(uε)A(t,x,∇uε) = S′(uε)A(x,t,∇uε)χ{|uε|≤k} a.e in Q.

The pointwise convergence of uε to u as ε tends to 0, the bounded character of S and (4.72) of

Lemma(4.3) imply that S′(uε)A(x,t,∇uε) converges to S′(u)A(x,t,∇u) weakly in
(
Lp
′(.)(Q)

)N
as

ε tends to 0, because S′(u) = 0 for |u| ≥ k a.e in Q and S′(vε)∇vε converges to S′(v)∇v weakly in

L2(Q) as ε tends to 0. The pointwise convergence of uε to u, the bounded character of S′, S′′ and

(4.72) of Lemma (4.3) allow to conclude that

S′′(uε)A(x,t,∇uε)∇Tk(uε) → S′′(u)A(x,t,∇u)∇Tk(u) weakly in L1(Q)

as ε → 0, and lemma (4.1) shows that

S′′(vε)∇εv∇Tk(vε) → S′′(v)∇v∇Tk(v) weakly in L1(Q).

The use of (4.56) to obtain that γ(uε)S′(uε) converges to γ(u)S′(u) in L1(Q), and we use (4.11),

(4.53) and we obtain that

fε1 (x,t,u
ε,vε)S′(uε) converges to f1(x,t,u,v)S

′(u) in L1(Q)

and

fε2 (x,t,u
ε,vε)S′(vε) converges to f2(x,t,u,v)S

′(v) in L1(Q).

As a consequence of the above convergence result, the position to pass to the limit as ε tends to 0

in equation (4.93) and (4.94), we conclude that (u,v) satisfies (4.4) and (4.5).

It remains to show that S(u) satisfies the initial condition (4.6) and S(v) satisfies the initial con-

dition (4.7). To this end, firstly remark that, S being bounded, (S(uε),S(vε)) is bounded in

(L∞(Q))2, (B1S (u
ε) ,B2S (v

ε)) is bounded in L∞(Q) × L∞(Q). Secondly, (4.93) and (4.94), the

above considerations on the behavior of the terms of this equation show that
∂B1S(u

ε)
∂t

is bounded in

L1(Q) + L(p−)
′
(]0,T[; W−1,p

′(.)(Ω)) and
∂B2S(v

ε)
∂t

is bounded in L1(Q) + L2(]0,T[; H10 (Ω)). As a con-

sequence, an Aubin’s type lemma ( [20], Corollary 4) implies that (B1S(u
ε),B2S(v

ε)) lies in a compact

set of (C(]0,T[; L1(Ω)))2. It follows that, on the one hand, B1S(u
ε)(t = 0) converges to B1S(u)(t = 0)

strongly in L1(Ω) and B2S(v
ε)(t = 0) converges to B2S(v)(t = 0) strongly in L

1(Ω). Due to (4.10),

to conclude that (4.6) and (4.7) holds true. As a conclusion of Step 3 and Step 5, the proof of

Theorem (4.1) is complete.

�



Int. J. Anal. Appl. 18 (5) (2020) 772

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

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	1. Introduction
	2. The Mathematical Preliminaries on Variable Exponent Sobolev Spaces
	2.1. Generalized Lebesgue-Sobolev Spaces.

	3. The Assumptions on The Data
	4. The Main Results
	4.1. The Existence Theorem 

	References