Applied Mathematics E-Notes, 15(2015), 121-136 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
Existence Of Solutions To Nonlinear th-Order
Coupled Klein-Gordon Equations With Nonlinear
Sources And Memory Terms
Khaled Zenniry, Amar Guesmiaz
Received 5 December 2014
Abstract
In this article, we consider a system of th-order derivatives of the dependent
variables of coupled Klein-Gordon equations to improve recent results obtained
in [10, 31, 33] using idea in [25]. Using the potential well method, we prove that
the solutions of (1) exist globally, under some conditions on the initial datum.
1
Introduction
In this paper, we consider a system of th-order derivatives of the dependent variables
of coupled Klein-Gordon equations
Rt
8 00
u1 + ( 1)
u1 + m21 u1 + 1 (t) 0 g1 (t s) u1 (x; s)ds + ju01 jr 2 u01
>
>
>
>
>
>
< = ju1 jp 2 u1 ju2 jp ;
(1)
Rt
>
00
2
0 r 2 0
>
u
+
(
1)
u
+
m
u
+
(t)
g
(t
s)
u
(x;
s)ds
+
ju
j
u
>
2
2
2
2
2
2
2
2
2
0
>
>
>
:
= ju2 jp 2 u2 ju1 jp
where mi ; i = 1; 2 are non-negative constants, r; p
2;
1. In a bounded domain
Rn Yaojun Ye [33] introduced related problem to (1) with = 1; i = 0; i =
1; 2, supplemented with the initial and Dirichlet boundary conditions. By using the
potential well method, global existence is discussed and asymptotic stability is also
given, by using multiplied method.
Erhan Piskin and Necat Polat [10] considered a system of class of nonlinear higherorder wave equations (1) with mi = gi = 0; i = 1; 2 and strong nonlinearity in sources.
Under suitable conditions on the initial datum, theorems of global existence and decay
rate are proved.
In (1), ui = ui (t; x); i = 1; 2; where x 2 is a bounded domain of Rn ; (n 1) with
a smooth boundary @ , t > 0. Our system is supplemented with the following initial
conditions
ui (x; 0) = ui0 (x) 2 H0 ( ); i = 1; 2;
(2)
Mathematics Subject Classi…cations: 35L05, 35B40, 35G31, 58J45.
LAMAHIS,University 20 Août 1955- Skikda 21000, Algeria
z Laboratory LAMAHIS,University 20 Août 1955- Skikda 21000, Algeria
y Laboratory
121
122
Existence of Solutions for a System of Klein-Gordon Equations
and boundary conditions
ui (x) =
u0i (x; 0)) = ui1 (x) 2 L2 ( ); i = 1; 2;
@ui
=
@
=
1
@
@
ui
1
= 0 for x 2 @
(3)
and i = 1; 2;
(4)
where is the outward normal to the boundary.
We mention here that
2
jr uj = (
and
where
=2
u)2 for pair value of
2
jr uj = jr(
jruj2 =
n
X
i=1
@u
@xi
(
1)=2
u)j2 for odd
2
and
u=
n
X
@2u
i=1
@x2i
:
This kind of systems (gi 6= 0; i = 1; 2) appears in the models of nonlinear viscoelasticity.
Viscoelastic materials have properties between two types, elastic materials and viscous
‡uids. This two types of materials are usually considered in basic texts on continuum
mechanics. At each material point of an elastic material the stress at the present
time depends only on the present value of the strain. On the other hand, for an
incompressible viscous ‡uid the stress at a given point is a function of the present
value of the velocity gradient at that point. Such materials have memory: the stress
depends not only on the present values of the strain and/or velocity gradient, but also
on the entire temporal history of motion.
The systems of nonlinear wave equations go back to Reed [27] who proposed a
system in three space dimensions, where this type of system was completely analyzed.
Existence and uniqueness of global weak solutions, asymptotic behavior for an analogous hyperbolic-parabolic system of related problems have attracted a great deal of
attention in the last decades, and many results have appeared. See in this directions
[5, 6, 7, 8, 12, 17, 22, 21, 24] and references therein.
We mention the work of [2], where authors studied the following system:
(
utt
u + jut jm 1 ut = f1 (u; v);
(5)
vtt
v + jvt jr 1 vt = f2 (u; v);
in
(0; T ) with initial and boundary conditions and the nonlinear functions f1
and f2 satisfying appropriate conditions. They proved under some restrictions on the
parameters and the initial data many results on the existence of a weak solution. They
also showed that any weak solution with negative initial energy blows up in …nite time
using the same techniques as in [11].
In [20], authors considered the nonlinear viscoelastic system
8
Rt
m 1
>
>
u + g(t s) u(x; s)ds + jut j
ut = f1 (u; v);
< utt
0
(6)
Rt
>
r 1
>
: vtt
v + h(t s) v(x; s)ds + jvt j
vt = f2 (u; v);
0
K. Zennir and A. Guesmia
for x 2
123
and t > 0 where
2( +1)
( +2)
f1 (u; v) = a ju + vj
(u + v) + b juj u jvj
f2 (u; v) = a ju + vj
(u + v) + b juj
2( +1)
( +2)
;
jvj v;
and they prove a global nonexistence theorem for certain solutions with positive initial
energy, the main tool of the proof is a method used in [28].
The non-critical case of (1) where gi = 0; m = 2; i = 1; 2, has been studied recently in [26]. M. A. Rammaha and Sawanya Sakuntasathien focus on the global
well-posedness of the system of nonlinear wave equations
8
< utt
: v
tt
u + djujk + ejvjl jut jm
v + d0 jvj + e0 juj
1
jvt jr
ut = f1 (u; v);
1
(7)
vt = f2 (u; v);
in a bounded domain
Rn , n = 1; 2; 3; and 0 < r; m < 1, with Dirichlet boundary
conditions. The nonlinearities f1 (u; v) and f2 (u; v) act as strong sources in the system.
Under some restriction on the parameters in the system, they obtained several results on
the existence and uniqueness of solutions. In addition, they proved that weak solutions
blow up in …nite time whenever the initial energy is negative and the exponent of the
source term is more dominant than the exponents of both damping terms. This last
result was extended by A. Benaissa et al. in [3] with positive initial energy, r; m > 1
and for n > 0.
The gender of our systems (mi 6= 0; i = 1; 2) has been proposed …rst by Segal [30]
in the next coupled Klein-Gordon equations which is considered in the study of the
quantum …eld theory and de…nes the motion of a charged meson in an electromagnetic
…eld
( 00
u1
u1 + m21 u1 + h1 u1 u22 = 0;
(8)
u002
u2 + m22 u2 + h2 u21 u2 = 0;
where m1 ; m2 ; h1 and h2 are non-negative constants.
When gi = 0; i = 1; 2; Yaojun Ye generalized the problem (8), where author studied
coupled nonlinear Klein-Gordon equations with nonlinear damping and source terms,
in a bounded domain with the initial and Dirichlet boundary conditions
(
u001
u1 + m21 u1 + aju01 j u01 = bju1 j u1 ju2 j
+2
;
u002
u2 + m22 u2 + aju02 j u02 = bju2 j u2 ju1 j
+2
;
(9)
where m1 ; m2 ; a; b are non-negative constants,
> 0 and
0. The existence of
global solutions is discussed by using the potential well method and the asymptotic
stability is also given by applying a Lemma due to V. Komornik [14].
REMARK 1.1. Noting here that our contribution is: We investigate the same
system in [33] with the presence of the viscoelastic terms and potential functions,
under additional condition (18), we prove that the solutions stay in the stable set (13).
124
2
Existence of Solutions for a System of Klein-Gordon Equations
Preliminaries
From now on, we denote by ci , i = 0; 1; 2; :::, used throughout this paper, various
positive constants which may be di¤erent at di¤erent occurrences and in the sequel,
for the sake of simplicity we will denote the t derivative value dv=dt by v 0 and d2 v=dt2
by v 00 .
We assume that, for i = 1; 2; the relaxation functions gi : R+ ! R+ and the
potential i : R+ ! R+ are nonincreasing di¤erentiable satisfying:
+1
Z
gi (s)ds, 1
gi (0) > 0, + 1 >
i (t)
Zt
gi (s)ds
li > 0, and
i (t)
> 0:
(10)
0
0
The following notation will be used throughout this paper
(
)(t) =
Z
0
t
(t
) k (t)
2
( )k2 d :
The following technical Lemma will play an important role.
LEMMA 2.1. For any v 2 C 1 (0; T; H ( )) we have
Z
=
Z
t
v(s)v 0 (t)dsdx
Z t
Z
1 d
1 d
2
(t) (g r v) (t)
(t)
g(s)
jr v(t)j dxds
2 dt
2 dt
0
Z
1
1
2
0
(t) (g r v) (t) +
(t)g(t)
jr v(t)j dxds
2
2
Z t
Z
1 0
1 0
2
(t) (g r v) (t) +
(t)
g(s)ds
jr v(t)j dxds:
2
2
0
(t)
g(t
s)
0
PROOF. It’s not hard to see
Z
=
=
Z
t
(t)
g(t s) v(s)v 0 (t)dsdx
0
Z t
Z
(t)
g(t s)
r v 0 (t)r v(s)dxds
0
Z t
Z
(t)
g(t s)
r v 0 (t) [r v(s) r v(t)] dxds
0
Z t
Z
(t)
g(t s)
r v 0 (t)r v(t)dxds:
0
(11)
K. Zennir and A. Guesmia
125
Consequently,
Z
Z t
(t)
g(t s) v(s)v 0 (t)dsdx
0
Z
Z t
1
d
2
(t)
g(t s)
jr v(s) r v(t)j dxds
2
dt
0
Z
Z t
d 1
2
jr v(t)j dx ds;
(t)
g(s)
dt
2
0
=
which implies,
Z
=
(t)
Z
t
g(t s) v(s)v 0 (t)dsdx
Z t
Z
2
g(t s)
jr v(s) r v(t)j dxds
(t)
0
1 d
2 dt
0
1 d
2 dt
(t)
Z
t
g(s)
0
Z
2
jr v(t)j dxds
Z t
Z
1
2
(t)
g 0 (t s)
jr v(s) r v(t)j dxds
2
0
Z
Z t
Z
1 0
1
2
(t)
+ (t)g(t) jr v(t)j dxds
g(t s)
jr v(s)
2
2
0
Z s
Z
1
2
g(s)ds
jr v(t)j dxds:
+ 0 (t)
2
0
2
r v(t)j dxds
This completes the proof.
The energy functional E(t) associated with our system is given by
2
1X 0 2
E(t) =
ku k + J(t)
2 i=1 i 2
(12)
where
2
J(t)
=
1X
1
2 i=1
i (t)
0
2
1X
+
2 i=1
Z
t
gi (s)ds kr ui k22
2
i (t)(gi
1X 2
r ui ) +
m kui k22
2 i=1 i
1
ku1 u2 kpp :
p
Now, we introduce the stable set as follows:
W = (u1 ; u2 ) 2 (H0 ( ))2 : I(t) > 0 and J(t) < d [ f(0; 0)g
(13)
126
Existence of Solutions for a System of Klein-Gordon Equations
where
I(t)
=
2
X
1
i (t)
+
i=1
t
0
i=1
2
X
Z
gi (s)ds kr ui k22
i (t)(gi r ui ) +
2
X
i=1
m2i kui k22
ku1 u2 kpp :
REMARK 2.2. We notice that the mountain pass level d given in (13) de…ned by
8
9
=
<
J ( (u1 ; u2 )) :
d = inf
sup
;
:(u ;u )2 H ( ) 2 nf(0;0)g 0
( 0 )
1
2
Also, by introducing the so called "Nehari manifold"
n
o
2
N = (u1 ; u2 ) 2 (H0 ( )) n f(0; 0)g : I (t) = 0 :
It is readily seen that the potential depth d is also characterized by
d=
inf
(u1 ;u2 )2N
J(t):
This characterization of d shows that
dist ((0; 0); N ) =
min
(u1 ;u2 )2N
k(u1 ; u2 )k(H
0
2
( ))
:
The notation k:k stands for the norm in L2 and we denote by k:kX the norm in
the space X. Also, the following imbedding will be used frequently without mention
kukp Ckr uk2 for u 2 H0 ( ) where
(
2 p < +1
if n = ; 2 ;
(14)
2n
if ; n 3 :
2 p n 2
We introduce the following de…nition of weak solution to (1)-(4)
DEFINITION 2.3. A pair of functions (u1 ; u2 ) is said to be a weak solution of
(1)-(4) on [0; T ] if u1 ; u2 2 Cw ([0; T ]; H0 ( )), u01 ; u02 2 Cw ([0; T ]; L2 ( )), (u10 ; u20 ) 2
H0 ( ) H0 ( ), (u11 ; u21 ) 2 L2 ( ) L2 ( ) and (u1 ; u2 ) satis…es
Z tZ
Z tZ
Z tZ
p 2
p
00
2
ju1 j u1 ju2 j dxds =
u1 dxds + m1
u1 dxds
0
0
0
Z tZ
Z tZ
+
r u1 r dxds +
ju01 jr 2 u01 dxds
0
0
Z tZ
Z s
g1 (t
)r u1 (x; )r d dxds
1 (t)
0
0
K. Zennir and A. Guesmia
and
Z tZ
0
p 2
ju2 j
p
u2 ju1 j
127
dxds =
Z tZ
u002
dxds +
m22
u2 dxds
Z tZ
ju02 jr
dxds +
0
0
+
Z tZ
Z tZ
r u2 r
Z tZ
Z s
g2 (t
2 (t)
0
0
2 0
u2
dxds
0
)r u2 (x; )r
d dxds
0
for all test functions ; 2 H0 ( )\L2 ( ) and almost all t 2 [0; T ] where Cw ([0; T ]; X)
denotes the space of weakly continuous functions from [0; T ] into Banach space X.
In order to state the local existence result, we introduce the following complete
metric space (the proof is similar to that in [29, 32, 31])
YT
=
f(u; v) : u; v 2 C ([0; T ]; H0 ( )
u0 ; v 0 2 C [0; T ]; L2 ( )
H0 ( )) ;
L2 ( )
THEOREM 2.4. Let (u10 ; u20 ) 2 (H0 ( ))2 and (u11 ; u21 ) 2 (L2 ( ))2 for i = 1; 2
be given. Suppose that r > 2 and p satis…es
(
1 p < +1
if n = ; 2 ;
(15)
1 p n4 2n if n 3 :
Then, under assumptions on two functions gi , i = 1; 2, the problem (1)-(4) has a unique
local solution (u1 (t; x); u2 (t; x)) 2 YT for T small enough.
3
Global Existence Result
LEMMA 3.1. Suppose that (10) and (15) hold. Let (u1 ; u2 ) be the solution of the
system (1)-(4). Then the energy functional is a non-increasing function, that is for all
t 0,
2
E 0 (t)
=
1X
2 i=1
2
0
i (t)(gi
r ui )
2
i (t)gi (t)kr
2
1X
+
2 i=1
0
(t)(gi r ui )
2
1X
2 i=1
1X
2 i=1
1X
2 i=1
0
(t)
(t)(gi r ui )
1X
2 i=1
t
0
2
0
Z
0
(t)
Z
0
ui k22
gi (s)ds kr ui k22
t
gi (s)ds kr ui k22 :
(16)
We will prove the invariance of the set W: That is for some t0 > 0 if (u1 (t0 ); u2 (t0 )) 2
W; then (u1 (t); u2 (t)) 2 W for t t0 and i = 1; 2. We begin with by the existence of
the potential depth in the next Lemma.
128
Existence of Solutions for a System of Klein-Gordon Equations
LEMMA 3.2. d is a positive constant.
PROOF. We have
2
X
2
J( (u1 ; u2 ))
=
2
i (t)
2
2
X
i=1
t
0
i=1
2
+
1
Z
2p
m2i kui k22
2
gi (s)ds kr ui k22 +
2
2
X
i (t)(gi
i=1
r ui )
ku1 u2 kpp :
p
Using (10) to get
J( (u1 ; u2 ))
K( );
where
2
K( ) =
2
2
X
i=1
li kr ui k22 +
2
2
2
X
i=1
2p
m2i kui k22
p
ku1 u2 kpp :
By di¤erentiating the second term in the last equality with respect to ; to get
d
K( ) =
d
For
1
2
X
i=1
2
X
li kr ui k22 +
i=1
m2i kui k22
2
2p 1
ku1 u2 kpp :
= 0 and
2
=2
1
2(p 1)
P2
i=1 li kr
! 2(p1 1)
P2
ui k22 + i=1 m2i kui k22
;
ku1 u2 kpp
then we have
d
K( ) = 0:
d
As
d
K(
d
2)
= 0; K(
1)
= 0;
and since
d2
K( )
d 2
=
< 0;
2
K. Zennir and A. Guesmia
129
we see that
sup j( )
sup K( ) = K(
0
0
=
! 2(p2 1)
P2
2
2
2
l
kr
u
k
+
m
ku
k
i
i
i
2
2
i
i=1
i=1
2
ku1 u2 kpp
!
2
2
X
X
2
2
2
li kr ui k2 +
mi kui k2
P2
2p
2(p 1)
i=1
i=1
! 2(p2p 1)
P2
2
2
2
ku
k
l
kr
u
k
+
m
i 2
i 2
i
i=1 i
i=1
ku1 u2 kpp
ku1 u2 kpp
! 2(p2p 1)
P2
P2
2
2
2
p 1
i=1 li kr ui k2 +
i=1 mi kui k2
:
p
ku1 u2 kp
P2
1 2(p2p1)
2
p
=
2)
2 2(p
2p
1)
It follows from the Holder inequality for some C > 0 and assumptions (10)
ku1 u2 kp
C 2 kr u1 k2 kr u2 k2
!
!
2
2
2
X
X
1 2 X
2
2
2
2
kr ui k2
C
li kr ui k2 +
mi kui k2 ;
2
i=1
i=1
i=1
ku1 k2p ku2 k2p
1 2
C
2
which implies that
ku1 u2 kp
P2
2
2
2
i=1 li kr ui k2 +
i=1 mi kui k2
P2
1 2
C :
2
Since p > 1, we obatin that
sup j( )
2
2p
2(p 1)
p
1
p
0
(p
1)
p
C (p
2p
1)
"P
2
i=1 li kr
# 2(p2p 1)
P2
ui k22 + i=1 m2i kui k22
ku1 u2 kp
= d > 0:
Then, by the de…nition of d; we conclude that d > 0 for p > 1:
LEMMA 3.3. W is a bounded neighborhood of 0 in H0 ( ).
130
Existence of Solutions for a System of Klein-Gordon Equations
PROOF. For u 2 W; and u 6= 0; we have
2
J(t)
=
1X
1
2 i=1
i (t)
=
1X 2
m kui k22
2 i=1 i
" 2
X
p 2
2p
+
i=1
2
X
i (t)(gi
i=1
p
2p
+
2
X
gi (s)ds kr ui k22 +
0
1
r ui )
#
2
X
0
i (t)(gi
i (t)(gi
1
m2i kui k22 + I(t)
p
i=1
Z t
(t)
gi (s)ds kr ui k22
i
i=1
r ui ) +
i=1
1X
2 i=1
1
ku1 u2 kpp
p
Z t
gi (s)ds kr ui k22
1
i (t)
r ui ) +
" 2
X
2
2
t
0
2
+
Z
2
X
i=1
m2i kui k22
#
:
(17)
By using (10), (17) becomes
p
J(t)
2
2p
p
1
i (t)
2p
2
X
i=1
2
2p
Z
t
0
i=1
2
p
2
X
gi (s)ds kr ui k22
li kr ui k22
min(l1 ; l2 )
2
X
i=1
kr ui k22 :
It follows that
2
X
i=1
kr ui k22
1
min(l1 ; l2 )
2p
p
2
J(t) <
1
min(l1 ; l2 )
Consequently, 8(u1 ; u2 ) 2 W; we have (u1 ; u2 ) 2 B; where
B=
(
2
(u1 ; u2 ) 2 (H0 ( )) :
2
X
i=1
kr
ui k22
2p
p
2
d = R:
)
<R :
This completes the proof.
In the following Lemma, we will see that if the initial data (or for some t0 > 0) is
in the set W , then the solution stays there forever.
K. Zennir and A. Guesmia
131
LEMMA 3.4. Suppose that (10), (15) and
p
C2
(2 min(l1 ; l2 ))
2pE(0)
p 2
(p 1)
< 1:
(18)
hold, where C is the best Poincare’s constant. If (u10 ; u20 ) 2 W and (u11 ; u21 ) 2
2
L2 ( ) , then the solution (u1 (t); u2 (t)) 2 W for t 0.
PROOF. Since (u10 ; u20 ) 2 W , we see that
I(t) =
2
X
i=1
kr ui0 k22 +
2
X
i=1
m2i kui0 k22
ku10 u20 kpp > 0:
Consequently, by continuity, there exists Tm T such that
Z t
2
2
X
X
I(u(t)) =
1
gi (s)ds kr ui k22 +
i (t)
0
i=1
+
2
X
i=1
m2i kui k22
i (t)(gi
r ui )
i=1
ku1 u2 kpp
0 for t 2 [0; Tm ] :
This gives
2
J(t)
=
1X
1
2 i=1
i (t)
Z
0
2
1X 2
m kui k22
2 i=1 i
" 2
X
p 2
1
2p
i=1
#
2
X
2
2
+
mi kui k2 +
+
=
i=1
p
2
2p
+
2
X
i=1
i=1
kr ui k22
gi (s)ds kr ui k22 +
1X
2 i=1
i (t)(gi
1
ku1 u2 kpp
p
Z t
2
X
gi (s)ds kr ui k22 +
i (t)
0
i=1
0
#
r ui )
(19)
i (t)(gi
r ui )
i (t)(gi
r ui )
i=1
1
I(u(t))
p
" 2
Z t
2
X
X
1
gi (s)ds kr ui k22 +
i (t)
i=1
m2i kui k22 :
By using (10) and the fact that
[0; Tm ],
2
X
2
t
Rt
0
gi (s)ds
1
min(l1 ; l2 )
1
min(l1 ; l2 )
2p
p
2
2p
p
2
R1
0
gi (s)ds; we easily see that, for t 2
J(t)
E(0):
1
min(l1 ; l2 )
2p
p
2
E(t)
132
Existence of Solutions for a System of Klein-Gordon Equations
We then exploit (10), (15) and from the Holder inequality for some C > 0. So we have
!
2
X
1
ku1 u2 kp ku1 k2p ku2 k2p C 2 kr u1 k2 kr u2 k2
C2
kr ui k22 :
2
i=1
for C = C(n; p; ):
Consequently, we have
ku1 u2 kpp
p
2
C
2
X
2p
i=1
p
2
2
X
C 2p
i=1
C
2p
ui k22
kr
kr ui k22
(2 min(l1 ; l2 ))
2
X
i=1
ui k22
li kr
!p
!p
1
i=1
p
kr ui k22
!
(p 1)
2p
p
!
2
X
(p 1)
E(0)
2
2
X
i=1
li kr
ui k22
!
;
where
(p 1)
2p
p
= C 2p (2 min(l1 ; l2 ))
p
E(0)(p
2
1)
:
Which means, by the de…nition of li ; i = 1; 2;
!
2
X
p
2
ku1 u2 kp
li kr ui k2
i=1
2
X
1
i (t)
t
gi (s)ds kr ui k22
0
i=1
2
X
Z
1
i (t)
Z
0
i=1
t
gi (s)ds kr ui k22 +
2
X
i (t)(gi
i=1
r ui ) +
2
X
i=1
m2i kui k22 :
Therefore, I(t) > 0 for all t 2 [0; Tm ], by taking the fact that
lim C 2p (2 min(l1 ; l2 ))
(p 1)
2p
p
p
t7!Tm
E(0)(p
2
1)
< 1:
This shows that the solution (u1 (t); u2 (t)) 2 W; for all t 2 [0; Tm ] : By repeating this
procedure Tm extends to T:
THEOREM 3.5. Suppose that (10), (15) and (18) hold. If (u10 ; u20 ) 2 W; (u11 ; u21 ) 2
2
L2 ( ) . Then the local solution (u1 ; u2 ) is global in time such that (u1 ; u2 ) 2 GT
where
GT
=
(u; v) : u; v 2 L1 R+ ; H0 ( )
0
0
1
and u ; v 2 L
+
2
R ;L ( )
H0 ( )
2
L ( )
:
K. Zennir and A. Guesmia
133
PROOF. In order to prove Theorem 3.5, it su¢ ces to show that the following norm
2
X
i=1
ku0i k22 +
2
X
i=1
kr ui k22 +
2
X
i=1
m2i kui k22
is bounded independently of t. To achieve this, we use (12), (16) and (19) to get
2
E(0)
1X 0
2
ku (t)k2
2 i=1 i
" 2
Z t
2
X
X
gi (s)ds kr ui k22 +
1
i (t)
E(t) = J(t) +
p
2
2p
i=1
p
2
2p
i=1
2
X
1
2
" 2
X
i (t)(gi
i=1
r ui )
2
m2i kui k22 +
i=1
Since I(t) and
#
2
X
+
+
0
li kr
1
1X 0
2
ku (t)k2 + I(t)
2 i=1 i
p
ui k22
+
2
X
i (t)(gi
i=1
r ui ) +
2
X
i=1
m2i kui k22
#
1
2
ku0i (t)k2 + I(t)
p
i=1
#
" 2
2
2
X
X
1X 0
p 2
2
m2i kui k22 +
kui (t)k2 :
li kr ui k22 +
2p
2
i=1
i=1
i=1
i (t)(g
2
X
i=1
ru)(t) are positive, hence
2
ku0i (t)k2
+
2
X
i=1
kr
ui k22
+
2
X
i=1
m2i kui k22
CE(0);
where C is a positive constant depending only on p and li .
This completes the proof.
Open problem Let us mention here that, it will be interesting to discuss the asymptotic stability of this problem where also, one can establish a general decay rate
estimate for the energy, which will depend on the behavior of both
and g under
following assumption
There exists a non-increasing di¤erentiable function i ; i = 1; 2 : R+ ! R+ satisfying i (t) > 0; gi0 (t)
0; and perhaps other conditions imposed by the
i (t)gi (t); 8t
nature of our system.
Acknowledgments. The authors want to thank the referee for his/her careful
reading of the proofs.
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