EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 2, 2024, 1321-1334
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Asymptotic Behavior of Global Solutions of an
Anomalous Gray-Scott Model

Maroua Mebarki

Department of Mathematics and Computer Science, Faculty of Technology and Sciences/
Amine Elokkal El Hadj Moussa Ag Akhamouk, P.O.Box 10034, Tamanrasset 11000, Algeria

Abstract. The object of this paper is to prove that existence global and asymptotic behavior of
solutions for anomalous coupled reaction diffusion system (Gray-Scott model) with homogeneous
Neumann boundary conditions. The existence and uniqueness of the local solution are given by
the Banach fixed point theorem. Further, the asymptotic behavior is investigated by technique
semi group estimates and the Sobolev embedding theorem .

2020 Mathematics Subject Classifications: 35K51, 35A01, 35B40

Key Words and Phrases: Parabolic system, fractional Laplacian, local and global existence,
asymptotic behavior

1. Introduction

The Gray-Scott system is a reaction-diffusion system. This means that it models a
process that consists of a reaction and diffusion. In the case of the Gray-Scott model that
reaction is a chemical reaction between two substances W and Z , both of which diffuse
over time. During the reaction W gets used up, while Z is produced.

The system is characterised by two parameters: f is the rate at which is replenished,
and k controls the rate at which Z is removed from the system. Varying these parameters
leads to a wide range of interesting patterns, some of which look quite familiar.

The Gray-Scott system models the chemical reaction W + 2Z → 3Z. This reaction
consumesW and produces Z. Consequently, the amount of both substances needs to be
controlled to maintain the reaction. This is done by adding W at the ”feed rate” f and
removing Z at the ”kill rate” k The removal of Z can also be described by another chemical
reaction:

Z → P. For this reaction P is an inert product, meaning it doesn’t react. In this case
the Parameter k controls the rate of the second reaction. Both substances diffuse over
time at the diffusion rates d1 and d2. The Gray-Scott system is defined by two equations
that describe the behavior of two reacting substances:

DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5017

Email address: maroua.mebarki@univ-tam.dz (M. Mebarki)

https://www.ejpam.com 1321 © 2024 EJPAM All rights reserved.



M. Mebarki / Eur. J. Pure Appl. Math, 17 (2) (2024), 1321-1334 1322


∂w
∂t = −d1(−∆)w − w2z + f(1− w)

∂z
∂t = −d2(−∆)z + w2z − (f + k)z

The variables in these equations w and z are the concentrations of the two reacting
substances W and Z. On the left-hand side of each equation is the time derivative of one
of these concentrations, describing the rate at which it changes. The right-hand sides of
the equations both contain three separate terms. The first describes the reaction between
the two substances. Since one W and two Z react, the corresponding term includes w to
the power of one and z to the power of two: w2z as W gets consumed by the reaction,
the term has a negative sign in the first equation. In the second equation it has a positive
sign, as is produced in the reaction.

The second term of the first equation describes the rate at which W is replenished
externally. This is necessary, as W would otherwise simply be used up. The feed rate is
given by the parameter f . f is multiplied by 1 − w to ensure that w is replenished at
a rate dependent on the current concentration, which never exceeds Z does not need to
replenished, since it is produced in the reaction. Booth, it needs to be removed in order
maintain the reaction. The rate of removal, the kill rate, is controlled by the parameter
k. To remove Z faster than W is added,k is added to f and multiplied by z, since the
removal of Z is also supposed to be dependent on its concentration. The last term in both
equations describe the diffusion of and, respectively.

In this current manuscript, we are interested in the fractional Gray Scott model which
arises in the modelling of autocatalytic reactions. We study the global existence and
asymptotic behavior of solutions to the system:

∂w
∂t = −d1(−∆)δw − w2z + f(1− w) in Ω× R+,

∂z
∂t = −d2(−∆)ϵz + w2z − (f + k)z in Ω× R+,

(1)

subjected with the boundary and initial conditions
∂w
∂η = ∂z

∂η = 0 in ∂Ω× R+,

w (., 0) = w0 (.) , z (., 0) = z0 (.) in Ω,

here Ω is an open bounded domain of class C1 in Rn , w (t, x) and z(t, x), t ≥ 0, x ∈ Ω

are real valued functions. w0 (.) and z0 (.) are non negatives, the constants d1, d2, k are
positive and 0 < ϵ < 1, 0 < δ < 1 and f ≥ 0.

The system, obtained by replacing the fractional Laplacian by the classical one, is
known as the chemical diffusion Gray Scott. This model was proposed by Gray and Scott
in 1983. Later on , the Gray Scott model has attracted significant attention. It has been
subject of a number of papers , for example Kirane [1], Hollis [4], Roth[9], Kouachi [10],
Lin[6] ,..., etc.



M. Mebarki / Eur. J. Pure Appl. Math, 17 (2) (2024), 1321-1334 1323

Our paper is organized as follows. In section 2, we present some preliminaries and
definitions which used in the following sections. In section 3, the definition of mild solution
of the system (1) and the theorem of local existence are obtained. The main results of
global existence and large time behavior for the solution are presented in section 4.

2. Notations and preliminary

In this section, we introduce some notations, definitions and lemmas which will be used
in the sequel. Where Ω is an open bounded domain of class C1 in Rn, we denote(−∆N )δ

the fractional power of the Laplacian in Ω with homogenous Neumann boundary condition
.

Let αm{m = 0, 1, ...,+∞} be the eigenvalues of the Laplacian operator in L2 (Ω) with
homogenous Neumann boundary condition and let Ψm be the corresponding eigenfunction

i.e.


(−∆N )δΨm = αδ

mΨm in Ω,

∂Ψm
∂t = 0 in ∂Ω,

and

D((−∆N )δ) = {w ∈ L2 (Ω) ;
∂w

∂η
=, 0

∥∥∥(−∆N )δw
∥∥∥
L2(Ω)

≺ +∞}

∥∥∥(−∆N )δw
∥∥∥
L2(Ω)

=
+∞∑
m=1

∣∣∣αδ
m ⟨w,Ψm⟩

∣∣∣2
so for w ∈ D((−∆N )δ) we get

(−∆N )δw =
+∞∑
m=1

αδ
m ⟨w,Ψm⟩Ψm

We obtain the following integration by parts formula∫
Ω

w(x)(−∆N )δz(x)dx =

∫
Ω

z(x)(−∆N )δw(x)dx, for w, z ∈ D((−∆N )δ) (2)

We will employ the following important inequalities of Strook and Varopoulos see( [8],
Theorem1 ) ∫

Ω

w(x)(−∆N )δw(x)dx ≥ 0, for w ∈ D((−∆N )δ) (3)

∫
Ω

wp−1(x)(−∆N )δw(x)dx ≥ 4(p− 1)

p2

∫
Ω

∣∣∣(−∆N )
δ
2w(x)

p
2

∣∣∣2 dx ≥ 0, p ≻ 1 (4)

for all w ∈ Lp (Ω) such that (−∆N )
δ
2w ∈ Lp (Ω)



M. Mebarki / Eur. J. Pure Appl. Math, 17 (2) (2024), 1321-1334 1324

Definition 1. For p ∈ (1,+∞), we denote Sp (resp. Tp) the realisation of (−∆)δ (resp. (−∆)ϵ)
with a homogeneous Neumann boundary condition in Lp (Ω) .

It is well known that−Sp (resp. − Tp) is a sectorial operator (see [4]); hence−Sp (resp. − Tp)
generates an analytic semigroup {e−tSp}t≥0

(
resp. {e−tTp}t≥0

)
Lemma 1. For λ ∈ [0, 1] and µ ∈ R, there exists a constant M (λ, µ) such that, for all
t ≻ 0, ∫ t

0
c(s)−λeµsds ≤


M (λ, µ) eµt, if µ ≻ 0,

M (λ, µ) (t+ 1), if µ = 0,
M (λ, µ) , if µ ≺ 0,

here
c(t) = min{t, 1}.

Proof. see [5]

Lemma 2. Let p, q, r ∈ [0, 1] , r ≤ p ≤ q and λ ∈ [0, 1] be such that 1
p = λ

r + 1−λ
q , we gain∥∥e−tSpw

∥∥
p
≤ e−αδ

1λtc(t)
−N
2δ

( 1
r
− 1

p
) ∥w∥r

Proof. see [3] By the interpolation inequality, we obtain∥∥e−tSpw
∥∥
p
≤

∥∥e−tSpw
∥∥λ
r

∥∥e−tSpw
∥∥1−λ

q
(5)

Practising the following inequalities∥∥e−tSpw
∥∥
r
≤ e−αδ

1t ∥w∥r (6)

and ∥∥e−tSpw
∥∥
q
≤ t

−N
2δ

( 1r− 1
q )

∥w∥r , (7)

we get ∥∥e−tSpw
∥∥
p
≤ e−αδ

1λtt
−N
2δ

(1−λ)( 1r− 1
q )

∥w∥r , (8)

here (1− λ)(1r −
1
q ) = (1r −

1
p).

Consequently, we gain∥∥e−tSpw
∥∥
p

≤ e−αδ
1λtt

−N
2δ

( 1r− 1
p )

∥w∥r (9)

≤ e−αδ
1λtc(t)

−N
2δ

( 1
r
− 1

p
) ∥w∥ r

where

c(t) = min{t, 1}.
Remark : An main result in [8] comes in our case ∀ξ ≻ 0,∃M(ξ) ∈ R+ such that∥∥e−tTpz

∥∥
∞ ≤ M(ξ)t

−N
(N+ξ)ϵ

)

∥z∥N
2
+ξ′ , for all z ∈ L∞ (Ω) , t ≻ 0. (10)

The relation (10) with ϵ = 1 injected with a successive iterations method have been
used in ([8] proposition 3.3 ) to prove that the solutions are bounded in C

(
Ω
)
.



M. Mebarki / Eur. J. Pure Appl. Math, 17 (2) (2024), 1321-1334 1325

3. Local existence

In this section, we investigate the local existence of mild solutions to the problem (1)-(2)

Lemma 3 (definition ). (Mild solution)
Let w0, z0 ∈ L∞ (Ω) and T ≻ 0. We say that (w, z) ∈C ([0, T ] ;L∞ (Ω)× L∞ (Ω)) is a

mild solution of (1)-(2) if w, z satisfy the followings integral equations for t ∈ [0, T ] :{
w(t) = e−d1tSpw0 +

∫ t
0 e

−d1(t−s)Sp(−w2z + f(1− w))ds,

z(t) = e−d2tTpz0 +
∫ t
0 e

−d2(t−s)Tp(w2z − (f + k)z)ds.
(11)

Theorem 1. local existence

Let w0, z0 ∈ C
(
Ω
)
, then there exist a maximal time Tmax ≻ 0 and a unique mild

solution (w, z) ∈C
(
[0, Tmax);C

(
Ω
)
× C

(
Ω
))

to the system (1)-(2) , with the alternative:
-either Tmax = +∞;
-or Tmax ≺ +∞ and limt→Tmax(∥w(t)∥∞ + ∥z(t)∥∞) = +∞.
Proof. ∀T ≻ 0, we define the Banach space:

BT := {(w, z) ∈ C
(
[0, T ];C

(
Ω
)
× C

(
Ω
))

; ∥(w, z)∥ ≤ 2 ∥(w0, z0)∥ = L},

where ∥.∥∞ := ∥.∥L∞(Ω) and∥.∥ is the norm of BT defined by:

∥(w, z)∥ := ∥w∥L∞([0,T ];L∞(Ω)) + ∥z∥L∞([0,T ];L∞(Ω)) .

Next, ∀ (w, z) ∈ BT ,, we define Φ (w, z) ;= (Φ1 (w, z) ,Φ2 (w, z)) where for t ∈ [0, T ]

Φ1 (w, z) = e−d1tSpw0 +

∫ t

0
e−d1(t−s)Sp(−w2z + f(1− w))ds

and

Φ2 (w, z) = e−d2tTpz0 +

∫ t

0
e−d2(t−s)Tp(w2z − (f + k)z)ds.

We will show the local existence by the Banach fixed point theorem.

• Φ : BT → BT :Let (w, z) ∈ BT . Practising the estimate (8) (with r = p = +∞),we
attain

∥Φ1 (w, z)∥∞ ≤ ∥w0∥∞ +

∫ t

0

∥∥−w2z(s)
∥∥
∞ ds+ f

∫ t

0
∥(1− w)(s)∥∞ ds

≤ ∥w0∥∞ + TL3 + fT + fTL.

Similarly, we obtain

∥Φ2 (w, z)∥∞ ≤ ∥z0∥∞ + TL3 + (f + k)TL.



M. Mebarki / Eur. J. Pure Appl. Math, 17 (2) (2024), 1321-1334 1326

Therefore we get,

∥Φ (w, z)∥∞ ≤ (∥w0∥∞ + ∥z0∥∞) + 2TL3 + (2f + k)TL+ fT

≤ 2(∥w0∥∞ + ∥z0∥∞)

by choosing T such that T ≤ 1
2(6L2+(2f+k))

.

Hence Φ (w, z) ∈ BT for T ≤ 1
2(6L2+(2f+k))

.

• Φ (w, z) is contraction map : for (w, z) , (w′′, z”) ∈ BT , we obtain

∥Φ1 (w, z)− Φ1 (w”, z”)∥∞ ≤
∫ t

0

∥∥−w2z(s) + w”2z”(s)
∥∥
∞ ds+ f

∫ t

0
∥(w − w”)(s)∥∞ ds

≤
∫ t

0

∥∥w2
∥∥ ∥z − z”∥+ ∥z”∥ (∥w∥+ ∥w”∥)(∥w − w”∥ ds+

f

∫ t

0
∥w − w”∥ ds

≤ 3TL2 + fT ∥(w, z)− (w”, z”)∥

by the same way,

∥Φ1 (w, z)− Φ1 (w”, z”)∥∞ ≤ 3TL2 + (f + k)T ∥(w, z)− (w”, z”)∥

So

∥Φ (w, z)− Φ (w”, z”)∥ ≤ 6TL2 + 2fT + kT

≤ 1/2 |∥(w, z)− (w”, z”)∥| .

for T ≤ 1
2(6L2+(2f+k))

.

Consequently, in view of the Banach fixed point theorem ç, Φ admits a fixed point
on B. Thus the system (1)-(2) has a mild solution.

The solution can be extended on a maximal interval [0, Tmax) where Tmax := sup {T ≻ 0; (w, z)} .
is a solution to (1)-(2)

4. Global existence and asymptotic behavior

In this section, we state and prove the main result using the ideas of [2] or [1].

Theorem 2. Let (w0, z0) ∈ C
(
Ω
)
×C

(
Ω
)
be such that w0 ≥ 0, z0 ≥ 0.Then there exists

a unique global solution (w, z) of (1)-(2) which satisfy :



M. Mebarki / Eur. J. Pure Appl. Math, 17 (2) (2024), 1321-1334 1327

·w(x, t) ≥ 0, z(x, t) ≥ 0; x ∈ Ω, t ≥ 0.
·w ∈ C

(
R+, C

(
Ω
))

, z ∈ C
(
R+, C

(
Ω
))

.
· limt→+∞ ∥z(t)∥∞ = 0; ∃w∞ ≥ 0.such that limt→+∞ ∥w(x, t)− w∞∥∞ = 0.
Proof. .
·Step1.
We define w+ = max(0, w) and w− = max(0,−w). We write w = w+ − w−, multiply

the first equation of system (1) by (−w−) and integrate over Ω; we get∫
Ω

∂w−

∂t
w−dx = −d1

∫
Ω
−Spw

−w−dx−
∫
Ω
(w−2z)w−dx+

∫
Ω
f(1− w−)w−dx.

By estimate (4) , we have

d

dt

∫
Ω
(w−)2dx ≤ 2[

∫
Ω
(w−2)(zw−)dx+

∫
Ω
fw−dx+

∫
Ω
f(w−)2dx.

Since (w, z) is a local solution on [0, Tmax), then w and z are bounded on [0, T ]
for T ≺ Tmax; Furthermore, there exist continuous functions m(t) and h(t) such that

∥z(t)∥∞ ≤ m(t) and ∥w(t)∥∞ ≤ h(t).
It holds that

d

dt

∫
Ω
(w−)2dx ≤ 2[(m(t)h(t) + f)

∫
Ω
(w−2)dx+ fh(t)]. (12)

As
∫
Ω(w

−2)(0)dx = 0 and f ≥ 0, Gronwall’s inequality [3] allows us to attain∫
Ω(w

−2)dx = 0; Consequently, w(x, t) ≥ 0.
By same manner, we gain∫

Ω

∂z−

∂t
z−dx = −d2

∫
Ω
−Tpz

−z−dx+

∫
Ω
w2(z−)2dx−

∫
Ω
(f + k)(z−)2dx.

Employing inequality (4) , we obtain

d

dt

∫
Ω
(w−)2dx ≤ 2[

∫
Ω
(w2 − (f + k))(z−)2dx.

It follows that

2[h2(t)− (f + k)]

∫
Ω
(z−)2dx.

By integration we have z− = 0 which gives z(x, t) ≥ 0.
· Step 2.
We will derive a uniform bound of ∥w(t)∥∞ .
Multiplying the first equation of (1) by wp−1 and integrating over Ω,
we get d

dt

∫
Ωwpdx ≤ 0 thanks to relation (5) and f = 0 . Hence, we get

∥w(t)∥∞ ≤ ∥w0∥∞ , ∀t ∈ [0, Tmax). (13)



M. Mebarki / Eur. J. Pure Appl. Math, 17 (2) (2024), 1321-1334 1328

Now, we integrate the first equation of (1) over Ω and practise the integration by parts
formula (3) which haven

∫
Ω

(−∆N )δw(x)dx = 0 and f = 0; we attain

∫
Ω

∂w

∂t
dx = −

∫
Ω

w2zdx ≤ 0

Therefore, the function t 7→
∫
Ωw(x, t)dx is nonincreasing. Since w ≥ 0. Then it admits

a limit as t → +∞ :

lim
t→+∞

1

|Ω|

∫
Ω
w(x, t)dx = w∞ ≥ 0.

We add the equations of system(1) and we integrate over Ω and putting f = 0 to obtain

d

dt

∫
Ω
(w + z)dx = −k

∫
Ω
zdx ≤ 0. (14)

We observe that the function t 7→
∫
Ω(w + z)dx ≥ 0 is nonincreasing. Accordingly

admits a limit; Also we obtain

lim
t→+∞

∫
Ω
z(x, t)dx = l ≥ 0.

Thence, z ∈ L∞ (
R+;L1 (Ω)

)
. Thanks to estimate (11) we can proceed analogously to

the proof of of( [8],proposition 3.3). Consequently, we get z ∈ C
(
R+, C

(
Ω
))

.
· Step 3.
Integrating the equation (15) over [0, T ], we have

−k

∫ t

0

∫
Ω
z(x, s)dxds =

∫
Ω
(w0 + z0)dx−

∫
Ω
(w + z)(x, t)dx

≤
∫
Ω
(w0 + z0)dx.

Also,
∫ t
0

∫
Ω z(x, s)dxds is finite. Moreover, z(x, t) is uniformly continuous in t.

In fact, let h ≻ 0 and 0 ≤ t ≺ t+ h ≺ Tmax, it follows that,

∥z(t+ h)− z(t)∥∞ ≤
∥∥∥(e−d2hTp − I)e−d2tTpz0

∥∥∥
∞

+

∫ t

0

∥∥∥(e−d2hTp − I)e−d2(t−h)Tpw2z(s)
∥∥∥
∞
ds

−(f + k)

∫ t

0

∥∥∥(e−d2hTp − I)e−d2(t−h)Tpz(s)
∥∥∥
∞
ds+

∫ t+h

t

∥∥∥e−d2(t+h−s)Tpw2z(s)
∥∥∥
∞
ds

−(f + k)

∫ t+h

t

∥∥∥e−d2(t+h−s)Tpz(s)
∥∥∥
∞
ds

= A1 +A2 +A3 +A4 +A5.



M. Mebarki / Eur. J. Pure Appl. Math, 17 (2) (2024), 1321-1334 1329

Employing ([6],lemma2.1), we get that for every θ ∈ (0.1)

A1 ≤ M2 (θ)h
θ
∥∥∥T θ

p e
−d2tTpz0

∥∥∥
∞

≤ M2 (θ)M1 (θ)h
θt−θe−d2αϵ

1t ∥z0∥∞

We also find

A2 ≤ M2 (θ)M1 (θ)h
θ

∫ t

0
(t− s)−θe−d2αϵ

1(t−s)
∥∥w2z(s)

∥∥
∞ ds.

Using lemma 1, we can observe that∫ t

0
(t− s)−θe−d2αϵ

1(t−s) ≤ M3 (θ,−d2α
ϵ
1) .

As z ∈ C
(
R+, C

(
Ω
))

we obtain ∥z(t)∥∞ ≤ C,∀t ≻ 0, here C is positive constant.
So, we have

A2 ≤ M2 (θ)M1 (θ)M3 (θ,−d2α
ϵ
1)Chθ ∥w0∥2 .

Similarly, it follows that

A3 ≤ M2 (θ)M1 (θ)M3 (θ,−d2α
ϵ
1)Chθ.

Using the relation see([7]) ∥∥∥e−d2tTpw
∥∥∥
∞

≤ R ∥w∥∞ ,

we obtain

A4 =

∫ h

0

∥∥∥e−d2τTpw2z(t− h− τ)
∥∥∥
∞
dτ

≤ MRh ∥w∥2∞ .

Therefore, ∀t ⪰ β ≻ 0

∥z(t+ h)− z(t)∥∞ ≤ M (θ, ϵ, β)hθ.

Consequently, limt→+∞
∫
Ω z(x, t)dx = 0.

Then, for θ ∈ (0.1) , applying T θ
p to both sides of the second equation of (12) and

estimate, we have∥∥∥T θ
p z(t)

∥∥∥
p
≤

∥∥∥T θ
p e

−d2tTpz0

∥∥∥
p
+

∫ t

0

∥∥∥T θ
p e

−d2(t−s)Tpw2z(s)
∥∥∥
p
ds.

Using ([3],theorem1.4.3), we obtain∥∥∥T θ
p z(t)

∥∥∥
p
≤ Rc(t)−θe−d2αϵ

1t ∥z0∥p +R |Ω|
1
P ∥w0∥2∞ ∥z0∥∞

∫ t

0
c(t− s)−θe−d2αϵ

1(t−s)ds.



M. Mebarki / Eur. J. Pure Appl. Math, 17 (2) (2024), 1321-1334 1330

Thus by Lemma (1) and ∀t ≥ β, we get
∥∥T θ

p z(t)
∥∥
p
≤ R (θ, p, β) .

Tereupon {z(t)}t≥β is uniformly bounded in D(T θ
p ); so by Sobolev’s imbeding theorem,

the compactness of {z(t)}t≥β in C
(
Ω
)
is assured.

Therefore, there exists a sequence {tj}j≥0, tj → +∞ such that z(tj) → z∗ in C
(
Ω
)
as

j → +∞.
Away we find limt→+∞ ∥z(t)∥∞ = 0.
Similarly, we can prove that {w(t)}t≥β is precompact in C

(
Ω
)
; So, there exists a

sequence {τj}j≥0, τj → +∞ such that w(τj) → w∗ in C
(
Ω
)
as j → +∞.

We have (w + z)(t) → w∗ in C
(
Ω
)
, then (w + z)(t) → w∗ in L1(Ω) as t → +∞.

Using the fact that limt→+∞
∫
Ω z(x, t)dx = 0 and limt→+∞

1
|Ω|

∫
Ωw(x, t)dx = w∞, we

gain (w + z)(t) → w∞ in L1(Ω) as t → +∞. By uniqueness of the limit, w∗ = w∞.

Theorem 3. Let (w, z) be the solution of (1)-(2). Therefore assume that k − w∞ ≻ 0.
Then there exist positive constants T and R such that

∥w(t)− w∞∥∞ ≤
{

R exp−κ(t−T ) if 2d1α
δ
1λ ̸= h(w∞)

R(t− T + 1) exp−κ(t−T ) if 2d1α
δ
1λ = h(w∞),

∥z(t)∥∞ ≤ R exp(−h(w∞)(t− T ), t ≥ T
where κ = min

{
h(w∞), 2d1α

δ
1λ

}
, h(w∞) = (w0 + ε)2 − k ≻ 0.

Proof. [Proof of Theorem]

For ε ≻ 0, there exists a constant T ≻ 0 such that fort ≥ T w∞ − ε ≺ w(t) ≺ w∞ + ε.
Putting 0 ≺ ε ≺ k − w∞.
We multiply the second equation of (1) by zp−1 and integrate over Ω; It follows that

d

dt

∫
Ω
zpdx ≤ p

∫
Ω
(w2 − (f + k))zpdx,∀t ≥ T,

in the light of relation (5) .
So

d

dt
∥z(t)∥pp ≤ p((w∞ + ε)2 − (f + k)) ∥z(t)∥pp , ∀t ≥ T.

Thus, for 1 ≤ p ≤ +∞, we have

∥z(t)∥p ≤ ∥z(T )∥p exp((w∞ + ε)2 − (f + k))(t− T ), t ≥ T. (15)

Which implies

∥z(t)∥∞ ≤ ∥z(T )∥∞ exp((w∞ + ε)2 − (f + k))(t− T ), t ≥ T. (16)

For the rate of convergence of ∥w(t)− w∞∥∞ to zero, we define as in [14] two bounded
linear operators I and G by



M. Mebarki / Eur. J. Pure Appl. Math, 17 (2) (2024), 1321-1334 1331

Iu := ⟨u⟩ , Gu := u− ⟨u⟩ ,here ⟨u⟩ := 1
|Ω|

∫
Ω u(x, t)dx.

Adding the two equations of problem (1) and integrating over Ω,we obtain if f = 0

⟨w(t)⟩ = −⟨z(t)⟩+ ⟨w0⟩+ ⟨z0⟩ − k

∫ t

0
⟨z(s)⟩ ds, as t → ∞.

It holds that

w∞ = ⟨w0⟩+ ⟨z0⟩ − k

∫ t

0
⟨z(s)⟩ ds.

Hence

|⟨w(t)− w∞⟩| ≤≺ ⟨z(t)⟩+ k

∫ t

0
⟨z(s)⟩ ds.

Using inequality (16) we get∫ ∞

t
⟨z(s)⟩ ds ≤ ⟨z(T )⟩

∫ ∞

t
exp((w∞ + ε)2 − (k))(s− T )ds

≤ R1 ⟨z(T )⟩ exp((w∞ + ε)2 − (k))(t− T ), t ≥ T,

where R1 =
1

k−(w∞+ε)2
.Out

|⟨Iw(t)− w∞⟩| ≤ R2 ⟨z(T )⟩ exp((w∞ + ε)2 − (k))(t− T ), t ≥ T, (17)

where R2 = max {1, kR1} .
Accordingly w(t) satisfies the integral equation for t ≥ T ,

w(t) = e−d1tSpw0 +

∫ t

0
e−d1(t−s)Sp(−w2z + f(1− w))(s)ds,

as f = 0, we get

w(t) = e−d1tSpw0 −
∫ t

0
e−d1(t−s)Sp(w2z)(s)ds,

= e−d1(t−T )Spw(T )−
∫ t

T
e−d1(t−s)Sp(−w2z)(s)ds,

Thus we have

Gw(t) = e−d1(t−T )SpGw(T )−
∫ t

T
e−d1(t−s)SpG(w2z)(s)ds.

Using Lemma 2 , we get the estimate∥∥∥e−d1(t−T )SpGw(T )
∥∥∥
p
≤ Re−d1αδ

1(t−T ) ∥w(T )∥p (18)

Now , to calculate A(t) =
∫ t
T

∥∥e−d1(t−s)SpG(w2z)(s)ds
∥∥
p
.



M. Mebarki / Eur. J. Pure Appl. Math, 17 (2) (2024), 1321-1334 1332

∫ t

T

∥∥∥e−d1(t−s)SpG(w2z)(s)ds
∥∥∥
p
≤ R

∫ t

T
c(t− s)

−N
2δ

( 1
q
− 1

p
)
e−2d1αδ

1λ(t−s) ∥z(s)∥q

Using the estimate (16), we find

A(t) ≤ R ∥z(T )∥q
∫ t

T
c(t− s)

−N
2δ

( 1
q
− 1

p
)
e−2d1αδ

1λ(t−s)e((w∞+ε)2−(k))(s−T )ds

≤ R ∥z(T )∥q
∫ t−T

0
c(t− T − τ)

−N
2δ

( 1
q
− 1

p
)
e−2d1αδ

1λ(t−T−τ)e(w∞+ε)2−(k))(τ)dτ

R ∥z(T )∥q e
(w∞+ε)2−(k))(t−T ) ×

∫ t−T

0
c(t− T − τ)

−N
2δ

( 1
q
− 1

p
)
e(((w0+ε)2−k)−2d1αδ

1λ)(t−T−τ)dτ

We note Aε(t) =
∫ t−T
0 c(t− T − τ)

−N
2δ

( 1
q
− 1

p
)
e(((w0+ε)2−k)−2d1αδ

1λ)(t−T−τ)dτ.

If we choose p and q satisfying 0 ≤
−N
2δ

( 1
q
− 1

p
)≺ 1 and use Lemma 1, we obtain

◦ When 2d1α
δ
1λ ≺ (w0 + ε)2 − k = h(w∞), we choose ε such that 0 ≺ ε ≺ h(w∞) −

2d1α
δ
1λ, so

Aε(t) ≤ R(
N

2δ
(
1

q
− 1

p
), h(w∞)− 2d1α

δ
1λ− ε)e(h(w∞)−2d1αδ

1λ−ε)(t−T ).

◦ When 2d1α
δ
1λ ≥ (w0 + ε)2 − k = h(w∞),

Aε(t) ≤ R(
N

2δ
(
1

q
− 1

p
), h(w∞)− 2d1α

δ
1λ− ε).

Hence
A(t) ≤ R ∥z(T )∥q e

−µ(t−T ) (19)

where

µ =

{
2d1α

δ
1λ if 2d1α

δ
1λ ≺ h(w∞)

(w0 + ε)2 − k if 2d1α
δ
1λ ≥ h(w∞)

Combining relation (19) and (20), we obtain

∥Gw(T )∥p ≤ R ∥(w, z)(T )∥p e
−µ(t−T ); t ≥ T. (20)

So, we get from (18) and (21)

∥w(t)− w∞∥p ≤ R ∥(w, z)(T )∥∞ e−µ(t−T ); t ≥ T.

Now, we have

w∞ −Re−µ(t−T ) ≺ w(t) ≺ w∞ +Re−µ(t−T ); t ≥ T.

Therefore, we can assert that ∀1 ≤ p ≤ +∞, ∥z(t)∥p ≤ R ∥z(T )∥p e−h(w∞)(t−T ), t ≥ T.



REFERENCES 1333

Hence, Gw(t)− w∞ can be estimated as

|Gw(t)− w∞| ≤ R |Gw(T )| e−h(w∞)(t−T ); t ≥ T.

and

∥Gw(t)∥∞ ≤
{

R exp−κ(t−T ) if 2d1α
δ
1λ ̸= h(w∞)

R(t− T + 1) exp−κ(t−T ) if 2d1α
δ
1λ = h(w∞),

where κ = min
{
h(w∞), 2d1α

δ
1λ

}
.

Accordingly

∥w(t)− w∞∥∞ ≤
{

R exp−κ(t−T ) if 2d1α
δ
1λ ̸= h(w∞)

R(t− T + 1) exp−κ(t−T ) if 2d1α
δ
1λ = h(w∞).

5. Conclusion

In this paper, we considered a Gray Scott system with anomalous diffusion described
by a fractional Laplacian power which accounts for a sub-diffusive situation. In addition
to the global existence of bounded solutions, we showed the convergence of the solution
(w∞, 0). (w∞): the final state of susceptible individuals and 0: the final state of infected
individuals. The exponent of the fractional Laplacian influences the asymptotic behavior
in time from w to w∞.

Acknowledgements

The authors would like to thank the anonymous referee for his/her comments that
helped us improve this article.

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