Applied Mathematics, 2010, 1, 529-533
doi:10.4236/am.2010.16070 Published Online December 2010 (http://www.SciRP.org/journal/am)
On the Exponential Decay of Solutions for Some
Kirchhoff-Type Modelling Equations with Strong
Dissipation
Yaojun Ye
Department of Math ematics and Informa tion Science, Zhejiang University of Science and Technology,
Hangzhou, China
Received August 24, 2010; revised October 19, 2010; accepted October 23, 2010
Abstract
This paper deals with the initial boundary value problem for a class of nonlinear Kirchhoff-type equations
with strong dissipative and source terms 22
2
() ||,,0
tt t
uuuaubuuxt




  in a bounded
domain, where ,0aband 2

 are constants. We obtain the global existence of solutions by construct-
ing a stable set in1
0()H and show the energy exponential decay estimate by applying a lemma of V. Ko-
mornik.
Keywords: Kirchhoff-type Equation; Initial Boundary Value Problem; Stable Set; Exponential Decay
Estimate
1. Introduction
Let be a bounded domain in n R with smooth boun-
dary. In this paper, we investigate the existence and
the energy exponential decay estimate of global solutions
for the initial boundary value problem of the following
Kirchhoff-type equation with strong dissipative and
source terms in a boun ded do m a i n

22
2,,0,
tt t
uuuaubuuxt



  (1.1)
01
(,0)(),(,0)(),,
t
ux uxux uxx (1.2)
(,) 0,,0,uxt xt (1.3)
where ,0ab and 2

are constants, ()
s

is a
1
C-class function on [0, ) satisfying
   
00
,,0,
s
smssd s   
 (1.4)
with 01m constant.
When 1n, the equation (1.1) describes a small am-
plitude vibration of an elastic string ([1]). The original
equation is
22
0
22
0
2
L
uuEhu u
hPdsf
tLx tx

 
 

 


where 0
x
L

and 0,t(,)uxt is the lateral dis-
placement at the space coordinate x and the time t,

is
the mass density, h is the cross-section area, L is the
length, 0 P is the initial axial tension,

is the resistance
modulus, E is the Young modulus and f is the external
force.
Many authors have studied the existence and unique-
ness of solutions of (1.1)-(1.3) by using various methods.
When ,0ab, and (), 1,
r
ssr 

 K. Nishihara and Y.
Yamada [2] have proved the existence and the polynomi-
al decay of global solution under the assumptions that the
initial data 0
u and 1 uare sufficiently small and 00u

.
However, the method in [2] can not be applied directly to
the case that the equations have the blow-up term
2
||uu
 . M. Aassila and A. Benaissa [3] extend the
global existence part of [2] to the case where () 0s


with


2
00u


 and the nonlinear dissipative term
2
||
tt
uu
 . K. Ono and K. Nishihara [4] have proved the
global existence and decay structure of solutions of
(1.1)-(1.3) without small condition of data using Galerkin
method. K. Ono [5] has obtained the global existence of
solutions for the problem (1.1)-(1.3) with dissipative term
t
u instead oft u

.
In the case 0a

, for large

and

0sr  , P.

Y. J. YE 530
D. Ancona and S. Spagnolo [6] proved that if

01 0
,n uuC R

 are small, then problem (1.1)-(1.3) has
a global solution. When

0,s  M. Ghisi and M. Gob-
bino [7] proved the existence and uniqueness of a global
solution

,uxt of (1.1)-(1.3) for small initial data
  


121
01 00
,uu HHH
with

2
00mu
and the asymptotic behavior
  
, ,,0,0
ttt
utu tutu


in
  

 
1212
01 00
,uu HHHL
as t, where either 0u

or

20u


.
The case () 0sr

 has been considered by M.
Hosoya and Y. Yamada [8] under the following condi-
tion: 2
0,5;0,4.
4nn n
 

They proved that, if the initial data are small enough,
the problem (1.1)-(1.3) has a global solution which de-
cays exponentially ast.
In this paper, we prove the global existence for the
problem (1.1) -(1.3) by applying the potential well theory
introduced by D. H. Sattinger [9] and L. Payne and D. H.
Sattinger [10]. Meanwhile, we obtain the exponential
decay estimate of global solutions by using the different
method from paper [8].
We adopt the usual notation and convention. Let m
H
denote the Sobolev space with the norm
2
2
() ()
|| ,
m
HL
m
uDu








0()
m
Hdenotes the closure in m
H
of 0()C. For sim-
plicity of notations, hereafter we denote by
p

the
Lebesgue space ()
p
L norm,  denotes 2()L

norm
and we write equivalent norm  instead of 1
0()H

norm 1
0()
.
H
Moreover,
M
denotes various positive
constants depending on the known constants and it may
be different at each appeara nc e.
2. Preliminary
In order to state and prove our main results, we first de-
fine the following functionals

2
0,
K
umu bu




2
0
2
mb
J
uuu




for

1
0
uH. Then we define the stable set S by
 


1
0,0, 0,SuHKuJud 
where
 


1
0
0
inf sup,0dJuuH



.
We denote the total energy functional associated with
(1.1)-(1.3) by
 
2
2
20
11
22
u
tb
Etusds u




 
 (2.1)
for


1
0,0uH t

, and
 
2
0
2
10
20
11
022
ub
Eu sdsu




 

is the total energy of the initial data.
Lemma 2.1 Let qbe a number with 2,q


2n

and 2
2,2.
2
n
qn
n


Then the re exists a cons -
tant C depending on

and q such that
1
0
1
0
()
,()
qH
uCu uH


 .
Lemma 2.2 [11] Let


:yt RR

be a nonincreas-
ing function and assume that there is a constant0A,
such that
() (),0,
sytdt Ayss




then 1 () (0),0.
t
yty et




We state a local existence result, which is known as a
standard one.
Theorem 2.1 Suppose that

satisfies
2
2,2;2 ,2.
2
n
nn
n


  
 (2.2)
If 12
01 0
(,)()()uu HL

 , then there exists 0T
such that the problem (1.1)-(1.3) has a unique local solu-
tion ()ut in the class






 

12
0
0, ;,0, ;.
t
uC THuC TL

  (2.3)
Lemma 2.3 Let (, )utx be a solutions of problem
(1.1)-(1.3). Then ()Et is a nonincreasing function for
0t and
 
.
t
dEtau t
dt   (2.4)
Proof By multiplying equation (1.1) by t u and inte-
grating over

, we get
 
0.
t
dEtau t
dt


Therefore,


Et is a nonincreasing function on t.
Lemma 2.4 Let

1
0
uH

, if (2.2) holds, then
0.d
Proof Since

22
0,
2
mb
J
uuu







Y. J. YE 531
so, we get

21
0.
d
J
umu bu
d



 



Let

0
dJu
d


, which implies that
1
12
2
12
0
.
u
b
mu






 
 

 
As 1

 , an elementary calculation shows that

2
20
dJu
d


.
Hence, we have from Lem ma 2.1 that
 
2
12
22
1
00
12
222
0
2
sup 2
20.
2
u
b
JuJ uu
m
bC
m









 












 

 





we get from t he definit i on of d that 0.d
In order to prove the existence of global solutions for
the problem (1.1)-(1. 3), we need the following Lemma.
Lemma 2.5 Supposed that (2.2) hold, If

2
01
,uSuL and

0Ed, then uS

, for
each


0, .tT
Proof Assume that there exists a number


*0,.tT
such that

ut Son
 
0, *t and

*utS. Then, in
virtue of the continuity of


ut, we see

*utS

.
From the definition of Sand the continuity of




J
ut
and
 
K
ut in t, we have either


*
J
ut d or


*0Kut

.
It follows from (1.4) and (2.1) that
   
 
2
0
***
2
*0.
mb
Jutut ut
Et Ed



 

(2.5)
So, the case
 
*
J
ut d is impossible.
Assume that



*0Kut holds, then we get that
 

2
2
0
*1 .
d
J
ut mu
d





We obtain from
 
*0
dJut
d


that 1

.
Since
  
2
10
2*2*0.
dJut mut
d




Consequently, we get from (2.5) that







  
1
0
sup ***
J
utJ utJutd







which contradicts the definition of d. Therefore, the
case (()) 0Kut

is impossible as well. Thus, we con-
clude that ()ut S

on [0, ).T
3. Main Results and Proof
Theorem 3.1 Suppose that (2.2) holds, and


ut is a
local solution of problem (1.1)-(1.3) on


0,T. If


2
01
,uSuL

 and

0Ed, then


,uxt is a
global solu tion of the problem (1.1)-(1.3 ).
Proof It suffices to show that
 
22
t
utu t
is bounded independently of t.
Under the hypotheses in Theorem 3.1, we get from
Lemma 2.5 that


ut S

on
 
0, .T So the following
formula holds on


0, .T
2
0
22
0
2
0
(())() ()
2
() ()
2
(2) () ,
2
mb
J
utut ut
mb
ut ut
mut






 
 


(3.1)
Therefore, we have from (3.1) that
22
0
2
(2) 1() ()
22
1()( ())()(0).
2
t
t
m
ut ut
utJutEt Ed






(3.2)
Hence, we get
22
0
2
()()max2,.
(2)
t
ut utd
m




 



The above inequality and the continuation principle lead
to the existence of global solution, that is, T

 .
Therefore, the solution


ut is a global solution of the
problem (1.1)-(1.3).
The following Theorem shows the exponential decay
estimate of global solutions for problem (1.1)-(1.3).
Theorem 3.2 If the hypo theses in Theor em 3.1 ar e va-
lid, then the global solutions of problem (1.1)-(1.3) has
the following expon ential decay property
 
1
0,
t
M
Et Ee


where 0M is a constant.
Proof Multiplying by u on both sides of the Equa-
tion (1.1) and integrating over [0, )T , we obtain that

Y. J. YE 532



22
2
0||,
T
tt t
Suuuua ubuudxdt






(3.3)
where 0.ST
Since
2.
TT
T
ttt St
SS
uudxdtuu dxudxdt
 

 (3.4)
So, substituting the Formula (3.4) into the right-hand
side of (3.3), we get that

222
2
2
2
0
2
21.
T
t
S
T
tt
S
T
T
tS S
b
uuuudt
uauu dxdt
uu dxbudt






















(3.5)
It follows from (3.2) that
   
2
00 0
22 2
0
222
utEt Ed
mm m
 
 
 
 
(3.6)
By exploiting Lemma 2.1 and (3.6), we easily arrive at
 
 
22
2
22
0
2,
2
bubC utbCutut
bCdu t
m









 






(3.7)
We obtain from (3.6) and (3.7) that
2
22
0
2
2
00
2
2
00
222
1()
(2)
222
()
(2)(2)
22 ().
(2)
bubC d ut
m
bCdE t
mm
bC dEt
mm








 








 
 

 
 










(3.8)
We derive from (1.4) that


222
0,
u
s
dsuu 
 
 (3.9)
It follows fro m (3.5), (3.8) and (3.9) that
 
2
2
00
2
2
21 2
2| |.
T
S
TT
tt tS
S
bC dEtdt
mm
uauudxdtuu dx










 








 
(3.10)
We have from Lemma 2.1 and (3.2) that


  
2 2
222
0
0
2
0
11
22
21
222
max ,1,
2
TT
tStS
T
tS
T
S
uu dxuu
m
Cuu
m
CEt MES
m
























(3.11)
Substituting the estimate (3.11) into (3.10), we conclude
that
 

2
2
00
2
22
21 2
2.
T
S
T
tt
S
bC dEtdt
mm
uauudxdtMES










 









(3.12)
We get from Lemma 2.1 and Lemma 2.3 that
 


22 2
2
22
222
22
.
TTT
tt t
SSS
udxdtudt Cudt
CC
ET ESES
aa

 
 (3.13)
From Young inequality, Lemma 2.1, Lemma 2.3 and
(3.6), We receive that



 

 
22
0
0
2
2
2.
2
TT
tt
SS
T
S
T
S
au udxdtauMudt
aEtdtMESET
m
aEtdtMES
m

 

 


 




 


(3.14)
Choosing small enough

such that
 
2
2
00 0
22
1,
22
abCd
mm m


 









then, substituting (3.13) and (3.14) into (3.12),

.
T
SEtdt MES
 (3.15)
Let T, then we have from (3.15) that

.
SEtdt MES
 
 (3.16)
Thus, we receive from (3.16) and Lemma 3.1 that
   
1
0,0,.
t
M
Et E et


 (3.17)
The proof of Theorem 3.2 is finished.
4. Acknowledgments
This Research was supported by Natural Science Foun-

Y. J. YE 533
dation of Zhejiang Province (No.Y6100016), The Sci-
ence and Research Project of Zhejiang Province Educa-
tion Commission (No. Y200803804 and Y200907298),
The Research Fundation of Zhejiang University of Sci-
ence and Technology (No. 200803) and the Middleaged
and Young Leader in Zhejiang University of Science an d
Technology (2008-2012).
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