The Asymptotic Behavior of Solutions for 3D Globally Modified Bénard Problem with Delay

(1)

ISSN Online: 2333-9721 ISSN Print: 2333-9705

DOI: 10.4236/oalib.1105163 Jan. 31, 2019 1 Open Access Library Journal

The Asymptotic Behavior of Solutions for 3D

Globally Modified Bénard Problem with Delay

Xia Hou

_,

Chaosheng Zhu

*

School of Mathematics and Statistics, Southwest University, Chongqing, China

Abstract

In this paper, we mainly study the existence and uniqueness of solutions and the asymptotic behavior of solutions for three-dimensional globally modified Bénard systems with delays under local Lipschitz conditions.

Subject Areas

Fluid Mechanics, Partial Differential Equation

Keywords

Bénard System, Delay, Galerkin Approximation, Asymptotic Behavior

1. Introduction

Let _{Ω ∈}_3_{be an open and bounded set with regular boundary}_Γ_{. We}

con-sider the following problem for the Bénard system with homogeneous Dirichlet boundary conditions on Ω:

(

)

(

)

(

)

(

)

(

)

(

)

( )

0

( ) ( )

0

( )

, in 0, ,

0, in 0, ,

0, 0, on 0, ,

,0 , ,0 , .

u _{u u} _u _{p f}

t

u g

t u u

u x u x x x x

ν ξω

ω _ω _ω

ω

ω ω

∂

 _{− ∆ + ⋅∇ +} _{+ ∇ =} _{+∞ × Ω}

 ∂ 

∂

 _{− ∆ + ⋅∇} ₌ _{+∞ × Ω}

 ∂ 

∇ ⋅ = +∞ × Ω

 = = +∞ × ∂Ω



 ₌ ₌ _{∈ Ω}



(1)

Here u x t

( ) ( )

, ,ω x t, _{are respectively the velocity, temperature of the fluid,}

( )

,

p x t is the pressure of the fluid, ν is the viscous coefficient of the fluid,

3

ξ ∈_ _{is a constant vector,} u₀,ω₀_{are the initial value. The external force}

terms _f _:__{× Ω →}_3_, _g_:__{× Ω →}__{are a given function.}

As we all know, Bénard system consists of the Navier-Stokes equations

How to cite this paper: Hou, X. and Zhu, C.S. (2019) The Asymptotic Behavior of Solutions for 3D Globally Modified Bénard Problem with Delay. Open Access Library Journal, 6: e5163.

https://doi.org/10.4236/oalib.1105163

Received: January 2, 2019 Accepted: January 28, 2019 Published: January 31, 2019

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

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DOI: 10.4236/oalib.1105163 2 Open Access Library Journal

coupled with a parabolic equation. This system is a well-known model in hy-drodynamics. It describes the behavior of the velocity, the pressure and the temper-ature for an incompressible flow. A detailed description of the physical background can be found in [1] and [2]. However, the well-posedness of the Bénard system is still an unsolved problem in the three-dimensional case. Therefore, it is particularly im-portant to modify equations for Bénard system. We consider the following glo-bally modified Bénard system:

( )

(

)

(

)

(

)

(

)

(

)

(

)

( )

0

( ) ( )

0

( )

, in 0, ,

0, in 0, ,

= 0, 0, on 0, ,

,0 , ,0 , .

N

u _{u F} _{u u} _u _{p f}

t

u g

t u u

u x u x x x x

ν ξω

ω _ω _ω

ω

ω ω

∂

 _{− ∆ +} _{⋅∇ +} _{+ ∇ =} _{+∞ × Ω}

 ∂ ∂

 − ∆ + ⋅∇ = +∞ × Ω

 ∂ 

∇ ⋅ = +∞ × Ω



 ₌ _{+∞ × ∂Ω}



= = ∈ Ω



(2)

Here the function F rN

( )

:+→+ is defined by

( )

: min 1, , , .

N N

F r r N

r

+

 

= _ _ ∈

  

In the modified term of Equation (2), F uN

( )

depends on the norm

( )

(

₂

)

3 3

L

u u ×

Ω

= ∇ _{, which in turn depends on}∇u_{over the whole domain}Ω

and not just at or near the point x∈ Ω_{under consideration. Essentially, it}

prevents large gradients dominating the dynamics and leading to explosions. The globally modified Bénard system can be better described when we add the appropriate delay term to the system. In this way, it can take into account not only the present state of the system but the history of the solution. Namely, we consider the following equations:

( )

(

)

(

( )

)

(

)

(

)

(

( )

)

(

)

(

)

(

)

( )

( ) ( )

( )

(

) ( )

(

)

(

)

1

2

0 0

1 2

, , in 0, ,

0, in 0, ,

0, 0, on 0, ,

, , , ,

, , , , , , in , .

N

u _{u F u u} _u _{p G t u t} _t

t

u G t t t

t u u

u x u x x x x

u t x t x t x t x h

ν ξω ρ

ω _ω _ω _ω _ρ

ω

τ ω τ ω

φ τ ω φ τ τ τ

∂

 _{− ∆ +} _{⋅∇ +} _{+ ∇ =} ₋ _{+∞ × Ω}

 ∂ 

∂

 _{− ∆ + ⋅∇} ₌ ₋ _{+∞ × Ω}

 ∂ 

∇⋅ = +∞ × Ω



= = +∞ × ∂Ω



 ₌ ₌ _{∈ Ω}



 = − = − − × Ω



(3)

Here τ ∈__{is the initial time,} G t u t₁

(

,

(

−

ρ

( )

t

)

,G t₂

(

,

ω

(

t−

ρ

( )

t

)

_{are the}

external force terms which depends on u t

(

−

ρ

( )

t

)

,

ω

(

t−

ρ

( )

t

)

, respectively.

( )

t 0

ρ ≥ _{is a delay function,}φ φ₁, ₂_{are respectively the velocity, temperature}

defined on

(

−h,0

)

_{. In addition,} ρ

( )

t ≤h h, >0_{. When} ω =0_{, the Equation (3)}

is the Navier-Stokes equations with delays (see [3] [4]).

(3)

globally modified Bénard system with delays under local Lipschitz conditions. This paper is organized as follows. In Section 2, we recall some definitions of function spaces and some related properties. In Section 3, we give the proof of the existence and uniqueness of weak solutions. In Section 4, we consider the asymptotic behavior of solutions.

2. Preliminary

Let

( )

(

)

{

}

( )

(

2

)

3

(

1( )

)

3 0

3

0 : 0 .

, .

L H

v C v

H cl V cl

∞

Ω Ω

= ∈ Ω ∇ ⋅ =

= =



 

We denote by

( )

⋅ ⋅, _{the scalar product,} ⋅ ₂_{the norm in}_H_{. For}

( )

(

₂

)

3

,

u v∈ L Ω _,

( )

u v, =

∑ ∫

3j= Ω₁ u x v x xj

( ) ( )

j d . We denote by

( )

⋅ ⋅, the scalar

product, ⋅ _{the norm in}_V_{. For}

(

1

( )

)

3 0

,

u v∈ H Ω ,

(

( )

)

3

, 1

, j jd

i j

i i

u v

u v x

x x

= Ω

∂ ∂ =

∂ ∂

∑ ∫

.

The dual space of V is V′_{. It follows that}_V ⊂_{H H}≡ ′⊂_V′_{, and the injections}

are dense and compact. We shall use ⋅∗ to denote the norm of V′, ⋅ ⋅,

represent the dual product between V and V′_._P_{is a Leray-Helmotz projection}

from

(

₂

( )

)

3

L Ω to H,  is a Leray-Helmotz projection from

(

₂

( )

)

3

L Ω to

( )

2

L Ω _{, suppose} A u₁ = − ∆P u_, A₂ω= − ∆ ω, where A A1, 2 are linear

conti-nuous operators. Let D A

( )

1 be the domain of the operator A1 in H, D A

( )

2

be the domain of the operator A2 in L2

( )

Ω . In addition,

(

)

1_{( )} 1_{( )}

(

)

0

1 , V V, , , 2 , H ,H , ,

A u v _′ = ∇ ∇u v A

ω η

_Ω − _Ω = ∇ ∇

ω η

where 1

( )

0

, , ,

u v V∈

ω η

∈H Ω .

Because of the regularity of the boundary ∂Ω_{, it is easy to prove that:}

( )

(

₂

( )

)

3 1

D A = H Ω _V,

( )

2

( )

1

( )

2 0

D A =H Ω H Ω , furthermore, we obtain

( )

1

( )

2

( )

1 , 2 0 .

D A ⊂ ⊂V H D A ⊂H Ω ⊂L Ω

The first eigenvalue of Stokes operator is defined by

{ }

2

2 \ 0

2

0 inf

v V

v v

λ ∈

= > , let λ1

be the minimum of the first eigenvalues of the operators A A1, 2. For all

, ,

u vω∈V_{, we put}

(

, ,

)

3_{, 1} j d

i j

i j i

v

b u v u x

x

ω = _Ω ₌ ∂ ω

∂

∑

∫

. We know from [1] that

(

, ,

)

b u vω _{is a trilinear continuous form on}V V V× × _{, and} b u v v

(

, ,

)

=0_for

u V∈ _,

(

1

( )

)

3 0

v∈ H Ω _{. For any} u v, ,ω∈V_{, it follows that (see}_[5]₎

(

, ,

)

12 12.

b u v

ω

≤C u v

ω ω

(4) Define

(

, ,

)

( )

(

, , , , ,

)

,

N N

b u v w =F v b u v w ∀u v w V∈

where bN is linear in u w, , but it is nonlinear in v. Obviously,

(

, ,

)

0, ,

N

(4)

defini-DOI: 10.4236/oalib.1105163 4 Open Access Library Journal

tion of FN, we can get

(

, ,

)

1 , , , .

N

b u v

ω

≤NC u

ω

∀u v

ω

∈V

(5) The following is a lemma, which is specifically proved in [5].

Lemma 2.1. For all u v V, ∈ _and N>0, the following two conclusions hold: 1) 0≤ u F uN

( )

≤N; 2) F uN

( )

−F vN

( )

≤ _N1 F u F v u vN

( ) ( )

N − .

For all 1

( )

0

, ,

u V v∈

ω

∈H Ω , let us define

(

, ,

)

3i₁ i d

i

v

c u v u x

x

ω ₌ ω

Ω

∂ =

∂

∑

∫

. It has

been given in [7], and c u v v

(

, ,

)

=0_for 1

( )

0

,

u V v H∈ ∈ Ω .

Let

( )

, ,

(

, , ,

)

, , ,

N N

B u v

ω

=b u v

ω

∀u v

ω

∈V

( )

(

)

1

( )

0

, , , , , , , .

C u v

ω

=c u v

ω

∀ ∈u V v

ω

∈H Ω

The following describes the properties of the external force terms

: , 1,2

i

G _×H→H i= _:

(c1) for any _{u H}_∈ _,_ω_∈_L2_,

( )

2

1 , : , 2 , :

G u⋅ →H G ⋅

ω

→L are measurable; (c2) there exists nonnegative functions p

( )

, 1,2,1

i loc

g L∈ _ i= ≤ ≤ +∞p _{, and a}

nondecreasing function L: 0,

(

∞ →

) (

0,∞

)

_{, such that for all}RT_i >0, if , T_i

u v R≤ _{, then:}

( )

_,

( )

_,

( )

12

( )

_, _{1,2, ;}

i

i i T i

G t u G t v− ≤L R g t u v i− = t∈_

(c3) there exists nonnegative functions 1

( )

_{, 1,2}

i loc

f ∈L _ i= _{, such that for any}

u H∈ _(or _L2_),

( )

2

( )

, , .

i i i

G t u ≤g t u + f t ∀ ∈t 

Supposing 2

(

)

2

(

2

)

1 Lp h H,0; , ,0;2 L p h L

φ

_∈ ′ ₋

φ

_∈ ′ ₋

,

{

}

2

0, 0

u

ω

∈ ×H L _{, where}

1 _{1 1}

p p+ ′= . We consider a delay function

ρ

∈C1

( )

 for any t∈ such that

( )

0≤ρ t ≤h _{, and there exists a constant}ρ_* _{for any}t∈_ _satisfying:

( )

t * 1

ρ′ ≤ρ < _.

We define

( )

1

( )

i i

g t_ ₌g

θ

− t

 , where θ

( )

s = −s ρ

( )

s _,

[

)

( )

)

: , ,

θ τ

+∞ →_

τ ρ τ

− +∞ is the differentiable and strictly increasing func-tion, we obtain

( )

(

)

(

)

( )

(

( )

)

( )

(

)

(

( )

)

( )

( ) ( )

( )

_{( )}

( )

( ) ( )

( )

( ) ( )

( )

2 2

1 1 1

2

1 1

2

1 1

2

1 1

*

2

1 1

*

, d d d

d d

1

( ) d d

1

1 _d _d

1

1 _d _{d ,}

1

T T T

T T

T T T

T T

G t u t t t g t u t t t f t t

g t t u t t t f t t

g s u s s f t t

s

g t u t t f t t

τ τ τ

τ τ

ρ

τ ρ τ τ

ρ

τ ρ τ τ

ρ ρ

ρ

ρ − −

− −

−

− ≤ − +

= − − +

= ⋅ +

′ −

≤ +

−

≤ +

−

∫

 



(5)

Similarly, when i=2_{, there is also an analogy estimate for the}ω term. Moreover, we call

{ }

u,ω ∈WT as a set of solution for Equation (3) in

(

0,T

)

,

when

(

)

(

)

(

2

)

(

2

(

1

( )

)

(

2

( )

)

0

0, ; 0, ; 0, ; 0, ; ,

T

W ₌ L T V L∞ T H _× L T H _Ω L∞ T L _Ω

 

and for any 1

( )

0

,

v V∈

ω

∈H Ω , the following is true:

( )

(

( )

)

(

) (

)

(

( )

)

(

) (

(

)

(

)

(

( )

)

1

2

d , , , , , , , ,

d

d _, _, _{, ,} _, _{, .}

d

N

u v u v b u u v v G s u s s v

t

c u G s s s

t

ν ξω ρ

ω ω ω ω ω ω ω ρ ω

 ₊ ₊ ₊ ₌ ₋

 

 + + = −

    

3. Existence and Uniqueness of Weak Solutions

Theorem 3.1. Under the conditions (c1)-(c3), assume that τ_∈_+_,

{

}

2

0, 0

u

ω

∈ ×H L and 2

(

)

1 Lp h H,0;

φ

_∈ ′ ₋

, 2

(

2

)

2 Lp h L,0;

φ

_∈ ′ ₋

. Then there

ex-ists a weak solution

{ }

u,ω _{of equation}₍₃₎_.

Proof. For simplicity, and without loss of generality, we assume τ =0_{. We}

consider the proof of the existence of weak solutions of Equation (3). We use Ga-lerkin method to prove the existence of solutions, which is standard (see [8] [9]).

Consider the following equations:

(

)

(

( )

)

(

)

(

( )

)

( )

(

)

1 1

2 2

0 0

1 2

, , ,

0 , 0 ,

, , , ,0 .

m

m m N m m m m m

m

m m m m m m

m m m m

m m m m m m

u _{A u} _{P B u u} _{P G t u t} _t

t

A C u G t t t

t

u P u

u P u h

ν ξω ρ

ω _ω _ω _ω _ρ

ω ω

φ ω φ ω

∂

 ₊ ₊ ₊ ₌ ₋

 ∂ ∂

 ₊ ₊ ₌ ₋

 ∂ 

= =



 = = ∈ −



 

(7)

Next, we need to make some priori estimates for the approximate solution

{

um,ωm

}

. So taking the inner product of the second equation of (7) with ωm,

and using c u

(

m, ,ω ω =m m

)

0, we obtain

( )

(

( )

)

( )

(

)

(

)

( )

(

)

(

)

( )

2 2

2

2 ₁ 2

2 1

2 2

2 1

1 d _, _,

2 d

1 _,

2 2

1 _, 1 _,

2 2

m m m

m m

t t G t t t

t

G t t t t

ω ω ω ρ ω

λ

ω ρ ω

λ

ω ρ ω

λ

+ = −

≤ − +

that is

( )

(

)

(

)

2

2 2

2 1

d 1 _, _.

d_t ωm + ωm ≤_λ G t ωm t−ρ t

Integrating over

( )

0,t , and using (6), we obtain for t∈

[

0,T2

]

( )

(

)

(

)

(

)

( )

2

2 2

0

2 2

0 ₀ 2

1

2 2

0

1 *

d

1 _, _d

1 _{d ,}

1

t

m m

t

m

t

T m

t s s

G s s s s

K g s s s

ω ω

ω ω ρ

λ

ω

λ ρ

+

≤ + −

≤ +

−

∫



(6)

where

₍

₎

_{( )}

( ) ( )

2

( )

2

0 2

2

0 ₀ 2 2 ₀ 2

1 * 1

1 _d 1 _d

1

T T

K ω _ρ g s φ s s f s s

λ ρ − λ

= + +

−

∫



∫

. It follows

from Gronwall lemma that

( )

₍

₎

2

( )

[

]

2 2

2

2 2

0

1 *

1

exp d , 0, .

1

T

m t KT g s s CT t T

ω

λ

ρ

 

 

≤ _ _= ∀ ∈

−

 



∫

  (9)

Taking the inner product of the first equation of (7) with um, from (9) and

(

, ,

)

0

N m m m

b u u u = _{, we obtain}

( )

(

)

(

( )

)

( )

(

( )

)

( )

(

)

(

)

( )

₂

2 2

1

2

2 2 2

2 1 1

1

1 1

2

2 2

1

1 1

1 d 2 d

, , ,

1 1 _,

4 4

1 _, _,

2

m m

m m m m

m m T

u t u t

t

t u t G t u t t u

t u t G t u t t u t

G t u t t u t C

ν

ξ ω ρ

νλ νλ

ξ ω ρ

νλ νλ

ν ξ

ρ

νλ νλ

+

≤ + −

≤ + + − +

≤ − + +

that is

(

)( )

(

)

2 2 ₂

2 2

1

1 1

d 2 _, 2 _.

d_t um um G t u tm t CT

ξ

ν ρ

νλ νλ

+ ≤ − +

Integrating over

( )

0,t , and using (6), we obtain for t∈

[

0,T1

]

( )

(

)

(

)

(

)

( ) ( )

2

1

2 2

0

2 2

2

0 ₀ 1 ₀

1 1

2 1

0

1 *

d

2 _, _d 2 _d

2 _{d ,}

1

t

m m

t t

m T

t

T m

u t u s s

u G s u s s s C s

K g s u s s

ν

ξ ρ

νλ νλ

νλ ρ

+

≤ + − +

≤ +

−

∫



(10)

where

(

)

( )

( ) ( )

( )

1 1

1 2

2 ₀ ₂

2

0 ₀ ₀ 1 1 ₀ 1

1 1 * 1

2 _d 2 _d 2 _d

1

T T

K u ξ C s _ρ g s φ s s f s s

νλ νλ ρ − νλ

= + + +

−

∫



∫

. It

follows from Gronwall lemma that

( )

₍

₎

1

( )

[ ]

1 1

2

1 1

0

1 *

2

exp d , 0, .

1

T

m T T

u t K g s s C t T

νλ

ρ

 

 

≤ _ _= ∀ ∈

−

 



∫

  (11)

From (10) and (11), a subsequence of

{ }

um is bounded in L∞

(

0, ;T H

)

and

(

)

2 _{0, ;}

L T V (where T =max ,

{

T T1 2

}

). Next from the first equation of (7) and

(5), we obtain that d

d

m

u

t is bounded in L2

(

0, ;T V′

)

. Therefore, there exists an

element _{u L}_∈ ∞

(

_{0, ;}_{T H}

)

_L2

(

_{0, ;}_{T V}

)

 and d 2

(

_{0, ;}

)

du L T Vt ∈ ′ , such that

(

)

(

)

(

)

2

in 0, ; ,

d _{d in 0, ; .}

d d

m m m

u u L T H

u u L T V

u u _L _{T V}

t t

∗ ∞

    

′ 



 

(7)

Obviously, u C∈

(

[

0,+∞

)

;H

)

(see [10]). By the compactness Aubin-Lions (see

[9]), we can deduce

(

)

(

)

2

in 0, ; ,

a.e.in 0, .

m m

u u L T H

u u T

 →

 

→ × Ω



Similarly, from (8) and (9), there is also an analogy conclusion for the ωm

term.

In summary, when tm= +∞,m≥1, there exists

{ }

(

)

2

(

)

(

2

( )

)

2

(

1

( )

)

0

, 0, ; 0, ; (0, ; 0, ;

u

ω

_∈ L∞ T H _L T V _× L∞ T L _Ω _L T H _Ω _{for any}

0

T> _{, such that}

{

} { }

(

)

(

( )

)

{

} { }

( )

(

)

2 2 2

2

, , in 0, ; 0, ; ,

, , in . . 0, .

m m

u u L T H L T L

u u H L a e t T

ω ω

 → × Ω

 

→ × Ω ∈



Since

{

um,ωm

}

weak converges to

{ }

u,ω in V H× 01

( )

Ω , we cannot deduce

that um → u , ωm → ω or F u tN

(

m

( )

)

→F u tN

(

( )

)

and

(

m, m

)

(

,

)

C u ω →C uω _{at least almost everywhere. So we need make a stronger}

estimate.

Taking the inner product of the first equation of (7) with A u1 m, and using (9),

we obtain

(

)

(

)

(

( )

)

(

)

(

( )

)

( )

(

)

(

)

( )

(

)

(

)

2

2 2

1

1 1 1 1

3 3

2 ₁ 2 ₁ ₁ ₁

2 2 2 2

4 2

1 1

2 ₂

1 1

2 ₂

2 4 2

1 1

1 d 2 d

, , , , ,

, , ,

1

8 4

2 _,

8

2 _, _,

2

m m

N m m m m m m m

m m m m m m

m

m m m m

m m

m m T m

u A u

t

b u u A u A u G t u t t A u

N C u Au A u G t u t t A u

u

CN u A u A u

G t u t t A u

A u CN u C G t u t t

ν

ξ ω ρ

ν _{ξ ω} ν

ν ν ρ

ν

ν ξ _ρ

ν ν

+

≤ + + −

≤ + + +

+ − +

≤ + + + −

that is

( )

(

)

(

)

2

2 ₂

2 2 4 2

1 1

d 2 4 _, _.

d_t um A um CN um CT G t u tm t

ξ

ν ρ

ν ν

+ ≤ + + −

(12) Integrating over

( )

s t, , and using (6), we obtain for all 0≤ ≤ ≤s t T

( )

(

( )

)

( )

₍

₎

_{( )}

( ) ( )

( )

2

2 ₂

2 ₄ 2

1

0 0 0

2 ₀

2 ₄ 2 2

1 1

0 0 0

* 2

1 1

0 0

* 2

2 4

d d , d

2 4

d d d

1

4 _d 4 _d

(1 )

: .

m

T T T

m m T m

T T

m m T

T T

m

u t

u s CN u r r C r G r u r r r

u s CN u r r C r g r r r

g r u r r f r r

u s

ρ

ξ _ρ

ν ν

ξ _φ

ν ν ρ

ν ρ ν

−

≤ + + + −

≤ + + +

−

+ +

−

= + Φ

∫



(8)

The following estimate Φ_{. Thanks to (10) and (11), we know that}u_m is bounded in _L2

(

_{0, ;}_{T V}

)

__L∞

(

_{0, ;}_{T H}

)

_{, therefore, there exists}

1 0

T

K > _{, such that}

for all integer m≥1

( )

₍

₎

_{( )}

( ) ( )

(

)

( ) ( )

( )

2

1

2 ₀

2 2

4

1 1

0 0 0

* 2

1 1

0 0

*

2 4

d d d

1

4 _d 4 _d _.

1

T T

m T

T T

m T

CN u r r C r g r r r

g r u r r f r r K

ρ

ξ _φ

ν ν ρ

ν ρ ν

−

+ +

−

+ + ≤

−

∫



 

Integrating (13) for s over

[ ]

0,t , we obtain

( )

2

( )

2 ₁

(

( )

2

)

₁ ₁

0 d sup₁ 0 d : ˆ .

T T

m m T m T T

m

t u t u s s TK u s s TK K

≥

≤

_∫

+  ≤

_∫

+  =

Consequently, u tm

( )

2≤1 ˆ

_ε

KT₁ for any t∈

[ ]

ε,T and any 0< <ε T .

On the other hand, using the uniformly Gronwall inequality for (12), we have

( )

(

)

(

)

4

1

2 1

2 ₂

2

1 4

2 4 _, _e _.

T CN

m T m T

K

u C G t u t t C

CN

ξ

_ρ

ν

 

′

≤_ + + − _ =

 



(14) Taking the inner product of the second equation of (7) with A2ωm, we obtain

(

)

(

( )

)

( )

(

)

(

)

( )

(

)

(

)

1

2 2

2

2 2 2

2

2 2 2 2

2 2 2

2

2 2 2

1 d 2 d

, , , ,

1 _, 1

4 4

1 _, 1 _,

4 4

m m

m m m m m

T m m m m

A t

c u A G t t t A

C u A G t t t A

CC A G t t t A

ω ω

ω ω ω ρ ω

+

≤ + −

≤ + + − +

′

≤ + + − +

where

(

)

2 2 2

2 2 2 1 2

, , .

4

c uω Aω ≤ u_∞ ∇ω Aω ≤C u ω Aω ≤C u ω + Aω

That is

( )

(

)

(

)

1

2

2 2 2

2 2

d ₂ _, _.

d_t

ω

m + A

ω

m ≤CCT′

ω

m + G t

ω

m t−

ρ

t

Similarly, integrating over

( )

s t, , we obtain for all 0≤ ≤ ≤s t T

( )

2

( )

2 ₂

,

m t m s KT

ω

≤

ω

+ 

(15) where

( )

_{( )}

( ) ( )

( )

1

2

0

2 2

0 0

* 2

2 1

0 0

*

2

d d

1

2 _d ₂ _d _.

1

T

T m

T T

m T

CC r r g r r r

g r r r f r r K

ρ

ω φ

ρ

ω ρ

−

′ +

−

+ + ≤

−

∫



 

Integrating (15) for s over

[ ]

0,t , we obtain

( )

2

(

( )

2

)

₂ ₂

0 1

ˆ

sup T d : .

m m T T

m

t ω t ω s s TK K

≥

≤

_∫

+  =

Consequently,

( )

2

2 1 ˆ

m t KT

ω

ε

(9)

On the other hand, using the uniformly Gronwall inequality for (14), we have

( )

(

)

(

)

2 1

2 2

2 , e T .

CC T

m m T

T

K

G t t t C

CC

ω

≤_ _′ +

ω

−

ρ

_ ′ = ′

 



(16) From the above discussion, we can deduce that um is bounded in L∞

(

ε

, ;T V

)

and 2

(

( )

)

1

, ;

L

ε

T D A . What’s more, when u0∈V , owing to um

( )

0  P um 0 0

u

≤ _{, we deduce that}u_m is bounded in

(

)

2

(

( )

)

1

0, ; 0, ;

L∞ T V _L T D A _{; when}

0

u ∈H, it follows that D A

( )

1 ⊂ ⊂V H and injections are compact. Supposing

there exists a positive sequence

{ }

εn 0, such that

(

)

2

(

)

(

)

2

(

( )

)

1

0, ; 0, ; , ; , ; .

u L_∈ ∞ T H L T V L∞

ε

T V L

ε

T D A

  

Therefore

(

)

( )

(

)

(

)

(

)

2

1

in 0, ; ,

in 0, .

m m m m

u u L T V

u u L T D A

u u L T V

u u T

∗ ∞

∞

   

→ 

 _→ _{× Ω}



 

As long as there exists um →u, in L2

(

ε

, ;T V

)

for T > >ε 0, we can get

(

)

a.e.in 0, ,

m

u → u +∞

( )

(

)

(

( )

)

a.e.in 0,

(

)

.

N m N

F u t →F u t +∞

Similarly, the same is true for the ωm term. In conclusion,

{ }

u,ω is a set of

solution for Equation (3).

Theorem 3.2. Under the previous all assumptions, there exists a unique

solu-tion of equasolu-tions (3).

Proof. Let

{

u1,ω1

}

and

{

u2,ω2

}

be two solutions of equations (3), and set

{ } {

u,ω = u u1− 2,ω ω1− 2

}

, according to the first equation of (3), we obtain

(

)

(

)

( )

(

)

(

)

(

( )

)

(

)

2 2

1 2

1 1 1 2

1 d _, _, _,

2 d

, , , ,

u u u u u u

t

G t u t t G t u t t u

ν ξ ω

ρ ρ

+ + +

= − − −



where 

(

u u u1, 2

)

, =F u b u u u F u b u u uN

( )

1

(

1, ,1

)

− N

( )

2

(

2, ,2

)

, using the

property of trilinear form b, we easily get

(

)

( )

(

)

(

( )

)

(

)

( )

(

)

1 2

1 1 1 2 2 1 2 2

, ,

, , , , , , .

N N N N

u u u

F u b u u u F u F u b u u u F u b u u u

= + − +



From Lemma 2.1, (4) and the Young inequality, we obtain

(

)

2 4 2

1, 2 , ₂ .

u u u ≤ν u +CN u



Therefore

( )

(

)

(

)

(

( )

)

(

)

2 2

2

2 2 2

4

1 1 1 2

1 d

2 d 2

1 _, _, _{, .}

2 2

u u

t

CN u u G t u t t G t u t t u

ν

ξ ω ρ ρ

+

(10)

To estimate on the last term for above formula. For fixed T1>τ,

[ ]

(

, ;1

)

i

u C∈

τ

T H _{, there exists}s∈

[

τ,T₁

]

_{, such that} u s_i

( )

≤R_T₁_{, in addition,}

using (c2), u=0, ,t∈ −

(

τ hτ

)

_{, the Hölder inequality, Young's inequality and (6),}

we obtain for any t∈

[ ]

τ,T1

( )

(

)

(

)

(

( )

)

(

)

( )

(

)

(

)

(

( )

)

( )

(

( )

)

( )

(

)

( )

(

( )

)

(

)

( )

(

)

_{( )}

( ) ( )

(

)

( )

1

1 1 1 2

1 2 1

1

2 2 2

*

1 1

2 * 1

2 ₂ ₂

*

1 1

* _* 2

, , , d

, , d

d

1 _d 1 _d

2 _{2 1}

1 1 _d 1 _d

2 1 _{2 1}

2 1

T

T T

T

T T

T

G s u s s G s u s s u s

G s u s s G s u s s u s s

L R g s u s s u s s

L R g s u s s s u s s

L R g s u s s u s s

L R

τ

τ τ

τ ρ τ τ

ρ ρ

ρ

ρ − _ρ

− − −

≤ − − −

≤ −

 ₋ 

 

≤ _ − + _

−

 

 

 ₋ 

 

≤ _ ⋅ + _

− ₋

 

 

= −

∫



∫

(

)

(

( )

)

( )

2 1

1 2 *

1 d .

T

g s u s s

τ ρ

+

∫



Therefore

( )

(

)

(

( )

)

1

2

2 2 ₄ 2 2 2 2

1 1 2 *

1 d 1 ₁ _.

2 d 2 2 2 _{2 1}

T

L R

u u CN u u g t u

t

ν _{ξ ω}

ρ

+ ≤ + + + +

−  (17)

Similarly, according to the second equation of (3), we obtain

(

) (

)

( )

(

)

(

)

(

( )

)

(

)

2 2

1 1 2 2

2 1 2 2

1 d _{, ,} _{, ,}

2 d

, , , .

c u c u

t

G t t t G t t t

ω ω ω ω ω ω

ω ρ ω ρ ω

+ + −

= − − −

Since

(

) (

)

(

2 1

)

1 1 2 2 1 2

1 2

2 2 2 2 2 2

1 2

2 2 2

, , , , , , , ,

2

,

2 T T

c u c u c u c u

u u

C u C u

u C C u

u C C C

ω ω ω ω ω ω ω ω

ω ω ω ω

ν _ω _ω _ω _ω

ν _ω _ω

∞ ∞

− = +

≤ ∇ + ∇

≤ +

≤ + + +

′

≤ + + +

where (16) is used, we can get

(

)

(

( )

)

(

( )

)

(

)

( )

(

)

(

( )

)

2 1

2 2 1

2

2 2

2 1 2 2

2 2 2

2 1 2 *

1 d 2 d

, , ,

2

1 .

2 _{2 1}

T T

T

T T

t

u C C C G t t t G t t t

L R

u C C C g t

ω

ν _ω _ω _ρ _ω _ρ _ω

ν _ω _ω

ρ ′

≤ + + + − − −

′

≤ + + + +

− 

(11)

From (17) and (18), we obtain

(

)

(

)

( )

(

)

(

( )

)

(

)

( )

(

)

(

( )

)

1

2 2 1

2 2 4 2

1 1 2 *

2 2

2 1 2 *

d ₁ ₁

d ₁

1 .

1

T

T T

L R

u CN g t u

t

L R

C C C g t

ω

ρ

ξ ω

ρ

 

 

+ ≤_ + + + _

−

 

 

 

 _′ 

+_ + + + + _

−

 

 



Because the coefficients of u2 and

ω

2 are bounded at the right of the equa-tion, the conclusion follows from Gronwall lemma, since u

( )

0 2 =0,

ω

( )

0 2 =0_.

4. Asymptotic Behavior of Solutions

In this section, we prove mainly the asymptotic behavior of the solution of Equ-ations (3) when t→ ∞.

Suppose that (c1)-(c2) hold with gi∈L∞

( )

 , assume also that:

(

)

2 2

1 1 * 2 g1

ν λ

−

ρ

> _∞_, 2

(

)

1 1 * g2

λ

−

ρ

> _∞_{, where} gi_∞:= gi L∞_{( )}_ , and let us

denote respectively by ε ε >1, 2 0 the unique solution of

(

)

1

1 1

1 *

2 e

0, 1

h

g ε

ε νλ

νλ ∞ ρ

− + =

−

(19)

(

)

2

2 1

1 *

e 0. 1

h

g ε

ε λ

λ ∞ ρ

− + =

−

(20)

Theorem 4.1. Under the above assumptions, for any

{

}

2 0, 0

u

ω

∈ ×H L _,

(

)

2

1 L h H,0;

φ

∈ − , 2

(

2

)

2 L h L,0;

φ

∈ − _{and any}τ ∈_, the solution

(

) (

)

{

u t u; , ,

τ

0

φ ω τ ω φ

1 , t; , ,0 2

}

of equations (3) satisfies

(

)

₍

₎

( )

1

1 1

1 1 1

2

0

2 2 1 2

0 1 0 1

1 *

2 1

1 1

2 e

; , , e d e

1

2 2

e e d e d ,

h

t s

h

t t

t s s

T

g

u t u u s s

f s s C s

ε

ε τ ε

ε ε ε

τ τ

τ

φ

νλ

ρ

ξ

νλ

− ∞

−

 

≤_ + _

−

 

 

+ _ + _

 

∫

(21)

(

)

₍

₎

( )

2

2 2

0

2 2 2 2

0 2 0 2

1 *

2 1

e

; , , e d e

1

1 e e d .

h

t s

h

t

t s

g

t s s

f s s

ε

ε τ ε

ε ε

τ

ω τ ω φ ω φ

λ ρ

λ

− ∞

−

 

≤_ + _

−

 

+

∫

(22)

Proof. Let

{

u t

( ) ( )

,

ω

t

}

=

{

u t u

(

; , ,

τ

0

φ ω τ ω φ

1

) (

, t; , ,0 2

)

}

be the solution of

eq-uations (3) corresponding to the initial data τ, , , ,u0 φ ω φ1 0 2. From

( )

(

)

(

)

2

2 ₂

2 2

1

1 1

d 2 2 _, _,

d_t u u CT G t u t t

ξ

ν ρ

νλ νλ

+ ≤ + −

we obtain

( )

(

)

( )

(

( )

)

( )

1

1 1 1 1

2

2 ₂

2 2

1 1

d e d

2 2

e e e , e

t

t t t t

T

u t t

u t C G t u t t u t

ε

ε ξ ε ε ε

ν ρ ε

νλ νλ

(12)

(

)

1

( )

1 1

(

( )

)

2

2 ₂

2

1 1 1

1 1

2 2

e t e t e t , .

T

u t C G t u t t

ε ξ ε ε

ε νλ ρ

νλ νλ

≤ − + + −

(23) Integrating over

[ ]

τ,t , we obtain

( )

(

)

(

)

( )

(

)

(

)

1 1 1

2 1 2 2 2 1 1 1 2 1 1

d _e _d _e 2 _e

d

2 e , d .

t _s t _s _s

T

s

u s s u s C

s

G s u s s s

ε ε ε

τ τ ε ξ ε νλ νλ ρ νλ  ≤ _ − +   + − _ 

∫

That is

( )

( ) (

)

( )

(

)

(

)

1 1 1

1 1

2

2 2 2

1 1

2 ₂

1

1 1

e e e d

2 _{e d} 2 _e _, _{d .}

t

t s

t _s t _s

T

u t u u s s

C s G s u s s s

ε ε τ ε

τ

ε ε

τ τ

τ ε νλ

ξ _ρ νλ νλ ≤ + − + + −

∫

Since

( )

(

)

(

)

( )

(

)

( )

(

)

( )

1 1 1 1 1

1 1 1

1 1

1 1 1

2 1 2 1 1 2 2 1 1 1 1 * * 0 2 2 1 1 1 1 * *

e , d

e d e d

e e

e d e d e d

1 1

e e

e d e d e d .

1 1

t _s

t _s t _s

h h

t _s _s t _s

h h h

t _s _s t _s

h

G s u s s s

g u s s s f s s

g g

u s s s s f s s

g g

u s s s s f s s

ε τ ε ε τ τ ε ε τ

ε ε ε

τ τ τ

ε τ ε

ε ε ε

τ τ ρ ρ φ τ ρ ρ φ ρ ρ ∞ ∞ ∞ − + ∞ ∞ − − ≤ − + ≤ + − + − − = + + − −

∫

(24)

Using (19), we can get

( )

( ) (

)

( )

1 1 1 1 1 1 2 1 1 1 2 2 2 1 1 2 ₂ 1

1 1 *

0 2

1

1 1

*

e

e e d

e

2 _{e d} 2 _e _d

1 e

e d e d

1

t

t _s

h

t _s t _s

T h t s s h u t

u u s s

g

C s u s s

g

s s f s s

ε

ε τ ε

τ ε ε ε τ τ ε τ ε ε τ

τ ε νλ

ξ

νλ νλ ρ

φ ρ ∞ + ∞ − ≤ + −  + + _{ −}    + +  − _

∫

( )

₍

₎

( )

(

)

( )

₍

(

₎

)

( )

1

1 1 1

2

1

1 1

1

1 1 1

2

1

2

2 1 2

1 1

1 * 1

0 2

1

1 1

1 * 1

2 ₀

2 ₁ 2

1

1 1 *

1 1

2 e ₂

e e d e d

1

2 e ₂

e d e d

1

2 e

2

e e d e d

1

2 e d .

h

t _s t _s

T h t s s h h

t _s _s

T _h

t _s

g

u u s s C s

g

s s f s s

g

u C s s s

f s s

ε

ε τ ε ε

τ τ

ε τ

ε ε

τ

ε τ

ε τ ε ε

τ

ε τ

ξ

τ ε νλ

νλ ρ νλ

φ

νλ ρ νλ

ξ

τ φ

νλ νλ ρ

(13)

( )

_{( )}

( )

(

)

( )

_{( )}

(

)

( )

1 1 1

2

1

1 1 1

1

1 1

1 1 1

2 2 2 2 1 0 2 1 1 1 1 1 * 0

2 1 2

1 1 * 2 1 1 1 2

e e e d

2 e _e _d 2 _e _e _d

e 1

2 e

e e d

1

2 2

e e d e d .

t

t t s

T

h

t

s t s

t h

h

t s

h

t t

t s s

T

u t u C s

g

s s f s s

g

u s s

f s s C s

ε τ ε ε

τ

ε τ

ε ε ε

ε τ

ε

ε τ ε

ε ε ε

τ τ ξ τ νλ φ νλ νλ ρ τ φ νλ ρ ξ νλ νλ − − + − ∞ − − ∞ − − ≤ + + + −   = _ + _ −     + _ + _  

∫

Similarly, from

( )

(

)

(

)

2

2 2

2 1

d 1 _, _,

dt ω +ν ω ≤λ G t ω t−ρ t

we obtain

( )

(

)

( )

(

( )

)

(

)

( )

(

( )

)

2

2 2 2

2 2 2 2 2 2 2 2 1 2 2

2 1 2

1

d e d

1

e e e ,

1

e e , .

t

t t t

t t

t t G t t t

t G t t t

ε

ε ε ε

ε ε

ω

ω ε ω ω ρ

λ

ε λ ω ω ρ

λ

≤ − + + −

≤ − + −

(25)

Integrating over

[ ]

τ,t _{, we obtain}

( )

(

)

(

)

( )

(

( )

)

2 2 2 2 2 2

2 1 2

1

d e d

d

1

e e , d .

t _s

t _s _s

s s

s

s G s s s s

ε τ

ε ε

τ

ω

ε λ ω ω ρ

λ   ≤ _ − + − _  

∫

Using (20) and (24), we can get

( )

( ) (

)

( )

(

( )

)

( ) (

)

( )

2

2 2 2

2

2 2 2

2

2 2

2

2 2

2 1 2

1

2 2 2 2

2 1 1 * 0 2 2 2 2 * e 1

e e d e , d

e 1

e e d e d

1 e

e d e d

1

t

t _s t _s

h

t _s t _s

h

t

s s

h

t

s s G s s s s

g

s s s s

g

s s f s s

ε

ε τ ε ε

τ τ

ε

ε τ ε ε

τ τ

ε τ

ε ε

τ ω

ω τ ε λ ω ω ρ

λ

ω τ ε λ ω ω

λ ρ φ ρ ∞ + ∞ − ≤ + − + −  ≤ + − + _{ −}    + +  − _

∫

( )

₍

₎

( )

(

)

( )

₍

(

₎

)

( )

2 2 2 2 2 2 2

2 2 2

2 1

1 *

0 2

2

2 2

1 * 1

0

2 2 2

2 2

1 * 1

e

e e d

1

e _e _d 1 _e _d

1

e ₁

e e d e d .

1 h t _s h t s s h h t s s h g s s g

s s f s s

g

s s f s s

ε

ε τ ε

τ

ε τ

ε ε

τ

ε τ

ε τ ε ε

τ

ω τ ε λ ω

λ ρ

φ

λ ρ λ

ω τ φ

λ ρ λ

(14)

( )

_{( )}

( )

(

)

( )

_{( )}

(

)

( )

2

2 2 2 2 2

2

2 2 2 2

2

0

2 2 2

2 2

1 * 1

0

2 2 2

2 2

1 * 1

e ₁

e e e d e e d

1

e 1

e e d e e d .

1

h

t

t t s t s

h h

t

t s t s

h

t

g

s s f s s

g

s s f s s

ε τ

ε τ ε ε ε ε

τ

ε

ε τ ε ε ε

τ ω

ω τ φ

λ ρ λ

ω τ φ

λ ρ λ

+

− ∞ − −

−

− ∞ −

−

≤ + +

−

 

= _ + _+

−

 

∫

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this pa-per.

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