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Research Article

Blow-Up Rate Estimates for a System of Reaction-Diffusion

Equations with Gradient Terms

Maan A. Rasheed , Hassan Abd Salman Al-Dujaly, and Talat Jassim Aldhlki

Department of Mathematics, College of Basic Education, Mustansiriyah University, Baghdad, Iraq

Correspondence should be addressed to Maan A. Rasheed; maan.rasheed.edbs@uomustansiriyah.edu.iq

Received 16 November 2018; Revised 12 January 2019; Accepted 22 January 2019; Published 10 February 2019

Academic Editor: Irena Lasiecka

Copyright © 2019 Maan A. Rasheed et al. is is an open access article distributed under the Creative Commons Attribution

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

cited.

is paper is concerned with the blow-up properties of Cauchy and Dirichlet problems of a coupled system of Reaction-Diusion

equations with gradient terms. e main goal is to study the inuence of the gradient terms on the blow-up prole. Namely, under

some conditions on this system, we consider the upper blow-up rate estimates for its blow-up solutions and for the gradients.

1. Introduction

In this paper, we consider the following problem:

𝑡=−|∇|𝑞1+V𝑝1,(,)∈×(0,)

V𝑡=V−|∇V|𝑞2+𝑝2(,)∈×(0,)

(,0)=0()≥0, ∈

V(,0)=V0()≥0, ∈,

(1)

where 1,2∈(1,∞);1,2∈(1,2);

=𝑛or 𝑅(a ball in 𝑛with radius ).

Moreover, for =

𝑅,and Vsatisfy the zero Dirichlet

boundary conditions:

(,)=V(,)=0, ∈,∈(0,);(2)

0,V0∈2()are both nonzero, satisfying the monotonicity

conditions:

0−∇0𝑞1+V0𝑝1≥0, ∈ (3)

V0−∇V0𝑞2+0𝑝2≥0, ∈ (4)

Moreover, in case of =

𝑅,(

0,V0)should satisfy

compatibility conditions:

0()=V0()=0, ∈ (5)

As an application to system (1), a single equation of this

system can be considered a simple model in population

dynamics, [1, 2]:

𝑡=−|∇|𝑞+𝑝,in ×{≥0}(6)

where ,>1.

Let the domain represent a territory where a biological

species live on. (,)refers to the spatial density of individ-

uals located near a point ∈at time ≥0.

In fact, the evolution of this density is the result of three

types of mechanisms: displacement, birth, and death. For

more details of deriving the evolution equation satised by

, see [1].

Basically, under dierent assumptions on the mecha-

nisms of accidental death, the corresponding term should

more generally be a nondecreasing function of the density

and its gradient ∇.

Moreover, homogeneous Dirichlet’s conditions can be

added to this model which, for instance, correspond to a

nonviable environment in the boundary zone.

It is expected that, with a large size initial function

(initial distribution of population 0), the density becomes

unbounded in a nite time >0. erefore, Chipot and

Weissler [3] studied the eect of the damping term in this

equation on global existence or nonexistence.

Hindawi

International Journal of Mathematics and Mathematical Sciences

Volume 2019, Article ID 9807876, 7 pages

https://doi.org/10.1155/2019/9807876

2 International Journal of Mathematics and Mathematical Sciences

e blow-up phenomena in Reaction-Diusion equa-

tions have been intensively studied; see, for instance, [4–

7]. One of the studied cases is the Cauchy problem of the

semilinear heat equation:

𝑡=+𝑝,(,)∈R𝑛×(0,),

(,)=0()≥0, ∈R𝑛(7)

e second studied case is zero Dirichlet problem of the

semilinear heat equation:

𝑡=+𝑝,(,)∈𝑅×(0,),

(,)=0, ∈𝑅

(,)=0()≥0, ∈𝑅

(8)

where >1.

For both cases (7) and (8), it has been proved in [8, 9] that

if the initial function is nonnegative and suitably large, then

blow-up occurs in a nite time. In [5, 10], it has been shown

that the upper blow-up rate estimate for this equation is as

follows:

(,)≤

(−)1/(𝑝−1) ,(,)∈×(0,).(9)

e blow-up properties of semiliear heat equations with

negative sign gradient terms (damping terms) have been

studied by some authors as in [3, 7, 11].

One of these equations is population model (6). For ⊆

𝑛, it is well known that blow-up can only occur if >;see

[3, 11, 12].

Moreover, if =

𝑅,thenblow-upoccursatthe

center of 𝑅, and this follows from the upper point-wise

estimate:

(,)≤

||𝛼,∈

𝑅\{0},∈[0,),(10)

where >2/(−1),for1<<2/(+1),

while >/(−),for2/(+1)≤<.

It is clear that /(−)>2/(−1),where>2/(+1)

erefore, the prole of blow-up solutions of (6) is

similar to that of problem (8), where < 2/(+1) (see

[12]), while if > 2/( + 1),thegradienttermcauses

more eect on the plow-up prole and it becomes more

singular.

Moreover, it has been proved in [4, 13, 14], that there are

positive constants and , such that the upper and lower

blow-up rate estimates for this equation, where <2/(+1),

take the following form:

(−)1/(𝑝−1) ≤(,)≤

(−)1/(𝑝−1) (11)

In [15–18], the coupled system of Reaction-Diusion equa-

tions was considered:

𝑡=+V𝑝1,

V𝑡=V+𝑝2,(12)

(,)∈×(0,),(13)

where 1,2>1;=𝑅or 𝑛.

It was shown that if the initial functions satisfy 0,V0≥0,

both being nonzero and large enough, then blow-up occurs

in a nite time.

For the Cauchy problem associated with (12), it was

proved in [16] that if

max ,≥

2,(14)

then blow-up occurs in a nite time, where

= 1+1

12−1,

= 2+1

12−1

(15)

Later, in [4, 15], it was proved that the upper blow-up rate

estimates of this system are as follows:

(,)≤1

(−)𝛼,(,)∈×(0,),

V(,)≤2

(−)𝛽,(,)∈×(0,).(16)

for some 1,2>0.

e system (1) has been studied in [19], where ∈3and

is a bounded convex domain and

=1=2== 1

1−;

=1=2;>>1. (17)

It has been shown that if a classical solution of this system

blows ups (becomes unbounded) in the W-norm, where

()=Ω2𝑝 +V2𝑝 (18)

then blow-up time for this problem can be estimated from

below as follows:

≥ 1

22

0(19)

where (0)=0=∫

Ω(2𝑝

0+V2𝑝

0)

and is a constant which depends on the data.

For the blow-up times and applications of other parabolic

systems with damping terms (such as Keller-Segel system

with Neumann and Robin boundary conditions), we refer to

[20, 21].

International Journal of Mathematics and Mathematical Sciences 3

In this paper, with some restricted conditions on system

(1), we show that the upper blow-up rate estimates for this

solution and its gradients terms take the following forms:

(,)≤1(−)−𝛼 ,

|∇(,)|≤1(−)−(1+2𝛼)/2 ,

V(,)≤2(−)−𝛽 ,

|∇V(,)|≤2(−)−(1+2𝛽)/2 ,

(20)

where (,) ∈ ×(0,) and 1,2> 0,,are given in

(15).

2. Local Existence and Blow-up

Set

1(V,∇)=V𝑝1−|∇|𝑞1,

2(,∇V)=𝑝1−|∇V|𝑞1(21)

Since the system (1) is uniformly parabolic and its equations

have the same principle parts and 1,2∈

1( × 𝑛),

also the growths of the nonlinearities in 1and 2with

respect to the gradient terms are subquadratic; 0,V0∈

2(), and satisfying (5), it follows that the local existence

and uniqueness of classical solution to the for system (1),

where =𝑅, with zero Dirichlet boundary conditions, are

guaranteed by standard parabolic theory (see eorem 7.1,

[22, 23]).

i.e., there exists >0,suchthat

∈2,1 (×(0,))∩×[0,). (22)

Also,thegradienttermsareboundedaslongasthecompo-

nents of the solution are bounded; see [23].

In case of =

𝑛, these results can also be extended to

the Cauchy problem associated with system (1) (see eorem

8.1, [22, 24]).

Moreover, from the monotonicity assumptions (3) and

(4) and since 0,V0are nonnegative, it follows by the

maximum principle [6] that in the interval of existence

the solutions of system (1) are nondecreasing in time and

nonnegative.

i.e., (,)≥0, V(,)≥0, ×(0,).

On the other hand, since the existence and uniqueness

of system (1) can only be locally guaranteed and according

to known blow-up results to the single equation (6), blow-up

may occur in this problem in a nite time. erefore, some

authors were interested in studying the blow-up properties

and numerical solutions of system (1) ; see for instance [19, 25].

3. Upper Blow-Up Rate Estimates

In the next theorem, we derive the upper blow-up rate

estimates for any blow-up solution of system (1) and its

gradients.

eorem 1. Assume that 1,2,1,and2satisfy the following

two conditions:

(i) max{,}≥/2,

(ii) 1<1<(2+2)/(2+1), 1<2<(2+2)/(2+1),

where ,are given in (15).

Let (,V)be a blow-up solution of the Cauchy (Dirichlet)

problem of system (1), with the above conditions, which blows

up at <∞.ere exist two positive constants 1and 2such

that upper blow-up rate estimates for (,V)and (∇,∇V)are as

follows:

(,)≤1(−)−𝛼 ,

|∇(,)|≤1(−)−(1+2𝛼)/2 ,

V(,)≤2(−)−𝛽 ,

|∇V(,)|≤2(−)−(1+2𝛽)/2 ,

(23)

in ×(0,).

Proof. For ∈(0,),set

𝑢()=sup

Ω×(0,𝑡] (,)+|∇(,)|2𝛼/(1+2𝛼),

V()=sup

Ω×(0,𝑡] V(,)+|∇V(,)|2𝛽/(1+2𝛽). (24)

Clearly, each of 𝑢,Vis continuous, nondecreasing, and

nonnegative function on (0,). Moreover, 𝑢→ ∞ or

V→ ∞ as →and that follows from (,V)blowing

up at .

It will be shown later that we can nd ∈(0,1)such that

≤−1/2𝛼

𝑢()1/2𝛽

V()≤1

(25)

for /2<<

So that consequently both 𝑢and Vdiverge as →.

In order to prove this theorem, we will use a rescaling

method as in [4] and the proof will have ve steps.

Step 1 (rescaling).If𝑢diverges as →, then we can apply

the following procedure.

Letting 0∈(0,),wecanchoose(1,1)∈×(0,

0]

such that

1,1+∇1,12𝛼/(1+2𝛼) ≥1

2𝑢0. (26)

Dene the new rescaled functions as follows:

𝛾

1,=2𝛼+1,2+1, (27)

𝛾

2,=2𝛽V+1,2+1, (28)

,∈𝛾×−1

2,−1

2, (29)

4 International Journal of Mathematics and Mathematical Sciences

where =(0)=−1/2𝛼

𝑢(0)is a scaling factor and

𝛾=∈𝑛:+1∈.(30)

It is clear that

𝛾ﬂ

𝑛 = 𝑛,

𝑅/𝛾 = 𝑅.(31)

Next, we aim to show that (𝛾

1,𝛾

2)is a solution of the system:

𝛾

1𝑠 −𝛾

1=−𝜇1∇𝛾

1𝑞1+𝛾

2𝑝1,

𝛾

2𝑠 −𝛾

2=−𝜇2∇𝛾

2𝑞2+𝛾

1𝑝2,(32)

where

1=2+2−(2+1)1,

2=2+2−2+12.(33)

From assumption (ii), we get 1,2>0.

Clearly,

𝛾

1𝑠 =2𝛼+2,

∇𝛾

1=2𝛼+1∇,

𝛾

1=2𝛼+2.

(34)

From (1) and (34), we get

1

(2𝛼+2) 𝛾

1𝑠 =1

(2𝛼+2) 𝛾

1+1

𝑞1(2𝛼+1) ∇𝛾

1𝑞1

+1

2𝑝1𝛽𝛾

2𝑝1.(35)

Hence, the rst equation of the system (32) can be obtained

by multiplying the last equation by (2𝛼+2).e same way can

be used to show that 𝛾

2satises the second equation of the

system (32).

Now, we restrict to ∈(−−21,0]to show that

𝛾

1,+∇𝛾

1,2𝛼/(1+2𝛼) ≤1, (36)

for (,)∈𝛾×(−−21,0].

From (34), we obtain

∇𝛾

1,2𝛼/(1+2𝛼) =2𝛼 |∇|2𝛼/(1+2𝛼) .(37)

Clearly,

(,)+|∇(,)|2𝛼/(1+2𝛼) ≤𝑢0,

(,)∈×0,1. (38)

From (32), (37), and (38), we get (36).

Moreover,

𝛾

2+∇𝛾

22𝛽/(1+2𝛽) ≤−𝛽/𝛼

𝑢0V0,(39)

for (,)∈𝛾×(−−21,0].

On the other hand, from (26), we obtain

𝛾

1(0,0)+∇𝛾

1(0,0)2𝛼/(1+2𝛼) ≥1

2(40)

If V→ ∞ as →, the same procedure can be repeated

by changing the roles of and.

Step 2 (Schauder’s estimates). In this step, we nd the interior

Schauder’s estimates of the functions 1,2on the sets

𝐾=∈𝛾,≤×[−,],(41)

where >0, =0,1.Assuming that 1and 2satisfy in 2𝑘

the condition

0≤𝛾

1+∇𝛾

12𝛼/(1+2𝛼) ≤,

0≤𝛾

2+∇𝛾

22𝛽/(1+2𝛽) ≤ (42)

Our claim is as follows: for any positive and small enough

values of ,,and,thereexistsaconstant = (,,)

such that

𝛾

1

𝐶2+𝜎,1+𝜎/2(𝑆𝑘)≤,

𝛾

2

𝐶2+𝜎,1+𝜎/2(𝑆𝑘)≤ (43)

From (42), we deduce that 𝛾

1,𝛾

2,∇𝛾

1,and∇𝛾

2are

uniformly bounded functions in 2𝑘.So, the functions

(𝛾

1)𝑝1,(𝛾

2)𝑝2,|∇𝛾

1|𝑞1,and|∇𝛾

2|𝑞2are uniformly bounded in

2𝑘.erefore, the right hand side of each equation in (32) is

uniformly bounded function in 2𝑘.By applying the interior

regularity theory (see [23]), we get locally uniform estimates

in 1+𝜎,(1+𝜎)/2-norms. Consequently, on the right hand side of

each equation in (32), we can obtain locally uniform estimates

in H¨

older norms 𝜎,𝜎/2. erefore, the parabolic interior

Schauder’s estimates (43) are held; see [23].

Step 3 (the proof of (25)). Suppose that the lower bound of

(25) is not held. So, there is a sequence 𝑗,suchthat𝑗→

as →∞,and

−1/2𝛼

𝑢𝑗1/2𝛽

V𝑗→0 →∞ (44)

us, 𝑢→ ∞ as 𝑗→ .

Now, for each 𝑗whichplaysthesameroleof0,asin

Step 1, we can scale about the corresponding point (∗

𝑗,∗

𝑗)

for each, where ∗

𝑗≤

𝑗.We get the corresponding rescaled

solution (𝛾𝑗

1,𝛾𝑗

2):

International Journal of Mathematics and Mathematical Sciences 5

𝛾𝑗

1,=2𝛼

𝑗𝑗+∗

𝑗,2

𝑗+∗

𝑗, (45)

𝛾𝑗

2,=2𝛽

𝑗V𝑗+∗

𝑗,2

𝑗+∗

𝑗. (46)

where 𝑗=(𝑗)=−1/2𝛼

𝑢(𝑗)is the scaling factor.

It is clear that (𝛾𝑗

1,𝛾𝑗

2)satises, as in Step 1, the following

problem:

𝛾𝑗

1𝑠 −𝛾𝑗

1=−𝜇1

𝑗∇𝛾𝑗

1𝑞1+𝛾𝑗

2𝑝1,

𝛾𝑗

2𝑠 −𝛾𝑗

2=−𝜇2

𝑗∇𝛾𝑗

2𝑞2+𝛾𝑗

1𝑝2,(47)

with

𝛾𝑗

1(0,0)+∇𝛾𝑗

1(0,0)2𝛼/(1+2𝛼) ≥1

2,

0≤𝛾𝑗

1+∇𝛾𝑗

12𝛼/(1+2𝛼) ≤1,

𝛾𝑗

2+∇𝛾𝑗

22𝛽/(1+2𝛽) ≤−𝛽/𝛼

𝑢𝑗V𝑗,

(48)

for (,)∈𝛾𝑗×(−−2

𝑗∗

𝑗,0],where

𝛾𝑗ﬂ

𝑛 = 𝑛,

𝑅/𝛾𝑗 = 𝑅.(49)

Clearly,

𝛾𝑗→ 𝑛 →∞. (50)

From (44) and (48), we see that

𝛾𝑗

2+∇𝛾𝑗

22𝛽/(1+2𝛽) → 0, → ∞. (51)

us 𝛾𝑗

2and ∇𝛾𝑗

2are bounded in 𝛾𝑗×(−−2

𝑗∗

𝑗,0]for all .

By applying Step 2, there is 𝑘independent of ,suchthat

the uniform Schauder’s estimates of (𝛾𝑗

1,𝛾𝑗

2)are as follows:

𝛾𝑗

1

𝐶2+𝜎,1+𝜎/2({𝑦∈Ω𝛾𝑗,|𝑦|≤𝑘}×[−𝑘,0]) ≤𝑘,

𝛾𝑗

2

𝐶2+𝜎,1+𝜎/2({𝑦∈Ω𝛾𝑗,|𝑦|≤𝑘}×[−𝑘,0]) ≤𝑘

(52)

Since (𝛾𝑗

1,𝛾𝑗

2)is dened on a compact set, by the Arzela-

Ascoli theorem, there exists a convergent subsequence, and

it is denoted by (𝛾𝑗

1,𝛾𝑗

2).

Since 1,2>0and ∇𝛾𝑗

1,∇𝛾𝑗

2are bounded, the limit

point (1,2)is a solution of the following system:

1𝑠 =1+𝑝1

2,

2𝑠 =2+𝑝2

1

(53)

𝑛×(−∞,0](54)

Since 𝛾𝑗

2→ 0, where →∞,itfollowsthat2≡

0𝑛×(−∞,0].

Consequently, from the second equation of (53), we get

1≡0, 𝑛×(−∞,0].(55)

us,

1(0,0)+∇1(0,0)2𝛼/(1+2𝛼) =0, (56)

which leads to a contradiction with (48), so the lower bound

is proved.

If we change the roles of and V, the upper bound of (25)

can be proved similarly as in the last proof.

Step 4 (estimates on doubling 𝑢).Since𝑢→ ∞ as →

,𝑢is a continuous function. For any 0∈(0,),thepoint

+

0can be dened as follows:

+

0=max ∈0,:𝑢()=2𝑢0. (57)

Clearly,

(,)+|∇(,)|2𝛼/(1+2𝛼) ≤2𝑢0,

(,)∈×0,+

0. (58)

Take =(0)=−1/2𝛼

𝑢(0).

We claim that there is 0<which is independent of 0

such that +

0−0

20≤, 0∈

2,,(59)

By supposing that this claim is not true, there is a sequence

𝑗→ ,as→∞such that

+

𝑗−𝑗

2

𝑗𝑗→ ∞, (60)

where

+

𝑗=max ∈𝑗,:𝑢()=2𝑢𝑗. (61)

For each 𝑗,where/2 < 𝑗<

+

𝑗<,∀,wecanchoose

0<∗

𝑗≤𝑗.

AsinStep3,wescaleaboutthecorrespondingpoint

(∗

𝑗,∗

𝑗), and we can get the corresponding rescaled functions

(𝛾𝑗

1,𝛾𝑗

2)with the scaling factor: 𝑗=(

𝑗)=

−1/2𝛼

𝑢(𝑗),

which satises (47) with the following conditions:

𝛾𝑗

1(0,0)+∇𝛾𝑗

1(0,0)2𝛼/(1+2𝛼) ≥1

2,

0≤𝛾𝑗

1+∇𝛾𝑗

12𝛼/(1+2𝛼) ≤2,

𝛾𝑗

2+∇𝛾𝑗

22𝛽/(1+2𝛽) ≤−𝛽/𝛼

𝑢𝑗V+

𝑗

(62)

,∈𝛾𝑗×−∗

𝑗

2

𝑗,+

𝑗−∗

𝑗

2

𝑗.(63)

6 International Journal of Mathematics and Mathematical Sciences

From (61) and (62), it follows that

𝛾𝑗

2+∇𝛾𝑗

22𝛽/(1+2𝛽) ≤2𝛽/𝛼−𝛽/𝛼

𝑢+

𝑗V+

𝑗.(64)

From (25), we conclude that

V()≤−2𝛽𝛽/𝛼

𝑢(),∈

2,. (65)

erefore, (64) becomes

𝛾𝑗

2+∇𝛾𝑗

22𝛽/(1+2𝛽) ≤2𝛽/𝛼

2𝛽 (66)

By applying Step 2, we use the Schauder estimates for

(𝛾𝑗

1,𝛾𝑗

2), and we can get a convergent subsequence in

2+𝜎,1+𝜎/2

𝑙𝑜𝑐 (𝑛×)to the solution of system (53) in 𝑛×.

us, we get a contradiction because under the assump-

tion (i), all nontrival solutions of system (53) blow up in a

nite time; see [17].

So, there is 0<such that

−2 0+

0−0≤, 0∈

2,. (67)

Step 5 (rate estimates).

For any 0∈(/2,),asinStep4,

dene 𝑢(1)=2𝑢(0),where1=+

0∈(0,).

By (67), we have

1−0≤−1/𝛼

𝑢0. (68)

We can g e t 2∈(,)by using 1as a new value of 0such

that 𝑢2=2𝑢1=4𝑢0. (69)

us,

2−1≤−1/𝛼

𝑢1=2−1/𝛼−1/𝛼

𝑢0.(70)

So, for any ≥0,wehave

𝑗+1 −𝑗≤2−𝑗/𝛼−1/𝛼

𝑢0, (71)

where the sequence 𝑗→ as →∞.

By adding the above inequalities, it follows that

−0≤

𝑗≥02−𝑗/𝛼−1/𝛼

𝑢0. (72)

us, (−0)≤(1−2−1/𝛼)−1−1/𝛼

𝑢(0).

Using (25) results in

V0≤−2𝛽𝛽/𝛼

𝑢0, 0∈

2,. (73)

us,

V0≤−2𝛽 1−2−1/𝛼−𝛽 𝛽−0−𝛽 ,

0∈

2,. (74)

From above, there exist positive two constants and such

that

𝑢0≤−0−𝛼 ,

0∈

2,,

V0≤−0−𝛽 ,

0∈

2,. (75)

From the last two inequalities and the denitions of

𝑢and V, it follows that there are 1and 2such that

(,)+|∇(,)|2𝛼/(1+2𝛼) ≤1(−)−𝛼 ,

V(,)+|∇V(,)|2𝛽/(1+2𝛽) ≤2(−)−𝛽 .(76)

Or, we can split the last estimates as follows:

(,)≤1(−)−𝛼 ,

|∇(,)|≤1(−)−(1+2𝛼)/2 ,

V(,)≤2(−)−𝛽 ,

|∇V(,)|≤2(−)−(1+2𝛽)/2

(77)

where (,)∈×(0,).

4. Conclusions and Future Studies

From eorem 1 and its proof, we can point out the following

conclusions:

(i) By (25), it follows that the blow-up in system (1) can

only occur simultaneously.

(ii) e gradient terms are bounded for any <.

(iii) e upper blow-up rate estimates for both of systems

(1) and (12) take the same forms. is means that, with

the two conditions of eorem 1, the gradient terms

in the system (1) do not eect ormake any changes on

the prole of blow-up solutions.

Next,wepointoutsomepossiblefutureresearchdirections:

(1) One may try to derive the blow-up rate estimates for

problem (1), in case one or both assumptions (i) and

(ii) of eorem 1 are not satised.

(2) For the semilinear system (12) dened in a ball,

and under some restricted assumptions on (0,V0)

(nonnegative and radial decreasing functions), it is

well known that the blow-up can only occur at

the center point (see [18]). However, it is unknown

whether and under which condition this result can be

extended to the system (1).

Data Availability

https://www.dr opbox.com/home?preview=Data+Availabil-

ity+Statement.pdf.

International Journal of Mathematics and Mathematical Sciences 7

Conflicts of Interest

e authors declare that they have no conicts of interest.

Acknowledgments

e authors would like to thank Mustansiriyah University

(http://www.uomustansiriyah.edu.iq) Baghdad-Iraq for its

support in the present work.

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