Applied Mathematics, 2014, 5, 7-15
Published Online January 2014 (http://www.scirp.org/journal/am)
http://dx.doi.org/10.4236/am.2014.51002
OPEN ACCESS AM
Global Attractor of Two-Dimensional Strong Damping
KDV Equation and Its Dimension Estimation
Cheng Zhang, Guoguang Lin
Mathematical of Yunnan University, Kunming, China
Received October 10, 2013; revised November 10, 2013; accepted November 17, 2013
Copyright © 2014 Cheng Zhang, Guoguang Lin. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In accordance of the Creative Commons Attribution License all Copyrights © 2014 are reserved for SCIRP and the owner of the
intellectual property Cheng Zhang, Guoguang Lin. All Copyright © 2014 are guarded by law and by SCIRP as a guardian.
ABSTRACT
Firstly, a priori estimates are obtained for the existence and uniqueness of solutions of two dimensional KDV
equations, and prove the existence of the global attractor, finally geting the upper bound estimation of the Haus-
dorff and fractal dimension of attractors.
KEYWORDS
KDV Equation; Strongly Damped; Existence; Global Attractor; Dimension Estimation
1. Introduction
Studies on the infinite dimension system with high dimension have obtained many achievements in recent years,
such as [1-5]. In the paper [6,7]. The authors study the estimates of global attractor for one-dimensional KDV
equation and its dimension. Based on these work, this paper further studies the global attractor of two-
dimensional KDV equations and its upper bound estimation of the Hausdorff and fractal dimension of
attractors.
The following form 2D-KDV equation is studied in this paper






2,, ,
t xxxx
uuuuvufxy xy    
(1.1)





,;,; ,,
xy
uxyt vxytxy (1.2)





0
,;0, ,,uxyu xyxy (1.3)





,; 0,,; 0,,uxyt uxytxy
 
 
(1.4)
where ,,

are positive constants. When 0


, the equation is the KDV equation.
The rest of this paper is organized as follows. In Section 2, we introduce basic concepts concerning global
attractor. In Section 3, we obtain the existence of the uniqueness global attractor, which has fractal and Haus-
dorff dimension.
In this paper, C denotes a positive constant whose value may change in different positions of chains of
inequalities.
2. Preliminaries
Denoting by
p
L
 the norm in


p
L, 1p

, for simplicity, we denote by  and   the norm in the
case 2p and p, respectively. Suppose that


2
HL

,


i
H

is a Hilbert space for the scalar

C. ZHANG, G. G. LIN
OPEN ACCESS AM
8
product
  

1
,,,, .
i
ijj
Hj
DD D
x


 


According to the Poincare inequality and (1.2) we can get
1.vCu


In fact,
1xyxx xyxxxyxxy
uvuvuvCu vCv Cu vCu  
Now, we can do priori estimates for Equation (1.1)
Lemma 1. Assume that







22
0
,,,fxyLu xyL  ,
22
2
C 


 then
 
22 22
22
22 2
022
22
2
,;,e1 e,
2
CC
tt
uxytuxyf
C
C







 
 
 
 
 


 






(2.1)
Certainly there exist

11 0tt
, such that


2
,; ,uxyt C (2.2)
Proof. We multiply u for both sides of Equation (1.1), we obtain
 










2
,,,,,,,
txxx x
uuuuuuuvuuufu    
(2.3)
where 
,,
xxx xxx
uuu u , we have

,0,
xxx
uu  (2.4)



22
222
,, ,
4
x
x
C
uvuuv uuuvuuCuuuu




 (2.5)

22 22
22
,,
4
C
uf ufuf
C


  (2.6)
Substituting (2.4)-(2.6) into (2.3) gets
22
222
22
1d
2d 2
C
uuf
tC



 


Using the Growall inequality, we can get
 
22 22
22
22 2
022
22
2
,;,e1 e
2
CC
tt
uxytuxyf
C
C


 





 




 





 ■
Lemma 2. Assume that







11
00 0
,,,fxyH uxyH  ,
22
2
C 


 then
  

2
222
02
,2
,;,e1 e,
tt
fxy C
uxytu xy





 
(2.7)

C. ZHANG, G. G. LIN
OPEN ACCESS AM
9
certainly, there also exist

22 0tt
, such that

2
3
lim, ;,
tuxytC  
(2.8)
Proof. We take parts of the scalar product in 2 L of (1.1) with u

:
 









2
,,,, ,,,
txxx x
uuu u uuuvuuufu       (2.9)
where 



,,
x
xx xxx
uuuu , thus


,0,
xxx
uu  (2.10)




,, ,
x
x
uvuuv uuuu 


 (2.11)
Noticing
12
33
,uCuu
 (2.12)
12
33
,uC uu  (2.13)
According to (12) and (13), Lemma 1 and Young inequality, we can obtain that


54 2
33
,,
x
uvuCu uuC 

 (2.14)

22
1
,,
22
f
ufuf u


  (2.15)
Using (2.10), (2.14) and (2.15), we can get
22
1d 1
2d2 2
uu fC
t



 
Using Growall inequality, we have

2
22 22
02
2
e1e
tt
fC
uu





 
■
Lemma 3. Assume that







22
00 0
,,,fxyH uxyH  ,
22
2
C 


 then
 

2
222
02
2
,;,e1 e,
tt
fC
uxytuxy





  (2.16)
Thus there exists

33 0tt

 , such that


4
,; ,uxyt C
(2.17)
Proof. We multiply 2u

for both sides of Equation (1.1), we obtain that
 










222 2222
,,,, ,,,
txxx x
uuuuuuuvuuufu     (2.18)
where


2
,0,
xxx
uu

 (2.19)
Noticing
13
244,uCuu
 (2.20)

C. ZHANG, G. G. LIN
OPEN ACCESS AM
10
12
233,uCu u  (2.21)
Using (2.20)-(2.21), we obtain that


3 19 2
222
4 12 3
,,
x
uvuC uuuCuuu 

  (2.22)
According to Lemma 1, Lemma 2 and Young inequality, we get that



2
22
,,
x
uvuuC 

 (2.23)

22
21
,,
22
f
uuf uf


 (2.24)
Substituting (2.19)-(2.24) into (2.18) gets
22 2
1d 1
2d2 2
uu fC
t



 
Using the Growall inequality, we can get

2
22 22
02
2
e1e
tt
fC
uu





 
■
Lemma 4. Assume that







22
00 0
,,,fxyH uxyH  ,
22
2
C 


 then

,; ,
Q
uxyt t  (2.25)
here Q and 2
0
0
H
u, 2
0
H
f
have relations.
Proof. We multiply 23 tu for both sides of Equation (1.1), we obtain that








 
2323232322323
,,,, ,,,
txxx x
utuutuutuuvtuutuftu    (2.26)
we have

2
1
2
23 2
1d
,,
2d
t
utut utu
t
 (2.27)

2
2232
,,ut utu   (2.28)

23
,0,
xxx
utu

 (2.29)

22
322
3
,,
62
f
ufuu f


 (2.30)

22
322
,,
6
uuuuu C



(2.31)







32 2
,,3 ,
xx
x
uvuu vuvuCuuuuu 

  (2.32)
Noticing
11
22
,uCuu
 (2.33)

C. ZHANG, G. G. LIN
OPEN ACCESS AM
11
1
324
4,uCu u

 (2.34)
3
225
5,uCu u  (2.35)
Taking (2.33)-(2.35) into (2.32) and using Young inequality, we have


2
32
,,
6
x
uvuu C



  (2.36)
namely,


2
23 2
,,
6
x
uv tutuC



  (2.37)
Taking (2.27)-(2.37) into (2.26), we obtain
222
d
dtutuCf t



So, we get
Q
ut


.■
From [8], we have
Theorem 2.1 Let E be a Banach space,




St are the semigroup operators.

:StEE,




StS


St

 ,


0SI, here I is unit operator.Set


St satisfy the following conditions:
1)

St is bounded. namely 0, E RuR, there exist a constant


CR, such that


E
Stu




0,CR t
.
2) There exist a bounded absorbing set 0 BE, namely BE

, there exist a constant 0
t, such that
 

00
StBB tt .
3) When 0t,


St is a completely continuous operator.
Then, the semigroup operators

St exist a compact global attractor
A
.
3. Global Attractor and Dimension Estimation
3.1. The Existence and Uniqueness of Solution
Theorem 3.1 Assume that




2
0
,fxy H and




2
00
,uxy H

,
22
2
C 


 there exists a unique
solution





2
0
,; 0,;,uxyt LTH


(3.1.1)
Proof. By the Galerkin method, we can easily obtain the existence of solutions. Next, we prove the
uniqueness of solutions.
Set 12 uu

, where

1, 2
i
ui are two solutions of (1.1)-(1.4). then

satisfies


2
112 20, t xxxuv uv
    (3.1.2)


d, 1,2,
iii ix
uvuuy i
 (3.1.3)

,;00,xy 

(3.1.4)
Take the inner product with

, we gets

22 2
112 2
1d ,0,
2d uv uv t
 

  (3.1.5)
Furthermore

C. ZHANG, G. G. LIN
OPEN ACCESS AM
12



222
112 2
22
21
222
21
d2,22
d
2dd,22
22,
xx
uv uv
t
uyuy
Cuu
 


 




 (3.1.6)
Noticing
13
44
,uCuu
 (3.1.7)
13
44
,uCuu

 (3.1.8)
11
22
,C

  (3.1.9)
So, we have
13 13
22 22
44 44
22 11
d22
dCuuCu u t



 
From Lemmas 1-3, we have
2211
,,,uCuCuCuC  
Using Young inequality, we obtain
22
d
dC t

 
Using Gronwall inequality, we have

2
22
0e 0
Ct



So, we can get 0.

■
3.2. Global Attractor
Theorem 3.2 Assume that




2
0
,fxy H and




2
00
,uxy H

,
22
2
C



 there exists a compact
global attractor
A
, such that
1)

,0StA At
2)




lim, 0
tdistStBA  
here, B is a bounded set in


2
0
H.


,inf, sup
E
yY
xX
distX Yxy




St are the semigroup operators.
Proof. Let us verify theorem 2.1 conditions (1), (2), (3). In Theorem 3.2 conditions, we know that there exist
the solution semigroup

St,


2
0
EH ,






22
00
:StHH

 . form Lemmas 1-3, we can get
that


2
0
BH  is a bounded set and B included in the ball


2
0
H
uR,
 

22
0
22
22
00120
,;0, .
HH
StuuxytuCfCtuB
This shows that

 
0St t is uniformly bounded in


2
0
H

. Furthermore, when 
123
max,,tttt, we
have
  
2
0
22
0234
,;2
H
StuuxytCC C

C. ZHANG, G. G. LIN
OPEN ACCESS AM
13
so, we can get that
  

2
2
0 0234
0
,;, 2
H
BuxytH uCCC
is bounded absorbing set of
semigroup

St.
From Lemma 4, we have

,>0,
Q
ut
t
  2
0
0.
H
uR

Since




32
00
HH

 is tightly embedded.
So the semigroup operator





22
00
:StHH


for 0t

 is continuous.
3.3. Dimension Estimation
Considering the following first variation equations







,;,;,; 0,
txyt Luxytxyt


 (3.3.1)



,;,;d,
x
vxytu xyty (3.3.2)

,;00,xy   (3.3.3)



,; 0, ,; 0xyt xyt 
 

 (3.3.4)
where



1
0
,;0xy H 


 









 
2
d
xxxx xx
Lutttt tvttyt     

It’s easy to prove that the equation has a unique solution.




1
0
,; 0,;xyt LTH 


.
Furthermore, Let 0
)(=)( utStu , )(=))((00 tutDS


, )(=))((*
00 tuutS
 , we can get 1
R, 2 R and T
are constants. There exist a constant


12
,,CCRRT such that for 0
u, 0

, t with

1
0
01
H
uR


,

1
0
02
HR

, tT, we have
 
 
1
10
0
2
*
0,
H
H
ut uttC 


  (3.3.5)
That suggests that


St is Frechet differential at


0,uxy
.
Let

12
,,,
N
VtVtVt be the solutions of the linear variational equations corresponding to the initial
value
 
112 2
0,0,,0
NN
VV V

  . We have
  

 
22
12 12
d20,
dNNN
VtVtV ttrLutQVtVtV t
t
(3.3.6)
here  represents the outer product, tr represents the trace,
N
Q means that the


2
L to the orthogonal
projection on the span
 




12
,,,
N
VtVtV t. So, from (3.3.8) we can obtain
 
22
0
2
12
,1
, sup supN
L
nn
NN
uA L
tVtVtVt




 (3.3.7)
where
N

is called Secondary index, namely





,,0
NNN
tttttt 

 
so

1
lim, 1
t
Nn
ttnN




eN
q
n

here
  


0
0
0
1
limsup infinfd.
t
NN
tuA
qtrLsuQ
t


 




Theorem 3.3 The global attractor
A
of Theorem 3.2 has finite fractal and Hausdorff dimension in

C. ZHANG, G. G. LIN
OPEN ACCESS AM
14


1
0
H,



00
,2,
HF
dAJdA J (3.3.8)
0
J
is a minimal positive integer of the following inequality
22
0
382
,
4
c aacabac
Ja
 
 (3.3.9)
here
6
16
5
,,.
62 22
CC
C
abCuCuuc u



 


 
Proof. From [9], we need to estimate





N
trLu tQ of the lower bound. Let 12 ,,,
N

  be the
orthogonal basis of subspace of

2
N
QL

,
 
 




2
=1
22
=1
d,
d, ,
N
Njxxxjxjxxjj j
j
N
jjjxjxx j
j
trL u tQvy
vu y






(3.3.10)
where
 
,,
jxjjx jjx
vvv

 
So, we can obtain
 
2
1
,,
2
j
xj xj
vv
 

Furthermore
 
22 2
1
11 11
,, ,
22 2
NNNN
jxjx jjxj
jj jj
vvCvCu
 
 

 
 

(3.3.11)






2
2
2
34 5
d,d, 2
,2
,2
,2, ,
jxxjj xxjxjxjxx
j xxjxjxjxx
j xxjxjxjxx
jjxjxjjxx
uyyu uu
Cy uuu
Cuuu u
Cu uCuuCuu
 
 
 
 

 



 

(3.3.12)







2
,22,2,2,
jxjxjx xj xxjjx xjxxj
uuu uu

   
hence




2
,2, ,
jxjxxxj
uu    (3.3.13)
 
2
1
,,,,
2
jjxxjxjx jxx
uu u     (3.3.14)
Taking (3.3.15)-(3.3.16) into (3.3.14), we can get

22
6
5
d, ,
2
jxx jjj
uy Cuuu  
 




 (3.3.15)
Set

,1,2,3,
jj
  are eigenvalues of uu


 and
j

are the corresponding eigenfunctions. Satisfying

2
1
2
222
21
,,1, 1,
2
jjjjj j
jCj
 








 (3.3.16)
so, we can get

C. ZHANG, G. G. LIN
OPEN ACCESS AM
15

 
2
2
16 6
1 1
5,
22
N N
N
j j
j j
trLutQNNCuC N uuCu


 

 
 

(3.3.17)
Let
6
16
5
,,,
622 2
CC
C
abCuCuuc u



 


  (3.3.18)
when
22
382
4
ca acabac
Na


we have





0
N
trLu tQ


so, we can obtain




00
,2.
HF
dAJdA J
■
Funding
This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant
11161057.
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