GLOBAL BEHAVIOUR OF SOLUTIONS TO NONLINEAR WAVE EQUATION

83

GLOBAL BEHAVIOUR OF SOLUTIONS TO NONLINEAR WAVE EQUATION

Şevket GÜR, Sema BAYRAKTAR

Sakarya Üniversitesi Fen Edebiyat Fakültesi Matematik Bölümü Sakarya, Türkiye E-posta:sgur@sakarya.edu.tr

ABSTRACT

This paper gives uniform stabilization of the energy of the solutions to nonlinear wave equation.

Key words: Global behaviour, nonlinear wave equation.

LİNEER OLAMAYAN DALGA DENKLEMİNİN ÇÖZÜMLERİNİN UZUN ZAMAN DAVRANIŞI

ÖZET

Bu çalışmada lineer olmayan dalga denkleminin çözümlerinin düzgün kararlılığı incelenmiştir.

Anahtar Kelimeler: genel davranış, lineer olmayan dalga denklemi.

1. INTRODUCTION

We consider the following initial boundary value problem

2

Ω, 0,

tt t t t

u    u  u u    u x  t  (1)

  , 0

0

  ,

_t

  , 0

1

  , Ω, u x  u x u x  u x x  (2)

0, Ω, 0.

u  x  t  (3)

where

Ω  R ,

ⁿ

n  3

bounded region with smooth boundary

 Ω

,

 and

and



positive constants.

84

This damped wave equation were suggested for a problem which is used to describe vibration with viscosity. Last term on the left-hand side of (1) is replaced

^{f u}  

was studied by Webb [1] in 1980. He proved the existence of a global strong solution to (1)-(3) for

n  1, 2,3

under four conditions regarding

^{f u}  

. In [2], the above conditions were weakened by taking out two of above four conditions in [1]. In [3], for

n  4

the existence of a global strong solution was obtained under some assumptions. Asymptotic behaviour of solutions to (1)-(3) under some assumptions was studied by Runzhang and Yacheng [5]. In this paper we aimed to give the global behaviour of solutions to problem with respect to coefficients of the nonlinear and damped terms.

Let us introduce certain notations used in this paper:



_p stands for the norm in

^L

^p

  ^Ω

^{. If}

^{p  2}

we write



instead of



₂ and

  ^,

^{is the}

innerproduct in

^L

²

  ^Ω

^{. 2}

   

0¹

 

Ω ,

1

Ω ,

H H H 

are the usual

Sobolev spaces.

Here is the important inequalities which we will use next.

Young's inequality with



:

 

p

1

q

q p

ab a b

p q





 

(4)

for each

a b , ,   0

with

,1 1

q p p

 p   



Poincare' inequality:

u  d

0

 u

(5)

for all

u  H

¹0

  Ω

where

d

₀is the Poincare coefficient which depends on the region

Ω

.

Sobolev inequality:

q 1

u  d  u

(6)

85

for al l

u  H

¹0

  Ω

and

q  2.

Ladyzhenskaya inequality:

1 1

2 2

4 2

u  d u  u (7)

for all

u  H

¹0

  Ω

.

d i

i

  ^{0,1, 2} 

are the constant related to inequalities.

Now let us remark that the following important lemma:

Lemma 1 (Lagnese,Haraux)

Let

E R :

^

 R

^ be a non-increasing function and assume that there exist a constant

T  0

such that

    ^,

t

E s ds TE t t R



  



 (8)

Then

    ⁰

^{1 Tt}

^,

E t  E e

^

  t T (9)

Proof. See [4]

2.

Behaviour of Solutions

Teorem 1.

Let

u

0

 H

0¹

  Ω , u

1

 L

²

  Ω

and

4  d

₂

 c d

_{1 0}

 0. Then

    ⁰

^{1 Tt0}

E t E e





(10)

Where T

₀

 0

86

Proof. Suppose that u x t is a solution of equation (1) and   ^,

satisfying the boundary conditions (2)-(3). Multiply the equation (1) by u and integrate over

_t

Ω . Due to boundary conditions we find

2 2 2 4

Ω

1 1

2

^t

2

^{t t}

0

d u u u u dx

dt ^     ^          ⁽¹¹⁾

Let us multiply the equation (1) by u and integrate over Ω we get

 

^{2 2 2 3}

Ω

, 0

t

2

t t

d d

u u u u u u udx

dt dt

 

        ⁽¹²⁾

By integrating (12) over (S,T) we obtain the relation

 

2 2 2 3

Ω

, | |

2

T T T

T T

t S S t t

S S S

u dt u u  u u dt  u udxdt

      

   ⁽¹³⁾

From (11) we have

   

^{2 4}

Ω

t t

d E t u u dx

dt       

where   ¹

²

¹

²

2

^t

2

E t  u   u is the energy integral of equation (1). Thus E t is nonincreasing. So  

 

²

 

^{2 3}

Ω

2 2 , | |

2

T T T

T T

t t S S t

S S S

E t dt  u dt  u u    u   u udxdt

   ⁽¹⁵⁾

87

Using the nonincreasing property of E t , the Cauchy-Schwarz  

inequality and definition of E t we have,  

   

 

       

2 2 2 2

0

2 2

0

0 0

1 1 1

, 2 2 2

1 1

max 1,

2 2

max 1, max 1,

t t t

t

u u u u d u u

d u u

d E t d E S

    

 

      

 

(16)

     

2

2 u E t E S S t

       (17)

     

2 0 2 4

4

0 0

T T

t t t

S S

u dt d u u dt

d d

E S E t E S

 



 

 

    

      

 

(18)

Using inequalities (4),(5),(6),(7) and the nonincreasing property of

 

E t we obtain the following estimates;

3 4 4

Ω Ω

4

Ω

T T T

t t

S S S

u udxdt u dxdt  u dxdt

       ⁽¹⁹⁾

4 2 2

2

4

Ω

T T

S S

u dxdt d u u dt

     

2 2 2

2 1 0

T

S

d  u ^ u u ^ dt

     

88

 

2

2 1 2 2 1 0

2

1 0 T

T T

S S

S

d  c u d t d  c d u dt d  c d E s ds

      

where c

₁

 u

₁ ²

  u

₀ ²

. Thus,

   

3

2 1 0

Ω

T T

t

S S

u udxdt E S d c d E s ds

       ⁽²⁰⁾

Using these estimates, we conclude from (15)

 2

2 1 0

   2 max 1,    

0

   

⁰

 

T

S

d  c d E t dt d E S  E S  E S d E S

      

 

0 ⁰

 

2 max 1, d

d   E S



 

        (21)

       

 

0 0

2 1 0

0

2 2 max 1,

2

T

S

E t dt d d E S

d c d T E S

   

 

        



 ₍₂₂₎

where

   

⁰

0 0

2 1 0

2 2 max 1,

2

T d d

d c d  

 

 

        

Letting T   this yields the following estimates:

 

0

 

S

E t dt T E S





 ⁽²³⁾

89 and from lemma we have

    ⁰

^{1 Tt0}

E t E e





.

3. KAYNAKLAR

[1]. Webb, G.F., “Existence and Asymptotic Behavior for a Strongly Damped Nonlinear Wave Equation”, Canadian Jounal of Mathematics, 32, 634- 643, 1980.

[2]. Yacheng, L., Dacheng,L., Initial value problem of equation

 

tt t

u    u    u f u

, Journal of Huazhang University of Science and Technology 16(6) ,169-173, 1988.

[3]. Yacheng, L., Wang Feng, Dacheng, L.,”Strongly damped nonlinear wave equation in arbitrary dimensions(I)”, Mathematical Applicata 8 (3),262-266, 1995.

[4]. Komornik, V., “Exact Controllability and Stabilization, The Multiplier Method”, J.Wiley Publ., 1994.

[5]. Runzhang,X., Yacheng, L.,” Asymptotic Behavior of Solutions for Initial Boundary Value Problem for Strongly Damped Nonlinear Wave Equations”, Nonlinear Analysis 69,2492-2495, 2008.