)1 ( cavalcanti 2002 berrimi and messaoudi cavalcanti 2003
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نعنهعربللاربىبلالباقاةتاتا
in 0,
0 1
, 0, , 0
( ,0) , ,0 ,t
u x t x t
u x u x u x u x x
, 1
n
iji j j i
A a xx x
g t 0
, ,t
ttu x t Au x t g t ( , ) 0,Au x d
)1 (
نعنهعربللاربىبلالباقاةتاتا
ij ija
1 2( ) '( ) , 0,g t g t g t t
CavalcantiCavalcanti 20022002
t
tto
u u g t u d
( )t
a x u 20u u
Berrimi and Messaoudi
2u u
نعنهعربللاربىبلالباقاةتاتا
00
( ) ( ) ( )t
ttu k u div a x g t u d
( ) ( ) ( ) 0t
b x h u f u
xxbxa ,0
CavalcantiCavalcanti 20032003
نعنهعربللاربىبلالباقاةتاتا
0
( ) ( )t
t tt ttu u u u g t u d
0tu 0
0
CavalcantiCavalcanti 20012001
0
نعنهعربللاربىبلالباقاةتاتا
t tt ttu u u u
0
( ) ( )t
tg t u d u
2p
b u u
damping terms
source term
Messaoudi and Tatar Messaoudi and Tatar
0
( ) ( )t
tg t u d u
2p
b u u
0
نعنهعربللاربىبلالباقاةتاتا
MessaoudiMessaoudi
pp
نعنهعربللاربىبلالباقاةتاتا
in 0,
0
, ,t
ttu x t Au x t g t ( , ) 0,Au x d
0
(0) 0, 1 ( ) 0g g s ds
:g
'( ) ( ) , 0
'( ), ( ) 0, '( ) 0, 0.
( )
g t t g t t
tk t t t
t
( )ijA a, , 1, 2, , a.e. inij jia a i j n x
2
, 1
( ) , , a.e. inn
nij i j
i j
a x x
0
( ) , , 1, 2, , a.e. inija x M i j n x
(A1)
(A2)
(A3)
(A4)
(A5)
)1 (
Decay of solutionsDecay of solutions
نعنهعربللاربىبلالباقاةتاتا
1 20 1 0( , )u u H L
1 2[0, ); , [0, ); .o tu C H u C L
2
20
1 1 1( ) 1 ( ) ( ( )) ( )( )
2 2 2
t
tt u g s ds B u t g u t
, 1
( ) ( )( ( ))
n
iji j i j
u t u tB u t a x dx
x x
0
( )( ) .t
g u t g t s B u t u s ds
Proposition
if
Then the solution
نعنهعربللاربىبلالباقاةتاتا
Decay of solutionsDecay of solutions
1 2F t t t t
0
,
( )( ( ) ( ))
t
t
t
t t uu dx
t t u g t u t u d dx
1 1 1'( ) ( ' )( ) ( ) ( ( )) ( ' )( ) 0
2 2 2t g u t g t B u t g u t
1 2F t t F t
1.
2.
Lemmas
(2)
(3)(5)
(4)
3. 10u H
2 2
00
1( )
tpcg t u t u d dx g u t
نعنهعربللاربىبلالباقاةتاتا
Decay of solutionsDecay of solutions
2 2
21
0
' 1 ( ( )) ( ),4
pt
k Ct t u dx t B u t t g u t
4. : tt t uu dx
(6)
نعنهعربللاربىبلالباقاةتاتا
ProofProof
2
, 1 0
' ' ( ( ))t t
tn
iji j i j
t t u dx t uu dx t B u t
u t u st g t s a x dxds
x x
(8)
0, 1 0
0 0
, 1 0 0
1 1( ) ( ( ))
2 2
1 1 1( ( )) ( )
2 2 2
( )
tn t
iji j i j
t t
t tn
iji j i j j
u t u sg t s a x dxds B u t B g t s u s ds
x x
B u t B g t s u s u t ds B g t s u t ds
u s u t u ta x g t s ds g t s ds
x x x
dx
(9)
نعنهعربللاربىبلالباقاةتاتا
0
22
, 1 , 10 0
1( )
2
4
t
t tn n
i j i ji i j j
B g t s u s u t ds
u s u t u s u tMg t s ds dx g t s ds dx
x x x x
(10)
2
, 1 0
0
1 .
tn
i j i i
u s u tg t s ds dx
x x
ng u t
(11)
نعنهعربللاربىبلالباقاةتاتا
00
1( ( ) ( )) 1 ( ).
2 2
t nMB g t s u s u t ds g u t
(12)
, 1 0 0
22
, 10
1
2
1 (1 )( ) .
2 2
t tn
iji j i j
t n
iji j i j
u t u ta x g t s ds g t s ds dx
x x
u t u tg s ds a x dx B u t
x x
(13)
نعنهعربللاربىبلالباقاةتاتا
0
1t
g s ds
, 1 0 0
2
0 0
(1 )1 ( ) ( ( )).
2 2
t tn
iji j i i j
u s u t u ta x g t s ds g t s ds dx
x x x
nM nMg Bu t B u t
(14)
, 1 00
20
0
1( ) ( ) 1 1 ( )
2
1(1 ) ( ( )).
2 2
tn
iji j i j
u t u s Mng t s a x dxds g u t
x x
n MB u t
(15)
2
2
0
( ) , 0,4
pt t
kCuu dx B u t u dx
k
0 / (1 )nM
نعنهعربللاربىبلالباقاةتاتا
Decay of solutionsDecay of solutions
5. 0
( ) : ( )( ( ) ( ))t
tt t u g t s u t u s dsdx
3 42 2
2
0
'( ) ( ( )) '
1 , 0,t
t
t t B u t g u t g u t
k g s ds t u dx
(16)
نعنهعربللاربىبلالباقاةتاتا
1 20 1 0( , ) ( ) ( )u u H L
0 0t
0t t
0
0( ) ,
t
t
s ds
t Ke t t
Let
be given Assume that (A1)-(A5) hold. Then, for each
, there exist strictly positive constants K and
,such that the solution of (1) satisfies, for all
TheoremTheorem
نعنهعربللاربىبلالباقاةتاتا
ProofProof
0
0 0
0 0
( ) ( ) 0, .tt
g s ds g s ds g t t (17)
2 22
2 0 1
4 12 2 2
31 1 2
'( ) { 1 } 1
1( ' )( ) ( ( ))
2 2
( )( ).
pt
k CF t g k t u dx
g u t t B u t
t g u t
(18)
نعنهعربللاربىبلالباقاةتاتا
0 0 2 02 2
1 4 11 , .
24 1 p
g k g gk C
0 0
2 1 22 2 2
4 1 2 1p p
g g
k C k C
2 2
1 2 0 1
12 2 2
{ 1 } 1 0,
04
pk Ck g k
k
(19)
343 2 1 1 2
10.
2k
نعنهعربللاربىبلالباقاةتاتا
342 1 1 2
42
0
31 1 2 3
1' ( )
2
1
2
.
t
g u t t g u t
t g t B u t u s dsdx
t g u t k t g u t
(20)
نعنهعربللاربىبلالباقاةتاتا
1 1 1 0'( ) ( ) ( ), .F t t F t t t (21)
1 1
00 0( ) ( ) , .
t
s ds
F t F t e t t
1 1
0 02 0 0( ) ( ) , .
t t
s ds s ds
t F t e Ke t t
(22)
(23)
نعنهعربللاربىبلالباقاةتاتا
00,t t ( ) and tt
Remark 3.1. This result generalizes and improves the results of [1-4]. In particular, it
allows some relaxation functions which satisfy
Remark 3.3. Estimates (23) is also true for
boundedness of
by virtue of continuity and
1 3/ 2. ' , 1 2pg ag
Remark 3.2. Note that the exponential and the polynomial decay estimates are only particular cases of (14). More precisely, we obtain exponential decay for
t a 1(1 ) , where 0 is a constant.t a t a
instead of
and polynomial decay for
