Long-Time Behavior of Solutions for a Class of Nonlinear Higher Order Kirchhoff Equation

(1)

http://www.sciencepublishinggroup.com/j/ajam doi: 10.11648/j.ajam.20190701.14

ISSN: 2330-0043 (Print); ISSN: 2330-006X (Online)

Long-Time Behavior of Solutions for a Class of Nonlinear

Higher Order Kirchhoff Equation

Guoguang Lin

*

, Ying Jin

School of Mathematics and Statistics, Yunnan University, Kunming, China

Email address:

*_{Corresponding author}

To cite this article:

Guoguang Lin, Ying Jin. Long-Time Behavior of Solutions for a Class of Nonlinear Higher Order Kirchhoff Equation. American Journal of Applied Mathematics. Vol. 7, No. 1, 2019, pp. 21-29. doi: 10.11648/j.ajam.20190701.14

Received: April 1, 2019; Accepted: May 10, 2019; Published: June 3, 2019

Abstract:

In this paper, we study a class of the long-time behavior of solutions to initial-boundary value problems for higher order equations with nonlinear source term and strong damping term. First of all, give some space and marks as well as the basic assumption of stress and nonlinear source term, take the inner product on both sides of the equation and obtain a priori estimate of the global smooth solution of the equation by using Holder inequality, Yong inequality, Poincare inequality and Gronwall inequality. Then prove the existence of the global solution of the equation by using the Galerkin finite element method. The uniqueness of the global solution of the equation is proved, and then the bounded absorption set of the solution semi-group is constructed by a priori estimate. It is proved that the solution semi-group is uniformly bounded and completely continuous in the interior, thus the global attractor family of the equation is obtained. Then the original equation is linearized, and the differentiability of the solution semi-group is proved, and the line is further proved. The decay of the volume element of the sexualization problem is studied, and the finite Hausdorff dimension and Fractal dimension of the global attractor family are obtained.

Keywords:

Kirchhoff Equation, The Existence and Uniqueness of Solutions, Global Attractor Family, Hausdorff Dimension, Fractal Dimension

1. Introduction

In this paper, we mainly study the higher order Kirchhoff equation with strong nonlinear damping term

( )

D u

( )

u

( )

u g

( )

u u f

( )

x M

u t

m m

m

tt+ −∆ + −∆ + =

2 2

β (1)

( )

, 0, =0, =1,2,⋅ ⋅⋅, −1, ∈∂Ω, >0

∂ ∂

= i m x t

v u t

x

u _i

i

(2)

( )

( ) ( )

( )

n

t x u x x R

u x u x

u ,0 = ₀ , ,0 = ₁ , ∈Ω⊂ (3)

where m>1,Ω is a bounded region on Rn with smooth Dirichlet boundary ∂Ω and g

( )

u2u is a non-linear source term.

Since Kirchhoff[1] proposed the Kirchhoff rope model for studying the free vibration of elastic strings in 1883, many

scholars have devoted themselves to the study of Kirchhoff

equation. In 1997, Kosuke Ono [2] studied the

initial-boundary value problem of the nonlinear Kirchhoff wave equation with strong dissipation term

u u u u Du

utt t

α γ

= ∆ − ∆

= 2

( )

u

t

( )

u

1 0

0 ,

0 = =

0

=

Ω ∂

u

by using the modified potential well method and concave method, it is proved that when the initial energy

(

)

( )

0 2

2 0

1 2 1

2 1 0

2 2 1

1

,u u Du u

u

E + +₊

+ − +

+

= γ α_α

α

γ related to

(2)

negative, the solution will explode in a certain time.

In 2007, Xiaoming Fan and Yaguang Wang [3] studied the low-order stochastic wave equation with nonlinear damping and white noise existence and fractal dimension of compact random attractors.

( )

( ) ( ) ( )

, , ɺ

tt t t

u +

α

u +h u u − ∆ +u f u t −g x t =q x W

In 2010, Zhijian Yang and Yunqing Wang [4] studied the long-term behavior of solutions of low-order Kirchhoff equation with strong damping term

( )

Du u u h

( ) ( ) ( )

u gu f x M

utt− ∆ −∆ t+ t + =

2

,

0 =

Ω ∂

u ,

( )

x u

( ) ( )

x u x u

( )

x u ,0 = 0 , t ,0 = 1 ,

where

( )

(

2

)

4 1

,

1 2

− ≤ ≤ +

= ₊

N m s s M

m

, when N =1 , the

equation does not contain dissipation term

-

∆

_u

_t to describe the small vibration of elastic strings. On this basis, in 2011, Zhijian Yang and Jianling Cheng [5] once again studied the long-term behavior of solutions of low-order Kirchhoff equation with strong damping:

( )

Du u u h

( ) ( ) ( )

u g xu f x M

utt− ∆ −∆ t+ t + , =

2

,

0 =

Ω ∂

u ,

( )

x u

( ) ( )

x u x u

( )

x u ,0 = 0 , t ,0 = 1 ,

where _M

( )

_s =1+_s2,_m≥1

m

. For more information on Kirchhoff equation, please refer to references [6-20].

On the basis of the lower-order Kirchhoff-type equation studied by previous scholars, this paper studies the global attractor family and its finite dimension estimation for the higher-order Kirchhoff-type equation with strong non-linear damping terms. The structure of this paper is as follows: The first part is mainly about some basic assumptions; the second part is to prove the existence and uniqueness of the global solution by using prior estimation and Galerkin method, and prove that the global solution is unique. In the third part, we first prove the differentiability of solution semi-groups, and then estimate the finite Hausdorff dimension and Fractal dimension of global attractor family.

2. Basic Assumptions

For narrative convenience, define the following spaces and marks:

D

=

∇ , ₌ 2

( )

_Ω

L

H ,

( )

Ω =

( )

Ω 1

( )

Ω

0

0 H H

Hm m _∩ _,

( )

Ω = +

( )

Ω

( )

Ω

+ 1

0

0 H H

Hmk mk _∩ _,_C_,_C ₍_i=₁_,₂_,⋅ ⋅⋅_,₂₁₎

i is a constant

greater than 0, E Hm k

( )

Hk

( ) (

k m

)

k = 0 Ω × 0 Ω, =0,1,2⋯

+ _,

when k=0, = × 2

( )

Ω

0

0 H L

E m .

Definitions (.,.) and 2

L

⋅ =

⋅ represent the inner product

and norm of H, respectively

2 ( , )u v u x v x dx u u( ) ( ) , ( , ) u .

Ω

=

_∫

=

Let A_k be a weak global attractor family from E₀ to E_k, and B0_k be a bounded absorption set from E0 to Ek ,

where k=1,2⋅⋅⋅,m.

Kirchhoff stress term M

( )

s satisfies the following condition:

( )

A M s

( )

∈

_C

2

( )

Ω ;

( )

2 0

0 1

1

, 0

1 , .

, 0

m k

m k d

D u

dt M s

d

D u

dt

µ

µ µ

µ

+



≥



< ≤ ≤ =

 _<



Nonlinear term g

( )

u2u satisfies the following condition:

( )

B Let g

( )

u2u be locally bounded and measurable,

( )

u ∈C

( )

Ω

g 2 2 .

3. Existence of Global Attractor Family

In this part, we first estimate the solution of problem (1) - (3) a priori, then prove the existence and uniqueness of global solution by Galerkin method. Finally, problem (1) - (3) generates solution semi-groups and obtains global attractor family by the properties of solution semi-groups.

Lemma 1 Assuming that the Kirchhoff stress term M

( )

s

and the non-linear term g

( )

u2u satisfy the conditions

( )

A

and

( )

B , f

( )

x ∈H ,

(

u0,u1

)

∈E_k

(

k=0,1,2,⋅ ⋅⋅,m

)

,

, 0 , >

+ =u

ε

u

ε

v _t respectively, the smooth solution

( )

u,v ∈Ek of the initial-boundary value problem (1) - (3)

satisfies the following inequality

( )

u,v 2 Dm ku2 Dkv2 (0)e t C

(

1 e t

)

,

Ek

γ γ

φ − −

+ ₊ _≤ ₊ ₋

=

therefore, there exists a non-negative real number C R( ₁) and 2( ) 0,

t=t Ω >

  

 

 ₋ ₋ ₋

− −

= − − ε

µ ε λ

β λ ε µ ε ε ε λ β

γ 2 ,2 ,2

2

min 1 1

2 1 2 0 2 1

m m

m

C

,

such that

( )

1

2

,

lim uv C C R

k

E

t→∞ ≤ γ =

(

2

)

t t> .

(3)

( )

−∆kv=

( )

−∆ku_t+

( )

−∆kεu and equation (1), we can get

( )

(

u M D u u ut g u u kv

)

(

f

( ) ( )

x kv

)

m m

m

tt+ −∆ + −∆ + , −∆ = , −∆

2 2

β (4)

The items in successive processing (4) are as follows:

( )

(

)

D v D v

(

D u D v

)

dt d v

utt k k k k , k

2 1

, ₋_∆ = 2−ε 2+_ε2

u D v

D v

D dt

d m k

m k

k ₋ + ₋ − +

≥ 2 2 2 2 ₁ 2

2 2

1 ε ε ε λ

(5)

according to hypothesis

( )

A , we can get

( )

( ) ( )

(

M Dmu ₋_∆mu ₋_∆ kv

)

,

2

( )

D u dt

d u D

M m m+k

= 2 2

2

1 +ε_M

( )

_Dm_u2 _Dm+k_u 2

u D dt

d m+k

≥ 2

2

1_µ + _Dm+k_u2

0

µ

ε (6)

according to Holder inequality, Young inequality and Poincare inequality, there are

( ) ( )

(

u kv

)

Dm kv

(

Dm ku Dm kv

)

t

m ₊ ₊ ₊

− =

∆ −

∆ , ,

- β 2 βε

β D v Dkv Dm ku

k m k

m

+ −

− +

≥ 2 2

2 2 1

2 1

2 4

4

ε β λ

β λ

β

(7)

by assuming

( )

B and Poincare inequality, we have

( )

(

g u2u, −∆kv

)

( )

D u g

( )

u D u

dt d u

g 2 k 2 2 k 2

2

1 ₊_ε

=

[

g

( )

u D u

]

C D u g

( )

u D u

dt

d m k k

m

k 2 1 1 2 2 2

2

2 2

1 ₋ λ ₊_ε

≥ − + ₍₈₎

because f ∈H, according to Holder inequality and Young inequality, can obtain

( ) ( )

(

f x v

)

f

( )

x Dkv

k m k

m

k 2 1 2 2

1 4

1

, βλ

λ β

−

− +

≤ ∆

− (9)

substitute formula (5) - (9) into formula (4), we have

( )

(

D v D u g u D u

)

dt

d k 2+µ m+k 2+ 2 k 2 _Dk_v m

2 2 1 ₂

2 

   

  

− −

+ βλ ε ε

(

m m

)

C −

− ₋ ₋

+ 1 1

2 1 2 0

-2εµ ε λ βε λ Dm+ku2

( )

u D u D f C

g k

k m

k ≤ ≤

+ ₋ 2

1 2

2 2

2

βλ

ε .

where 2 2 ₁ ₁ 0

1 2

0− − − >

− −m m

Cλ

ε β λ ε µ

ε , and we can get

( )

t

( )

t C dt

d

≤ +γφ

φ (10)

where 2εµ0−ε2λ1− −βε2− 1λ1− >0

m m

C ,φ

( )

t = Dkv2+µ Dm+ku2+g

( )

u2 Dku2_{, then by Gronwall inequality, there are}

( ) ( )

t eγt C

(

eγt

)

γ φ

φ _≤ ₀ − ₊ ₁₋ − ₍₁₁₎

Because µ>1,min{1,µ}=1, so we can get

( )

u v Dm ku Dkv e t C

(

e t

)

Ek

γ γ

φ − −

+ ₊ _≤ ₊ ₋

= (0) 1

, 2 2 2 (12)

where k=0,1,2,⋅ ⋅⋅,m. Therefore, there exists a nonnegative real numberC(R₁) and t=t₂(Ω)>0, such that

( )

1

2

,

lim uv C C R

k

E

t→∞ ≤ γ =

(

2

)

t

(4)

Lemma 1 is proved.

Theorem 1 (Existence and uniqueness of solutions) Under Lemma 1, then the initial boundary value problem (1) - (3) has a unique smooth solution

( )

u,v ∈L∞

(

[

0,+∞

)

;Ek

)

.

Proof Let u v, be the two solutions of equation (1), let w= −u v be introduced into equation (1) and the inner product with

t

w , can obtain

( )

2

( )

2

( )

( ) ( )

2 2 =0

  



 ₊ ₋

∆ − + ∆ −

   

 

− ∆ −

+ t t

m m

m

tt M D u u M D v v w g u u g v v w

w β ， (14)

Formula (14) is dealt with item by item as follows:

(

)

2

2 1

, t t

tt w

dt d w

w = (15)

( )

(

t t

)

m

w w, -∆

β 2

t m

w D β

= (16)

according to assume

( )

A and differential mean value theorem, we can get

( )



  



 ₋_∆

   

 

− ∆

− t

m m

w v v

D M u u

D

M ，

2 2

( )

D w dt d u D

M m 2 m 2

2 1

≥ _M

_{( )}

(

_D _u _D _v

)

_D _w _D _v _D _w

t m m m m m ₊

′

− ξ _∞ D w

dt d m 2

2

µ

≥ m m _t

w D C w D C

2 2

2 ₋

−

(17)

by hypothesis

( )

B , Holder inequality, Poincare inequality and differential mean value theorem, we have

( ) ( )

(

g u u g v v，wt

)

2 2

−

(

g

( )

u w wt

)

(

g

( )

(

u v

)

wv,wt

)

' 2

+ +

= ，

ξ

t m

m

w C C w D C

C 2 ₄ 2 ₃ 2

1 3 2 4

2 2 2

2 

 

 

+ −

   

 

+ −

≥ ε _λ− ε ₍₁₈₎

substitute formula (15) - (18) into formula (14), we conclude that

(

)

2

2 2

2

t

m m

t

D

w

C

D

w

dt

d

−

+













₊

_µ

_β

2

1 2 4 1 3 2

w D C

C

C m λ m µ m

µε λ

µ

µ 

 

 

+ +

− − −

(

2

)

2 0

4

3+ ≤

−C C

ε

wt

because when

β

is large enough, if 2β−C2 ≥0, then

2

2)

2

( t

m

w D C

−

β can be omitted temporarily, so the above formula can be written as

2

2 m

t d

w D w

dt µ

 

+

 

  ≤

2 1

2 4 1 3 2

w D C

C

C m _λ m µ m

µε λ

µ

µ 

 

 

+

+ − −

(

₂

)

2

4

3 C wt

C +

ε

+ .

If we take σ =min

  

  

+

   

 

+

+ − _λ− _ε

µε λ µ µ

2 4 3 1 2 4 1 3 2

,C C C

C

C m m _,

then we have

( )

t + N

( )

t ≤0

N dt

d _σ

(19)

Where

( )

2 2

w

D

w

t

N

=

_t

+

µ

m , by using the Gronwall

inequality, we get

( )

t ≤N

( )

0 _e− =0

N σt ₍₂₀₎

so w=0, the uniqueness is proved.

Theorem 2[9] Let E be a Banach space, and the

semi-group S t

( )

：E→E satisfy the following conditions:

(1) The semi-group S

( )

t is uniformly bounded in E, i. e.

0

>

∀R , and there exists a constant C

( )

R , so that when

R

u_E≤ , there is

( )

t u ≤C

( )

R

(

∀t∈

[

0,+∞

)

S _E (21)

(2) There exists a bounded absorption set B₀ in E; (3) S

( )

t

(

t≥0

)

is a fully continuous operator. Then semi-group S

( )

t has compact global attractorA₀.

By replacing Banach space E in Theorem 2 with Hilbert space E_k, the following theorem of global attractor family is obtained:

(5)

problem (1) - (3) satisfies the condition of lemma 1, then the

problem (1) - (3) has a global attractor family Ak_{, that is there}

exists a compact set A_k such that

(1)S

( )

t Ak=Ak,t>0;

(2) dist

(

S

( )

t B Ak

) (

B Ek

)

t→∞ , =0, ∀ ⊂

lim , where

( )

(

)

( )

_→ →∞

  

  

− =

∈ ∈

∞

→ dist S t B A S t x y t

k k

A y

E B

x k

t , sup inf 0,

lim (22)

( )

t

S is the solution semi-group generated by problem (1) - (3).

Proof It is necessary to verify the hypotheses of Theorem 2 (1), (2), (3). Under the hypothesis of Theorem 2, there exists a solution semi-group S

( )

t :

( )

t Ek Ek

S ： → (23)

By Lemma 1, for ∀Bk ⊂Ek and contained in

{

u Hm+k( )_Ω + vHk( )_Ω

}

≤Rk

0

0 bounded set

( )(

t

u

v

)

u

_{( )}

v

_{( )}

(

u

_{( )}

v

_{( )}

)

R

S

_E _Hmk _Hk _Hmk _Hk k

k

2 2

0 2

2 2 0 0

0 0

0

+

≤

+

≤

=

+

+ _Ω _Ω + _Ω _Ω (24)

where t≥0，

(

u0,v0

)

∈Bk, this shows that

{ }(

S

( )

t t≥0

)

is uniformly bounded in E_k. By Lemma 1, there is further

( )(

t

u

v

)

u

_{( )}

v

_{( )}

R

S

_E _Hmk _Hk k

k

2 2

0 0

≤

+

=

+

+ _Ω _Ω

t

≥

t

0

=

t

0

( )

R

k (25)

so B0k is bounded absorption set of semi-group S

( )

t .

Because Ek ⊂E0 is compact embedded, that is, bounded set

in Ek(k=1,2,⋅ ⋅⋅,m) is compact set in E0 , so operator

semi-group S

( )

t is totally connected to t≥0 , then the

equation has a global attractor family

( )

0 ( 1,2, , )

0

0 S t B k m

B

A k

t k

k = = = ⋅ ⋅⋅

≥ ≥ ∪

∩

τ τ

ω .

4. Dimension Estimation

In this part, we first linearize the equation (1), then prove the Frechet differentiability of the solution semi-group, and finally prove the Haudorff dimension and Fractal dimension of the finite dimension of the global attractor family.

Linearization of equation (1) - (3) is obtained

( )

D u

( )

U M

( )

D u

(

D u D U

)

( )

u M

Utt+ m −∆ m +2 m m , m −∆ m

2 ' 2

( )

₊

(

'

( )

22 2₊

( )

2

)

₌0

∆ −

+ Ut g u u g u U

m

β

(26)

( )

x,t =0

U , x∈∂Ω,t>0 (27)

( )

x,0 =ξ

U ,Ut

( )

x,0 =η (28) where (ξ,η)∈Ek,(u,ut)=S(t)(u0,u1) is the solution of problem (1) - (3) obtained by (u0,u1)∈Ak.

Given (u0,u1)∈Ak,S(t):Ek →Ek, we can prove that for

any (ξ,η)∈Ek, the linear initial-boundary value problem (26)

- (28) has a unique solution (U(t),U_t(t))∈L∞([0,+∞);E_k). Lemma 2 For any t>0,R>0 , the mapping

k k E

E t

S(): → is Frechet differentiable on Ek . The

differential on ϑ=(u₀,u₁)T is a linear operator on

T t T

t U t U

F:(ξ,η) →( (), ()) , where U(t),Ut(t) are solutions of the problem (26) - (28).

Proof Let ϕ₀ =

(

u₀,v₀

)

T∈E_k , ϕ~₀ =

(

u₀+ξ,v₀+η

)

T∈E_k ,

and R R

k

k E

E ≤ 0 ≤

0 ,

ϕ

~

ϕ

, we define

( )

u,vT=S

( )

tϕ₀ ,

( )

0

~ ~

,

~_v _S_t_ϕ

u T= , can obtain Lipchitz property of bounded set

( )

t

S on Ek, i. e.

( )

ϕ₀

( )

ϕ~₀2

( )

ξ,η 2

E ct

Ek e k

t S t

S − ≤ (29)

let θ=u~−u−U be the solution of equation

( )

D u

( )

h

M t

m m

m

tt+ −∆ θ+β −∆ θ =

θ 2 (30)

( )

0 =θ

( )

0 =0

θ t (31)

Let s Dmu s~ Dmu~

, 2

2

=

= , then we can get

( )

(

M s M s

)( )

u M

( )

s

(

D u D U

)

( )

u h₌ ₋ ₋_∆m ₊ m m ₋_∆ m

, 2

~

~ '

( ) ( ) ( )

u u g u u

(

g u u g

( )

u

)

U

g 2 2 ' 2 2 2

2 ~

~ ₊ ₊

−

+ , (32)

by taking the inner product of both sides of equation (30) and

θ

t, we get

( )

θ β θ

( )

θ

θ t t

m m

t D D h

dt d s M dt

d

, 2

1 2

1 2 2 2

= +

+ , (33)

(6)

theorem, we can obtain

( ) ( )

( )

'( )

(

)

( )

2 ,

ɶ ɶ ，

m _m _m m

t

M s M s u

_M

s

_{D D}

u U u

θ

 

− +

 



−∆



( )

(

)

(

) (

)

( )

(

)

( )

' '

1 ɶ , ɶ, ɶ ɶ, 2 , ,

m m

m m m m m m

t t

s s u u u u u s u U u

M

α α

D D



θ



M

D D



θ



= + − −  +  



−∆

 

−∆



( ) 2 2

5

ɶ

m m t

C M′′ς

_D

m

u

_D

m

u

_D

m

u

_D

u

_D

θ

≤

+

'( ) 2

ɶ

m m t

s m

u

M

D

θ

+

( )

2 m m mɶ m

t

M s′

_D

θ

_D

u

_D

u

_D

θ

+ ( ) 2

2 m m m m m m m

t t

M′ s

_D

u

_D

u

_D

θ

_D

θ

_D

u

_D

u

_D

θ

+ +

θt m m

D u D C6 2

≤

(

θ

)

θt

m m m

D D u D

C +

+ 2

7

(

θ

)

θt

m m m

D D u D

C +

+ 2

8 (34)

according to hypothesis

( )

B , differential mean value theorem and Holder inequality, we can get

( ) ( ) ( )

(

( )

)

(

g u u gu~ u~ g u 2u g u U,

θ

t

)

2 2 2 ' 2 2

+ +

− C g

( )

ϖ u θt g

( )

u u θ θt g

( )

u θ θt

2 2

2 ' 2

''

9 2

∞ ∞

∞ + +

≤ ≤C

(

u +

θ

)

θ

t

2

10 (35)

so according to (34) - (35) formula and using Young's inequality, we have

( )h (C C) C D D C Dmu

t m m

m t

t

4 11 2 2

1 2 10 8 7

2 2

2 4

3 ₊ ₊

    

  

+ + +

≤ θ β θ

λ ε β θ

ε θ

， (36)

Formula (36) is now substituted for formula (33), and according to hypothesis

( )

A , can obtain

(

D

)

C

dt

d m

t 12

2 2

≤ +µ θ

θ

(

θt µ Dmθ

)

2 2

+ +C13 Dmu 4

(37)

Then by using Gronwall inequality, we conclude that

τ θ

µ

θt + Dm ≤C eC t∫0t Dmu d

4 14

2 2

15

( )

ξ_,η

4 16 17

T E t

C

k

e C

≤ (38)

so from formula (38) we can get in Ek, let B

( )

t =

(

U

( ) ( )

t ,Ut t

)

, when

( )

, 0

2

→ η ξ T

, we have

( ) ( ) ( )

( )

≤

− −

η ξ

ϕ ϕ

, ~

2 2

T E

E

k k

t B t t

( )

, 2 0

18

19 ξηT →

E t

C

k

e

C (39)

Lemma 2 is proved.

Theorem 4 Under the condition of Theorem 3, the global attractor A_k of the problem (1) - (3) has Hausdorff dimension and

Fractal dimension, and

( )

1 ,

( )

7

6 6

H k F k

d A < n d A < n.

Proof Let A= −∆, equation (1) can be written as

( )

u u f

( )

x g

u A u A u A M

u m m t

m

tt  + + =

    

+ 2

2 2

β (40)

let Ψ =R_εφ=( , ),u v φ=( ,u u_t),v= +u_t εu, :

{ } {

, ,

}

t t

u

_u

u

_u

u

R

ε → +ε be an isomorphic mapping, so equation (40) can be written

as

( )

t ε g f

(7)

where

{

uut u

} ( )

g

{

g

( )

u u

}

T f

{

f

( )

x

}

T T , 0 , , 0 ,

, + Ψ = 2 =

= Ψ ε ,           − + − − = Λ I A I A u A M I I m m m ε β ε βε ε ε 2 2

2 ₎ ₎

(

( .

: ( ) ( )

t F f ε g

Ψ = Ψ = − Λ Ψ − Ψ (42)

Lemma 2 shows that S

( )

t :Ek →Ek is a Frechet differential, then the linearized equation (42) can be written as

0 ) ( ' _Ψ ₌ + Λ

+ P g P

Pt ε (43)

where Pt =Ft(Ψ),P

(

UUt U

) ( )

g

(

g

( )

u u g

( )

u

)

U

)

2 2 2 ' ' 2 , 0 , , + Ψ = ⋅ +

= ε . U is the solution of equation (41).

For a fixed (u0,v0)∈Ek, let ξ1,ξ2,⋯,ξn be n elements in Ek. Let U1(t),U2(t),⋯,Un(t) be n solutions of linear

equation (43) with initial values U1(0)=ξ1,U2(0)=ξ2,⋯,Un(0)=ξn. By direct calculation, we can obtain

2 '

1( ) 2( ) ( ) 1 2 exp(₀ ( ( )) ( ) )

k k

t

n n n

U t U t U t trF Q d

E ξ ξ ξ E τ τ τ

Λ Λ

Λ Λ Λ⋯ = Λ Λ⋯

_∫

Ψ ⋅ (44)

where Λ denotes the outer product, tr denotes the trace of the operator, and Qn(τ) denotes the orthogonal projection of

subspaces generated by Ek to U1(t),U2(t),⋯,Un(t).

For a given time τ , let ωj(τ)=(ζj(τ),ηj(τ))T,j=1,2,⋯,n be the standard orthogonal basis of space )} ( , ), ( ), (

{U1 t U2 t U t

span ⋯ n . Define the inner product on Ek

)) , ( ) , (( )) , ( ), ,

((ζ η ζ η = Dm+kζ Dm+kζ + Dkη Dkη . (45) To sum up, we can get

, )) ( ), ( )) ( ( ( )) ( ), ( ) ( )) ( ( ( ) ( )) ( ( 1 1 k k E j j n j t E j j n

j t n

n t F Q F Q trF τ ω τ ω τ τ ω τ ω τ τ τ τ ∑ Ψ = ∑ Ψ ⋅ = ⋅ Ψ = = (46)

where ( ( ( )) ( ), ( )) ( , ) ( '( ) , ).

j j j j E j j t g

F Ψτ ω τ ω τ _k =− Λ_εω ω − Ψω ω From Poincare inequality and hypothesis

( )

A , we can obtain )) , ( ), ) ) ( ( , (( ) , ( 2 2 2 j j j j m j j m m j j j

j ω εζ η M A u βε A ζ ε ζ βA η εη ζ η

ω ε =− − − + + − Λ −

η

ε

η

β

η

ζ

ε

η

ζ

βε

ζ

ε

j k j k m j k j k j k m j k m m j k m D D D D D D u A M D 2 2 2 2 2 2 2 2 2 2 ) ( 2 ) ( 2 ) ( 1 + − + + + − + + − ≤ + + + +

η

ε

η

λ

βε

β

ζ

ε

λ

ε

βε

j k j k m m j k m m m D D u A M D u A M 2 2 2 1 2 2 2 2 2 1 2 2 2 2 ) 1 ) ( 2 ( 2 2 ) ( 1 ₊ + − − + − − − + + ≤ + − ) ( 2 2 2

20 ζ η

j k j k m D D C ₊ −

≤ + ₍₄₇₎

(8)

  

  

− − − − +

− −

+ +

−

=_min ₍_βε _ε _λ− ₁ ₍ ₎ ₂_ε_),₍₂_β ₍ ₎ _βε ₁₎_λ _ε2 ₂_ε

1 2

2 2

2

1 2 20

m m

m

m _u

A M u

A M

C .

According to the definition of inner product over Ek, Holder inequality and hypothesis

( )

B , we can get

) , ) ( ( '

j j

g Ψω ω ₌

(

0,

(

g'

( )

u2_⋅2u2₊g

( )

u2

)

η_j

)

,

(

ζ _j,η_j

)

( )

(

)

(

η ηj

)

k j k

D u g u u g

D ' 2_⋅2 2₊ 2 ,

=

( )

ηj

k

D u g u u

g' 2 2 2 2

2 

  



 ₊

− ≥

∞ ∞

ηj k

D

r 2

−

≥ (48)

According (46),(47) and (48), we have

2 2 2

20 21

( ( ( )) ( ), ( )) ( ) ( )

2 2

k

m k k k

t j j E j j j

C

F r r

k

j

C

D

τ ω τ ω τ +ζ η η

ζ

η

Ψ ≤ − + + ≤ − + + (49)

Because T j n

j j

j(τ)=(ζ (τ),η (τ)) , =1,2,⋯,

ω are the

standard orthogonal bases of space

)} ( , ), ( ), (

{U1 t U2 t U t

span ⋯ n , so

2 2

1

j j

ζ + η = (50)

then have

2 21

1 1

( ( ( )) ( ), ( ))

2 k

n n

k

t j j E j

j j

nC

F τ ω τ ω τ r Dη

= =

Ψ ≤ − +

∑

(51)

For almost all t , there is ,

1 1

1 2

∑

≤

∑

= − =

n j

s j n

j j

k

Dη λ so there

have

∑

+ − ≤ ⋅ Ψ

= − n j

s j n

t r

C n Q

trF

1 1 21

2 ) ( )) (

( τ τ λ (52)

Let

0

0 0

1

( ) sup sup ( ( ( ) ) ( ) ) j

j

t

n t n

E A

q t trF S Q d

t

η η

τ τ τ

∈ Ψ ∈

≤

=

∫

Ψ ⋅ (53)

and have

lim sup ( )

n _t n

q q t

→∞

= (54)

then from formulas (53) and (54), we can see that

.

2 1

1 21₊ _∑

− ≤

= − n j

s j

n r

C n

q λ (55)

Therefore, the Lyapunov exponent of Ak is uniformly

bounded

1 21

1 2

1

2

n s

n j

j

n r

C

σ σ σ λ−

=

+ + +⋯ ≤ − +

∑

₍₅₆₎

then there exists a s∈

[ ]

0,1, such that

( )

21 1 1 21

1 1

2 7

n n

s s

j j j

j j

n n

q ₊

C

r λ − r λ −

C

= =

≤ − + ∑ _≤ ∑ _≤ (57)

where λj are the eigenvalues of m k

A , and

m

λ λ

λ1 < 2 <⋅ ⋅⋅< , then there have

1 21

21 1

21

2 5

1

2 14

n s

n j

j

n r

q n

n

C

_C

C

λ

− =

 

≤ − _ − ∑ __{≤ −}

  (58)

so

( )

1

1 6

m ax

j n n

j q

q

+

≤ ≤ ≤

(59)

Then we can get conclusion dH

( )

A n dF

( )

A n

6 7 ,

6

1 _<

< .

The proof is complete.

5. Conclusions

On the basis of previous studies on global attractors, this paper studies the global attractor family on space

) , , 2 , 1

(k m

E_k = ⋅ ⋅⋅ , and further studies the dimension

estimation of global attractor family. However, the proof of the global attractor family in this paper is not sufficient, so the proof of the global attractor family needs to be further improved. It is suggested to further study and expand the global attractor family on this basis.

Acknowledgements

(9)

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