Advances in Pure Mathematics, 2011, 1, 276-279
doi:10.4236/apm.2011.15049 Published Online September 2011 (http://www.SciRP.org/journal/apm)
Exponential Decay Rate of the Perturbed Energy of the
Wave Equation with Zero Order Term
Hamchi Ilhem
Department of Mat hematics, University of Batna, Algeria
Received April 30, 2011; revised May 26, 2011; accepted June 10, 2011
Abstract
In this paper, we consider the wave equation with zero order term. We use the compactness uniqueness ar-
gument and some result of I. Lasiecka and D. Tataru in [4] to prove, directly, the exponential decay rate of
the perturbed energy.
Keywords: Perturbed Energy, Compactness Uniqueness Argument
1. Introduction
Consider the boundary feed-back system
 
0
1
01
=0 in ,
=0 on ,
=0 on ,
0=,0= in .
tt
t
t
yyqy Q
y
yby
yyy y

 









(1)
Here is a bounded domain of with
smooth boundary and

is a partition of


*nn


01 ,

such that and
0
 01 =.  , >0T


=0,QT,


=0,
ll T . 1

=0,1l:b


,qq


and are two positiv e bounded fu nctions, that
is there exists four positive constants  such
that for all and
for all
:q
0<bb

0<( )qq
x
,,bb
1,( )xb

x
q

.x

Fix a point 0
x
of n . Set 0 =()=xxhhx

and

=sup():.Rhx

x Assume that for some constant
we have
0>0h
 
01
=:.0 and =:.
0
.
x
hxh   h 
For all >0,

we define the perturbed energy of the
system (1) by for all
Here is the usual energy defined by
 
=,Et Ett

 0.t
E


222
1
=d
2t
Etyqyy x


 and
()= d,
t
tyMyx 

 where

=2 .1.
M
yhyny 
In the ca se of V. Komornick and E. Zuazua in
[3] have shown that
=0,qE

decays exponentially. When
0q

, there is some difficulty to obtain this result since
we have a lower order term with respect to the energy in
some multiplier estimate (see (3.5) in [3]). The purpose
of this paper is to overcome this kind of difficulty, where
we prove an useful estimate then by the compactness
uniqueness argument we absorb the lower order term.
Finally, we employ some result of I. Lasiecka and D.
Tataru in [4].
In all this paper, is a generic positive constant in-
dependent of the initial data and it may change from line
to line.
C
2. Exponential decay rate of
E

Using the multiplier method we can show that the energy
of the system (1) is a decreasing function. That is, for
all we have
Et0,

2
1
d
== d0.
d
tt
E
Et by
t
 (2)

The proof of the main theorem involves two lemmas.
Lemma 1 For all two positive constants and S
such that 0
T
>TST

, where 0 is some sufficiently
large positive constant, we have for
T

sufficiently small
 
2
0
3d.
4
T
S
CTE TESCyQ




 (3)
Here d=dd.Qxt
Proof. First we have from the definition of E







11EtE tEt

  , (4)
where

is the constant verifying

H. ILHEM277
 
=d
t
tyMyxEt 


.
On the other hand , we have (s ee (3.5) in [3])
 
2d.
t
EtCEtCqy x



 
Integrate over


,ST use (4) with

sufficiently
small and by the decreasing of , we find
E
 

2
01d.
1
T
S
ESETES
CTE TCyQ






 

 

Therefore
 
2
01d.
1
T
S
CTE TESCyQ








With

sufficiently small we find the desired result.
To absorb the lower order term 2d
T
S
yQ

 from the
estimate (3) we apply the compactness uniqueness argu-
ment (see for example [1]).
Lemma 2 For , where is sufficiently
large, we have 0
>TST0 T
22
1
d
TT
t
SS
yQCby


 d, (5)
where d=dd.t
Proof. First, we have from (3)
 
2
00
d,
T
S
CC
ETESy Q
TT





with sufficiently large, we obtain
0
T
 
2
0
d.
T
S
C
ETESy Q
T




 (6)
On the other hand , by (2)
 

2
1
2
1
=d=
1d.
1
TT
tt
SS
T
t
S
ESETE ttETby
ET by









d
Then, with

sufficiently small, we find
 
 

2
1
2
1
1
11d
1
42d,
3
T
t
S
T
t
S
ES ES
ET by
ET by




 







 


by (6), we find
 
22
01
dd
TT
t
SS
C
ESESCy QCby
T



,


 
with sufficiently large, we find
0
T

22
1
dd
TT
t
SS
ESCbyy Q







 . (7)
Now, we come back to the proof of (5).
It is sufficient to prove (see [1]] that, for some
large enoug h, we have 0
T
00
22
00
1
dd
TT
t
yQCby

.



We argue by contradiction. There exists a sequence of
solutions


k
y
of system (1) such that
02
01
limd= 0
T
kt
kby  

 (8)
and
02
0
d=1 for all ,
T
k
yQ k

 (9)
where represents the energy of .
k
Ek y
Let, for all , k

 
=
kk k
Et Ett

 

, where


=d
kkt
tyMy 

k
x

.
If we apply (7) with and
0, we obtain by (8), (9) and (4) that
 
=,
k
Et Et

=0S
=TT




0
k
E is
bounded and therefore there exists a subsequence


k
y
such that


1
00
weakly in0,;,
k
yy LTH



and

 
20
weaklyin 0,.
k
yy LT
Using (8) and passing to the limit, we obtain




0
00
01
=0 in 0,,
=0 on 0,,
==0 on 0,.


 





tt
t
yyqyT
yT
yby T
Hence we find
If then
=0y02
0
limd=0.
T
k
kyQ  
 This contradicts
(9).
If 0y

, then is solution of =t zy


0
=0 in0,,

 
tt
zzqzT

H. ILHEM 278



0
01
=0 on 0,,
=0 on 0,.




zT
zT
Then, for sufficiently large, . So, is a
solution of 0
T=0zy




0
00
01
=0 in 0,,
=0 on 0,,
=0 on 0,.

 



yqy T
yT
yT
If we multiply the first equation by , integrate over
and use the first Green’s formula we find
y
Q

022
0
d=0,
T
yqyQ



thus . =0y
We give, now, the proof of the main result.
Theorem 3 For any initial data

01
,
y
yD, the en-
ergy perturbed E

is exponentially stable.
Proof. If we insert (5) in (3) we find
 
2
0
1
3d,
4
T
t
S
CTE TESCby




 
But
  

2
1
d= d=
1.
1
TT
tt
SS
byEttE SET
ES



 

 
Then
 
01
1
C
CTE TES








If we choose sufficiently large we find
such that 0
T0< <1r
 
1,ET ES
r

 (10)
for all
0
This imply that
.TST
 
0
for all .ET ESTST

 (11)
If we apply (10) repeatedly on the intervals
, , we get

00
,1mT mT
 
=0,1,m
   
 

0
00
0
11
1
=1 .
r
Em TEmT
r
Em TEmT
r


 

1
Put

1
=r ps s
r
 and

0
=
m


11
.
mmm
s
ps s


Using lemma 3.3 in [4] to obtain for all m

,
m
s
Sm
where


St is the solution of the system




 
1=
0= 0.
t
St rSt
SE






0
Here we have used
  



1=StI pStSt 
 
  
1
1=1 r
I
ISt
r





,
0
rSt.
The last system have the solution
then



1
=0
rt
St eE





1
00 for all .
rm
EmT eEm



Let then
0,tT
00
= where 0,tT mTT




0

this imply that
000
= where =2.tmTT T T 


Then by (11)





00
=.Et EmTEmT
 



So






20
11
00
000
tT
t
rr
TT
Et EmTeEeE

 


 
 .
Thus




0
0 for ,





t
Et cEetT

21
=


cer and
0
1
=. 

r
T where
Remark 4 We can treat exactly in the same way the
situation of the second order hyperbolic equation with
variable coefficients, linear zero order term and polyno-
mial growth of the nonlinear feedback near the origin. In
this case, we use the Riemann geometric approach to
handle the case of the variable coefficients principal part
(see [2]] to show, directly, that we have an exponential
or polynomial decay rate of the perturbed energy func-
tional defined for all >0

by
 
1
2
=,Et EttEt





where

depends on the behavior of non linear d amping
at the origin.
3. References
,
s
EmT

to find [1] S. Feng and X. Feng, “Nonlinear Internal Damping of

H. ILHEM 279
Wave Equations with Variable Coefficients,” Acta
Mathematica Sinica, Vol. 20, No. 6, 2004, pp. 1057-1072.
doi:10.1007/s10114-004-0394-3
[2] Y. Guo and P. F. Yao, “Stabilization of Euler-Bernoulli
Plate Equation with Variable Coefficients by Nonlinear
Boundary Feedback,” Journal of Mathematical Analysis
and Applications, Vol. 317, No. 1, 2006, pp. 50-70.
doi:10.1016/j.jmaa.2005.12.006
[3] V. Komornick and E. Zuazua, “A Direct Method for
Boundary Stabilization of the Wave Equation,” Journal
de Mathématiques Pures et Appliquées, Vol. 69, 1990, pp.
33-54.