∫ ∫ ∫ ∫

(1)

Menemui Matematik (Discovering Mathematics) Vol. 43, No. 1: 27 - 39 (2021)

Growth of Solutions for a Coupled Viscoelastic Kirchhoff System with Distributed Delay Terms

Erhan Pişkin^1,a), Hazal Yüksekkaya^1,b) and Nadia Mezouar^2,3,c)

1Department of Mathematics, Dicle University, Diyarbakir, Turkey,

2Department of Mathematics, Faculty of Economics Sciences, Mascara, Algeria,

3Department of Mathematics, Analysis and Control of PDE’s Laboratory, Sidi Bel Abbes, Algeria.

a) [email protected], ^b)[email protected], ^c)[email protected]

ABSTRACT

In this work, we are concerned with a coupled nonlinear viscoelastic Kirchhoff system with distributed delay terms and source terms. Under suitable conditions, we prove the exponential growth of solutions.

Keywords: Exponential growth, Distributed delay, Nonlinear source.

INTRODUCTION

In this paper, we investigate the following viscoelastic Kirchhoff system with distributed delay terms and source terms:

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

2 1

2

0 1

2 1

2

0 3

4 2

0 0 2

0

,

, , , , ,

,

, , , , ,

, 0, v , 0, x ,

, , , v , , , , 0, ,

, 0

t

tt t

t

tt t

t

t t

u M u u g t s u s ds u x t

u x t d f u v x t R

v M v v h t s v s ds v x t

v x t d f u v x t R

u x t x t

u x t f x t x t k x t x t

u x u x





   



   



+

−   + −  +

+ − = 

−   + −  +

+ − = 

= = 

− = − = 

=



( ) ( )

( )

0

( ) ( )

1¹

( )

, , 0 , x ,

, 0 , v , 0 , x ,

t t

u x u x

v x v x x v x









 = 

 = = 



(1)

where  is a bounded domain in Rⁿ, with a smooth boundary . ₁, ₃ 0, ₁, ₂ are the time delay with 0  ₁ ₂, ₂, ₄are L^ functions, and g, h are differential functions. ^{M s}

( )

is a C¹ nonnegative function on R⁺ defined as M s

( )

=m0+s^ with m₀ 0, 0 and  0, to simplify our calculations we take ^{M s}

( )

^{= +}¹ ^s^ in the problem (1).

From mathematically point of view, ‘’Growth’’ phenomenon gives us importance knowledge to understand the asymptotic behaviour of the equation when time arrives at infinity. In recent years, there has been published much work concerning the wave equation with time delay or time varying delay. Our aim is to study the exponential growth of solutions for viscoelastic Kirchhoff system with distributed delay terms.

(2)

Menemui Matematik Vol. 43(1) 2021 28

The viscous materials are the converse of elastic materials that possess the ability to dissipate and store mechanical energy. Because of the mechanical properties of these viscous substances are of great significance when they seem in many applications of natural sciences [14].

Time delays often appear in many practical problems such as thermal, economic phenomena, biological, chemical and physical. The authors, in [2], indicated that a small delay in a boundary control could turn a well-behave hyperbolic system into a wild one, hence, delay becomes a source of instability. Besides, sometimes it can also be improved the performance of the system.

Rahmoune et al. [14], considered the following Klein-Gordon system with strong damping, nonlinear source and distributed delay terms:

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

2 1

2

1 1 0 1

2 1

2

2 2 0 3

4 2

, , , , ,

t

tt t t

t

tt t t

t

u m u u u g t s u s ds u

u x t d f u v x t R

v m v v v h t s v s ds v

v x t d f u v x t R



 

   

 

   

+

 + −  −  + −  +



+ − = 



 + −  −  + −  +

+ − = 





(2)

where m₁, m , ₂  ₁, ₂ 0. The authors investigated the exponential growth of solutions for the problem (2) under suitable conditions.

The viscoelastic wave equation of the form:

utt−  +u



₀^tg t

(

− 

) ( )

u  d +h u

( )

t = f u

( )

^{, x}^{, t>0,} (3) has been investigated extensively by some authors (see [3], [6], [7], [15], and references therein).

In [4], Mezouar proved the global existence and decay properties of solutions for the following viscoelastic Kirchhoff equation:

  ( ) ( ) ( )

( ( ) ) ( ( ( ) ) )

2

0

1 1 , 2 2 ,

0.

l t

t tt tt

t t

u u M u u u h t s u s ds

u g u x t g u x t t

   

−   −  + − 

+ + + −

=



(4) Moreover, in [5], she established the global existence and exponential decay of solutions for generalized coupled Kirchhoff system with a time varying delay term.

In [12], Pişkin considered the following system of viscoelastic wave equations with weak damping terms:

( ) ( ) ( )

1 1

0

2 2

0

, , .

t

tt t

t

tt t

u u g t u d u f u v

v v g t v d v f u v

  

 −  + −  + =



 −  + −  + =





(5)

He obtained the global nonexistence of solutions for the problem (5). Moreover, in [11], Pişkin proved the blow up of solutions for coupled nonlinear Klein-Gordon equations with weak damping terms. Also, Pişkin, in [13], established the decay estimates of the solution by using Nakao’s inequality and proved the blow up of solution in a finite time with negative initial energy.

Motivated by previous works, we prove the exponential growth of solutions of the system (1) both Kirchhoff term ( ^M

(

^^u ²

)

) that depends on ( u ² ) and viscoelastic term (



₀^t^{g t}

(

^{− }^s

) ( )

^{u s ds}) with distributed delay terms ( ²

( ) ( )

1

2 u x t_t , d



   − 



) in the system, and

in a similar way, we benefit from the study of [14]. We investigated that the solutions of system (1) grows exponentially under suitable conditions.

(3)

The content of this paper is organized as follows: In section 2, we provide some assumptions and lemmas that will be used later. In section 3, we prove our main result that we will get the growth result for the system (1).

PRELIMINARIES

In this part, we give some assumptions and lemmas which will be used throught this work.

Firstly, we make the following assumptions:

(A1) , h: Rg ₊ →R₊ are differential decreasing functions such that

( ) ( )

0 1

0 2

0, 1- 0,

0, 1- 0.

g t g s ds l

h t g s ds l



  = 



  = 





(6) (A2) There exist constants  ₁, ₂ such that

( ) ( ) ( )

¹2

( )

' , t 0,

' , t 0.

g t g t

h t h t



 − 

  − 

 (7) (A3)    2, 4:



1, 2



→R are L^ functions so that, for all ¹

 2,

( ) ( )

2 1

4 3

2 1

2 ,

2 1

2 .

d d



    

 −  

 

 − 

  

 





(8)

Concerning the functions f u v1

( )

, and f2

( )

u v, , we take the source terms as follows:

( )

⁽ ⁾

( )

⁽ ⁾

( )

2 1 2

1 1 1

2 1 2

2 1 1

, ,

p p p

f u v a u v u v b u u v f u v a u v u v b v v u

+ +

 = + + +



= + + +

 (9)

where a₁, b₁ 0.

We have the following lemmas.

Lemma 1 ([14]) There exists a function ^{F u v}

( )

^, such that

( ) ( ) ( ) ( )

( )

⁽ ⁾

1 2

2 2 2

1 2

, 1 , ,

2 2

1 2

2 2

0,

p p

F u v uf u v vf u v

p

a u v b uv

p

+ +

= +  + 

 

= +  + + 



(10)

where

( ) ( )

1 , , 2 , ,

F F

f u v f u v

u v

 =  =

 

taking a₁ = =b₁ 1 for convenience.

Lemma 2 ([8]) There exist two positive constants c₀ and c₁ such that

(

⁰

) (

²⁽ ²⁾ ²⁽ ²⁾

) ( )

^,

(

¹

) (

²⁽ ²⁾ ²⁽ ²⁾

)

^.

2 2 2 2

p p p p

c c

u v F u v u v

p p

+ + + +

+   +

+ + (11)

Lemma 3 ([10]) For ^{ }^, ^C¹

(

^{R R}+^,

)

we have

(4)

( ) ( )

²

( )( ) ( )( ) ( 0 ( ) ) 2

1 1 1

* '

2 2 2

t t

dx t t t d t s ds

      dt    



 

= − + −  − 

 

(12)

where

(

^{ }

)( )

^t ⁼



₀^t^

(

^t⁻^s

) ( )

^ ^t ⁻^

( )

^s ²^ds^.

GROWTH RESULT

In this part, we prove the exponential growth of solutions of the problem (1). In order to do so, we first introduce, as in [9], the new variables:

( ) ( )

, , , , ,

t t

y x t u x t z x t v x t

  

= −

thus we have,

( ) ( )

, , , , , , 0,

, 0, , , ,

t

y x t y x t

y x t u x t

     



+ =

= (13) and

( ) ( )

, , , , , , 0,

, 0, , , .

t

z x t z x t

z x t v x t

     



+ =

= (14) Hence, problem (1) is equivalent to

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

2 1

2

0 1

2 1

2

0 3

4 2

,

,1, , , , x , 0,

,

,1, , , , x , 0,

, , , , , , 0,

t

tt t

t

tt t

t t

u M u u g t s u s ds u x t

y x t d f u v t

v M v v h t s v s ds v x t

z x t d f u v t

y x t y x t

z x t z x t







   



   

    

 −   + −  +



+ =  



 −   + −  +



+ =  



+ =

 + =





(15)

with the initial and boundary condition

( ) ( )

( ) ( ) ( ) ( )

0 0

0 1

, 0, v , 0, x ,

, , , 0 , , z , , , 0 , ,

, 0 , u , 0 ,

, 0 , v , 0 ,

t t

u x t x t

y x f x x k x

u x u x x u x

v x v x x v x

     

= = 

 = =

 = =

 = =



(16)

where

(

x, , ,  t

)



( ) (

0,1   1, 2

) (

 0,

)

. We define the functional space ˆ as

( ) ( ) ( ) ( ) ( ( ) ) ( )

( ( ) ) ( )

1 2 1 2 2

0 0 1 2

2

1 2

ˆ 0,1 ,

0,1 , .

H L H L L

L

 

 =          

  

Theorem 4 ([14]) Suppose that (6), (7) and (8) hold. Let us

1 4 , n 3,

2 1, n=1,2.

p n n p

−   − 

−

 −

(17)

(5)

Then, for any initial data,

(

u0, , v , v , , ku1 0 1 f0 0

)

, the problem (15) has a unique solution, in

(  

^0, ^;

)

C T  for some T0.

Lemma 5 Suppose that (6), (7), (8) and (17) hold, let

(

^{u v y z}^{, , ,}

)

be a solution of (15), then ^{E t}

( )

is nonincreasing,

( ) ( )

⁽ ⁾

( )

⁽ ⁾

( ) ( ) ( )

( )

2 2 2 1 2 1

2 2

1 2

1 1 1 1

2 2 2 1 2 1

1 1 1 1 1

2 2 2 2 2 ,

,

t t

E t u v u v

l u l v g u h v M y z

F u v dx

 

 

+ +



= + +  + 

+ +

+  +  +  +  +

−



(18)

satisfies

( )  12 ( ) ( ) 12 ( ) ( ) 

2 2 2 2

3 2 2

' ,1, , ,1, ,

0,

t t

E t c u v ^ y x t d dx ^ z x t d dx

         

 

 − + + +



   

₍₁₉₎

where

( )

²

 ( ) ( ) ( ) ( ) 

1

1 2 2

2 4

, 0 , , , , , , .

M y z ^ y x t z x t d d dx

           

=

  

 + (20)

Proof. We multiply the first and the second equation in (15) respectively by u_t, v_t and integrating over , we obtain

( )

⁽ ⁾

( )

⁽ ⁾

( ) ( ) ( )

( ) ( )

2 1 2 1

2 2 2 1 2 1

2 2

1 2

2

1 2

2

3 4

2

1 1 1 1

2 2 2 1 2 1

1 1 1 1

2 2 2 2 ,

,1, , ,1, ,

1 1

2 ' 2

1 1

' ,

2 2

t t

d u v u v

dt

d l u l v g u h v F u v dx

dt

u u y x t d dx

v v z x t d dx

g u g t u

h v h t v

 



 

 

    



+ +



 

 + +  +  

 + + 

 

 

  +  +  +  − 

 

 

= − −

− −

+  − 



 

(21)

and by using the initial and boundary conditions in (15), we have

(6)

( ) ( )

( )

( ) ( )

( ( ) )

( ) ( )

2 1

2 1 2 1

1 2

0 2 1 0 2

2 2

1 , , ,

2

1 2

2

1 , 0, ,

2

1 ,1, ,

2 1 2

1 ,1, ,

2

t

d y x t d d dx

dt

yy d d dx y x t d dx y x t d dx

d u

y x t d dx



 



      

   

   

  

   



= −

=

−

=

−

  

 



 

(22)

and

( ) ( )

( )

( ) ( )

( ( ) )

( ) ( )

2 1

1 2

0 4 1 0 4

2 4

1 , , ,

2

1 2

2

1 , 0, ,

2

1 ,1, ,

2 1 2

1 ,1, , ,

2

t

d z x t d d dx

dt

zz d d dx z x t d dx z x t d dx

d v

z x t d dx



 



      

   

   

  

   



= −

=

−

=

−

  

 



 

(23)

then

( ) ( ) ( ) ( )

( ) ( ( ) )

( ) ( )

( ) ( ) ( )

( ) ( ( ) )

( ) ( )

2 1

2 1 2

1 2 1

2 1

2

1 2

2 2

2

2 2 2

3 4

2 2

4

2 4

,1, , 1 '

2

1 1

2 2

1 ,1, ,

2

,1, , 1 '

2

1 1

2 2

1 ,1, , .

2

t t

t

t t

t

d E t u u y x t d dx g u

dt

g t u d u

y x t d dx

v v z x t d dx h u

h t v d v

z x t d dx



     

  

   

     

  

   



= − − + 

−  +

−

− − + 

−  +

−

 



 



 

(24)

From (21)-(23), we obtain (18). Moreover, utilizing Young’s inequality, (6), (7) and (8) in (24), we get (19).

Next, we define the functional

(7)

( ) ( ) ( )

⁽ ⁾

( )

⁽ ⁾

( ) ( ) ( )

( )

⁽⁽ ⁾⁾ ⁽⁽ ⁾⁾

2 2 2 1

2 1 2 2

1 2

2 2 2 2

1 1 1

2 2 2 1

1 1 1

2 1 2 2

1 1 1

2 2 2 ,

1 2 .

2 2

t t

p p

H t E t u v u

v l u l v

g u h v M y z

u v uv

p



 

+

+ +

= − = − − − 

+

−  −  − 

+

−  −  −

 

+ +  + + 

(25)

Theorem 6 Suppose that (6)-(8) and (17) hold. Suppose further that ^E

( )

⁰ ^^0,then the solution of problem (15) grows exponentially.

Proof. By (18), we have

^{E t}

( )

^^E

( )

⁰ ^^0. (26) Hence,

( ) ( )

( )

( ) ( )

( )

2 1 2 1

2 2

3 2

2 2

3 4

' '

,1, ,

,1, , .

t

H t E t

c u y x t d dx

c v z x t d dx



   



= −

 +

+ +

 

(27)

Therefore

( )  12 ( ) ( ) 12 ( ) ( ) 

2 2

3 2 4

' max ,1, , , ,1, , 0

H t c ^ y x t d dx ^ z x t d dx

         

 



   

 ⁽²⁸⁾

and

( ) ( )

( )

⁽⁽ ⁾⁾ ⁽⁽ ⁾⁾

( )

⁽⁽ ⁾⁾ ⁽⁽ ⁾⁾

2 2 2 2

1

2 2 2 2

0 0

1 2

2 2

2 2 .

p p

H H t

u v uv

p

c u v

p

+ +

 

 

 +  + + 

 

 +  + 

(29)

We set

( ) ( ) ( ) (

¹ ² ³ ²

)

t t 2

K t H t  uu vv dx  u  v dx

 

= +



+ +



+ (30)

where 0to be given later.

We multiply the first and second equations on (15) respectively by u, v and with a derivative of (30), we obtain

( ) ( ) ( ) ( )

( )

2 2

1 1

2 2 2 2

0 0

2 4

2 2 2 2

' '

,1, , ,1, ,

2 .

t t

p p

K t H t u v u v

u u dx v v dx

u g t s u s dsdx v h t s v s dsdx

uy x t d dx vz x t d dx

u v uv

 

 

 

         



 

+ +

= + + −  + 

−   −  

+  −  +  − 

− −

 

+  + + 

 

   

(31)

By using Young’s inequality, we obtain

(8)

( ) ( )

( ( ) )

( ) ( )

2 1

2 1 2 1

2

1 2

2 2 1

,1, ,

,1, , .

4

uy x t d dx d u

y x t d dx



    

   

    





 +

 



 

(32)

Therefore

( ) ( )

( ( ) )

( ) ( )

2

1 2 1 2 1

4

2

2 4

2 4 2

,1, ,

,1, , .

4

vz x t d dx d v

z x t d dx



    

   

    





 +

 



 

(33)

Moreover,

( ) ( )

( ) ( ( ) ( ) )

( ( ) )

( ( ) ) ( )

0

2 0

2

0 ,

2 2

t

g t s ds u u s dxds g t s ds u u s u t dxds

g s ds u

g s ds u g u



  



−  

= −   − 

+ 

  − 

 



(34)

hence

( ) ( )

( ) ( ( ) ( ) )

( ( ) )

( ( ) ) ( )

0

2 0

2

0 .

2 2

t

h t s ds v v s dxds h t s ds v v s v t dxds

g s ds v

h s ds v g v



  



−  

= −   − 

+ 

  − 

 



(35)

By (31),

( ) ( ) ( ) (

⁽ ⁾ ⁽ ⁾

)

( ) ( )

( ( ) ) ( ( ) )

( ) ( ) ( )

( )

2 2

1 1

2 1

2 2 2 1 2 1

2 2

0 0

2 2

1 2 2 4

2 2 1

2 4 2

2 2

2 2 2

' '

1 1

2 2

,1, ,

2 4

,1, ,

2 4

2

t t

p p

K t H t u v u v

g s ds u h s ds v

d u d v

g u y x t d dx

h v z x t d dx

u v uv

 

 



 



       

      



      





+ +



 + +

 + + −  + 

    

−  −   + −   

− −

−  −

+ + +

 

( )

2 2

2 .

p p +

 + 

 

(36)

(9)

Thus by choosing  ₁, ₂ such that

1 3 2 3

1 1

4 c 4 c 2



 ⁼  ⁼ and by using (28) in (36), we obtain

( )   ( ) ( ) (

⁽ ⁾ ⁽ ⁾

)

( ) ( )

( ( ) ) ( )

( )

2 1

2 2 2 1 2 1

2 2

0 0

2 2

3

2 4

3

2 2 2 2

' 1 '

1 1

2 2

1

2 2

1

2 2

2 .

t t

p p

K t H t u v u v

g s ds u h s ds v

d u g u

c

d v h v

c

u v uv

 



  

 

     



     





+ +

 − + + −  + 

   

−  −   −  −  

− − 

 

+  + + 

 



(37)

By (25) and for 0 a 1, we have

( )

( )( ) ( )

( )( )

( ) (

⁽ ⁾ ⁽ ⁾

)

( )( ) ( ( ) )

( )( )( )

( )( ) ( )

2 2 2 2

2 2

2 1 2 1

2 0

2 2

2 2 1

2 1 2 1

1

2 1 1

2 1 2 1

2 1 , .

p p

t t

t

u v uv

a u v uv

p a H t

p a u v

p a

u v

p a g s ds u

p a h s ds v

p a g u

p a h u

p a M y z

 



 



 



+ +

 + + 

 

 

=  + + 

+ + −

+ + − +

+ −

+  + 

+

+ + − − 

+ + − 

+ + −



(38)

Substituting in (37), we obtain

(10)

( )   ( ) ( )( ) ( )

( )( )

( ) (

⁽ ⁾ ⁽ ⁾

)

( )( ) ( ( ) ) ( )

( ( ) ) ( ( ) )

( )( )

2 2

1 1

2 2

2 1 2 1

2

0 0

2

0 0

2 2

2 4

3 3

' 1 ' 2 1 1

2 1 1

1

2 1 1 1 1

2

2 1 1 1 1

2

1 1

2 2

2 1

t t

K t H t p a u v

p a

u v

p a g s ds g s ds u

p a h s ds h s ds v

d u d v

c c

p a M y

 

 

 

 



       

 



+ +

 − +  + − +  +

 + − 

+  + −   + 

  

+  + − − − −  

  

+  + − − − −  

− −

+ + −

 

( ) ( )( ) ( ( ) ( ) )

( )

( )( ) ( )

2 2 2 2

, 2 1 1

2

2 2 2 1 .

p p

z p a g u h u

a u v uv p a H t

  

 ⁺₊ ⁺₊ 

 

+  + − −   + 

 

+  + + + + −

Utilizing Poincare’s inequality, we get

( )   ( ) ( )( ) ( )

( )( )

( ) (

⁽ ⁾ ⁽ ⁾

)

( )( ) ( ( ) ) ( )( )

( ( ) )

( )( ) ( ( ) ) ( )( )

( ( ) )

2 1

2 2

2 1 2 1

2 0

2 2

2 0

4

' 1 ' 2 1 1

2 1 1

1

2 1 1 2 1 1

2

2 1 1 2 1 1

2

t t

t

K t H t p a u v

p a

u v

p a g s ds p a u

c d u

p a g s ds p a v

c d

 



 

 



  





  



+ +

 − +  + − +  +

 + − 

+  + −   + 

  

+  + − − −  + − −  

−  

 

 

  

+  + − − −  + − −  

−  

 

 



( )( ) ( ) ( )( ) ( ( ) ( ) )

( )

( )( ) ( )

2

2 2 2 2

0 2 2 2 2

2 1 , 2 1 1

2

2 2 1 .

p p

v

p a M y z p a g u h u

c a u v

p a H t

   



+ +

 

+ + − +  + − −   + 

 

+  + 

+ + − (39)

Here, by taking a0 small enough which gives

( )( )

1 p 2 1 a 1 0

 = + − −  and

( )( )

( )

2 1 1 0

1

p a



 + − 

 − 

 + 

 

also we suppose that



⁰

( )

⁰

( )  ( ( )( )( ) )

¹

1

2 1 1

max , .

1 1

2 1 2 2

p a

g s ds h s ds

p a



  + − −

 =

+ − − +

 

(40)

Choosing  so large that

(11)

( )( )

( ) ( ) ( )( )

(

1²

( ) )

2 0

2

2 1 1 2 1 1

2 2

0,

t

p a g s ds p a

c ^ s ds





 

 

= + − − −  + − − 

−





( )( )

( ) ( ) ( )( )

(

1²

( ) )

3 0

4

2 1 1 2 1 1

2

0.

p a th s ds p a

c ^ s ds





 

 

= + − − −  + − − 

 

−





After fixing  and a,we assign  small enough such that

4 1 0,

 = − and

( ) ( )

⁽⁽ ⁾⁾ ⁽⁽ ⁾⁾

2 2 2 2

1

2 2 2 2 .

2 2

p p

K t c u v

p

+ +

 

 +  +  (41) Therefore, for some  0, the estimate (39) becomes

( )  ( )

⁽ ⁾ ⁽ ⁾



( ) ( ) ( )

₍⁽ ₎⁾ ₍⁽ ₎⁾

 

2 2 2 2 2 1 2 1

2 2 2 2

'

, .

t t

p p

K t H t u v u v u v

g u h v M y z u v

 



  

+ +

 + + +  +  +  + 

 

+  +  + + + 

(42) From (11), for some ₁0,

( )  ( )

⁽ ⁾ ⁽ ⁾



( ) ( ) ( )

₍⁽ ₎⁾ ₍⁽ ₎⁾

 

2 2 2 2 2 1 2 1

1

2 2 2 2

1 2 2 2 2

'

, 2

t t

p p

K t H t u v u v u v

g u h v M y z u v uv

 



  

+ +

 + + +  +  +  + 

 

+  +  + + + + 

(43) and

^{K t}

( )

^^K

( )

⁰ ^^0,^t^^0. (44) Now, we use Young’s and Poincare’s inequalities, hence from (30), we obtain

( ) ( ) ( ) ( )

( ) ( )

 

( )

⁽ ⁾ ⁽ ⁾

( )

⁽ ⁾ ⁽ ⁾

( ) ( )

₍⁽ ₎⁾ ₍⁽ ₎⁾

2 2

1 3

2 2

2 2 2 2 2 1 2 1

2 2 2 2

t t 2

t t

p p

K t H t uu vv dx u v dx

H t uu vv dx u v

c H t u v u v u v

g u h v u v

 

   

 

 



+ +

 

= + + + + 

 

 + + +  + 

 

  + + +  +  +  +  

 

  + + +  +  +  +  

 

+  +  + + 

 



(45)

for some c0. Utilizing the inequalities in (42) and (45) we get the differential inequality ^{K t}^'

( )

^^^{K t}

( )

^, (46) where 0, depending only on  and c.

A simple integration of (46) gives us

^{K t}

( )

^^K

( )

⁰ ^e^^t^{for any}^t^^0. (47) From (41) and (47), we obtain

(12)

( )

2 2 2 2

2 ^p 2 2 ^p 2 ^t, 0.

p p

u ⁺₊ + v ⁺₊ Ce^  t Hence, we completed the proof.

CONCLUSION

In recent years, there has been published much work concerning the wave equation with constant delay or time-varying delay. However, to the best of our knowledge, there was no growth of solutions for the coupled viscoelastic Kirchhoff system with distributed delay terms. We have been proved the growth of solutions for problem (1) under the sufficient conditions. This improves and extends many results in the literature.

ACKNOWLEDGEMENT

The authors are grateful to DUBAP (ZGEF.20.009) for research funds.

REFERENCES

Cavalcanti, M. M., Cavalcanti, D. P., Filho, J. S. and Soriano, J. A. (2001), Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping, Differ.

Integral Equ., 14: 85-116.

Fridman, E., Nicaise, S., Valein, J. (2010), Stabilization of second order evolution equations with unbounded feedback with time-dependent delay, SIAM J. Control Optim, 48(08): 5028- 5052.

Kafini, M., and Messaoudi, S. A. (2008), A blow-up result in a Cauchy viscoelastic problem, Appl. Math. Lett., 21: 549-553.

Mezouar, N., Boulaaras, S. (2020), Global existence and decay of solutions for a class of viscoelastic Kirchhoff equation, Bull. Malays. Math. Sci. Soc., 43: 725-755.

Mezouar, N., Boularas, S. (2020), Global existence and exponential decay of solutions for generalized coupled non-degenerate Kirchhoff system with a time varying delay term, Bound. Value Probl., 2020:90, https://doi.org/10.1186/s13661-020-01390-9.

Messaoudi, S. A. (2003), Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr., 260: 58-66.

Messaoudi, S. A. (2006), Blow up of solutions with positive initial energy in a nonlinear viscoelastic wave equations, J. Math. Anal. Appl., 320: 902-915.

Messaoudi, S. A., Said-Houari, B. (2010), Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, J. Math. Anal. Appl., 365: 277-287.

Nicaise, S. And Pignotti,C. (2008), Stabilization of the wave equation with boundary or internal distributed delay, Differ. Integral Equ., 21: 935-958.

Ouchenane, D., Zennir, K., Bayoud, M. (2013), Global nonexistence of solutions for a system of non-linear viscoelastic wave equation with degenerate damping and source terms, Ukr.

Math. J., 65: 723-739.

Pişkin, E. (2014), Blow-up of solutions for coupled nonlinear Klein.Gordon equations with weak damping terms, Math. Sci. Lett., 3: 189-191.

Pişkin, E. (2015), Global nonexistence of solutions for a system of viscoelastic wave equations with weak damping terms, Malaya J. Mat., 3: 168-174.

Pişkin, E. (2014), Uniform decay and blow-up of solutions for coupled nonlinear Klein.Gordon equations with nonlinear damping terms, Math. Methods Appl. Sci., 37: 3036-3047.

(13)

Rahmoune, A., Ouchenane, D., Boulaaras, S. and Agarwal, P. (2020), Growth of solutions for a coupled nonlinear Klein-Gordon system with strong damping, source and distributed delay terms, Adv. Differ. Equ., 2020: 335.

Wang,Y. (2009), A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy, Appl. Math. Lett., 22: 1394-1400.