Int. J. Anal. Appl. (2023), 21:25

Asymptotic Behavior of Some Parabolic Equations and Application in Image

Restoration

Fatima Zohra Zeghbib1,∗, Abir Bounaama1, Messaoud Maouni1, Halim Zeghdoudi2

1Laboratory of LAMAHIS, Department of Mathematics, Université 20 août 1955-Skikda, El-Hedaiek

B.P. 26, Skikda 21000, Algeria
2Department of mathematics, University of Badji Mokhtar - Annaba, Algeria

∗Corresponding author: fz.zeghbib@univ-skikda.dz

Abstract. In this paper, we consider some nonlinear parabolic problem involving the well known

p−laplacian and some operator having exponential growth with respect to the gradient. We start
by dealing the asymptotic behavior for some evolution equation then we give some numerical results

with an application in image processing.

1. Introduction

Image processing has always been a challenging problem, this field has become "hot". In recent

years, image processing has been a very active field of computer application and research [9].

The most active topics in this field is image restoration because it allow to recovery lost information

from the observed degraded image data.

In [4, 6, 19, 24, 25] the authors have studied the partial differential equation (PDE) and fractional

partial differential equation (FPDE) methods in image processing and proved the fundamental tools

for image diffusion and restoration.

In 1987, the Perona Malik is the first attempts to derive a model from an image within a PDE

framework in [21].

Then, by using Perona Malik the authors were concluded a nonlinear diffusion model (anisotropic

model).

Received: Feb. 2, 2023.

2020 Mathematics Subject Classification. 46E30, 35K55, 35K85, 94A08.
Key words and phrases. Orlicz-Sobolev spaces; parabolic equations in Orlicz spaces; parabolic inequalities; image

processing.

https://doi.org/10.28924/2291-8639-21-2023-25
ISSN: 2291-8639

© 2023 the author(s).

https://doi.org/10.28924/2291-8639-21-2023-25


2 Int. J. Anal. Appl. (2023), 21:25

In [1], the authors have studied a diffusion model, this model is a combination of fast growth with

respect to low gradient and slow growth when the gradient is large which used for restoration in

image processing.

Then, in [14] the researchers showed some class of nonlinear parabolic inequalities in Orlicz spaces.

The authors presented a novel model for image denoising and compared the results with the model

of Perona-Malik and the method of the total variation (see [18]).

And, in [3] the authors proposed a novel parabolic equations for image restoration and enhace-

ment.They proved the existence of solution, established a nonnegative weak solution obtained as

limit of approximation and give some application in image processing.

Also, in [15] we find some optimal control problem for the Perona-Malik equation. The authors

obtained existence results and approximation of an optimal control problem.

In [8], authors studied the minimisations problem numerically with anistropic diffusion and obtained

the results in image restoration.

Let Ω be a regular open bounded subset of RN with N ≥ 2. And let Q be the cylinder
Ω × (0,T ) with some given T > 0.
We consider the following nonlinear parabolic problem:


∂u
∂t

+ A (u) = f in Q

aα,β (x,t,∇u) .η = 0, On ∂Q = ∂Ω × (0,T )
u (x, 0) = u0 in Ω,

(1.1)

where

A(u) = −div
(
aα,β (x,t,∇u)

)
, (1.2)

with aα,β (x,t,∇u) satisfying the following expression

aα,β (x,t,∇u) = exp (α |∇u|)∇uχ{|∇u|≤β} +
∇u
|∇u|

logγ (1 + |∇u|) χ{|∇u|>β} + κ |∇u|
p−2 ∇u. (1.3)

For instance, if we take β = γ = 0 and p = 1 in problem (1.1), so we obtain the curvature-driven

diffusion (see [22]).

If α = 0, β = 1, γ = 0 and κ = 0 is treated in [7].

And, if we make κ = 0 and β ≥ 0 then, the problem (1.1) has been recently studied in ( [1], [20])
and successfully used in image processing.

Actually, nonlinear partial differential equations of type (1.1) can be considered as Perona–Malik

equations see [21].

In this work, we will study the problem (1.1) when β = +∞, κ > 0 and p → +∞. More precisely ,



Int. J. Anal. Appl. (2023), 21:25 3

we will show that the problem

(Pp)




∂u
∂t
−div (aα (x,t,∇u)) = f in Q

aα (x,t,∇u) .η = 0, On ∂Q = ∂Ω × (0,T )
u (x, 0) = u0 in Ω,

(1.4)

with

aα (x,t,∇u) = exp (α |∇u|)∇u + κ |∇u|p−2 ∇u, for p > 2, (1.5)

admits at least one solution up ∈ W 1,xLAα (Q) where Aα(t) = t
2 exp(αt). Next, we study the

asymptotic behavior of the solution up as p → +∞, we show that the limit of up satisfies some
parabolic obstacle problem.

In our work, we use an application illustrating that the problem can be used for denoising filter in

image processing.

For recent works which involving the partial differential equations with nonstandard growth and with

applications in image processing, the reader can refereed to [8], [17], [18] and [13].

This work is organized as follows: in the next section, we present somme lemmas and spaces; in

section 3, we obtain main results; in section 4, we get some numerical results.

2. Preliminaries

In this section, we shall give some corollaries and definitions which will be used throughout this

work.

2.1. N− Functions. Let A : R+ → R+ be an N− function, i.e. A is continuous and convex with
A > 0, for t > 0, A(t)

t
→ 0 as t → 0 and A(t)

t
→ +∞ as t → +∞.

Equivalently, A admits the representation: A(t) =
t∫

0

a(s)ds, where a : R+ →R+ is nondecreasing,

right continuous, with a(0) = 0, a(t) > 0 for t > 0 and a(t) tends to ∞ as t →∞.

The N− function Ā conjugate to A is defined by Ā(t) =
t∫

0

ā(s)ds, where ā : R+ →R+is given by

ā(t) = sup{s : a(s) ≤ t} ( [2]).
The N-function is said to satisfy the ∆2 condition (∃ k > 0 : A(t) ≤ kA(t),∀t ≥ 0), so for some

k > 0 we obtain,

A(2t) ≤ kA(t), ∀t ≥ 0, (2.1)

when (2.1) holds only for some t > 0 then, A is said to satisfy the ∆2 condition near infinity.

We will extend these N-functions into even functions on all R.

2.2. The Orlicz spaces. Let Ω be an open subset of RN. The Orlicz space LA (Ω) is defined as the
set of (equivalence classes of) real-valued measurable functions u on Ω such that:∫

Ω

A

(
u(x)

λ

)
dx < +∞ for some λ > 0. (2.2)



4 Int. J. Anal. Appl. (2023), 21:25

LA (Ω) is Banach space under the norm

‖u‖A,Ω = inf


λ > 0,

∫
Ω

A

(
u(x)

λ

)
dx ≤ 1


 . (2.3)

The closure in LA (Ω) of the set of bounded measurable functions with compact support in Ω is

denoted by EA(Ω). The equality EA(Ω) = LA (Ω) holds if only if A satisfies ∆2 condition, for all t or

for t large according to whether Ω has infinite measure or not.

The dual of EA(Ω) can be identified with LA (Ω) by means of the pairing
∫
Ω

uvdx, and the dual

norm of LA (Ω) is equivalent to ‖.‖A,Ω.
The space LA (Ω) is reflexive if and only if A and Ā satisfy the ∆2 condition, for all t or for t large,

according to whether Ω has infinite measure or not.

2.3. The Orlicz-Sobolev Spaces. Now, we turn to the Orlicz-Sobolev space, W 1LA (Ω) (respectively

W 1EA(Ω)) is the space of all functions u and its distributional derivatives up to order 1 lie in LA
(respectively EA(Ω)). It is a Banach space under the norm

‖u‖1,A =
∑
|k|≤1

∥∥Dku∥∥
A
. (2.4)

Thus, W 1LA (Ω) and W 1EA(Ω) can be identified with sub-spaces of product of N + 1 copies of

LA. Denoting this product by ΠLA, we will use the weak topologies σ(ΠLA, ΠEĀ) and σ(ΠLA, ΠLĀ).

We say that un converges to u for the modular convergence in W 1LA (Ω) if for some λ > 0∫
Ω

A

(
Dkun −Dku

λ

)
dx → 0 for all |k| ≤ 1. (2.5)

This implies convergence for σ(ΠLA, ΠLĀ).

If A satisfies ∆2 condition on R+, then modular convergence coincides with norm convergence.

2.4. Duality in Orlicz-Sobolev space. Let W−1LA (Ω) denote the space of distributions on Ω which

can be written as sums of derivatives of order < 1 of functions in LĀ. It is a Banach space under the

usual quotient norm.

If the open set Ω has the segment property then the space D(Ω̄) is dense in W 1LA (Ω) for the

modular convergence and thus for the topology σ(ΠLA, ΠLĀ) ( [12]). Consequently, the action of a

distribution in W−1LĀ (Ω) on an element of W
1LA (Ω) is well defined.

2.5. Inhomogeneous Orlicz-Sobolev spaces. Let Ω be an abounded open subset of RN, T > 0, and
set Q = Ω × (0,T ). Let A be an N−function. For each k ∈ NN, denote by Dkx the distributional
derivatives on Q of order k with respect to the variable x ∈ RN. The inhomogeneous Orlicz-Sobolev
spaces of order 1 are defined as follows

W 1,xLA (Q) =
{
u ∈ LA (Q) : Dkxu ∈ LA (Q) , ∀|k| ≤ 1

}
, (2.6)



Int. J. Anal. Appl. (2023), 21:25 5

and

W 1,xEA(Q) =
{
u ∈ EA (Q) : Dkx ∈ EA (Q) , ∀|k| ≤ 1

}
. (2.7)

The latest space is a subset of the first one. They are Banach spaces under the norm

‖u‖ =
∑
|k|=1

∥∥Dkxu∥∥A,Q . (2.8)
We can easily show that they form a complementary system when Ω satisfies the segment property.

These spaces are considered as subspaces of the product spaces ΠLA (Q) which has N + 1 copies.

We shall also consider the weak topologies σ(ΠLA, ΠEĀ) and σ(ΠLA, ΠLĀ). If u ∈ W
1,xLA (Q)

then the function t → u(t) = u(.,t) is defined on (0,T ) with values in W 1LA(Ω). If, further,
u ∈ W 1,xEA(Q) then u(t) is a W 1EA(Ω) valued and is strongly measurable. Furthermore, the
following continuous imbedding holds: W 1,xEA(Q) ⊂ L1(0,T ; W 1EA(Ω)). The space W 1,xLA (Q) is
not in general separable, if u ∈ W 1,xLA (Q), we cannot conclude that the function u(t) is measurable
from (0,T ) into W 1LA(Ω). However, the scalar function t →

∥∥Dkxu(t)∥∥A,Ω , is in L1(0,T ) for all
|k| ≤ 1.

2.6. Duality in inhomogeneous Orlicz-Sobolev spaces. We denote by F = W−1,xLĀ (Q) the space

F =


f = ∑

|k|=1

Dkx fk : fk ∈ LĀ (Q)


 . (2.9)

This space will be equipped with the usual quotient norm:

‖f‖ = inf
∑
|k|=1

‖fk‖Ā,Q , (2.10)

where the inf is taken on all possible decomposition f =
∑
|k|=1

Dkx fk : fk ∈ LĀ (Q). The space

F0 = W
−1,xLĀ (Q) is then given by

F0 =


f = ∑

|k|=1

Dkx fk : fk ∈ EĀ (Q)


 . (2.11)

The following corollary will be useful in the proof of our existence theorem.

Corollary 2.1. ( [10]). Let A be an N−function. Let (un) be a sequence of W 1,xLA (Q) such that

un → u weakly in W 1,xLA (Q) for σ(ΠLA, ΠEĀ) (2.12)

and ∂un
∂t

is bounded in W−1,xLĀ (Q) +M(Q), where M(Q) is the space of measures defined on Q.

Then un → u strongly in L1loc(Q).



6 Int. J. Anal. Appl. (2023), 21:25

3. The Main Results of the Existence

Theorem 3.1. Let f ∈ L∞(Q), and u0 that |∇u0| ≤ 1. Then the problem

(Pp)




up ∈ W 1,xLAα(Q)〈
∂up
∂t
,v
〉

+
∫
Q

∇up exp(α |∇up|)∇vdxdt +
∫
Q

|∇up|p−2 ∇up∇vdxdt =
∫
Q

f vdxdt,

for v ∈ W 1,xLAα(Q) ∩L
2(Q) such that ∂v

∂t
∈ W−1,xLĀα (Q) + L

2(Q),

(3.1)

admits at least one solution up ∈ W 1,xLAα(Q) such that up → u for the modular convergence where
u is solution of the following parabolic inequality:

(P )




|∇u| ≤ 1〈〈
∂v
∂t
,u −v

〉〉
+
∫
Q

a(x,∇u) (∇u −∇v) dxdt ≤
∫
Q

〈f ,u −v〉dxdt + 1
2

∫
Ω

(u0 −v(x, 0))2 dx

for v ∈ W 1,xLAα(Q) ∩L
2(Q) such that ∂v

∂t
∈ W−1,xLĀα (Q) + L

2(Q) and |∇v| ≤ 1.

Remark 3.1. Since
{
v ∈ W 1,xLAα(Q) ∩L

2(Q) : ∂v
∂t
∈ W−1,xLĀα (Q) + L

2(Q)
}

⊂
C
(

([0,T ]) ,L2 (Ω)
)
(see, [11]) the least term of problem (P ) is well defined.

Proof. Step 1

Let us consider the following approximate problem:

(Pp,n)




∂unp
∂t
−div(∇unp exp(α

∣∣∇unp∣∣)) −div(∣∣∇unp∣∣p−2 ∇unp) + 1n (unp −M) exp (α ∣∣unp −M∣∣) = f in Q
unp(x, 0) = u0 in Ω

exp(α
∣∣∇unp∣∣)∂unp∂n + ∣∣∇unp∣∣p−2 ∂unp∂n = 0 on ∂Ω × (0,T ),

(3.2)

where M = max(‖f‖∞ ,‖u0‖∞). As it is done in [11], one can is seen easily that the problem (Pp,n)
admits at least one solution unp ∈ W 1,xLAα(Q) furthermore∥∥unp∥∥∞ ≤ max{‖f‖∞ ,‖u0‖∞} . (3.3)

By choosing unp −M as test function in (Pp,n), we obtain
∫
Q

∣∣∇unp∣∣2 exp(α ∣∣∇unp∣∣)dxdt ≤ C, and
via (3.3) it follows that ∥∥unp∥∥1,Aα ≤ M′,
and thanks to (Pp,n), we deduce that

∂unp
∂t

is bounded in W−1,xLĀα(Q) + L
∞ (Q) . Thanks to corollary

(2.1) we have unp → up in L1 (Q) as n → +∞ almost everywhere convergence in Q.
Arguing as in [10] and [5], we pass to the limit in (Pp,n) to obtain

(Pp)




∂up
∂t
−div(∇up exp(α |∇up|)) −div(|∇u|p−2 ∇u) = f in Q

u(x, 0) = u0 in Ω

exp(α |∇up|)
∂up
∂n

+ |∇u|p−2 ∂up
∂n

= 0 on ∂Ω × (0,T ),
(3.4)

with up ∈ W 1,xLAα(Q) ∩L
∞ (Q) and ‖up‖∞ ≤ M

′′.

Step 2: A priori estimates



Int. J. Anal. Appl. (2023), 21:25 7

Choosing v = up as test function in (Pp) we obtain:〈〈
∂up
∂t

,up

〉〉
+

∫
Q

|∇up|2 exp(α |∇up|)dxdt +
∫
Q

|∇up|p dxdt =
∫
Q

f updxdt,

which gives by using Young’s inequality

1

2

∫
Ω

u2p(x,t)dx−
1

2

∫
Ω

u20dx+

∫
Q

|∇up|p dxdt+
∫
Q

|∇up|2 exp(α |∇up|)dxdt ≤
1

2

∫
Q

f 2dxdt+
1

2

∫
Q

u2pdxdt

1

2

∫
Ω

u2p(x,t)dx +
1

2

∫
Ω

u20dx +

∫
Q

|∇up|p dxdt +
∫
Q

|∇up|2 exp(α |∇up|)dxdt ≤ c +
1

2

t∫
0

∫
Ω

u2pdxdt,

by Gronwall’s lemma (see [23]), we get

∫
Ω

u2p(x,t)dx +

∫
Q

|∇up|2 exp(α |∇up|)dxdt +
∫
Q

|∇up|p dxdt ≤ C.

Consequently, since up is bounded in W 1,xLAα(Q)∩L
2(Q) so there exist some u ∈ W 1,xLAα(Q)∩

L2(Q) such that ( for a subsequence still denoted by up)

up → u weakly in W 1,xLAα(Q) ∩L
2(Q).

Step 3:

To obtain |∇u| ≤ 1, we will use the estimate∫
Q

|∇up|p dxdt ≤ C. (3.5)

Let q < p, we have∫
Q

|∇up|q dxdt =
∫

|∇up|≤1

|∇up|q dxdt +
∫

|∇up|>1

|∇up|q dxdt

≤ meas(Q) +
∫

|∇up|>1

|∇up|q dxdt ≤ meas(Q) + C,

which gives ∫
Q

|∇up|q dxdt ≤ M,

by letting p →∞ for q fixed, we obtain∫
Q

|∇u|q dxdt ≤ M.

Now, let k > 1, we get∫
|∇u|≥k

|∇u|q dxdt ≤ M =⇒ meas{|∇u| ≥ k}≤
M

kq
=⇒ meas{|∇u| ≥ 1} = 0,



8 Int. J. Anal. Appl. (2023), 21:25

which gives

|∇u| ≤ 1.

Step 4: Modular convergence of up → u in W 1,xLAα (Q) :
Let wµ = uµ + e−µtu0, where uµ is the mollifier function defined in [16] with respect to time of

u and the function wµ have the following properties:

∂wµ
∂t

= µ (u −wµ) ; wµ (0) = u0, (3.6)

with

uµ = µ

t∫
−∞

u(x,s)χ(0,T ) exp(µ (s − t))ds, (3.7)

∇wµ = µ
t∫

−∞

∇u(x,s)χ(0,T ) exp(µ (s − t))ds + exp (−µt)∇u0. (3.8)

By using |∇u| < 1 and |∇u0| < 1, we get:

|∇wµ| ≤ µ
t∫

0

exp(µ (s − t))ds + exp (−µt) = [exp(µ (s − t))]t0 + exp (−µt) = 1.

This implies that

|∇wµ| ≤ 1.

Now, we proof that up → u in W 1,xLAα (Q), for the modular convergence as p → +∞.
For this, we will denote by ε (p,µ,θ) function with all quantities such that

lim
µ→+∞

lim
θ→1

lim
p→+∞

ε (p,µ,θ) = 0, (3.9)

and we will respect the order of the parameters p,θ,µ. Similarly, we will write ε (p) , or ε (p,µ) that

the limits are made only on the specified parameters. Firstly take vp = up−θwµ for 0 < θ < 1 as test
function in (Pp), which belong to W 1,xLAα (Q), we get〈

∂up
∂t

,vp

〉
+

∫
Q

∇up exp (α |∇up|)∇(up −θwµ) dxdt (3.10)

+

∫
Q

|∇up|p−2 ∇up∇(up −θwµ) dxdt =
∫
Q

f (up −θwµ) dxdt.

On the other hand, by using the monotonicity of the p−Laplacien, we deduce that;∫
Q

|∇up|p−2 ∇up∇(up −θwµ) dxdt ≥ θp−1
∫
Q

|∇wµ|p−2 ∇wµ (∇up −θ∇wµ) dxdt, (3.11)

by using the Holder’s inequality

|I1 (p,µ,θ)| =

∣∣∣∣∣∣
∫
Q

|∇wµ|
p−2 ∇wµ (∇up −θ∇wµ) dxdt

∣∣∣∣∣∣ ≤

∫
Q

|∇wµ|
p
dxdt




1
p′

∫
Q

|∇up −θ∇wµ|
p
dxdt




1
p

,



Int. J. Anal. Appl. (2023), 21:25 9

which implies
∫
Q

|∇up −θ∇wµ|p dxdt




1
p

≤


2p


∫
Q

(|∇up|p + θp |wµ|p)dxdt






1
p

≤ 2M
1
p .

And finally, we obtain

I1 (p,µ,θ) ≤ ε (p) .

On the other hand〈〈
∂up
∂t

,zp

〉〉
=

〈〈
∂up
∂t

,up −θwµ
〉〉

=

〈〈
∂up
∂t
−θ

∂wµ
∂t

,up −θwµ
〉〉

+ θ

〈〈
∂wµ
∂t

,up −θwµ
〉〉

= J1 + θJ2.

With

J1 =

〈〈
∂up
∂t
−θ

∂wµ
∂t

,up −θwµ
〉〉

=

∫
Ω

(up −θwµ)2 dx −
∫
Ω

(u0 −θwµ)2 dx,

we deduce that

J1 ≥−
∫
Ω

(u0 −θwµ)2 dx ≥−(1 −θ)2
∫
Ω

u20dx + ε (µ) = ε (µ,θ) .

For what concerns J2, we deduce that

J2 =

〈〈
∂wµ
∂t

,up −θwµ
〉〉

= µ

∫
Q

(u −wµ) (up −θwµ) dxdt

lim
θ→1

lim
p→+∞

J2 = lim
θ→1

lim
p→+∞

〈
∂wµ
∂t

,up −θwµ
〉
≥ 0.

Finally, we get

lim
µ→+∞

lim
θ→1

lim
p→+∞

∫
Q

a (x,∇up) (∇up −θ∇wµ) dxdt ≤ 0. (3.12)

Since ∫
Q

[a (x,∇up) −a (x,∇u)] [∇up −∇u] dxdt −
∫
Q

a (x,∇up) (∇up −θ∇wµ) dxdt

= −
∫
Q

a (x,∇up)∇udxdt −
∫
Q

a (x,∇u) [∇up −∇u] dxdt + θ
∫
Q

a (x,∇up)∇wµdxdt.

Since a (x,∇up) is bounded in
(
LĀα (Q)

)N
, we have a (x,∇up) → h weakly for σ(ΠLĀα, ΠEAα)

consequently∫
Q

[a (x,∇up) −a (x,∇u)] [∇up −∇u] dxdt −
∫
Q

a (x,∇up) (∇up −θ∇wµ) dxdt

= −
∫
Q

h∇udxdt + θ
∫
Q

h∇wµdxdt + ε (p)



10 Int. J. Anal. Appl. (2023), 21:25

= −
∫
Q

h∇udxdt +
∫
Q

h∇wµdxdt + ε (p,θ) = ε (p,θ,µ) .

Which gives∫
Q

[a (x,∇up) −a (x,∇u)] [∇up −∇u] dxdt −
∫
Q

a (x,∇up) (∇up −θ∇wµ) dxdt = ε (p,θ,µ) ,

by using (3.12), we obtain∫
Q

[a (x,∇up) −a (x,∇u)] [∇up −∇u] dxdt → 0 as p → +∞.

By strict monotonicity of a (., .), we obtain that ∇up →∇u a.e in Q.
Finally a (x,∇up) → h = a(x,∇u), weakly for σ(ΠLĀα, ΠEAα), consequently∫

Q

[a (x,∇up) −a (x,∇u)] [∇up −∇u] dxdt

=

∫
Q

a (x,∇up)∇updxdt −
∫
Q

a (x,∇up)∇udxdt −
∫
Q

a (x,∇u) [∇up −∇u] dxdt,

because ∇up →∇u weakly in
(
LĀα (Q)

)N
, we get

lim
p→+∞

∫
Q

[a (x,∇up) −a (x,∇u)] [∇up −∇u] dxdt

= lim
p→+∞

∫
Q

a (x,∇up)∇updxdt −
∫
Q

a (x,∇u)∇udxdt = 0.

Since a (x,∇up)∇up = Aα (|∇up|) we get

lim
p→+∞

∫
Q

Aα (|∇up|) dxdt =
∫
Q

Aα (|∇u|) dxdt.

Thanks to fact that

Aα

(
|∇up −∇u|

2

)
≤

1

2
(Aα (|∇up|) + Aα (|∇u|)) .

By using Vitali’s theorem, we obtain∫
Q

Aα

(
|∇up −∇u|

2

)
dxdt → 0 as p → +∞.

Which shows that ∇up converges to ∇u for the modular convergence in LAα(Q).
Step 5: The passage to the limit

Let us consider v ∈ W 1,xLAα(Q)∩L
2(Q) = W 1,xLAα(Q) such that |∇v| < 1,

∂v
∂t
∈ W−1,xLĀα(Q)+

L2(Q) and 0 < θ < 1. Using up −θv as test function in (Pn), the fact that〈〈
∂up
∂t

,up −θv
〉〉

+

∫
Q

∇up exp(α |∇up|)∇(up −θv) dxdt (3.13)



Int. J. Anal. Appl. (2023), 21:25 11

+

∫
Q

|∇up|p−2 ∇up∇(up −θv) dxdt ≤
∫
Q

f (up −θv) dxdt,

we have 〈〈
∂up
∂t
−θ

∂v

∂t
,up −θv

〉〉
+ θ

〈〈
∂v

∂t
,up −θv

〉〉
+

∫
Q

∇up exp(α |∇up|)∇(up −θv) dxdt +
∫
Q

|∇up|p−2 ∇up∇(up −θv) dxdt ≤
∫
Q

f (up −θv) dxdt,

since ∫
Q

|∇up|p−2 ∇up∇(up −θv) dxdt ≥ θp−1
∫
Q

|∇v|p−2 ∇v∇(up −θv) dxdt,

which gives

θ

〈〈
∂v

∂t
,up −θv

〉〉
+

∫
Q

a(x,∇up) (∇up −θ∇v) dxdt + θp−1
∫
Q

|∇v|p−2 ∇v∇(up −θv) dxdt

≤
∫
Q

〈f ,up −θv〉dxdt +
∫
Ω

(u0 −θv)2 dx.

Since a (x,∇u) belongs to
(
LĀα (Q)

)N
, and using Fatou’s lemma in the first term of the last side

gives

lim inf
p→+∞

∫
Q

a(x,∇up) (∇up −θ∇v) dxdt ≥
∫
Q

a(x,∇u) (∇u −θ∇v) dxdt,

then, we can easily pass to the limit as θ → 1 and p tend to infinity, we obtain〈〈
∂v

∂t
,u −v

〉〉
+

∫
Q

a(x,∇u) (∇u −∇v) dxdt ≤
∫
Q

〈f ,u −v〉dxdt +
1

2

∫
Ω

(u0 −v(x, 0))2 dx.

Which completes the proof. �

4. Numerical results

We consider the following model problem:

(P 1)




∂up
∂t
−div(∇up exp(α |∇up|)) −div(|∇u|p−2 ∇u) = f in Q

u(x, 0) = u0 in Ω

exp(α |∇up|)
∂up
∂n

+ |∇u|p−2 ∂up
∂n

= 0 in ∂Ω × (0,T ),
(4.1)

where u0 represents the input image. We apply finite differences method to this problem. We denote

respectively by h and ∆t the spatial and time steps sizes.

In what follows, we take h = 1 and we define for every field p = (p1,p2) ∈R2, the discrete divergence
approximation:

divi,j(p) =




p1(i, j) −p1(i − 1, j) if 1 < i < n
p1(i, j) if i = 1

p1(i − 1, j) if i = n
+




p2(i, j) −p2(i, j − 1) if 1 < j < n
p2(i, j) if j = 1

−p2(i, j − 1) if j = m,
(4.2)



12 Int. J. Anal. Appl. (2023), 21:25

where n and m is an integer greater than 2.

One can write the following scheme:

uk+1 (i, j) = uk (i, j) + ∆t
[

(div (dα(x,∇u) + div(q (x,∇u)))k (i, j)
]
, 1 ≤ k ≤ N (4.3)

where dα(x,∇u) = ∇up exp(α |∇up|), q (x,∇u) = |∇u|p−2 ∇u, u(tk,xi,yj) = uk(i, j), xi =
ih, yj = jh, tk = k∆t, and ∆t =

T
N
.

In our numerical tests we take ∆t = T
N

= 0.1, and we compute the PNSR (Peak Signal to Noise

Ratio ) quotient of every image.

In Figs. 1-3, we give some examples by taking α = 0.25, σ is the standard deviation of the distribution

which performs an edge-preserving average filter on the image and with different values of p.

We give in Fig. 4, tests with different values of α, with p = 40.

Noisy image with salt&pepper = 0.008 p = 40,PSNR = 24.2487

Noisy image with salt&pepper = 0.08 p = 40,PSNR = 16.0574

fig 1.



Int. J. Anal. Appl. (2023), 21:25 13

Noisy image with σ = 1 p = 900E900,PSNR = 15.2328

Noisy image with σ = 1 p = 900E900,PSNR = 16.4051

fig 2.



14 Int. J. Anal. Appl. (2023), 21:25

Noisy image with σ = 0.9 p = 3,PSNR = 19.4112

p=300, PSNR=25.1073 p = 900E900,PSNR = 25.1176

fig 3.



Int. J. Anal. Appl. (2023), 21:25 15

Noisy image with σ = 0.9 α = 3,PSNR = 13.9802

α = 0.25, PSNR=20.1509 α = 10E − 15,PSNR = 18.5800

fig 4.

In Numerical tests, we show the better value of a α which gives a good restored image is equal to

0.25, so we should not take α close to 0 and no more than 0.25.

5. Conclusion

In this article, we presente a parabolic model for image denoising and restoration, with their theo-

retical results and numerical results.

This model preserve the contours of image more than other models.

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

cation of this paper.



16 Int. J. Anal. Appl. (2023), 21:25

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	1. Introduction
	2. Preliminaries
	2.1. N- Functions
	2.2. The Orlicz spaces
	2.3. The Orlicz-Sobolev Spaces
	2.4. Duality in Orlicz-Sobolev space
	2.5. Inhomogeneous Orlicz-Sobolev spaces
	2.6. Duality in inhomogeneous Orlicz-Sobolev spaces

	3. The Main Results of the Existence
	4. Numerical results
	5. Conclusion
	References