CUBO, A Mathematical Journal
Vol. 23, no. 03, pp. 385–409, December 2021

DOI: 10.4067/S0719-06462021000300385

Entropy solution for a nonlinear parabolic problem
with homogeneous Neumann boundary condition
involving variable exponents

U. Traoré1

1 Laboratoire de Mathématiques et

Informatique (LAMI), Université Joseph

KI-ZERBO 03, BP 7021 Ouaga 03,

Ouagadougou, Burkina Faso.

urbain.traore@yahoo.fr

ABSTRACT

In this paper we prove the existence and uniqueness of an
entropy solution for a non-linear parabolic equation with ho-
mogeneous Neumann boundary condition and initial data in
L1. By a time discretization technique we analyze the ex-
istence, uniqueness and stability questions. The functional
setting involves Lebesgue and Sobolev spaces with variable
exponents.

RESUMEN

En este artículo probamos la existencia y unicidad de una
solución de entropía para una ecuación parabólica no lineal
con condiciones de borde Neumann homogéneas y data ini-
cial en L1. Usando una técnica de discretización del tiempo,
analizamos las preguntas de existencia, unicidad y estabili-
dad. El contexto funcional involucra espacios de Lebesgue y
Sobolev con exponentes variables.

Keywords and Phrases: Nonlinear parabolic problem, variable exponents, entropy solution, Neumann-type

boundary conditions, semi-discretization.

2020 AMS Mathematics Subject Classification: 35K55, 35K61, 35J60, 35Dxx.

Accepted: 12 August, 2021

Received: 19 December, 2020

©2021 U. Traoré. This open access article is licensed under a Creative Commons

Attribution-NonCommercial 4.0 International License.

http://cubo.ufro.cl/
http://dx.doi.org/10.4067/S0719-06462021000300385
https://orcid.org/0000-0002-9729-4724


386 U. Traoré CUBO
23, 3 (2021)

1 Introduction and main result

Let Ω be a smooth bounded open domain of Rd, (d ≥ 3) with Lipschitz boundary ∂Ω, T is a fixed
positive number, in this paper we study the existence and uniqueness of an entropy solution for

the following nonlinear parabolic problem

(P)




∂u

∂t
− diva(x,∇u) + b(u) = f in QT = ]0,T[×Ω,

a(x,∇u) .η = 0 on
∑

T = ]0,T[×∂Ω,

u(0, .) = u0 in Ω,

where f ∈ L1(QT ), b : R → R, a(x,ξ) : Ω × Rd → R is Carathéodory function and verifying some
assumptions which will be given later, η denotes the unit vector normal on ∂Ω.

The usual weak formulations of parabolic problems in the case where the initial data are in L1 do

not ensure existence and uniqueness of solutions. For this reason, new formulations and types of

solutions are given in order to obtain existence and uniqueness. For that, three notions of solution

have been adopted: solutions named SOLA (Solution Obtained as the Limit of Approximations)

defined by A. Dall’Aglio (see [10]); renormalized solutions defined by R. DiPerna and P.-L. Lions

(see [12]); and entropy solutions defined by Ph. Bénilan et al. in [8]. In this paper, we will be

interested in the entropy formulation.

The stationary version of the problem for the problem (P) has been already studied by Bonzi et

al. (cf. [9]), where they proved the existence and uniqueness of an entropy solution for the initial

data in L1.

The study of parabolic equations with variable exponents is a very active field (see [1, 2, 20, 21,

23, 27, 29]), in these papers, the authors consider the homogeneous Dirichlet boundary conditions,

which permit them to use many results in the generalized Sobolev space W1,p(.)(Ω) and the many

results concerned the differential equation in the literature to achieve there works. In particular

in the case of p(x)-Laplace, where b ≡ 0, Bendahmane et al. (see [6]) have proved the existence
and uniqueness of renormalized solution. We can also point out that the well-posedness of triply

nonlinear degenerate elliptic- parabolic-hyperbolic problems: b(u)t − diva(x,∇ϕ(u)) +ψ(u) = f in
a bounded domain with homogeneous Dirichlet boundary conditions by K. H. Karlsen et al. in [3].

Unfortunately, in this paper, due to the Neumann boundary condition, we cannot use the ideas

developed in these papers and also some functional analysis results which play and important role

in the a priori estimation, in particular the famous Poincaré inequality.

To overcome these difficulties we apply a time discretization of given continuous problem by the

Euler forward scheme. Let’s recall that this method has been used in the literature for the study



CUBO
23, 3 (2021)

Entropy solutions for nonlinear parabolic problems with... 387

of some nonlinear parabolic problems, we refer for example to [7, 13, 16, 17] for some details. This

scheme is usually used to prove existence of solutions as well as to compute numerical approxima-

tions.

In this paper, our assumptions are the following:
 p(.) : Ω → R is a continuous function such that1 < p− ≤ p+ < +∞, (1.1)

where p− := ess inf
x∈Ω

p(x) and p+ := ess sup
x∈Ω

p(x) and

b : Ω → R is a continuous, nondecreasing function, surjective such that b(0) = 0. (1.2)

Also, we assume that a(x,ξ) : Ω × RN → RN is Carathéodory such that:

• there exists a positive constant C1 with

|a(x,ξ)| ≤ C1
(
j(x) + |ξ|p(x)−1

)
(1.3)

for almost every x ∈ Ω and for every ξ ∈ RN, where j is a nonnegative function in Lp
′(.)(Ω) with

1

p(x)
+

1

p′(x)
= 1;

• there exists a positive constant C2 such that for every x ∈ Ω and every ξ1, ξ2 ∈ Rd with
ξ1 ̸= ξ2, the following two inequalities hold

(a(x,ξ1) − a(x,ξ2)) .(ξ1 − ξ2) > 0 (1.4)

a(x,ξ) .ξ ≥ C2|ξ|p(x). (1.5)

The rest of the paper is organized as follows: after some preliminary results in Section 2, we

introduce the Euler forward scheme associated with the problem (P) in Section 3. We analyze

the stability of the discretized problem and we study the existence of an entropy solution to the

parabolic problem (P) in the Section 4.

2 Preliminaries

We define the Lebesgue space with variable exponent Lp(.) (Ω) (see [11]) as the set of all measurable

functions u : Ω → R for which the convex modular

ρp(.) (u) :=

∫
Ω

|u|p(x) dx

is finite.

If the exponent is bounded, i.e., if p+ < +∞, then the expression

∥u∥p(.) := inf
{
λ > 0 : ρp(.) (u/λ) ≤ 1

}



388 U. Traoré CUBO
23, 3 (2021)

defines a norm in Lp(.) (Ω) , called the Luxembourg norm.

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.

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

uvdx

∣∣∣∣ ≤
(

1

p−
+

1

(p−)′

)
∥u∥p(.) ∥v∥p′(.) , (2.1)

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

Let

W1,p(.) (Ω) :=
{
u ∈ Lp(.) (Ω) : |∇u| ∈ Lp(.) (Ω)

}
,

which is Banach space equipped with the following norm

∥u∥1,p(.) := ∥u∥p(.) + ∥∇u∥p(.) .

The space
(
W1,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) . We have the following result.

Proposition 2.1 (see [14, 28]). 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 → +∞) ;

(v) ρp(.)
(
u/∥u∥p(.)

)
= 1.

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

ρ1,p(.) (u) =

∫
Ω

|u|p(x) dx +
∫
Ω

|∇u|p(x) dx.

Proposition 2.2 (see [25, 26]). If u ∈ W1,p(.) (Ω) , 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(.)

< ρp(.) (u) < ∥u∥
p−
1,p(.)

;

(iii) ∥u∥1,p(.) < 1 (respectively = 1;> 1) ⇔ ρ1,p(.) (u) < 1 (respectively = 1;> 1) .



CUBO
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Entropy solutions for nonlinear parabolic problems with... 389

Put

p∂ (x) := (p(x))
∂
=




(N − 1)p(x)
N − p(x)

, if p(x) < N

∞, if p(x) ≥ N.

Proposition 2.3 (see [26]). Let p ∈ C
(
Ω̄
)

and p− > 1. If q ∈ C (∂Ω) satisfies the condition

1 < q (x) < p∂ (x) ∀x ∈ ∂Ω,

then, there is a compact embedding W1,p(.) (Ω) ↪→ Lq(.) (∂Ω) .
In particular, there is a compact embedding W1,p(.) (Ω) ↪→ Lp(.) (∂Ω) .

Following [29], we extend a variable exponent p : Ω → [1,+∞) to QT = [0,T] × Ω by setting
p(t,x) = p(x) for all (t,x) ∈ QT .
We may also consider the generalized Lebesgue space

Lp(.) (Q) =

{
u : Q → R measurable such that

∫∫
Q

|u(t,x)|p(x) d(t,x) < ∞
}

endowed with the norm

∥u∥Lp(.)(QT ) := inf

{
λ > 0,

∫∫
QT

∣∣∣∣u(t,x)λ
∣∣∣∣p(x) d(t,x) < 1

}
,

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

For a measurable set U in Rd, meas(U) denotes its measure, Ci and C will denote various positive

constants. For a Banach space X and a < b, Lq(a,b;X) is the space of measurable functions

u : [a,b] → X such that (∫ b
a

∥u∥qX dt

)1
q

:= ∥u∥Lq(a,b;X) < ∞. (2.2)

For a given constant k > 0 we define the cut-off function Tk : R → R by

Tk(s) :=


 s if |s| ≤ kk sign(s) if |s| > k

with

sign(s) :=




1 if s > 0

0 if s = 0

−1 if s < 0.

Let Jk : R → R+ defined by
Jk(x) =

∫ x
0

Tk(s)ds

(Jk is a primitive of Tk). We have (see [15])〈
∂v

∂t
,Tk(s)

〉
=

d

dt

(∫
Ω

Jk(v)dx

)
in L1(]0,T[)



390 U. Traoré CUBO
23, 3 (2021)

which implies that ∫ t
0

〈
∂v

∂t
,Tk(s)

〉
=

∫
Ω

J(v(t))dx −
∫
Ω

J(v(0))dx

For all u ∈ W1,p(.) (Ω) we denote by τ (u) the trace of u on ∂Ω in the usual sense.
In the sequel, we will identify at the boundary, u and τ (u) .

Set

T 1,p(.) (Ω) =
{
u : Ω → R, measurable such that Tk (u) ∈ W1,p(.) (Ω) , for any k > 0

}
.

Proposition 2.4 (see [8]). Let u ∈ T 1,p(.) (Ω) . Then there exists a unique measurable function
v : Ω → RN such that ∇Tk (u) = vχ{|u|<k}, for all k > 0. The function v is denoted by ∇u.
Moreover, if u ∈ W1,p(.) (Ω) then v ∈

(
Lp(.) (Ω)

)N
and v = ∇u in the usual sense.

We denote by T 1,p(.)tr (Ω) (cf. [4, 5, 18, 19]) the set of functions u ∈ T 1,p(.) (Ω) such that there
exists a sequence (un)n∈N ⊂ W

1,p(.) (Ω) satisfying the following conditions:

i) un → u a.e. in Ω.

ii) ∇Tk (un) → ∇Tk (u) in
(
L1 (Ω)

)N
for any k > 0.

iii) There exists a measurable function v on ∂Ω, such that un → v a.e. on ∂Ω.

The function v is the trace of u in the generalized sense introduced in [4, 5]. In the sequel, the

trace of u ∈ T 1,p(.)tr (Ω) on ∂Ω will be denoted by tr (u) . If u ∈ W1,p(.) (Ω) , tr (u) coincides with
τ (u) in the usual sense. Moreover u ∈ T 1,p(.)tr (Ω) and for every k > 0, τ (Tk (u)) = Tk (tr (u)) and
if φ ∈ W1,p(.) (Ω) ∩ L∞ (Ω) then (u − φ) ∈ T 1,p(.)tr (Ω) and tr (u − φ) = tr (u) − tr (φ) .

3 The semi-discrete problem

In this section, we study the Euler forward scheme associated with the problem (P):

(Pn)




Un − τdiv a(x,∇Un) + τb(Un) = τfn + Un−1 in Ω

a(x,∇Un) .η = 0 on ∂Ω,

U0 = u0 in Ω

where Nτ = T, 0 < τ < 1, 1 ≤ n ≤ N and

fn(.) =
1

τ

∫ nτ
(n−1)τ

f(s, .)ds in Ω.



CUBO
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Entropy solutions for nonlinear parabolic problems with... 391

Definition 3.1. An entropy solution to the discretized problems (Pn) is a sequence (Un)0≤n≤N
such that U0 = u0 ∈ L1 (Ω) and Un is defined by induction as an entropy solution to the problem


Un − τdiv a(x,∇Un) + τb(Un) = τfn + Un−1 in Ω

a(x,∇Un) .η = 0 on ∂Ω

i.e. Un ∈ T 1,p(.)tr (Ω), b(Un) ∈ L1(Ω), and for every k > 0

τ

∫
Ω

a(x,∇Un).∇Tk(Un −φ)dx+
∫
Ω

(τb(Un) +Un)Tk(U
n −φ)dx ≤

∫
Ω

(τfn +U
n−1)Tk(U

n −φ)dx
(3.1)

for all φ ∈ W1,p(.)(Ω) ∩ L∞(Ω).

We have the following result

Lemma 3.2. Let hypotheses (1.3) − (1.5) be satisfied. If (Un)0≤n≤N is an entropy solution of
problems (Pn), then Un ∈ L1(Ω) for all n = 1, . . . ,N.

Proof. For n = 1, we take φ = 0 in (3.1), to get,

τ

∫
Ω

a(x,∇U1).∇Tk(U1)dx +
∫
Ω

(τb(U1) + U1)Tk(U
1)dx ≤

∫
Ω

(τf1 + u0)Tk(U
1)dx,

which is equivalent to

τ

∫
Ω

a(x,∇Tk(U1))∇Tk(U1)dx +
∫
Ω

τb(U1)Tk(U
1)dx +

∫
Ω

U1Tk(U
1)dx ≤

∫
Ω

(τf1 + u0)Tk(U
1)dx,

(3.2)

By the assumption (1.5) and the properties of the function b, we have

τ

∫
Ω

a(x,∇Tk(U1))∇Tk(U1)dx +
∫
Ω

τb(U1)Tk(U
1)dx ≥ 0,

then it follows that ∫
Ω

U1Tk(U
1)dx ≤ kτ ∥f1∥1 + k ∥u0∥1 .

Since
N∑

n=1

τ ∥fn∥1 ≤ ∥f∥1 .

Then, it follows that ∫
Ω

U1Tk(U
1)dx ≤ k(∥f∥1 + ∥u0∥1). (3.3)

Since

lim
k→0

U1
Tk(U

1)

k
= |U1|.

Then dividing (3.3) by k and letting k → 0; we deduce by Fatou’s lemma that∥∥U1∥∥
1
≤ (∥f∥1 + ∥u0∥1) (3.4)



392 U. Traoré CUBO
23, 3 (2021)

Theorem 3.3. Let hypotheses (1.3) − (1.5) be satisfied. Then for all N ∈ N, the problems (Pn)
have unique entropy solution Un ∈ T 1,p(.)tr (Ω) ∩ L1(Ω) for all n = 1, . . . ,N.

Proof. The problem (P1) can be rewritten in the following form

−τdiva(x,∇u) + b(u) = F1 in Ω

a(x,∇u).η = 0 on ∂Ω

with

b(s) := τb(s) + s, F1 := τf1 + u0.

From the assumption (H2), we have F1 ∈ L1(Ω), and using the properties of b, we obtain b is a
continuous, nondecreasing function, surjective such that b(0) = 0. Hence, using [9, Theorem 4.3],

we have the existence of unique entropy solution U1 ∈ T 1,p(.)tr (Ω), b
(
U1
)
∈ L1(Ω).

Thanks to Lemma 3.2, by induction, we deduce that for n = 2, . . . ,N, the problem

u − τdiva(x,∇u) + τα(u) = τfn + Un−1 in Ω
a(x,∇u) .η = 0 on ∂Ω,

has an unique entropy solution Un ∈ T 1,p(.)tr (Ω) ∩ L1(Ω), b(Un) ∈ L1(Ω).

4 Stability

This section is devoted to the a priori estimates for the discrete entropy solution (Un)1≤n≤N. These

result are essentials for the study of the convergence of the Euler forward scheme.

Theorem 4.1. Let hypotheses (1.3)−(1.5) be satisfied. Then there exist positive constants C(u0,f),
C(u0,f,Ω) depending on the data but not on N such that for all n = 1, . . . ,N, we have

1. ∥Un∥1 ≤ C(u0,f)

2. τ
∑n

i=1

∥∥b(Ui)∥∥
1
≤ C(u0,f)

3.
∑n

i=1

∥∥Ui − Ui−1∥∥
1
≤ C(,u0,f)

4. τ
∑n

i=1 ρp(.)(∇Tk(U
i)) ≤ kC(u0,f)

5. τ
∑n

i=1

∫
{|Ui|≤k} |∇U

i|p−dx ≤ kC(u0,f,Ω)

Proof. 1 and 2. For φ = 0 as a test function in (3.1), we have

τ

k

∫
Ω

a(x,∇Tk(Ui))∇Tk(Ui)dx +
∫
Ω

Ui
Tk(U

i)

k
dx +

∫
Ω

τb(Ui)
Tk(U

i)

k
dx

≤ τ ∥fi∥1 +
∥∥Ui−1∥∥

1
dx.



CUBO
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Entropy solutions for nonlinear parabolic problems with... 393

Since ∫
Ω

a(x,∇Tk(Ui))∇Tk(Ui)dx ≥ 0.

Then, it follows that∫
Ω

Ui
Tk(U

i)

k
dx +

∫
Ω

τb(Ui)
Tk(U

i)

k
dx ≤ τ ∥fi∥1 +

∥∥Ui−1∥∥
1
.

Then letting k → 0 and using Fatou’s lemma, we deduce that

∥∥Ui∥∥
1
+ τ

∥∥b(Ui)∥∥
1
≤ τ ∥fi∥1 +

∥∥Ui−1∥∥
1
. (4.1)

Now, we sum (4.1) from i = 1 to n to obtain

∥Un∥1 + τ
n∑

i=1

∥∥b(Ui)∥∥
1
≤ ∥f∥1 + ∥u0∥1 (4.2)

which give, the inequalities 1 and 2.

3. For k ≥ 1, we take φ = Th(Ui −sign(Ui −Ui−1)), (h > 1) as a test function in (3.1), then letting
h → ∞, for k ≥ 1, we obtain,

τ lim
h→∞

I(k,h) +
∥∥Ui − Ui−1∥∥

1
≤ τ

(
∥fi∥1 +

∥∥b(Ui)∥∥
1

)
where

I(k,h) :=
∫
Ω

a(x,∇Ui)∇Tk(Ui − Th(Ui − sign(Ui − Ui−1)))dx

=

∫
Ωk,h∩Ω(k)

a(x,∇Ui)∇Uidx

and

Ωk,h :=
{
|Ui − Th(Ui − sign(Ui − Ui−1))| ≤ k

}
Ω(k) =

{
|Ui − sign(Ui − Ui−1)| > h

}
.

Then by the hypothesis (1.3) , we have

lim
h→∞

I(k,h) ≥ 0.

Then, it follows that ∥∥Ui − Ui−1∥∥
1
≤ kτ

(
∥fi∥1 +

∥∥b(Ui)∥∥
1

)
. (4.3)

Then, summing (4.3) from i = 1 to n and by the stability result 2, we obtain the stability result 3.

4. We take φ = 0 as a test function in 3.1 to get

τ

(∫
Ω

|a(x,∇Tk(Ui))∇Tk(Ui)dx
)

≤ kτ(∥fi∥1 +
∥∥b(Ui)∥∥

1
) + k

∥∥Ui − Ui−1∥∥
1
.



394 U. Traoré CUBO
23, 3 (2021)

Therefore, using the assumption (1.5) it follows that

τρp(x)(∇Tk(Ui)) ≤ C3[kτ(∥fi∥1 +
∥∥b(Ui)∥∥

1
) + k

∥∥Ui − Ui−1∥∥
1
]. (4.4)

Now, summing (4.4) from i = 1 to n and using the stability results 1, 2, 3, we get

τ

n∑
i=1

ρp(x)(∇Tk(Ui)) ≤ C3k

[
∥f∥1 + τ

n∑
i=1

∥∥b(Ui)∥∥
1
+

n∑
i=1

∥∥Ui − Ui−1∥∥
1

]
≤ kC(f,u0). (4.5)

5. According to (4.5), we get from the above estimate

τ

n∑
i=1

∫
{|Ui|≤k}

|∇Ui|p(x)dx ≤ kC(u0,f). (4.6)

Now, note that∫
{|Ui|≤k}

|∇Ui|p−dx =
∫
{|Ui|≤k, |∇Ui|> 1N }

|∇Ui|p−dx +
∫
{|Ui|≤k |∇Ui|≤ 1N }

|∇Ui|p−dx

≤
∫
{|Ui|≤k, |∇Ui|> 1N }

|∇Ui|p−dx +
1

N
meas(Ω)

≤
∫
{|Ui|≤k}

|∇Ui|p(x)dx +
1

N
meas(Ω).

By the inequalities above, thanks to (4.6), we obtain

τ

n∑
i=1

∫
{|Ui|≤k}

|∇Ui|p−dx ≤ kC(u0,f) +
n

N
meas(Ω)

≤ kC(u0,f) + meas(Ω) ≤ k(C(u0,f) + meas(Ω)) (4.7)

for all k ≥ 1.

5 Convergence and existence result

In this section, we prove the existence of an entropy solution of problem (P). First of all, we

introduce the appropriate spaces for the entropy formulation of the nonlinear parabolic problem

(P).

We define the space:

V =
{
v ∈ Lp−(0,T;W1,p(·)(Ω)) : ∇v ∈ (Lp(·)(QT ))d

}
,

and

T 1,p(·)(QT ) =
{
u : Ω × (0,T]; measurable | Tk(u) ∈ Lp−(0,T;W1,p(·)(Ω))

with ∇Tk(u) ∈ (Lp(·)(QT ))d for every k > 0
}
.



CUBO
23, 3 (2021)

Entropy solutions for nonlinear parabolic problems with... 395

Definition 5.1. An entropy solution to problem (P) is a function u ∈ T 1,p(·)(QT )∩C(0,T;L1(Ω))
such that and for all k > 0 we have

∫ t
0

∫
Ω

a(x,∇u)∇Tk(u − φ) +
∫ t
0

∫
Ω

b(u)Tk(u − φ)

≤ −
∫ t
0

〈
∂φ

∂s
,Tk(u − φ)

〉
+

∫
Ω

Jk(u(0) − φ(0)) −
∫
Ω

Jk(u(t) − φ(t))

+

∫ t
0

∫
Ω

fTk(u − φ)

for all φ ∈ L∞(Q) ∩ V ∩ W1,1(0,T;L1(Ω)) and t ∈ [0,T].

Our main result is

Theorem 5.2. Let hypotheses (H1)−(H3) be satisfied. Then the nonlinear parabolic problem (P)
has an entropy solution.

Proof. The proof is divided into two steps

Step 1: The Rothe function. We introduce a piecewise linear extension:




uN(0) := u0,

uN(t) := Un−1 + (Un − Un−1)t−t
n−1

τ

(5.1)

for all t ∈]tn−1, tn], n = 1, · · · ,N, in Ω and a piecewise constant function




uN(0) := u0,

uN(t) := Un, ∀t ∈]tn−1, tn], n = 1, · · · ,N, in Ω,

(5.2)

where tn := nτ and (Un)1≤n≤N an entropy solution of (Pn).

By Theorem 3.3, for any N ∈ N; the solution (Un)N∈N of problems (Pn) is unique. Thus, uN and
uN are uniquely defined. Consequently, by the Theorem 4.1, we deduce the existence of a constant

C(T,u0,f) not depending on N such that for all N ∈ N, we have

∥∥uN − uN∥∥
L1(QT )

≤
1

N
C(T,u0,f)∥∥uN∥∥

L1(QT )
≤ C(T,u0,f)∥∥uN∥∥

L1(QT )
≤ C(T,u0,f) (5.3)∥∥∥∥∂uN∂t

∥∥∥∥
L1(QT )

≤ C(T,u0,f)∥∥b(uN)∥∥
L1(QT )

≤ C(T,u0,f)



396 U. Traoré CUBO
23, 3 (2021)

Moreover combining Proposition 2.1 and Young inequality, we get

∥∥∇Tk(UN)∥∥p−p(x) ≤ max
{
ρp(x)(∇Tk(UN)),ρ1,p(x)(∇TkUN)

p−
p+

}
≤ ρp(x)(∇Tk(UN)) + ρ1,p(x)(∇TkUN)

p−
p+

≤ ρp(x)(∇Tk(UN)) +
p−
p+
ρp(x)(∇Tk(UN)) + 1 −

p−
p+

(5.4)

≤ 2ρp(x)(∇Tk(UN)) + 1.

Thanks to Poincaré-Wirtinger inequality, we have

∥∥Tk(UN)∥∥p(x) ≤ Cmeas(Ω)∥∥∇Tk(UN)∥∥p(x) + k ∥1∥p(x) ,
which implies that

∥∥Tk(UN)∥∥p−p(x) ≤ 2p−−1 ((Cmeas(Ω))p− ∥∥∇Tk(UN)∥∥p−p(x) + kp− ∥1∥p−p(x)) , (5.5)
then it follows that,

∥∥Tk(UN)∥∥p−1,p(x) ≤ 2p−−1 [(Cmeas(Ω))p− (2ρp(x)(∇Tk(UN)) + 1) + kp− ∥1∥p−p(x)] (5.6)
+2ρp(x)(∇Tk(UN)) + 1.

Therefore,

∫ T
0

∥∥Tk(UN)∥∥p−1,p(.) dt ≤ 2p−−1
[
(Cmeas(Ω))p−

(
2

∫ T
0

ρp(.)(∇Tk(UN))dt + T

)

+Tkp− ∥1∥p−
p(x)

N∑]
+ 2

∫ T
0

ρp(.)(∇Tk(UN))dt + T

≤ 2p−−1
[
(Cmeas(Ω))p−

(
2

N∑
n=1

∫ nτ
(n−1)τ

ρp(.)(∇Tk(UN))dt + T

)

+Tkp− ∥1∥p−
p(.)

N∑]
+ 2

N∑
n=1

∫ nτ
(n−1)τ

ρp(.)(∇Tk(UN))dt + T (5.7)

≤ 2p−−1
[
(Cmeas(Ω))p−

(
2

N∑
n=1

τρp(.)(∇Tk(Un)) + T

)

+Tkp− ∥1∥p−
p(.)

N∑]
+ 2

N∑
n=1

τρ1,p(.)(Tk(U
n)) + T.

Consequently from stability result 4 it follows that

∥∥Tk(uN)∥∥Lp− (0,T ;W 1,p(x)(Ω)) ≤ C(T,k,u0,f,p−). (5.8)
Lemma 5.3. Let hypotheses (1.3) − (1.5) be satisfied. Then the sequence (uN)N∈N converges in
measure and a.e. in QT .



CUBO
23, 3 (2021)

Entropy solutions for nonlinear parabolic problems with... 397

Proof. Let ε,r,k be positive numbers. For N,M ∈ N, we have the inclusion

{
|uN − uM| > r

}
⊂

{
|uN| > k

}
∪
{
|uM| > k

}
∪

{
|uN| ≤ k, |uM| ≤ k, |uN − uM| > r

}
.

Firstly, we have

meas
{
|uN| > k

}
≤

1

k

∥∥uN∥∥
L1(QT )

≤
1

k
C(T,u0,f). (5.9)

Similarly, we have

meas
{
|uM| > k

}
≤

1

k

∥∥uN∥∥
L1(QT )

≤
1

k
C(T,u0,f). (5.10)

Therefore, for k large enough, we have

meas(
{
|uM| > k

}
∪
{
|uM| > k

}
) ≤

ε

2
. (5.11)

Secondly, by the Proposition 2.1 and Young inequality, we have

∥∥∥∇Tk(uN )∥∥∥
Lp(.)(QT )

≤ max

{(∫ T
0

∫
Ω

|∇Tk(uN )|p(x)dxdt
) 1

p−
;

(∫ T
0

∫
Ω

|∇Tk(uN )|p(x)dxdt
) 1

p+

}

≤
(∫ T

0

∫
Ω

|∇Tk(uN )|p(x)dxdt
) 1

p−
+

(∫ T
0

∫
Ω

|∇Tk(uN )|p(x)dxdt
) 1

p+

and also, we have∫ T
0

∫
Ω

|∇Tk(uN )|p(x)dxdt =
∫ T
0

ρp(.)(Tk(∇u
N
)) =

N∑
n=1

∫ nτ
(n−1)τ

ρp(.)(∇Tk(U
N
))dt

≤
N∑

n=1

τρp(.)(∇Tk(U
n
)).

Therefore, using the stability result 4 and Proposition 2.1, it follows∥∥∥∇Tk(uN )∥∥∥
(Lp(x)(QT ))

d
≤ (kC(u0, f))

1
p− + (kC(u0, f))

1
p+ . (5.12)

Since by the Poincaré-Wirtinger inequality, we have∥∥∥Tk(uN )∥∥∥
Lp(x)(QT )

≤ Cmeas(Ω)
∥∥∥∇Tk(uN )∥∥∥

Lp(x)(QT )
+ k ∥1∥

Lp(x)(QT )
,

then by (5.12), we get∥∥∥Tk(uN )∥∥∥
Lp(x)(QT )

≤ Cmeas(Ω)
(
(kC(u0, f))

1
p− + (kC(u0, f))

1
p+

)
+ k ∥1∥ |Lp(x)(QT ). (5.13)

Hence, the sequences (Tk(uN ))N∈N are bounded in Lp(.)(QT ). Then, there exists a subsequence, still

denoted by (Tk(uN ))N∈N, that is a Cauchy sequence in Lp(.)(QT ) and in measure. Thus, there exists

N0 ∈ N such that for all N, M ≥ N0, we have

meas
({

|uN | ≤ k, |uM | ≤ k, |uN − uM | > r
})

<
ε

2
. (5.14)

Then, by (5.11) and (5.14), (uN )N∈N converges in measure. Therefore there exists an element u ∈ M(QT )

such that

u
N → u a.e. in QT .



398 U. Traoré CUBO
23, 3 (2021)

Now, by (5.12)

(∇Tk(uN))N∈N is uniformly bounded in, (Lp(.)(QT ))d. (5.15)

Hence there exists a subsequence, still denoted by

(∇Tk(uN))N∈N converges weakly to an element V in Lp(.)(QT ).

Since

Tk(u
N) converges weakly to Tk(u) in Lp(.)(QT ).

Then

∇Tk(uN) converges weakly to ∇Tk(u) in (Lp(.)(QT ))d. (5.16)

and by (5.8) we conclude that

Tk(u) ∈ Lp−(0,T;W1,p(.)(Ω)) for all k > 0.

In the sequel, we need the following Lemma (see [22]).

Lemma 5.4. Let (vn)n≥1 be a sequence of measurable functions in Ω. If (vn)n≥1 converges in

measure to v and is uniformly bounded in Lp(.)(Ω) for some 1 << p(.) ∈ L∞(Ω), then (vn)n≥1 → v
strongly in L1(Ω).

Now, we have the following result

Lemma 5.5. Let hypotheses (1.3) − (1.5) be satisfied. Then

(i) (∇Tk(uN))N∈N converges in measure to ∇Tk(u);

(ii) (a(x,Tk(uN)))N∈N converges strongly to a(x,∇Tk(u)) in (L1(QT ))d and weakly in (Lp
′(.)

(QT ))
d.

Proof. (i) Let h ≥ 1, from the Hölder type inequality, we have

meas
{
|∇Tk(uN) − ∇Tk(u)| > h

}
≤

1

h

∫
QT

|∇Tk(uN) − ∇Tk(u)|dxds

≤
1

h

(
1

p−
+

1

p+

)∥∥∇Tk(uN) − ∇Tk(u)∥∥p(.) ∥1∥p′(.) (5.17)
≤

1

h

(
1

p−
+

1

(p−)′

)(∥∥∇Tk(uN)∥∥p(.) + ∥∇Tk(u)∥p(.))∥1∥p′(.) .
So by (5.15), meas

{
|∇Tk(uN) − ∇Tk(u)| > h

}
→ 0 as h → ∞ for any fixed k > 0 and the proof

of (i) is complete.

As a consequence of (i), up to a subsequence, we can assume that ∇Tk(uN) → ∇Tk(u) a.e in QT .
(ii) Since a(x,ξ) is continuous with respect to ξ ∈ RN, then by (i) we deduce that

(a(x,Tk(u
N)))N∈N converges in measure to a(x,∇Tk(u)) and a.e. in QT .



CUBO
23, 3 (2021)

Entropy solutions for nonlinear parabolic problems with... 399

Moreover, using the hypotheses (1.3) and (5.12) one shows that (a(x,∇Tk(uN)))N∈N is uniformly
bounded in (Lp

′(.)(QT ))
d.

Consequently, in the one part thanks to Lemma 5.4 it follows that (a(x,Tk(uN)))N∈N → a(x,∇Tk(u))
strongly in

(
L1(QT )

)d
.

On the other part, we can extract a subsequence still denoted by (a(x,∇Tk(uN)))N∈N such
that a(x,∇Tk(uN)) ⇀ ζk in (Lp

′(.)(QT ))
d. Since each of the convergence implies the weak L1-

convergence, ζk can be identified with a(x,∇Tk(u)), thus a(x,∇Tk(u)) ∈ (Lp
′(.)(QT ))

d. This com-

pletes the proof.

Lemma 5.6. (uN)N∈N converges a.e. in ΣT .

Proof. We know that the trace operator is compact from W1,1 (Ω) into L1 (∂Ω) , then there exists

a constant C such that∫ T
0

∥∥Tk(uN(t)) − Tk(u(t))∥∥L1(∂Ω) dt ≤ C
∫ T
0

∥∥Tk(uN(t)) − Tk(u(t))∥∥W 1,1(Ω) dt.
Since W1,p(.) (Ω) ↪→ W1,1 (Ω) for all p(.) ≥ 1, then by the Hölder type inequality, we deduce that

Tk(u
N(t)) → Tk(u) in L1 (ΣT ) and a.e. on ΣT .

So, there exists A ⊂ ΣT such that Tk(uN(t)) converges to Tk(u(t)) on ΣT \ A with meas(A) = 0.
For every k > 0, we set

Ak = {(t,x) ∈ ΣT : |Tk(u(t))| < k} , and B = ΣT \
∞⋃
k=1

Ak.

We have, by Hölder’s inequality

meas (B) ≤
1

k

∫
B

|Tk (u)|dσ

≤
1

k

∫ T
0

∥Tk(u)∥L1(∂Ω) dt

≤
1

k

∫ T
0

∥Tk (u)∥W 1,1(Ω) dt (5.18)

≤
1

k

∫ T
0

∫
Ω

(|Tk (u) | + |∇Tk (u) |)

≤
1

k

(
1

p−
+

1

(p−)′

)
∥1∥Lp′(x)(QT )

(
∥Tk (u)∥Lp(x)(QT ) + ∥∇Tk (u)∥(Lp(x)(QT ))d

)
.

Thanks to (5.12) and (5.13), for all k > 0, we have

∥∥Tk (uN)∥∥Lp(x)(Q) + ∥∥∇Tk (uN)∥∥(Lp(x)(Q))d ≤ 2(k 1p− + k 1p+ ) (5.19)
×max

{
C(u0,p+,f,g)

1
p+ ,C(u0,p+,f,g)

1
p+

}



400 U. Traoré CUBO
23, 3 (2021)

We now use the Fatou’s lemma in (5.19) to get

∥Tk (u)∥Lp(x)(Q) + ∥∇Tk (u)∥(Lp(x)(Q))d ≤ 2
(
k

1
p− + k

1
p+

)
×max

{
C(u0,p+,f,g)

1
p+ ,C(u0,p+,f,g)

1
p+

}
,

and (5.18) becomes

meas (B) ≤ 2
(

1

k
1− 1

p−

+
1

k
1− 1

p+

)
max

{
C(u0,p+,f,g)

1
p+ ,C(u0,p+,f,g)

1
p+

}
. (5.20)

Therefore, we get by letting k → ∞ in (5.20) that meas (B) = 0.
Let us now define on ∂Ω, the function v by

v(t,x) = Tk(u(t))(x) if (x,t) ∈ Ak.

We take (x,t) ∈ ΣT \ (A ∪ B); then there exists k > 0 such that (x,t) ∈ Ak and we have

uN (t,x) − v (t,x) = (uN (t,x) − Tk(uN(t))(x)) + (Tk(uN(t))(x) − Tk(u(t))(x)).

Since (x,t) ∈ Ak, we have
∣∣Tk(uN(t))(x)∣∣ < k from which we deduce that Tk(uN(t))(x) = uN (t,x) .

Therefore,

uN (t,x) − v (t,x) = (Tk(uN(t))(x) − Tk(u(t))(x)) → 0, as N → ∞.

This means that
(
uN
)

converges to v a.e. on ΣT .

Lemma 5.7. The sequence (uN)N∈N converges to u in C(0,T;L1(Ω)).

Proof. Let (tn = nτN)Nn=1 and (t
m = mτM)

M
n=1 be two partitions of the interval [0,T] and let

(uN(t),uN(t)), (uM(t);uM(t)); be the semi-discrete solutions defined by (5.1), (5.2) and corre-

sponding to the respective partitions. Let φ ∈ L∞(Ω) ∩ V ∩ W1,1(0,T;L1(Ω)). We rewrite (3.1)
in the forms ∫ t

0

〈
∂uN

∂s
,Tk(u

N − φ)
〉
ds +

∫ t
0

∫
Ω

a(x,∇uN).∇Tk(uN − φ)dxds

+

∫ t
0

∫
Ω

b(uN)Tk(u
N − φ)dxds

≤
∫ t
0

∫
Ω

fNTk(u
N − φ)dxds (5.21)

and ∫ t
0

〈
∂uM

∂s
,Tk(u

M − φ)
〉
ds +

∫ t
0

∫
Ω

a(x,∇uM).∇Tk(uM − φ)dxds

+

∫ t
0

∫
Ω

b(uM)Tk(u
M − φ)dxds

≤
∫ t
0

∫
Ω

fMTk(u
M − φ)dxds (5.22)



CUBO
23, 3 (2021)

Entropy solutions for nonlinear parabolic problems with... 401

where
fN(t,x) = fn(x) ∀t ∈]tn−1, tn]

fM(t,x) = fm(x) ∀t ∈]tm−1, t
m

]

Let h > 1, in inequality (5.21) we take φ = Th(uM) and in inequality (5.22) we take φ = Th(uN).

Summing both inequalities, we get, for k = 1,∫ t
0

〈
∂(uN − uM)

∂s
,T1(u

N − uM)
〉
ds + IN,M(h)

+

∫ t
0

∫
Ω

b(uN)T1(u
N − Th(uM))dxds

+

∫ t
0

∫
Ω

b(uM)T1(u
M − Th(uN))dxds

≤
∫ t
0

〈
∂(uN − uM)

∂s
,T1(u

N − uM)
〉
−
〈
∂uN

∂s
,T1(u

N − Th(uM))
〉
ds (5.23)

−
∫ t
0

〈
∂uM

∂s
,T1(u

M − Th(uN))
〉
ds

+

∫ t
0

∫
Ω

[fNT1(u
N − Th(uM)) + fMT1(uM − Th(uN))]dxds

where

IN,M(h) =

∫ t
0

∫
Ω

a(x,∇uN).∇T1(uN − Th(uM))dxds

+

∫ t
0

∫
Ω

a(x,∇uM).∇T1(uM − Th(uN))dxds.

We have∣∣∣∣
∫ t
0

〈
∂(uN − uM)

∂s
,T1(u

N − uM)
〉
ds

∣∣∣∣ ≤
∥∥∥∥∂(uN − uM)∂s

∥∥∥∥
L1(QT )

∥∥T1(uN − uM)∥∥L∞(QT )
≤ 2C(T,f,u0)

∥∥T1(uN − uM)∥∥L∞(QT ) .
Since

lim
N,M→∞

∥∥T1(uN − uM)∥∥L∞(QT ) = 0.
Then it follows that

lim
h→∞

lim
N,M→∞

∫ t
0

〈
∂(uN − uM)

∂s
,T1(u

N − uM)
〉
ds = 0. (5.24)

Similarly, we show that

lim
h→∞

lim
N,M→∞

(∫ t
0

〈
∂uN

∂s
,T1(u

N − Th(uM))
〉
+

〈
∂uM

∂s
,T1(u

M − Th(uN))
〉
ds

)
= 0

lim
h→∞

lim
N,M→∞

∫ t
0

∫
Ω

[fNT1(u
N − Th(uM)) + fMT1(uM − Th(uN))]dxds = 0



402 U. Traoré CUBO
23, 3 (2021)

and

lim
h→∞

lim
N,M→∞

∫ t
0

∫
Ω

b(uN)T1(u
N − Th(uM))dxds +

∫ t
0

∫
Ω

b(uM)T1(u
M − Th(uN))dxds = 0.

Then, letting N, M → ∞ and h → ∞, in (5.23)we get

lim
h→∞

lim
N,M→∞

∫ t
0

〈
∂(uN − uM)

∂s
,T1(u

N − uM)
〉
ds + lim

h→∞
lim

N,M→∞
IN,M(h) ≤ 0. (5.25)

Since 〈
∂v

∂t
,Tk(v)

〉
=

d

dt

∫
Ω

Jk(v) in L1(]0,T[),

inequality (5.25) becomes

lim
N,M→∞

∫
Ω

J1(u
N(t) − uM(t))dx + lim

h→∞
lim

N,M→∞
IN,M(h) ≤ 0. (5.26)

Now, we show that

lim
h→∞

lim
N,M→∞

IN,M(h) ≥ 0.

We consider the decomposition

IN,M(h) =

4∑
i=1

Li(h),

where

Li(h) =

∫ t
0

∫
Ωi(h)

a(x,∇uN).∇T1(uN − Th(uM))dxds

+

∫ t
0

∫
Ωi(h)

a(x,∇uM).∇T1(uM − Th(uN))dxds

and

Ω1(h) =
{
|uN| ≤ h, |uM| ≤ h

}
Ω2(h) =

{
|uN| ≤ h, |uM| > h

}
Ω3(h) =

{
|uN| > h, |uM| ≤ h

}
Ω4(h) =

{
|uN| > h, |uM| > h

}
.

On the one hand, thanks to assumption (1.4) we have

L1(h) =

∫ t
0

∫
Ω11(h)

[a(x,∇uN) − a(x,∇uM)].∇(uN − uM)dxds ≥ 0.

Therefore

lim
h→∞

lim
N,M→∞

L1(h) ≥ 0.

On the other hand, we have

L2(h) =

∫ t
0

∫
Ω12(h)

a(x,∇uN).∇uNdxds

+

∫ t
0

∫
Ω22(h)

a(x,∇uM).∇(uM − uN)dxds

≥ −
∫ t
0

∫
Ω22(h)

a(x,∇uM).∇uNdxds,



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Entropy solutions for nonlinear parabolic problems with... 403

where

Ω12(h) =
{
|uN| ≤ h, |uM| > h, |uN − hsign(uM)| ≤ 1

}
,

Ω22(h) =
{
|uN| ≤ h, |uM| > h, |uN − uM| ≤ 1

}
.

Now, taking φ = Th(uN) in (5.21), we deduce that

lim
h→∞

lim
N→∞

∫ t
0

∫
{h≤|uN |≤h+k}

a(x,∇uN).∇uN = 0.

This implies

lim
h→∞

lim
N→∞

∫ t
0

∫
{h≤|uN |≤h+k}

|∇uN|p(x) = 0, k > 0. (5.27)

By the Young inequality, we have∣∣∣∣∣
∫ t
0

∫
Ω22(h)

a(x,∇uM).∇uNdxds

∣∣∣∣∣
≤

∫ t
0

∫
Ω22(h)

|∇uM|p(x)−1|∇uN|dxds

≤
∫ t
0

∫
{h≤|uM |≤h+1}

1

p′(x)
|∇uM|p(x)dxds +

∫ t
0

∫
{h−1≤|uN |≤h}

1

p(x)
|∇uM|p(x)dxds

≤
∫ t
0

∫
{h≤|uM |≤h+1}

1

p′−
|∇uM|p(x)dxds +

∫ t
0

∫
{h−1≤|uN |≤h}

1

p−
|∇uM|p(x)dxds.

Thus (5.27) gives

lim
N,M→∞

∫ t
0

∫ t
0

∫
Ω22(h)

a(x,∇uM).∇uNdxds = 0,

which implies that

lim
h→∞

lim
N,M→∞

L2(h) ≥ 0.

Similarly, we show that

lim
h→∞

lim
N,M→∞

(L3(h) + L4(h)) ≥ 0.

Therefore

lim
h→∞

lim
N,M→∞

IN,M(h) ≥ 0.

Thus (5.26) becomes

lim
N,M→∞

∫
Ω

J1(u
N(t) − uM(t))dx = 0. (5.28)

Since

1

2

∫
{|uN −uM |≤1}

|uN(t) − uM(t)|2dx +
∫
{|uN −uM |≥1}

|uN(t) − uM(t)|dx ≤
∫
Ω

J1(u
N(t) − uM(t));



404 U. Traoré CUBO
23, 3 (2021)

we have ∫
{|uN −uM |≥1}

|uN(t) − uM(t)|dx

=

∫
{|uN −uM |≤1}

|uN(t) − uM(t)|dx +
∫
{|uN −uM |≥1}

|uN(t) − uM(t)|dx

≤ CΩ

(∫
{|uN −uM |≤1}

|uN(t) − uM(t)|2dx

)1
2

+

∫
{|uN −uM |≥1}

|uN(t) − uM(t)|dx

≤ C2(Ω)
(∫

Ω

J1(u
N(t) − uM(t))dx

)1
2

+

∫
Ω

J1(u
N(t) − uM(t))dx.

By (5.26), we deduce that (uN)N∈N is a Cauchy sequence in C(0,T;L1(Ω)). Hence (uN)N∈N
converges to u in C(0,T;L1(Ω)).

Step 2: Existence of entropy solution. Now, we prove that the limit function u is an entropy

solution of the problem (P). Since uN(0) = U0 = u0 for all N ∈ N, we have u(0, .) = u0, and
inequality (5.21) implies∫ t

0

〈
∂uN

∂s
,Tk(u

N − φ) − Tk(uN − φ)
〉
ds +

∫ t
0

∫
Ω

a(x,∇uN).∇Tk(uN − φ)dxds

+

∫ t
0

∫
Ω

b(uN)Tk(u
N − φ)dxds (5.29)

≤
∫ t
0

〈 φ
∂s
,Tk(u

N − φ) − Tk(uN − φ)
〉
ds +

∫
Ω

Jk(u
N(0) − φ(0))dx −

∫
Ω

Jk(u
N(t) − φ(t))dx

+

∫ t
0

∫
Ω

fNTk(u
N − φ)dxds.

Let k = k + ∥φ∥∞ . Then∫ t
0

∫
Ω

a(x,∇uN).∇Tk(uN − φ)dxds =
∫ t
0

∫
Ω

a(x,∇Tk(u
N)).∇Tk(Tk(u

N) − φ)dxds

=

∫ t
0

∫
Ω

[a(x,∇Tk(u
N)).∇Tk(u

N)

−a(x,∇Tk(u
N)).∇φ]1Q(N,k)dxds,

where Q(N,k) =
{
|Tk(u

N) − φ| ≤ k
}
. Thus, the inequality (5.29) becomes

∫ t
0

〈
∂uN

∂s
,Tk(u

N − φ) − Tk(uN − φ)
〉
ds −

∫ t
0

∫
Ω

a(x,∇Tk(u
N)).∇φ1Q(N,k)

+

∫ t
0

∫
Ω

[a(x,∇Tk(u
N)).∇Tk(u

N)]1Q(N,k) +
∫ t
0

∫
Ω

b(uN)Tk(u
N − φ)dxds (5.30)

≤ −
∫ t
0

〈
∂φ

∂s
,Tk(u

N − φ)
〉
ds +

∫
Ω

Jk(u
N(0) − φ(0))dx −

∫
Ω

Jk(u
N(t) − φ(t))dx

+

∫ t
0

∫
Ω

fNTk(u
N − φ)dxds.



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Entropy solutions for nonlinear parabolic problems with... 405

On the one hand, thanks to Lemma 5.5 a(x,∇Tk(u
N)) converges weakly to a(x,∇Tk(u)) in(

Lp
′(.) (Ω)

)d
. Therefore,

lim
N→∞

∫ t
0

∫
Ω

a(x,∇Tk(u
N)).∇φ1Q(N,k) =

∫ t
0

∫
Ω

a(x,∇Tk(u)).∇φ1Q(k), (5.31)

where Q(k) =
{
|Tk(u) − φ| ≤ k

}
. Moreover, a(x,∇Tk(u

N)).∇Tk(u
N) is nonnegative and converges

a.e. in QT to a(x,∇Tk(u)).∇Tk(u) (see Lemma 5.5). Therefore by Fatou’s lemma, we obtain

lim inf
N→∞

∫ t
0

∫ t
0

∫
Ω

[a(x,∇Tk(u
N)).∇Tk(u

N)]1Q(N,k)dxds ≥
∫ t
0

∫ t
0

∫
Ω

[a(x,∇Tk(u)).∇Tk(u)]1Q(k)dxds.

For the fourth term of (5.30), we have∫ t
0

∫
Ω

b(uN)Tk(u
N −φ)dxds =

∫ t
0

∫
Ω

(b(uN)−b(φ))Tk(uN −φ)dxds+
∫ t
0

∫
Ω

b(φ)Tk(u
N −φ)dxds.

The quantity (b(uN) − b(φ))Tk(uN − φ) is is nonnegative and since for all s ∈ R, s 7→ b(s) is
continuous, we obtain

(b(uN) − b(φ))Tk(uN − φ) → (b(u) − b(φ))Tk(uN − φ) a.e. in Ω.

Then, it follows by Fatou’s lemma that

lim inf
N→∞

∫ t
0

∫
Ω

(b(uN) − b(φ))Tk(uN − φ)dxds ≥
∫ t
0

∫
Ω

(b(u) − b(φ))Tk(u − φ)dxds.

We have b(φ) ∈ L1(QT ). Since Tk(uN − φ) converges weakly−∗ to Tk(u − φ) and b(φ) ∈ L1(QT ),
it follows that

lim inf
N→∞

∫ t
0

∫
Ω

b(φ)Tk(u
N − φ)dxds ≥

∫ t
0

∫
Ω

b(φ)Tk(u − φ)dxds.

By Lemma 5.7 , we deduce that uN(t) → u(t) in L1(Ω) for all t ∈ [0,T], which implies that∫
Ω

Jk(u
N(t) − φ(t))dx →

∫
Ω

Jk(u(t) − φ(t))dx ∀t ∈ [0,T]. (5.32)

We follow the method used in the proof of equality (5.24) to show that

lim
N→∞

∫ t
0

〈
∂uN

∂s
,Tk(u

N − φ) − Tk(uN − φ)
〉
ds = 0. (5.33)

Finally, letting N → ∞ and using the above results, the continuity of b and the facts that

fN → f in L1(QT ),

Tk(u
N − φ) → Tk(u − φ) in L

∞(QT ),

we deduce that u is an entropy solution of the nonlinear parabolic problem (P).



406 U. Traoré CUBO
23, 3 (2021)

6 Conclusion

In this paper we prove the existence and uniqueness of an entropy solution for a non- linear

parabolic equation with homogeneous Neumann boundary conditions and initial data in L1 by a

time discretization technique.

This method turns out to be better suited for the study of parabolic problems under Neumann-

type boundary conditions. However, this technique assumes that the associated elliptic problem is

well posed. This study opens up new perspectives, we could always in the context of the Sobolev

space with variable exponents look at the problem with measure data or consider the function b

as maximal monotone graph.



CUBO
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Entropy solutions for nonlinear parabolic problems with... 407

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	Introduction and main result
	Preliminaries
	The semi-discrete problem
	Stability
	Convergence and existence result
	Conclusion