CUBO, A Mathematical Journal

Vol. 24, no. 02, pp. 187–209, August 2022

DOI: 10.56754/0719-0646.2402.0187

Numerical analysis of nonlinear parabolic
problems with variable exponent and L1 data

Stanislas Ouaro
1, B

Noufou Rabo
1

Urbain Traoré
1

1Laboratoire de Mathématiques et

Informatique (LAMI), Unité de

Formation et de Recherche en Sciences

Exactes et Appliquées, Université Joseph

KI-ZERBO, 03 BP. 7021 Ouagadougou

03, Burkina Faso.

ouaro@yahoo.fr B

rabonouf@gmail.com

urbain.traore@yahoo.fr

ABSTRACT

In this paper, we make the numerical analysis of the mild

solution which is also an entropy solution of parabolic

problem involving the p(x)−Laplacian operator with L1−

data.

RESUMEN

En este art́ıculo, realizamos el análisis numérico de

la solución mild que también es una solución de en-

troṕıa del problema parabólico involucrando el operador

p(x)−Laplaciano con datos en L1.

Keywords and Phrases: Elliptic-parabolic, numerical iterative method, variable exponent, mild solution, renor-

malized solution.

2020 AMS Mathematics Subject Classification: 65M12, 65N22, 35K55, 35K65, 46E35.

Accepted: 17 December, 2021

Received: 06 June, 2021

c©2022 S. Ouaro et al. This open access article is licensed under a Creative Commons

Attribution-NonCommercial 4.0 International License.

http://cubo.ufro.cl/
https://doi.org/10.56754/0719-0646.2402.0187
mailto:ouaro@yahoo.fr
https://orcid.org/0000-0003-0671-2378
https://orcid.org/0000-0003-3659-0906
https://orcid.org/0000-0002-9729-4724
mailto:ouaro@yahoo.fr
mailto:rabonouf@gmail.com
mailto:urbain.traore@yahoo.fr


188 S. Ouaro, N. Rabo & U. Traoré CUBO
24, 2 (2022)

1 Introduction

We consider a bounded open domain Ω ⊂ Rd (d ≥ 2) with a Lipschitz boundary denoted by ∂Ω.

Let T > 0 and p : Ω → (1, ∞) be a continuous function. In this paper, one of our main goals

is the numerical approximation of the mild solution of the following nonlinear parabolic problem

involving the p(x)−Laplacian operator























∂u

∂t
− div(|∇u|p(x)−2∇u) = f in Q ≡ Ω × (0, T ),

u = 0 on ∂Ω × (0, T ),

u(x, 0) = u0 in Ω,

(1.1)

where u0 ∈ L
1(Ω), f ∈ L1(Q). The assumptions on the variable exponent p(x) will be specified

later.

Partial differential equations with nonlinearities involving non-constant exponents have attracted

an increasing amount of attention on recent years. Their study is an interesting topic which raises

many mathematical difficulties (see [1, 2, 14, 16, 27, 30]). There are many results devoted to ques-

tions on existence and uniqueness of solutions to problems like (1.1), we refer for example the reader

to the bibliography [3, 4, 5, 9, 24, 29] and references therein. Many of these models have already

been analyzed for constant exponents of nonlinearity (see the references therein), but it seems to be

more realistic to assume the exponent to be variable. From numerical point of view, in the classical

evolution problem case where p(x) ≡ p, the numerical analysis was firstly considered in [7, 22].

Afterward, Jäger and Kačur [18] and Kačur [20] studied the numerical approximation. Inspired by

these works, Maitre [23] proposed a numerical scheme to approximate the mild solutions. On the

other side, for problems with variable exponent, in recent years, there are some papers devoted to

their numerical analysis (see for example [8, 10, 12, 13, 17, 19, 26]). Thus, in [13] the authors used

a quasi-Newton minimization method to approach the solution of the p(x)−Lapacian problems; in

[12], they present an inverse power method to compute the first homogeneous eigenpair. In [26],

an interior penalty discontinuous Galerkin method has been used by the authors to approximate

the minimizer of a variational problem related to the p(x)−Laplacian. Other authors use finite

elements to approximate the solution (see [10]). Nevertheless, there are scarcely papers about the

numerical analysis of nonlinear parabolic problems with variable exponent (see for example [11]).

The importance of investigating the problem (1.1) lies in their occurrence in modeling various

physical problems involving strong anisotropic phenomena related to electrorheological fluids (an

important class of non-Newtonian fluids, see [27]) which are characterized by their ability to change

the mechanical properties under the influence of the exterior electromagnetic field. Other important

applications are related to image processing, elasticity [30], the processes of filtration in complex

media, stratigraphy problems and also mathematical biology. The study of problem (1.1) involves

using of generalized Lebesgue and Sobolev spaces i.e., Lp(.) and W 1,p(.) respectively (see [15]).



CUBO
24, 2 (2022)

Numerical analysis of nonlinear parabolic problems with variable... 189

Throughout this paper we assume that the exponent p(.) appearing in (1.1) is a continuous function

p : Ω → (1, ∞) such that:























∃C > 0 : |p(x) − p(y)| ≤
C

− log |x − y|
for every x, y ∈ Ω with |x − y| ≤

1

2

2d

d + 2
< p− := min

x∈Ω p(x) ≤ p
+ := max

x∈Ω p(x) < ∞.

(1.2)

The first condition says that p(.) belongs to the class of log-Hölder continuous functions. These

assumptions are used to obtain several regularity results for Sobolev spaces with variable exponents;

in particular, C∞(Ω) is dense in W 1,p(.)(Ω) and W
1,p(.)
0 (Ω) = W

1,p(.)(Ω) ∩ W
1,1
0 (Ω).

Our paper was inspired by the work of Maitre (see [23]) where the author studied the numerical

analysis of an elliptic-parabolic problem in the context of constant exponent setting.

The rest of this paper is organized as follows: in Section 2, we give some results for the study

of (1.1). In Section 3, we recall the notion of mild solution. In Section 4, we proceed to the

numerical study, where we show the existence and uniqueness of solution of numerical scheme for

the approximation of mild solution and the study of the convergence of this numerical scheme. We

conclude this section by numerical tests.

2 Preliminaries

We first recall in what follows some definitions and basic properties of generalized Lebesgue-Sobolev

spaces with variable exponent. We define the Lebesgue space with a variable exponent p(.) by

Lp(.)(Ω) =
{

u : Ω → R; u is measurable with ρp(.)(u) < ∞
}

,

where

ρp(.)(u) =

∫

Ω

|u(x)|p(x)dx,

is called a modular. We define a norm, the so-called Luxemburg norm, on this space by the formula

|u|p(.) = inf

{

µ > 0 : ρp(.)

(

u

µ

)

≤ 1

}

.

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

uv dx

∣

∣

∣

∣

≤

(

1

p−
+

1

p′−

)

|u|p(.)|v|p′(.)

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



190 S. Ouaro, N. Rabo & U. Traoré CUBO
24, 2 (2022)

We define also the variable Sobolev space

W 1,p(.)(Ω) =
{

u ∈ Lp(.)(Ω) : |∇u| ∈ Lp(.)(Ω)
}

.

On W 1,p(.)(Ω) we may consider the following norm

‖u‖1,p(.) = |u|p(.) + |∇u|p(.).

The space (W 1,p(.)(Ω), ‖u‖1,p(.)) is a separable and reflexive Banach space. Next, we define

W
1,p(.)
0 (Ω) as the closure of C

∞
0 (Ω) in W

1,p(.)(Ω) under the norm

‖u‖ := |∇u|p(.).

The space (W
1,p(.)
0 (Ω), ‖u‖) is a separable and reflexive Banach space. For the interested reader,

more details about Lebesgue and Sobolev spaces with variable exponent can be found in [15] (see

also [21]).

Since Ω is bounded and p : Ω → (1, ∞) is log-Hölder continuous, the Poincaré inequality holds (see

[28])

|u|p(.) ≤ C|∇u|p(.), ∀ u ∈ W
1,p(.)
0 (Ω),

where C is a constant which depends on Ω and on the function p.

An important role in manipulating the generalized Lebesgue and Sobolev spaces is played by

modular ρp(.) of the space L
p(.). We have the following result (see [28]).

Lemma 2.1. If un, u ∈ L
p(.) and p+ < ∞, then the following relations hold:

(1) |u|p(.) > 1 ⇒ |u|
p
−

p(.)
≤ ρp(.)(u) ≤ |u|

p
+

p(.)
;

(2) |u|p(.) < 1 ⇒ |u|
p
+

p(.)
≤ ρp(.)(u) ≤ |u|

p
−

p(.)
;

(3) |u|p(.) < 1 (respectively = 1; > 1) ⇐⇒ ρp(.)(u) < 1 (respectively = 1; > 1);

(4) |u|p(.) → 0 (respectively → ∞) ⇐⇒ ρp(.)(u) → 0 (respectively → ∞);

(5) ρp(.)
(

u/|u|p(.)
)

= 1.

Following [4], we extend a variable exponent p : Ω → [1, +∞) to Q = [0, T ] × Ω by setting

p(t, x) := p(x) for all (t, x) ∈ Q. We also consider the generalized Lebesgue space

Lp(.)(Q) =

{

u : Q → R measurable such that

∫∫

Q

|u(x, t)|p(x) d(x, t) < ∞

}

endowed with the norm

‖u‖Lp(.) := inf

{

µ > 0 :

∫∫

Q

∣

∣

∣

∣

u(x, t)

µ

∣

∣

∣

∣

p(x)

d(x, t) < 1

}



CUBO
24, 2 (2022)

Numerical analysis of nonlinear parabolic problems with variable... 191

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

Now, we recall the main results for the study of (1.1).

In order to approximate the mild solution of (1.1), let us recall that Ouaro and Traoré have studied

in [25] the existence and uniqueness of weak energy and entropy solutions of the following stationary

problem associated to the problem (1.1)














u − div a(x, ∇u) = f in Ω,

u = 0 on ∂Ω,

(2.1)

where Ω ⊂ Rd is a bounded domain with smooth boundary and f ∈ L1(Ω). For the vector field

a(x, ξ) : Ω × Rd → Rd, in addition to be Carathéodory, is the continuous derivative with respect

to ξ of the mapping A : Ω × Rd → Rd, i.e. a(x, ξ) = ∇ξA(x, ξ) such that:

A(x, 0) = 0 for almost every x ∈ Ω. (2.2)

There exists a positive constant C1 such that

|a(x, ξ)| ≤ C1(j(x) + |ξ|
p(x)−1), (2.3)

for almost every x ∈ Ω and for every ξ ∈ Rd where j is a non-negative function in Lp
′(.)(Ω), with

1

p(x)
+

1

p′(x)
= 1.

The following inequalities hold

(a(x, ξ) − a(x, η)).(ξ − η) > 0, (2.4)

for almost every x ∈ Ω and for every ξ, η ∈ Rd, with ξ 6= η and

1

C
|ξ|p(x) ≤ a(x, ξ).ξ ≤ Cp(x)A(x, ξ), (2.5)

for almost every x ∈ Ω, C > 0 and for every ξ ∈ Rd.

The exponent appearing in (2.3) and (2.5) is defined as follows.














p(.) : Ω → R is a measurable function such that

1 < p− := ess infx∈Ω p(x) ≤ p
+ := ess supx∈Ω p(x) < ∞.

(2.6)

For more details, see [24, 25].

As example of models with respect to above assumptions, we can give the following.

Set A(x, ξ) =
1

p(x)
|ξ|p(x), a(x, ξ) = |ξ|p(x)−2ξ. Then, we get the p(x)−Laplace operator

div (|∇u|p(x)−2∇u).

Note that the weak solution of (2.1) is defined as follows.



192 S. Ouaro, N. Rabo & U. Traoré CUBO
24, 2 (2022)

Definition 2.2. A weak solution of (2.1) is a function u ∈ W
1,1
0 (Ω) such that a(., ∇u) ∈

(

L1loc(Ω)
)d

and
∫

Ω

a(., ∇u).∇ϕ dx +

∫

Ω

uϕ dx =

∫

Ω

fϕ dx, (2.7)

for all ϕ ∈ C∞0 (Ω).

A weak energy solution is a weak solution such that u ∈ W
1,p(.)
0 (Ω).

Now, we recall one of main results.

Theorem 2.3. Assume that (2.2)–(2.6) hold and f ∈ L∞(Ω). Then there exists a unique weak

energy solution of (2.1).

We also recall a useful result needed in this paper (see [23]).

Lemma 2.4 ([23]). Let X be a Banach space and C a convex subset of X, containing 0. Let T̄ be

a non-expansive map on C such that T̄ (C) ⊂ C, admitting a unique fixed point x∗ in C. Let λk be

a sequence of (0, 1) verifying

lim
k→∞

λk = 1,
∏

k≥0

λk = 0,
∑

k≥0

|λk+1 − λk| < ∞.

Then the sequence (xk) generated by the iterative scheme

x0 ∈ C, xk+1 = λk+1T̄(x
k) (2.8)

verifies limk→∞ x
k − T̄(xk) = 0. Consequently, if all subsequences of (xk) have in turn a subse-

quence converging to a point of C, then the whole sequence (xk) converges toward x∗.

Recall that a self-mapping T̄ of C is non-expansive if

‖T̄(x) − T̄(y)‖ ≤ ‖x − y‖ for all x, y ∈ C.

In the next section, we give the definition of mild solution.

3 Notion of mild solution

Let f ∈ L1(0, T ; L1(Ω)), u0 ∈ L
1(Ω) and ε > 0 be given. We consider the time discretization of

problem (1.1) by an implicit Euler scheme



















uεn+1 − u
ε
n

tn+1 − tn
− div(|∇uεn+1|

p(x)−2∇uεn+1) = f
ε
n+1 in D

′(Ω) for n = 0, . . . , N − 1,

uεn+1 ∈ W
1,p(.)
0 (Ω) ∩ L

∞(Ω);

(3.1)



CUBO
24, 2 (2022)

Numerical analysis of nonlinear parabolic problems with variable... 193

where



















































































































N ∈ N∗, 0 = t0 < t1 < · · · < tN ≤ T is a partition of [0, T ].

fεn ∈ L
∞(Ω) for n = 1, . . . , N such that

N
∑

n=1

∫ tn

tn−1

‖f(t) − fεn‖L1(Ω)dt → 0 as ε → 0,

maxn=1,...,N(tn − tn−1) → 0, T − tN → 0 as ε → 0, u
ε
0 ∈ L

∞(Ω) such that

‖u0 − u
ε
0‖L1(Ω) → 0 as ε → 0,

with uε the piecewise constant function defined by

uε(t) = uεn on (tn−1, tn] with n = 1, . . . , N; u
ε(0) = uε0.

(3.2)

Definition 3.1. A mild solution of (1.1) is a function u ∈ C([0, T ]; L1(Ω)) with u(0) = u0 ∈ L
1(Ω)

such that, for all ε > 0, there exists (t0, t1, . . . , tN ; f
ε
1 , f

ε
2 , . . . , f

ε
N) and u

ε
0 verifying (3.2); and for

which there exists (uε1, . . . , u
ε
N) verifying (3.1) such that ‖u(t) − u

ε
n‖L1(Ω) ≤ ε for all t ∈ (tn−1, tn],

n = 1, . . . , N.

Remark 3.2. In this paper, for the sake of simplicity and readability, we chose to present the

constant step subdivision algorithm, i.e. that we set tn+1 − tn = h =
T
N

for all n = 0, . . . , N − 1.

However, the techniques developed thereafter can be adapted to a varying step subdivision without

difficulty.

Note that using the nonlinear semigroups theory [6], Ouaro and Ouédraogo have proved in [24]

the existence and uniqueness of mild solutions of the following parabolic problem























∂u

∂t
− div a(x, ∇u) = f in Q ≡ Ω × (0, T ),

u = 0 on ∂Ω × (0, T ),

u(x, 0) = u0 in Ω,

where u0 ∈ L
1(Ω) and f ∈ L1(Q). The assumptions on the vector field are the same than those

given in (2.2)–(2.5) and those on the variable exponent p(x) are the same as (2.6). Thanks to their

paper, one has the existence and uniqueness of the mild solution of problem (1.1).



194 S. Ouaro, N. Rabo & U. Traoré CUBO
24, 2 (2022)

4 Numerical study

4.1 Numerical scheme

We are now interested in the numerical resolution of (3.1). Let f1, f2, . . . , fN, u0 be some functions

satisfying (3.2), we use the following iterative scheme (proposed by Maitre in [23]) to get uεn+1

from uεn.














Let u
ε,0
n+1 = u

ε
n ∈ L

∞(Ω), solve for k = 0, 1, . . . ,

u
ε,k+1
n+1 − ρ div(|∇u

ε,k+1
n+1 |

p(x)−2∇u
ε,k+1
n+1 ) = λku

ε,k
n+1 −

ρ

h
(λku

ε,k
n+1 − u

ε
n) + ρf

ε
n+1,

(4.1)

where ρ > 0 is a given parameter and (λk) is a sequence of (0, 1) such that

lim
k→∞

λk = 1,
∏

k≥0

λk = 0,
∑

k≥0

|λk+1 − λk| < ∞. (4.2)

For example, we can take λk = 1 −
1

k + 1
.

Remark 4.1. For the sake of simplicity, we could take ρ = h, but in this paper our idea is to

build a non-expansive map and use the Halpern algorithm to approach the solution of (3.1). In the

numerical simulation one will give examples where ρ = h.

4.2 Existence and uniqueness of solution of (4.1)

In this section, we state and prove the well-posedness of our scheme.

Definition 4.2. For any n = 0, . . . , N − 1, ε > 0 and uεn ∈ L
∞(Ω), a weak solution of (4.1) is a

sequence
(

u
ε,k+1
n+1

)

k≥0
such that u

ε,k+1
n+1 ∈ W

1,p(.)
0 (Ω) ∩ L

∞(Ω) for all k = 0, 1, . . . , and

∫

Ω

u
ε,k+1
n+1 ϕ dx + ρ

∫

Ω

|∇u
ε,k+1
n+1 |

p(x)−2∇u
ε,k+1
n+1 .∇ϕ dx =

∫

Ω

gεn,kϕ dx, (4.3)

for all ϕ ∈ W
1,p(.)
0 (Ω), where

gε,kn := λku
ε,k
n+1 −

ρ

h
(λku

ε,k
n+1 − u

ε
n) + ρf

ε
n+1.

Theorem 4.3. Let ε > 0. For any n = 0, . . . , N − 1 let u
ε,0
n+1 = u

ε
n ∈ L

∞(Ω) and fεn+1 ∈ L
∞(Ω).

Then, problem (4.1) admits a unique weak solution u
ε,k+1
n+1 ∈ W

1,p(.)
0 (Ω) for all k = 0, 1, . . .

Furthermore, for k = 0, 1, . . . , u
ε,k+1
n+1 ∈ L

∞(Ω).

Proof. Let ε > 0 and fix n. For k = 0 we rewrite problem (4.1) as














u
ε,1
n+1 − ρ div(|∇u

ε,1
n+1|

p(x)−2∇u
ε,1
n+1) = g

ε,0
n in Ω

u
ε,1
n+1 = 0 on ∂Ω,

(4.4)



CUBO
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Numerical analysis of nonlinear parabolic problems with variable... 195

where

gε,0n =
[

λ0

(

1 −
ρ

h

)

+ 1
]

uεn + ρf
ε
n+1.

Consider the energy functional Jρ on W
1,p(.)
0 (Ω) associated to (4.4) given by

Jρ(U) =
1

2

∫

Ω

U2dx + ρ

∫

Ω

|∇U|p(x)

p(x)
dx −

∫

Ω

gε,0n U dx.

We will establish that Jρ(U) has a minimizer u
ε,1
n+1 in W

1,p(.)
0 (Ω).

Note that Jρ is well-defined and Gateaux differentiable on W
1,p(.)
0 (Ω), since W

1,p(.)
0 (Ω) →֒ L

2(Ω)

thanks to (1.2).

For ‖U‖
W

1,p(.)
0 (Ω)

≥ 1 we have from the continuous embedding of W
1,p(.)
0 (Ω) in L

p
−

(Ω) and gε,0n ∈

L∞(Ω),

Jρ(U) =
1

2

∫

Ω

U2dx + ρ

∫

Ω

|∇U|p(x)

p(x)
dx −

∫

Ω

gε,0n U dx ≥
ρ

p+
‖U‖

p
−

W
1,p(x)
0 (Ω)

− C‖U‖
W

1,p(x)
0 (Ω)

.

As p− > 1, then Jρ is coercive. Jρ(U) is lower bounded and furthermore weakly lower semi-

continuous; therefore, admits a global minimizer u
ε,1
n+1 ∈ W

1,p(.)
0 (Ω) which is a weak solution to

(4.4). The global minimizer u
ε,1
n+1 is also unique.

It remains to show that u
ε,1
n+1 ∈ L

∞(Ω). To do this, let us show that ‖u
ε,1
n+1‖∞ ≤ ‖g

ε,0
n ‖∞.

As u
ε,1
n+1 is a weak solution of (4.4), we have

∫

Ω

u
ε,1
n+1ϕ dx + ρ

∫

Ω

|∇u
ε,1
n+1|

p(x)−2∇u
ε,1
n+1.∇ϕ dx =

∫

Ω

gε,0n ϕ dx, (4.5)

for all ϕ ∈ W
1,p(.)
0 (Ω).

Let τ ∈ R+. Then, u
ε,1
n+1 − τ ∈ W

1,p(.)
0 (Ω) and

(

u
ε,1
n+1 − τ

)+

∈ W
1,p(.)
0 (Ω).

Note that for r ∈ R, r+ := max(r, 0) and r− := min(r, 0).

Taking
(

u
ε,1
n+1 − τ

)+

as a test function, it follows from (4.5) that

∫

Ω

u
ε,1
n+1(u

ε,1
n+1 − τ)

+ dx + ρ

∫

Ω

|∇u
ε,1
n+1|

p(x)−2∇u
ε,1
n+1.∇(u

ε,1
n+1 − τ)

+ dx =

∫

Ω

gε,0n (u
ε,1
n+1 − τ)

+ dx.

Setting Aτ =
{

x ∈ Ω : u
ε,1
n+1 ≥ τ

}

, we have

ρ

∫

Ω

|∇u
ε,1
n+1|

p(x)−2∇u
ε,1
n+1.∇(u

ε,1
n+1 − τ)

+ dx = ρ

∫

Aτ

|∇u
ε,1
n+1|

p(x)−2∇u
ε,1
n+1.∇(u

ε,1
n+1 − τ) dx

= ρ

∫

Aτ

|∇u
ε,1
n+1|

p(x)dx ≥ 0.

Therefore,
∫

Ω

u
ε,1
n+1(u

ε,1
n+1 − τ)

+ dx ≤

∫

Ω

gε,0n (u
ε,1
n+1 − τ)

+ dx.



196 S. Ouaro, N. Rabo & U. Traoré CUBO
24, 2 (2022)

As Ω is a bounded open domain, we have

∫

Ω

[(u
ε,1
n+1 − τ)

+]2 dx ≤

∫

Ω

(gε,0n − τ)(u
ε,1
n+1 − τ)

+ dx.

Taking τ = ‖gε,0n ‖∞, then g
ε,0
n − τ ≤ 0 a.e. in Ω.

Therefore, we have (u
ε,1
n+1 − τ)

+ = 0 a.e. in Ω for all τ = ‖gε,0n ‖∞ which is equivalent to saying

u
ε,1
n+1 ≤ ‖g

ε,0
n ‖∞ a.e. in Ω.

It remains to prove that u
ε,1
n+1 ≥ −‖g

ε,0
n ‖∞ a.e. in Ω. To do this we take (u

ε,1
n+1 + τ)

− as test

function in (4.5) and use the same argument as previously. Thus, setting C = ‖gε,0n ‖∞ implies

that u
ε,1
n+1 ∈ L

∞(Ω).

In short u
ε,1
n+1 ∈ W

1,p(.)
0 (Ω) ∩ L

∞(Ω).

By induction, we deduce in the same manner that the problem (4.1) has a unique weak solution
(

u
ε,k+1
n+1

)

k≥0
such that u

ε,k+1
n+1 ∈ W

1,p(.)
0 (Ω) ∩ L

∞(Ω) for all k ∈ N.

4.3 Study of the convergence

We begin with the following lemma which provides a crucial L∞ uniform bound for the sequence
(

u
ε,k
n+1

)

k≥0
.

Lemma 4.4. Let ε > 0 and fix n. If ρ ≤ h, there exists M > 0 independent of k such that

‖u
ε,k
n+1‖∞ ≤ M.

Proof. Let M = max
(

‖u
ε,0
n+1‖∞, ‖hf

ε
n+1 + u

ε
n‖∞

)

.

Now let us show by induction that ‖u
ε,k
n+1‖∞ ≤ M. We first note that ‖u

ε,0
n+1‖∞ ≤ M.

One assumes that ‖u
ε,k
n+1‖∞ ≤ M, and one shows that ‖u

ε,k+1
n+1 ‖∞ ≤ M.

As u
ε,k+1
n+1 ∈ L

∞(Ω) and verifies

u
ε,k+1
n+1 − div

(

ρ|∇u
ε,k+1
n+1 |

p(x)−2∇u
ε,k+1
n+1

)

= λku
ε,k
n+1 −

ρ

h
(λku

ε,k
n+1 − u

ε
n) + ρf

ε
n+1,

then, from the previous proof, it is established that for all k = 1, 2, . . . ,

‖u
ε,k+1
n+1 ‖∞ ≤

∥

∥

∥
λku

ε,k
n+1 −

ρ

h
(λku

ε,k
n+1 − u

ε
n) + ρf

ε
n+1

∥

∥

∥

∞
.

Since ρ ≤ h, we then obtain using the induction assumption

‖u
ε,k+1
n+1 ‖∞ ≤

(

1 −
ρ

h

)

M +
ρ

h
‖hfεn+1 + u

ε
n‖∞ ≤ M.

Thanks to M defined in the above proof we have the following convergence result.



CUBO
24, 2 (2022)

Numerical analysis of nonlinear parabolic problems with variable... 197

Theorem 4.5. Assume that conditions in Theorem 4.3 are satisfied. Then, for ρ ≤ h, the iterative

scheme (4.1) converges, i.e.

u
ε,k
n+1 −→ u

ε
n+1 strongly in L

1(Ω) as k → +∞,

where uεn+1 verifies (3.1).

Proof. Thanks to Lemma 4.4, we can write (4.1) as

1

λk+1
ū
ε,k+1
n+1 − ρ div

(

|∇
1

λk+1
ū
ε,k+1
n+1 |

p(x)−2∇
1

λk+1
ū
ε,k+1
n+1

)

= ū
ε,k
n+1 −

ρ

h
(ū

ε,k
n+1 − u

ε
n) + ρf

ε
n+1, (4.6)

where we put ū
ε,k
n+1 = λku

ε,k
n+1 and ū

ε,k+1
n+1 = λk+1u

ε,k+1
n+1 .

Let A(u) = −div(|∇u|p(x)−2∇u). We identify the operator A : L1(Ω) → L1(Ω) associated with

the p(x)−Laplacian problem (1.1) with its graph i.e.

G(A) =
{

(u, v) ∈ L1(Ω) × L1(Ω); v ∈ A(u)
}

.

Therefore, A is T −accretive as soon as u is an entropy solution of problem (2.1) where a(x, ∇u) =

(|∇u|p(x)−2∇u). For more details, see [6] and [24, Proposition 4.3]. A is called T −accretive

if ‖(u − û)+‖1 ≤ ‖(u − û + ρ(v − v̂))
+)‖1, for any (u, v), (û, v̂) ∈ A, ρ > 0; equivalently, if

∫

{u>û}

(v − v̂) +

∫

{u=û}

(v − v̂)+ ≥ 0 for any (u, v), (û, v̂) ∈ A.

Hence, (4.6) yields

(I + ρA)

(

1

λk+1
ū
ε,k+1
n+1

)

= ū
ε,k
n+1 −

ρ

h
(ū

ε,k
n+1 − u

ε
n) + ρf

ε
n+1. (4.7)

To complete the proof of Theorem 4.5, we use the following technical lemma.

Lemma 4.6. Let ρ ≤ 2h and M defined in the above proof such that CM =
{

u ∈ L1(Ω), ‖u‖∞ ≤ M
}

.

The iteration operator

T̃(ū) = (I + ρA)−1
(

ū −
ρ

h
(ū − uεn) + ρf

ε
n+1

)

is an L1-non-expanding operator from CM to CM .

Proof. The fact that T̃ maps CM to CM is easily seen thanks to the proof of the Lemma 4.4 and

(4.7). Now let (ū, v̄) ∈ C2M . One has from the T −accretiveness of A on L
1(Ω) that (I + ρA)−1 is

a T −contraction in L1(Ω) (see [6]), thus, a contraction. Therefore,

‖T̃(ū) − T̃(v̄)‖1 =
∥

∥

∥
(I + ρA)−1

(

ū −
ρ

h
(ū − un) + ρfn+1

)

− (I + ρA)−1
(

v̄ −
ρ

h
(v̄ − un) + ρfn+1

)
∥

∥

∥

1

≤
∥

∥

∥

(

1 −
ρ

h

)

ū −
(

1 −
ρ

h

)

v̄
∥

∥

∥

1
.



198 S. Ouaro, N. Rabo & U. Traoré CUBO
24, 2 (2022)

Since ρ ≤ 2h, we obtain

‖T̃(ū) − T̃(v̄)‖1 ≤ ‖ū − v̄‖1.

Consequently, from (4.7) one has the iteration ū
ε,k+1
n+1 = λk+1T̃(ū

ε,k
n+1) where T̃ is a non-expansive

operator in L1(Ω) defined as in Lemma 4.6. Now, we are going to apply the Lemma 2.4 with

X = L1(Ω) and C = CM which is clearly a convex subset of L
1(Ω) containing 0. The uniqueness

of a fixed point is verified thanks to Theorem 2.3. Indeed a fixed point u∗ of T̃ verifies

u∗ − ρ div (|∇u∗|p(x)−2∇u∗) = u∗ −
ρ

h
(u∗ − uεn) + ρf

ε
n+1.

Thus, u∗ − h div (|∇u∗|p(x)−2∇u∗) = uεn + hf
ε
n+1. From Theorem 2.3 this equation has a unique

solution and from the definition of mild solution it is uεn+1.

To conclude the proof of convergence of (4.1), we point out that each subsequence of ū
ε,k
n+1 has a

convergent subsequence to an element of CM , using the L
∞ bound of ū

ε,k
n+1 and the monotonicity of

(|∇ū
ε,k
n+1|

p(x)−2∇ū
ε,k
n+1), to the equation (4.6). Applying Lemma 2.4, we conclude that the sequence

ū
ε,k
n+1 converges strongly in L

1(Ω) toward uεn+1. The same occurs for u
ε,k
n+1 =

1

λk
ū
ε,k
n+1.

4.4 Convergence when ε → 0 toward a solution of (1.1)

Note that for a mild solution we do not need to show the convergence in time since it is included in

its definition: once convergence in k is achieved for uεn+1, then, by the definition of mild solution,

uεn+1 approaches u
ε(t) on (tn, tn+1] up to ε. Thus, our scheme converges to the mild solution when

ε goes to zero.

We can state also the following result.

Proposition 4.7. Let u0 ∈ L
∞(Ω), f ∈ L∞(Q) and u the unique mild solution of (1.1). Then u is

a weak solution of (1.1). By a weak solution we understand a solution in the sense of distributions

that belongs to the energy space, i.e.,

u ∈ V :=
{

v ∈ Lp
−

(0, T ; W
1,p(.)
0 (Ω)); |∇v| ∈ L

p(.)(Q)
}

,

∂u

∂t
− div(|∇u|p(x)−2∇u) = f in D

′

(Q), u(., 0) = u0.
(4.8)

Remark 4.8. Note that a proof of the above proposition exists in [24]. Here, we use L∞ uniform

boundedness and the strong convergence in L1(Ω) of the solution of our numerical scheme to prove

Proposition 4.7.

Moreover, these two results lead to the L∞ uniform boundedness of the weak solution.

Proof of Proposition 4.7. Let u be the mild solution of (1.1). For n = 0, . . . , N − 1, uεn+1 is the

unique weak solution of (3.1). We have
∫

Ω

uεn+1 − u
ε
n

h
ϕ dx +

∫

Ω

|∇uεn+1|
p(x)−2∇uεn+1.∇ϕ dx =

∫

Ω

fεn+1ϕ dx, (4.9)



CUBO
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Numerical analysis of nonlinear parabolic problems with variable... 199

∀ϕ ∈ W
1,p(.)
0 (Ω) ∩ L

∞(Ω) and















































• 0 = t0 < · · · < tN = T such that tn − tn−1 = h ≤ ε for n = 1, . . . , N,

•

N
∑

n=1

∫ tn

tn−1

‖f(t) − fεn‖L1(Ω) dt ≤ ε ⇒ ‖f
ε
n‖L∞(Ω) ≤ ‖f(t)‖L∞(Ω),

•

N
∑

n=1

h‖fεn‖L∞(Ω) ≤

∫ T

0

‖f(., t)‖L∞(Ω) dt,

• ‖u0 − u
ε
0‖L1(Ω) ≤ ε ⇒ ‖u

ε
0‖L∞(Ω) ≤ ‖u0‖L∞(Ω).

(4.10)

Note that relations in (4.10) are equivalent to relations in (3.2).

Let us set uε(t) = u
ε
n+1 ∀ t ∈ (tn, tn+1], uε(0) = u

ε
0 and fε(t) = f

ε
n+1, ∀ t ∈ (tn, tn+1].

Lemma 4.4, Theorem 4.5 and the above relations in (4.10) imply that

‖uε‖L∞(Q) ≤ C(‖u0‖L∞(Ω); ‖f‖L∞(Q)). (4.11)

Let ζ be the function defined by ζ(r) =
r2

2
that satisfies ζ(r) − ζ(r̃) ≤ (r − r̃)r.

Taking ϕ = uεn+1 as test function in (4.9) and integrating over (tn, tn+1] and summing over n =

0, . . . , N − 1, we get

∫

Ω

ζ(uε(t)) dx +

∫

Q

|∇uε|
p(x) dx dt ≤

∫

Q

fεuε dx dt +

∫

Ω

ζ(uε0) dx.

Thanks to the uniform boundedness of uε in ε and as u
ε
0 ∈ L

∞(Ω), we have

∫

Q

|∇uε|
p(x) dx dt ≤ C.

Moreover,

∫ T

0

‖∇uε‖
p
−

Lp(.)(Ω)
dt ≤

∫ T

0

max





∫

Ω

|∇uε|
p(x);

(
∫

Ω

|∇uε|
p(x)

)
p−

p+



 dt.

Hence,
∫ T

0

‖uε‖
p
−

W
1,p(.)
0 (Ω)

dt ≤ C.

As a consequence, there exists a subsequence still denoted (uε)ε>0, such that

uε ⇀ u, weakly-* in L
∞(Q),

uε ⇀ u, weakly in L
p
−

(0, T ; W
1,p(.)
0 (Ω)),

|∇uε|
p(.)−2∇uε ⇀ Φ, weakly in

(

Lp
′(.)(Q)

)d

.

Using the monotonicity method we show that Φ = |∇u|p(.)−2∇u a.e. in Q.

Now, let ũε be the piecewise linear function defined by

ũε(t) = u
ε
n +

t − tn
h

(uεn+1 − u
ε
n) for t ∈ [tn, tn+1], n = 0, . . . , N − 1.



200 S. Ouaro, N. Rabo & U. Traoré CUBO
24, 2 (2022)

The function ũε verifies (ũε)t (t) =
uεn+1 − u

ε
n

h
and ũε → u in L

∞(0, T ; L1(Ω)). Hence,

u ∈ C([0, T ]; L1(Ω)).

Integrating (4.9) over (tn, tn+1) and summing over n = 0, . . . , N − 1, we find

−

∫ T

0

∫

Ω

ϕtũε dx dt −

∫

Ω

ϕ(0)uε0 dx +

∫ T

0

∫

Ω

(

|∇uε|
p(x)−2∇uε

)

.∇ϕ dx dt

=

∫ T

0

∫

Ω

fεϕ dx dt. (4.12)

Using the convergence results and passing to the limit in (4.12) as ε → 0, we get the result.

Remark 4.9. For u0 ∈ L
1(Ω), f ∈ L1(Q) the unique mild solution u of (1.1) is also an entropy

solution. Indeed, since L∞ is dense in L1, we consider two sequences of functions (fm)m≥1 ⊂

L∞(Q) and (u0m)m≥1 ⊂ L
∞(Ω) satisfying















fm → f in L
1(Q), u0m → u0 in L

1(Ω), as m → ∞,

‖fm‖L1(Q) ≤ ‖f‖L1(Q), ‖u0m‖L1(Ω) ≤ ‖u0‖L1(Ω).

(4.13)

Then, we get the following approximate problem of (1.1).























∂um
∂t

− div(|∇um|
p(x)−2∇um) = fm in Q,

um = 0 on ∂Ω × (0, T ),

um(x, 0) = u0m in Ω.

(4.14)

Thanks to [24], for each m = 1, 2, . . . , we can find a unique mild solution um ∈ C([0, T ]; L
1(Ω))

for problem (4.14) which verifies the L1−contraction principle, i.e. the following estimate holds

for almost all t ∈ (0, T ),

‖um(., t)‖L1(Ω) ≤ ‖u0m‖L1(Ω) +

∫ t

0

‖fm(., s)‖L1(Ω) ds

≤ ‖u0‖L1(Ω) +

∫ t

0

‖f(., s)‖L1(Ω) ds.

By Proposition 4.7, and following the proof of [24, Theorem 5.1] we get the result.

Note that this entropy solution is equivalent to the renormalized solution of (1.1). Indeed, in

[29], Zhang and Zhou have proved thanks to the assumptions (1.2) the existence and uniqueness

of renormalized and entropy solutions of (1.1). In their paper, they have showed the equivalence

between entropy and renormalized solutions.



CUBO
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Numerical analysis of nonlinear parabolic problems with variable... 201

4.5 Numerical tests

4.5.1 Implementation

We know that solving the equation (4.1) is equivalent to solve the following minimization problem

for n = 0, 1, . . . , N − 1 and k = 0, 1, . . .

u
ε,k+1
n+1 = argminv ∈WJ(v), (4.15)

where,

W :=
{

v ∈ W
1,p(.)
0 (Ω) ∩ L

∞(Ω)
}

and the functional J is

J(v) =
1

2

∫

Ω

v2 dx + ρ

∫

Ω

1

p(x)
|∇v|p(x) dx −

(

1 −
ρ

h

)

λk

∫

Ω

u
ε,k
n+1v −

ρ

h

∫

Ω

uεnv dx

−ρ

∫

Ω

fεn+1v dx. (4.16)

We formulate a basic procedure for solving problem (4.15) following the split Bregman technique

(see [17]). We solve the minimization problem by introducing an auxiliary variable b. We have

min
v

{

1

2

∫

Ω

v2 dx + ρ

∫

Ω

1

p(x)
|b|p(x) dx −

(

1 −
ρ

h

)

λk

∫

Ω

u
ε,k
n+1v dx −

ρ

h

∫

Ω

uεnv dx

− ρ

∫

Ω

fεn+1v dx subject to b = ∇v

}

. (4.17)

By adding one quadratic penalty function term, we convert equation (4.17) to an unconstrained

splitting formulation as follow.

min
v,b

{

1

2

∫

Ω

v2 dx + ρ

∫

Ω

1

p(x)
|b|p(x) dx +

γ

2

∫

Ω

|b − ∇v|2 dx −
(

1 −
ρ

h

)

λk

∫

Ω

u
ε,k
n+1v dx

−
ρ

h

∫

Ω

uεnv dx − ρ

∫

Ω

fεn+1v dx

}

, (4.18)

where γ is a positive parameter which controls the weight of the penalty term. Similar to the split

Bregman iteration, we propose the following scheme.















































(vl+1, bl+1) = argminv,b

{

1

2

∫

Ω

v2 dx + ρ

∫

Ω

1

p(x)
|b|p(x) dx +

γ

2

∫

Ω

|b − ∇v − δl|2 dx

−
(

1 −
ρ

h

)

λk

∫

Ω

u
ε,k
n+1v dx −

ρ

h

∫

Ω

uεnv dx − ρ

∫

Ω

fεn+1v dx

}

,

δl+1 = δl + ∇vl+1 − bl+1.

(4.19)

Alternatively, this joint minimization problem can be solved by decomposing into several subprob-

lems.



202 S. Ouaro, N. Rabo & U. Traoré CUBO
24, 2 (2022)

4.5.2 Subproblem v with fixed b and δ

Given the fixed variable bl and δl, our aim is to find the solution of the following problem

vl+1 = argminv

{

1

2

∫

Ω

v2 dx +
γ

2

∫

Ω

|bl − ∇v − δl|2 dx −
(

1 −
ρ

h

)

λk

∫

Ω

u
ε,k
n+1v dx

−
ρ

h

∫

Ω

uεnv dx − ρ

∫

Ω

fεn+1v dx

}

. (4.20)

We know that solve (4.20) is equivalent to solve the following optimality condition.

v − γ∆v = γ∇.(δl − bl) +
(

1 −
ρ

h

)

λku
ε,k
n+1 +

ρ

h
uεn + ρf

ε
n+1. (4.21)

Since the discrete system is strictly diagonally dominant with Neumann boundary condition, the

most natural choice is the Gauss-Seidel method.

4.5.3 Subproblem b with fixed v and δ

Similarly, we solve

bl+1 = argminb

{

ρ

∫

Ω

1

p(x)
|b|p(x) dx +

γ

2

∫

Ω

|b − ∇vl+1 − δl|2 dx

}

(4.22)

In two dimensional space.

Here, setting b = (bx, by) and δ = (δx, δy).

Then, the resolution of (4.22) is equivalent to solve the following optimality condition.















ρ|b|p(x,y)−2bx + γ(bx − ∇xv
l+1 − δlx) = 0

ρ|b|p(x,y)−2by + γ(by − ∇yv
l+1 − δly) = 0,

(4.23)

where ∇v = (∇xv, ∇yv).

If bx and by are not zero, then,

bx =
∇xv

l+1 + δlx
∇yvl+1 + δly

by. (4.24)

Substituting (4.24) into (4.23), we obtain

sign(by)T |by|
p(x,y)−1 + γ(by − ∇yv

l+1 − δly) = 0, (4.25)

where T = ρ

(

(

∇xv
l+1 + δlx

∇yvl+1 + δly

)2

+ 1

)

p(x,y)−2
2

. Here, sign is defined as follows.

sign(ω) :=















1 if ω > 0,

0 if ω = 0,

−1 if ω < 0.



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Numerical analysis of nonlinear parabolic problems with variable... 203

Note that

sign(bx) = sign(∇xv
l+1 + δlx) (4.26)

and

sign(by) = sign(∇yv
l+1 + δly). (4.27)

So, (4.25) can be expressed as

sign(∇yv
l+1 + δly)T |by|

p(x,y)−1 + γ(by − ∇yv
l+1 − δly) = 0. (4.28)

Unfortunately, we cannot obtain the explicit solution of the equation (4.28). We can use Newton

method to get an approximate solution. If by is solved, bx can be easily determined using (4.24)

and (4.26).

4.5.4 Applications

In the following numerical simulation the iteration process stops when the following condition is

satisfied

‖uk+1n+1 − u
k
n+1‖2

‖uk+1n+1‖2
≤ stop := 10−5, (4.29)

where ‖.‖2 is the Euclidean norm and u
k
n+1 the vector approaching, at iteration k, the space-

discretization of un+1. After stopping the iterations at k = klast, we denote un+1 = u
klast
n+1 and

switch to the next time step.

Note that for implementation, finite difference method is used to approximate the partial deriva-

tives. Moreover, for sake of simplicity, the domain Ω will be a square. The domain Ω will be

subdivided into N2x uniform squares.

For numerical simulation, we will use the following parameters

Nx = 80 and h = 0.02.

Let us recall that h is the time step. The space step is easily computed thanks to Nx and Ω.

Example 4.10. In this example, we take Ω = (0, 1) × (0, 1), T = 1, p(x, y) = 2, and f =

xy(1 − x)(1 − y) + 2t((1 − y)y + (1 − x)x). As initial condition, we set

u0(x, y) = 0.

Let us note that with these data p, u0 and f, the exact solution is

u(x, y, t) = txy(1 − x)(1 − y).



204 S. Ouaro, N. Rabo & U. Traoré CUBO
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0

1

0.01

0.02

1

0.03

u
e
x

0.04

0.8

Exact solution at t=1.000

y

0.05

0.5 0.6

x

0.06

0.4
0.2

0 0

0

1

0.01

0.02

1

0.03u

0.04

0.8

Numerical solution at t=1.000

y

0.05

0.5 0.6

x

0.06

0.4
0.2

0 0

Figure 1: left: u(x, y, t) = txy(1 − x)(1 − y) right: For ρ = h and γ = 0.02

.

0

1

0.01

0.02

1

0.03

u
e
x

0.04

0.8

Exact solution at t=1.000

y

0.05

0.5 0.6

x

0.06

0.4
0.2

0 0

0

1

0.01

0.02

1

0.03u

0.04

0.8

Numerical solution at t=1.000

y

0.5

0.05

0.6

x

0.06

0.4
0.2

0 0

Figure 2: left: u(x, y, t) = txy(1 − x)(1 − y) right: For ρ = h/2 and γ = 0.02

Figure 1 shows the exact solution and the numerical solution for γ = 0.02 and ρ = h. While,

Figure 2 shows the exact solution and the numerical solution for γ = 0.02 and ρ = h/2.

As we can see, we always get a good numerical approximation of the solution even if ρ varies.

Denoting uh the numerical solution and u the exact solution of Example 4.10, with ρ = h and

γ = 0.02, we get the following table of the error approximation.

t 0.1 0.2 0.3 0.4 0.5

‖uh − u‖1 2.5099.10
−5 5.6941.10−5 7.9789.10−5 1.0717.10−4 1.345.10−4

t 0.6 0.7 0.8 0.9 1

‖uh − u‖1 1.6192.10
−4 1.8930.10−4 2.1668.10−4 2.4406.10−4 2.7144.10−4



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Numerical analysis of nonlinear parabolic problems with variable... 205

Example 4.11. In this example, we set Ω = (0, 1) × (0, 1), T = 5, p(x, y) = 2 +
|x|

2
, and f = 1.

As initial condition we set

u0(x, y) = 0.

As parameters we set ρ = h and γ = 0.02.

1

0

1

x

0.02

0.5

Numerical solution at t=1.000

0.8

0.04

y

0.6

0.06u

0.4

0.08

0.2

0.1

0

0.12

0

1

0

1

x

0.02

0.5

Numerical solution at t=5.000

0.8

0.04

y

0.6

0.06u

0.4

0.08

0.2

0.1

0

0.12

0

Figure 3: Numerical solution for p(x, y) = 2 +
|x|

2
, ρ = h and γ = 0.02.

Figure 3 shows the numerical solution at t = 1 and at t = 5. One can see that both figures are the

same.

Example 4.12. In this example, we take Ω = (−1, 1) × (−1, 1), T = 5, p(x, y) =
9

5
−

x2

2
and

f =







1 if x ≥ 0

0 if x < 0.

As the initial condition, we set

u0(x, y) = e
(1−x2)(1−y2) − 1.

We use the same parameters ρ and γ as previously.

Figure 4 shows the numerical solution at t = 1 and t = 5.



206 S. Ouaro, N. Rabo & U. Traoré CUBO
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0

1

0.02

0.04

0.5 1

u

0.06

Numerical solution at t=1.000

0.5

y

0.08

0

x

0.1

0
-0.5

-0.5
-1 -1

0

1

0.02

0.04

0.5 1

u

0.06

Numerical solution at t=5.000

0.5

y

0.08

0

x

0.1

0
-0.5

-0.5
-1 -1

Figure 4: Numerical solution for p(x, y) =
9

5
−

x2

2
, ρ = h and γ = 0.02.

We remark that the exponents p(x) considered in the three examples satisfy the condition 1.2.

Also, note that the choice of γ results from the knowledge of the explicit solution of the Example

4.10. Indeed, knowing the explicit solution, we choose γ so as to obtain a better approximation of

this explicit solution. This leads to the choice of γ = 0.02.

Conclusion and discussion

Inspired by the work of Maitre (see [23]), we have in this paper made a numerical analysis of the

mild solution of parabolic problem involving the p(x)−Laplacian operator. Using the works of

Zhang and Zhou (see [29]), and Ouaro and Ouédraogo (see [24]), we have shown that the mild

solution is also an entropy solution which is equivalent to the renormalized solution. For the

numerical tests, we have used the split Bregman iteration.

In a forthcoming paper, we will make a comparison of the solutions of our numerical scheme (4.1)

to those of the classical backward Euler scheme.



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Numerical analysis of nonlinear parabolic problems with variable... 207

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	Introduction
	Preliminaries
	Notion of mild solution
	Numerical study
	Numerical scheme
	Existence and uniqueness of solution of (4.1)
	Study of the convergence
	Convergence when 0 toward a solution of (1.1)
	Numerical tests
	Implementation
	Subproblem v with fixed b and 
	Subproblem b with fixed v and 
	Applications