The Long-Term Dynamic Behavior of Solutions to a Class of Generalized Higher-Order Kirchhoff-Type Coupled Wave Equations

In this paper, we study the long-term dynamic behavior of a class of generalized high-order Kirchhoff-type coupled wave equations. Firstly, the existence of uniqueness global solution of this kind of equations in E_k space is proved by prior estimation and Galerkin method; Then, through using Rellich-Kondrachov compact embedding theorem, it is proved that the solution semigroup S( t) has the family of the global attractors A_k in space E_k; Finally, through linearization method, proves that the operator semigroup S( t) Frechet differentiable and the attenuation of linearization problem volume element. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractors A_k.

• Kirchhoff Equation
• Existence and Uniqueness of Solutions
• Global Attractor Family
• Dimension Estimation

1. Introduction

In this paper, we study the long-term dynamic behavior of a class of generalized high-order Kirchhoff-type coupled wave equations:

u t t + M ( ‖ ∇ m u ‖ 2 + ‖ ∇ m v ‖ 2 ) ( − Δ ) m u + β ( − Δ ) m u t + g ( u t , v ) = f 1 ( x ) , (1)

v t t + M ( ‖ ∇ m u ‖ 2 + ‖ ∇ m v ‖ 2 ) ( − Δ ) 2 m v + β ( − Δ ) 2 m v t + g ( u , v t ) = f 2 ( x ) , (2)

the boundary conditions:

∂ i u ∂ n i = 0 , i = 0 , 1 , 2 , ⋯ , m − 1 , x ∈ ∂ Ω , t > 0 , (3)

∂ j v ∂ v j = 0 , j = 0 , 1 , 2 , ⋯ , 2 m − 1 , x ∈ ∂ Ω , t > 0 , (4)

the initial conditions:

u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , v ( x , 0 ) = v 0 ( x ) , v t ( x , 0 ) = v 1 ( x ) , x ∈ Ω , (5)

where Ω is a bounded domain in R n with smooth boundary ∂ Ω , u 0 ( x ) , u 1 ( x ) is a known function, g ( u , v ) , f i ( x ) , i = 1 , 2 are nonlinear source term and the external force interference terms, m > 1 , β is real number.

Recently, the global attractor and its dimension estimation for Kirchhoff type equations have been favored by many scholars. Many scholars have done a lot of research on this kind of problems and obtained good results [1] [2] [3].

Lin Guoguang, Gao Yunlong [1] studied the longtime behavior of solution to initial boundary value problem for a class of strongly damped higher-order Kirchhoff type equation:

u t t + ( − Δ ) m u t + ( α + β ‖ ∇ m u ‖ 2 ) q ( − Δ ) m u + g ( u ) = f ( x ) ,     ( x , t ) ∈ Ω × [ 0 , + ∞ ) ,

they got the existence and uniqueness of the solution by the Galerkin method and obtained the existence of the global attractor in H 0 m ( Ω ) × L 2 ( Ω ) according to the attractor theorem, besides, the estimation of the upper bound of Hausdorff dimension for the attractor was established.

Guoguang Lin, Ming Zhang [2] studied the initial boundary value problem for a class of Kirchhoff-type coupled equations:

u t t − M ( ‖ ∇ u ‖ 2 + ‖ ∇ v ‖ 2 ) Δ u − β Δ u t + g 1 ( u , v ) = f 1 ( x ) ,

v t t − M ( ‖ ∇ u ‖ 2 + ‖ ∇ v ‖ 2 ) Δ v − β Δ v t + g 2 ( u , v ) = f 2 ( x ) ,

they obtained the existence of the global attractor and a precise estimate of upper bound of Hausdorff dimension.

Lin Guoguang, Yang Lujiao [3] studied the long-time properties of solutions of generalized Kirchhoff-type equation with strongly damped terms:

u t t + M ( ‖ ∇ m u ‖ p p ) ( − Δ ) 2 m u + β ( − Δ ) 2 m u t + g ( u ) = f ( x ) ,

by assuming the nonlinear source terms g ( u ) and Kirchhoff stress term M ( s ) , the author verified the appropriateness of the solution and proved the existence of the global attractor, obtained the upper boundary estimation of the Hausdorff dimension and Fractal dimension of a family of the global attractor.

For more significant research results about the global attractor and its dimension estimation of Kirchhoff equation, please refer to the literature [4] - [18].

2. Existence and Uniqueness of Solutions

The following symbols and assumptions are introduced for the convenience of statement:

V m = H m ( Ω ) ∩ H 0 1 ( Ω ) , V 2 m = H m ( Ω ) ∩ H 0 1 ( Ω ) , V 4 m = H 4 m ( Ω ) ∩ H 0 1 ( Ω ) , ‖   ⋅   ‖ = ‖   ⋅   ‖ L 2 ( Ω ) , E 0 = V m × V 0 × V 2 m × V 0 , E 1 = V 2 m × V 0 × V 4 m × V 0 , E k = V m + k × V k × V 2 m + 2 k × V 2 k , V 0 = L 2 ( Ω ) .

In order to obtain our results, we consider system (1)-(5) under some assumptions on M ( s ) and g ( u , v ) . Precisely, we state the general assumptions:

(A1) M ( s ) ∈ C 2 ( [ 0 , + ∞ ) , R ) is not decreasing function and for positive constants δ 0 , δ 1 ,

(1) ε + 1 ≤ δ 0 ≤ M ( s ) ≤ δ 1 ,

(2) M ( s ) is a non-negative Lipschitz function, L is associated with the Lipschitz constant M ( s ) , M ( s ) = M ( ‖ ∇ m u ‖ 2 + ‖ ∇ m v ‖ 2 ) .

(A2) For any u , v , p , q ∈ V , g ( u t , v ) , g ( u , v t ) ∈ C 2 ( ℝ ) , there exist α ≥ 0 , ε ≥ 0 , C ( α , ε ) ≥ 0 , such that

( g ( u t , v ) , p ) + ( g ( u , v t ) , q ) ≥ α ( ‖ p ‖ 2 + ‖ q ‖ 2 ) − ε ( ‖ u ‖ 2 + ‖ v ‖ 2 ) − C ( α , ε ) ( ‖ u t ‖ 2 + ‖ v t ‖ 2 ) .

Lemma 1 Assuming (A1)-(A2) are true, letting ( u 0 , p 0 , v 0 , q 0 ) ∈ E 0 , f 1 ( x ) , f 2 ( x ) ∈ L 2 ( Ω ) , then there is a solution ( u , p , v , q ) for problem (1)-(5) which has the following properties:

(i) ( u , p , v , q ) ∈ L ∞ ( ( 0 , + ∞ ) ; E 0 ) ;

(ii)

y ( t ) ≤ y ( 0 ) e − 2 ε 0 k 1 t + k 1 2 ε ε 0 ( ‖ f 1 ( x ) ‖ 2 + ‖ f 2 ( x ) ‖ 2 ) , (6)

where y ( t ) = ‖ ∇ m u ‖ 2 + ‖ p ‖ 2 + ‖ ∇ 2 m v ‖ 2 + ‖ q ‖ 2 .

(iii) There are normal numbers C ( R 0 ) and t 0 = t 0 ( Ω ) > 0 , such that

‖ ( u , p , v , q ) ‖ E 0 2 = ‖ ∇ m u ‖ 2 + ‖ p ‖ 2 + ‖ ∇ 2 m v ‖ 2 + ‖ q ‖ 2 ≤ C ( R 0 ) . (7)

Proof: Let p = u t + ε u inner product with Equation (1),

( u t t + M ( ‖ ∇ m u ‖ 2 + ‖ ∇ m v ‖ 2 ) ( − Δ ) m u + β ( − Δ ) m u t + g ( u t , v ) , p ) = ( f 1 ( x ) , p ) , (8)

according to the hypothesis (A1), using the Young inequality, Holder inequation, Poincare inequality, etc., there are

( u t t , p ) = 1 2 d d t ‖ p ‖ 2 − ε ‖ p ‖ 2 + ε 2 ( u , p ) , (9)

( M ( ‖ ∇ m u ‖ 2 + ‖ ∇ m v ‖ 2 ) ( − Δ ) m u , p ) ≥ δ 2 d d t ‖ ∇ m u ‖ 2 + ε δ 0 ‖ ∇ m u ‖ 2 , (10)

( β ( − Δ ) m u t , p ) = β ‖ ∇ m u t ‖ 2 + β ε 2 d d t ‖ ∇ m u ‖ 2 , (11)

( f 1 ( x ) , p ) ≤ ε 2 ‖ p ‖ 2 + 1 2 ε ‖ f 1 ( x ) ‖ 2 , (12)

where δ = δ 0 or δ 1 .

Similarly, letting q = v t + ε v inner product with Equation (2), the treatment of each item is similar to (9)-(11), and the above results are sorted out,

1 2 d d t ( ( δ + β ε ) ( ‖ ∇ m u ‖ 2 + ‖ ∇ 2 m v ‖ 2 ) + ‖ p ‖ 2 + ‖ q ‖ 2 ) − 3 ε 2 ( ‖ p ‖ 2 + ‖ q ‖ 2 ) + ε 2 [ ( u , p ) + ( v , q ) ] + ε δ 0 ( ‖ ∇ m u ‖ 2 + ‖ ∇ 2 m v ‖ 2 ) + β ( ‖ ∇ m u t ‖ 2 + ‖ ∇ 2 m v t ‖ 2 ) + ( g ( u t , v ) , p ) + ( g ( u , v t ) , q ) = 1 2 ε ( ‖ f 1 ( x ) ‖ 2 + ‖ f 2 ( x ) ‖ 2 ) , (13)

using the Young inequality, the Poincare inequality, and the assumptions (A2), The individual items in Equation (13) are treated as follows:

ε 2 [ ( u , p ) + ( v , q ) ] ≥ − ε ( ‖ u ‖ 2 + ‖ v ‖ 2 ) − ε 3 4 ( ‖ p ‖ 2 + ‖ q ‖ 2 ) , (14)

by the Poincare inequality has

− ε ( ‖ u ‖ 2 + ‖ v ‖ 2 ) ≥ − ε ( λ 1 − m + λ 1 − 2 m ) ( ‖ ∇ m u ‖ 2 + ‖ ∇ 2 m v ‖ 2 ) , (15)

where λ 1 is the first eigenvalue with homogeneous Dirichlet boundary conditions of − Δ , in the same way

− C ( α , ε ) ( ‖ u t ‖ 2 + ‖ v t ‖ 2 ) ≥ − C ( α , ε ) ( λ 1 − m + λ 1 − 2 m ) ( ‖ ∇ m u t ‖ 2 + ‖ ∇ 2 m v t ‖ 2 ) , (16)

1 2 d d t ( ( δ + β ε ) ( ‖ ∇ m u ‖ 2 + ‖ ∇ 2 m v ‖ 2 ) + ‖ p ‖ 2 + ‖ q ‖ 2 ) + ( α − 3 ε 2 − ε 3 4 ) ( ‖ p ‖ 2 + ‖ q ‖ 2 ) + ε ( δ 0 − 2 ( λ 1 − m + λ 1 − 2 m ) ) ( ‖ ∇ m u ‖ 2 + ‖ ∇ 2 m v ‖ 2 ) + ( β − C ( α , ε ) ( λ 1 − m + λ 1 − 2 m ) ) ( ‖ ∇ m u t ‖ 2 + ‖ ∇ 2 m v t ‖ 2 ) , ≤ 1 2 ε ( ‖ f 1 ( x ) ‖ 2 + ‖ f 2 ( x ) ‖ 2 ) , (17)

where α > 3 ε 2 + ε 3 4 , δ 0 > 2 ( λ 1 − m + λ 1 − 2 m ) , β > C ( α , ε ) ( λ 1 − m + λ 1 − 2 m ) .

Let y ¯ ( t ) = ( δ + β ε ) ( ‖ ∇ m u ‖ 2 + ‖ ∇ 2 m v ‖ 2 ) + ‖ p ‖ 2 + ‖ q ‖ 2 ,

y ( t ) = ‖ ∇ m u ‖ 2 + ‖ ∇ 2 m v ‖ 2 + ‖ p ‖ 2 + ‖ q ‖ 2 , there are normal numbers k 1 = min { 1 , δ + β ε } , such that

y ¯ ( t ) ≥ k 1 y ( t ) ≥ 0 , (18)

let ε 0 = min { α − 3 ε 2 − ε 3 4 , ε ( δ 0 − 2 ( λ 1 − m + λ 1 − 2 m ) ) } , there are

d d t y ( t ) + 2 ε 0 k 1 y ( t ) ≤ 1 ε ( ‖ f 1 ( x ) ‖ 2 + ‖ f 2 ( x ) ‖ 2 ) , (19)

by Gronwall inequality,

y ( t ) ≤ y ( 0 ) e − 2 ε 0 k 1 t + k 1 2 ε ε 0 ( ‖ f 1 ( x ) ‖ 2 + ‖ f 2 ( x ) ‖ 2 ) , (20)

so there are normal numbers C ( R 0 ) and t 0 = t 0 ( Ω ) > 0 , such that

‖ ( u , p , v , q ) ‖ E 0 2 = ‖ ∇ m u ‖ 2 + ‖ p ‖ 2 + ‖ ∇ 2 m v ‖ 2 + ‖ q ‖ 2 ≤ C ( R 0 ) . (21)

Lemma 1 is proved.

Lemma 2 Assuming (A1)-(A2) are true, ( u 0 , p 0 , v 0 , q 0 ) ∈ E k , f 1 ( x ) ∈ H m ( Ω ) , f 2 ( x ) ∈ H 2 m ( Ω ) , then there is a solution ( u , p , v , q ) for problem (1)-(5), which has the following properties:

(i) ( u , p , v , q ) ∈ L ∞ ( ( 0 , + ∞ ) ; E k ) ;

(ii)

y 1 ( t ) ≤ y 1 ( 0 ) e − 2 ε 1 k 2 t + k 2 2 ε ε 1 ( ‖ ∇ k f 1 ( x ) ‖ 2 + ‖ ∇ 2 k f 2 ( x ) ‖ 2 ) , (22)

where y ( t ) = ‖ ∇ m u ‖ 2 + ‖ p ‖ 2 + ‖ ∇ 2 m v ‖ 2 + ‖ q ‖ 2 .

(iii) There are normal numbers C ( R k ) and t 0 k , such that

‖ ( u , p , v , q ) ‖ E k 2 = ‖ ∇ m + k u ‖ 2 + ‖ ∇ 2 m + 2 k v ‖ 2 + ‖ ∇ k p ‖ 2 + ‖ ∇ 2 k q ‖ 2 ≤ C ( R k ) . (23)

Proof: Let ( − Δ ) k p inner product with Equation (1),

( u t t + M ( ‖ ∇ m u ‖ 2 + ‖ ∇ m v ‖ 2 ) ( − Δ ) m u + β ( − Δ ) m u t + g ( u t , v ) , ( − Δ ) k p ) = ( f 1 ( x ) , ( − Δ ) k p ) , (24)

according to the hypothesis (A1), using the Young inequality, Holder inequation, Poincare inequality, etc., there are

( u t t , ( − Δ ) k p ) ≥ 1 2 d d t ‖ ∇ k p ‖ 2 − ( ε + ε 2 2 ) ‖ ∇ k p ‖ 2 − ε 2 λ 1 − m 2 ‖ ∇ m + k u ‖ 2 , (25)

( M ( ‖ ∇ m u ‖ 2 + ‖ ∇ m v ‖ 2 ) ( − Δ ) m u , ( − Δ ) k p ) ≥ δ 2 d d t ‖ ∇ m + k u ‖ 2 + ε δ 0 ‖ ∇ m + k u ‖ 2 , (26)

( β ( − Δ ) m u t , ( − Δ ) k p ) = β ‖ ∇ m + k u t ‖ 2 + β ε 2 d d t ‖ ∇ m + k u ‖ 2 , (27)

where δ = δ 0 or δ 1 .

Similarly, letting ( − Δ ) 2 m q inner product with Equation (2), the treatment of each item is similar to (25)-(27), using Young inequality, Poincare inequality, and assumption (A2), the individual terms are treated as follows:

( g ( u t , v ) , ( − Δ ) k p ) + ( g ( u , v t ) , ( − Δ ) 2 k q ) ≤ C 1 2 ε ( λ 1 − m + λ 1 − 2 m ) ( ‖ ∇ m + k u t ‖ 2 + ‖ ∇ 2 m + 2 k v t ‖ 2 ) + ε C 1 2 ( ‖ ∇ k p ‖ 2 + ‖ ∇ 2 k q ‖ 2 ) , (28)

( f 1 ( x ) , ( − Δ ) k p ) + ( f 2 ( x ) , ( − Δ ) 2 k 2 q ) ≤ ε 2 ( ‖ ∇ k p ‖ 2 + ‖ ∇ 2 k q ‖ 2 ) + 1 2 ε ( ‖ ∇ k f 1 ( x ) ‖ 2 + ‖ ∇ 2 k f 2 ( x ) ‖ 2 ) . (29)

Then sort out the result of appeal and other inner product items, get

1 2 d d t ( ( δ + β ε ) ( ‖ ∇ m + k u ‖ 2 + ‖ ∇ 2 m + 2 k v ‖ 2 ) + ‖ ∇ k p ‖ 2 + ‖ ∇ 2 k q ‖ 2 ) + ε 2 − ε C 1 − 3 ε 2 ( ‖ ∇ k p ‖ 2 + ‖ ∇ 2 k q ‖ 2 ) + ( ε δ 0 − ε 2 2 ( λ 1 − m + λ 1 − 2 m ) ) ( ‖ ∇ m + k u ‖ 2 + ‖ ∇ 2 m + 2 k v ‖ 2 ) + ( β − C 1 2 ε ( λ 1 − m + λ 1 − 2 m ) ) ( ‖ ∇ m + k u t ‖ 2 + ‖ ∇ 2 m + 2 k v t ‖ 2 ) ≤ 1 2 ε ( ‖ ∇ k f 1 ( x ) ‖ 2 + ‖ ∇ 2 k f 2 ( x ) ‖ 2 ) , (30)

using the Young inequality, Poincare inequality, and assumption (A2), the individual terms in Equation (30) are treated as follows:

( g ( u t , v ) , ( − Δ ) k p ) + ( g ( u , v t ) , ( − Δ ) 2 k q ) ≤ C 1 2 ε ( λ 1 − m + λ 1 − 2 m ) ( ‖ ∇ m + k u t ‖ 2 + ‖ ∇ 2 m + 2 k v t ‖ 2 ) + ε C 1 2 ( ‖ ∇ k p ‖ 2 + ‖ ∇ 2 k q ‖ 2 ) , (31)

( f 1 ( x ) , ( − Δ ) k p ) + ( f 2 ( x ) , ( − Δ ) 2 k 2 q ) ≤ ε 2 ( ‖ ∇ k p ‖ 2 + ‖ ∇ 2 k q ‖ 2 ) + 1 2 ε ( ‖ ∇ k f 1 ( x ) ‖ 2 + ‖ ∇ 2 k f 2 ( x ) ‖ 2 ) , (32)

where ε − C 1 − 3 > 0 , δ 0 > ε 2 ( λ 1 − m + λ 1 − 2 m ) , β > C 1 2 ε ( λ 1 − m + λ 1 − 2 m ) .

Let y ¯ 1 ( t ) = ( δ + β ε ) ( ‖ ∇ m + k u ‖ 2 + ‖ ∇ 2 m + 2 k v ‖ 2 ) + ‖ ∇ k p ‖ 2 + ‖ ∇ 2 k q ‖ 2 ,

y 1 ( t ) = ‖ ∇ m + k u ‖ 2 + ‖ ∇ 2 m + 2 k v ‖ 2 + ‖ ∇ k p ‖ 2 + ‖ ∇ 2 k q ‖ 2 , there are normal numbers k 2 = min { 1 , δ + β ε } , such that

y ¯ 1 ( t ) ≥ k 2 y 1 ( t ) ≥ 0 , (33)

let ε 1 = min { ε 2 − ε C 1 − 3 ε 2 − ε 3 4 , ε δ 0 − ε 2 2 ( λ 1 − m + λ 1 − 2 m ) } , get

d d t y 1 ( t ) + 2 ε 1 k 2 y 1 ( t ) ≤ 1 ε ( ‖ ∇ k f 1 ( x ) ‖ 2 + ‖ ∇ 2 k f 2 ( x ) ‖ 2 ) , (34)

by Gronwall inequality,

y 1 ( t ) ≤ y 1 ( 0 ) e − 2 ε 1 k 2 t + k 2 2 ε ε 1 ( ‖ ∇ k f 1 ( x ) ‖ 2 + ‖ ∇ 2 k f 2 ( x ) ‖ 2 ) , (35)

so there are normal numbers C ( R k ) and t 0 k > 0 , such that

‖ ( u , p , v , q ) ‖ E k 2 = ‖ ∇ m + k u ‖ 2 + ‖ ∇ 2 m + 2 k v ‖ 2 + ‖ ∇ k p ‖ 2 + ‖ ∇ 2 k q ‖ 2 ≤ C ( R k ) (36)

Lemma 1 is proved.

Theorem 1 Assuming (A1)-(A2) is true, ( u 0 , p 0 , v 0 , q 0 ) ∈ E k , f 1 ( x ) ∈ V k , f 2 ( x ) ∈ V 2 k , then the initial boundary value problem (1)-(5) has a unique solution

( u ( x , t ) , p ( x , t ) , v ( x , t ) , q ( x , t ) ) ∈ L ∞ ( ( 0 , + ∞ ) ; E k ) . (37)

Proof: According to literature [9] and Galerkin method, combining with lemma 1 and lemma 2, we can easily obtain the existence of solutions.

Next, prove the uniqueness of the solution:

Assuming ( u 1 , p 1 , v 1 , q 1 ) , ( u 2 , p 2 , v 2 , q 2 ) ∈ E k are the two solutions of the problem (1)-(5), letting u ¯ = u 1 − u 2 , v ¯ = v 1 − v 2 , obtain that

{ u ¯ t t + M ( s 1 ) ( − Δ ) m u 1 − M ( s 2 ) ( − Δ ) m u 2 + β ( − Δ ) m u ¯ t + g ( u 1 t , v 1 ) − g ( u 2 t , v 2 ) = 0 , v ¯ t t + M ( s 1 ) ( − Δ ) 2 m v 1 − M ( s 2 ) ( − Δ ) 2 m v 2 + β ( − Δ ) 2 m v ¯ t + g ( u 1 , v 1 t ) − g ( u 2 , v 2 t ) = 0 , u ¯ ( x , 0 ) = 0 , u ¯ t ( x , 0 ) = 0 , v ¯ ( x , 0 ) = 0 , v ¯ t ( x , 0 ) = 0 , x ∈ Ω , ∂ i u ¯ ∂ n i = 0 , i = 0 , 1 , 2 , ⋯ , m − 1 , x ∈ ∂ Ω , t > 0 , ∂ j v ¯ ∂ v j = 0 , j = 0 , 1 , 2 , ⋯ , 2 m − 1 , x ∈ ∂ Ω , t > 0 , (38)

where s 1 = ‖ ∇ m u 1 ‖ 2 − ‖ ∇ m v 1 ‖ 2 , s 2 = ‖ ∇ m u 2 ‖ 2 − ‖ ∇ m v 2 ‖ 2 .

Let u ¯ t , v ¯ t inner product with the first two equations in (38) and obtain,

{ 1 2 d d t ‖ u ¯ t ‖ 2 + ( M ( s 1 ) ( − Δ ) m u 1 − M ( s 2 ) ( − Δ ) m u 2 , u ¯ t ) + β ‖ ∇ m u ¯ t ‖ 2 = ( g ( u 2 t , v 2 ) − g ( u 1 t , v 1 ) , u ¯ t ) , 1 2 d d t ‖ v ¯ t ‖ 2 + ( M ( s 1 ) ( − Δ ) 2 m v 1 − M ( s 2 ) ( − Δ ) 2 m v 2 , v ¯ t ) + β ‖ ∇ 2 m v ¯ t ‖ 2 = ( g ( u 2 , v 2 t ) − g ( u 1 , v 1 t ) , v ¯ t ) , (39)

using the Young inequality, Poincare inequality, as well as the assumptions (A2), for processing on the type of individual items as follows:

( M ( s 1 ) ( − Δ ) m u ¯ + M ( s 1 ) ( − Δ ) m u 2 − M ( s 2 ) ( − Δ ) m u 2 , u ¯ t ) ≥ M ( s ) 2 d d t ‖ ∇ m u ‖ 2 − ( M ( s 1 ) ( − Δ ) m u 2 − M ( s 2 ) ( − Δ ) m u 2 , u ¯ t ) ≥ M ( s ) 2 d d t ‖ ∇ m u ‖ 2 − L ( ( ‖ ∇ m u 1 ‖ + ‖ ∇ m u 2 ‖ ) ‖ ∇ m u ¯ ‖     + L ( ‖ ∇ m v 1 ‖ + ‖ ∇ m v 2 ‖ ) ‖ ∇ m v ¯ ‖ ) ‖ ( − Δ ) m u ¯ 2 ‖ ‖ u ¯ t ‖ ≥ C 2 ( ‖ ∇ m u ¯ ‖ + ‖ ∇ 2 m v ¯ ‖ ) ‖ u ¯ t ‖ ≥ C 2 ( ‖ ∇ m u ¯ ‖ 2 + ‖ ∇ 2 m v ¯ ‖ 2 + ‖ u ¯ t ‖ 2 2 ) , (40)

similarly, we obtain

( M ( s 1 ) ( − Δ ) 2 m v 1 − M ( s 2 ) ( − Δ ) 2 m v 2 , v ¯ t ) ≥ C 3 ( ‖ ∇ m u ¯ ‖ 2 + ‖ ∇ 2 m v ¯ ‖ 2 + ‖ v ¯ t ‖ 2 2 ) , (41)

| ( g ( u 1 t , v 1 ) − g ( u 2 t , v 1 ) + g ( u 2 t , v 1 ) − g ( u 2 t , v 2 ) , u ¯ t ) | ≤ | ( g ′ ( ξ t , v 1 ) | u ¯ t | + g ′ ( u 2 t , η ) | v ¯ | , u ¯ t ) | ≤ ‖ g ′ ( ξ t , v 1 ) ‖ ∞ ‖ u ¯ t ‖ 2 + ‖ g ′ ( u 2 t , η ) ∞ ‖ ‖ | v ¯ | ‖ ‖ u ¯ t ‖ ≤ C 4 ( ‖ v ¯ ‖ 2 2 + 3 ‖ u ¯ t ‖ 2 2 ) , (42)

| ( g ( u 1 , v 1 t ) − g ( u 2 , v 2 t ) , v ¯ t ) | ≤ C 5 ( 3 ‖ v ¯ t ‖ 2 2 + ‖ u ¯ ‖ 2 2 ) , (43)

Through the (40)-(43), finally will become

1 2 d d t ( ‖ u ¯ t ‖ 2 + ‖ v ¯ t ‖ 2 + M ( s 1 ) ( ‖ ∇ m u ¯ ‖ 2 + ‖ ∇ 2 m v ¯ ‖ 2 ) ) + β ( ‖ ∇ m u ¯ t ‖ 2 + ‖ ∇ 2 m v ¯ t ‖ 2 ) ≤ ( C 2 + C 3 ) ( ‖ ∇ m u ¯ ‖ 2 + ‖ ∇ 2 m v ¯ ‖ 2 ) + C 2 + 3 C 4 2 ‖ u ¯ t ‖ 2     + C 3 + 3 C 5 2 ‖ v ¯ t ‖ 2 + C 4 2 ‖ v ‖ 2 + C 5 2 ‖ u ‖ 2 ≤ C 6 ( ‖ ∇ m u ¯ ‖ 2 + ‖ ∇ 2 m v ¯ ‖ 2 ) + C 7 ( ‖ u ¯ t ‖ 2 + ‖ v ¯ t ‖ 2 ) . (44)

Let y 2 ( t ) = ‖ u ¯ t ‖ 2 + ‖ v ¯ t ‖ 2 + M ( s 1 ) ( ‖ ∇ m u ¯ ‖ 2 + ‖ ∇ 2 m v ¯ ‖ 2 ) , there are normal numbers k 3 = min { C 7 , C 8 } , where C 6 ≤ C 8 M ( s 1 ) , such that

d d t y 2 ( t ) ≤ k 3 y 2 ( t ) , (45)

using the Gronwall inequality, we have

y 2 ( t ) ≤ y 2 ( 0 ) e k 4 t = 0 , (46)

y 2 ( t ) = ‖ u ¯ t ‖ 2 + ‖ v ¯ t ‖ 2 + M ( s 1 ) ( ‖ ∇ m u ¯ ‖ 2 + ‖ ∇ 2 m v ¯ ‖ 2 ) = 0 , (47)

so

u ¯ = v ¯ = 0. (48)

Theorem 1 is proved.

3. The Family of Global Attractors and Dimension Estimation

Theorem 2 [9] Assume E 0 is a Banach space, { S ( t ) } t ≥ 0 is the operator semigroup, S ( t ) : E → E , S ( t + τ ) = S ( t ) ⋅ S ( τ ) ( ∀ t , τ ≥ 0 ) , S ( 0 ) = I , where I is the identity operator. If S ( t ) satisfies

1) Semigroup S ( t ) is uniformly bounded in E, i.e. ∀ R > 0 , exists a constant C ( R ) such that when ‖ u ‖ E ≤ R , there is ‖ S ( t ) u ‖ E ≤ C ( R ) ( ∀ t ∈ [ 0 , + ∞ ) ) ;

2) There exists a bounded absorbing set B in E, that is, for any bounded set B ⊂ E , there exists a constant t 0 > 0 , such that

S ( t ) B ⊂ B ( t > t 0 ) , (49)

3) { S ( t ) } t ≥ 0 is completely continuous operator.

Then operator semigroup S ( t ) has compact global attractor A.

Theorem 3 Let S ( t ) is a solution semigroup generated by the initial boundary value problems (1)-(5) under the hypothesis of lemma 1 and lemma 2, then the initial boundary value problems (1)-(5) have the family of global attractors. There are compact sets satisfying:

A k ⊂ E k ⊂ E 0 and A k = ω ( B 0 k ) = ∩ τ ≥ 0 ∪ t ≥ τ S ( t ) B 0 k ¯ , k = 1 , 2 , ⋯ , m .

where B 0 k = { ( u , p , v , q ) ∈ E k : ‖ ∇ m + k u ‖ 2 + ‖ ∇ k p ‖ 2 + ‖ ∇ 2 m + 2 k v ‖ 2 + ‖ ∇ 2 k q ‖ 2 ≤ R 0 k } ,

1) S ( t ) A k = A k , t > 0 ,

2) A k attracts all bounded sets of E k , that is, any bounded set B 0 k ⊂ E k , having

lim t → ∞ d i s t ( S ( t ) B k , A k ) = 0 , where d i s t ( S ( t ) B k , A k ) = sup x ∈ B 0 k inf y ∈ A k ‖ S ( t ) − y ‖ E k .

then compact set A k are called the family of global attractors of semigroup S ( t ) .

Proof: From lemma 1, lemma 2, for any bounded set B 0 k ⊂ E k and B 0 k ⊂ { ‖ ( u , p , v , q ) ‖ E k ≤ R k } , the equation has solution semigroups S ( t ) : E k → E k , and

‖ S ( t ) ( u 0 , p 0 , v 0 , q 0 ) ‖ E k 2 = ‖ u ‖ V m + k 2 + ‖ p ‖ V k 2 + ‖ v ‖ V 2 m + 2 k 2 + ‖ q ‖ V 2 k 2 ≤ C ( R k ) , (50)

where t 0 ≥ 0 , ( u 0 , v 0 ) ∈ B 0 k , shows that S ( t ) t ≥ 0 is uniformly bounded in E k ;

Further,

B 0 k = { ( u , p , v , q ) ∈ E k : ‖ ∇ m + k u ‖ 2 + ‖ ∇ k p ‖ 2 + ‖ ∇ 2 m + 2 k v ‖ 2 + ‖ ∇ 2 k q ‖ 2 ≤ R 0 k } is a

bounded absorption set of semigroup S ( t ) ; E k is compactly embedded in E 0 , i.e., the bounded set in E k is a compact set in E 0 , so the operator semigroup S ( t ) is completely continuous operator. Then there exists a global attractor family of equations

A k = ω ( B 0 k ) = ∩ τ ≥ 0 ∪ t ≥ τ S ( t ) B 0 k ¯ , k = 1 , 2 , ⋯ , m .

Theorem 3 is proved.

After the family of global attractors are obtained, in order to estimate the Hausdroff dimension and Fractal dimension of the family of global attractors, the initial boundary value problem (1)-(5) is linearized and obtain that

{ U t t + M ′ ( ‖ ∇ m u ‖ 2 + ‖ ∇ m v ‖ 2 ) [ ( ∇ m u , ∇ m U ) + ( ∇ m v , ∇ m V ) ] ( − Δ ) m u + M ( ‖ ∇ m u ‖ 2 + ‖ ∇ m v ‖ 2 ) ( − Δ ) m U + β ( − Δ ) m U t + ∂ g ( u t , v ) ∂ u t U t + ∂ g ( u t , v ) ∂ v V = 0 , U t t + M ′ ( ‖ ∇ m u ‖ 2 + ‖ ∇ m v ‖ 2 ) [ ( ∇ m u , ∇ m U ) + ( ∇ m v , ∇ m V ) ] ( − Δ ) m u + M ( ‖ ∇ m u ‖ 2 + ‖ ∇ m v ‖ 2 ) ( − Δ ) m U + β ( − Δ ) m U t + ∂ g ( u t , v ) ∂ u t U t + ∂ g ( u t , v ) ∂ v V = 0 , U ( x , 0 ) = ξ 1 , U t ( x , 0 ) = ξ 2 , V ( x , 0 ) = η 1 , V t ( x , 0 ) = η 2 , U ( x , 0 ) | x ∈ Ω = V ( x , 0 ) | x ∈ Ω = 0 , t > 0. (51)

where ( ξ 1 , ξ 2 , η 1 , η 2 ) ∈ E 0 , ( u , p , v , q ) = S ( t ) ( u 0 , p 0 , v 0 , q 0 ) is the solution of the initial boundary value problem (51).

Given ( u 0 , p 0 , v 0 , q 0 ) ∈ A k , S ( t ) : E k → E k , for any ( ξ 1 , ξ 2 , η 1 , η 2 ) ∈ E k , there exists a unique solution ( U ( t ) , P ( t ) , V ( t ) , Q ( t ) ) ∈ L ∞ ( 0 , T ; E k ) to the linear initial boundary value problem (51).

Lemma 3 For any t > 0 , R > 0 , the mapping S ( t ) : E k → E k is Frechet differentiable. The derivative on ρ 0 = ( u 0 , p 0 , v 0 , q 0 ) T is a linear operator on E k ,

D S ( t ) ρ 0 : ( ξ , ζ , η , σ ) → ( U , P , V , Q ) ,

where ( U ( t ) , P ( t ) , V ( t ) , Q ( t ) ) is the solution of the problem (51).

Proof: suppose ρ 0 = ( u 0 , p 0 , v 0 , q 0 ) ∈ E k , ρ ˜ 0 = ( u 0 + ξ , p 0 + ζ , v 0 + η , q 0 + σ ) ∈ E k with ‖ ρ 0 ‖ E k ≤ R k , ‖ ρ ˜ 0 ‖ E k ≤ R k , we denote

S ( t ) ρ 0 = ρ = ( u 1 , p 1 , v 1 , q 1 ) , S ( t ) ρ ˜ 0 = ( u 2 , p 2 , v 2 , q 2 ) .

First, we can prove a Lipschitz property of S ( t ) on the bounded sets on E k , that is

‖ S ( t ) ρ 0 − S ( t ) ρ ˜ 0 ‖ E k 2 ≤ e c t ‖ ( ξ , ζ , η , σ ) ‖ E k 2 . (52)

We now consider the difference θ = u 2 − u 1 − U , ω = v 2 − v 1 − V is the solution to problem (53),

{ θ t t + M ( ‖ ∇ m u 1 ‖ 2 + ‖ ∇ m v 1 ‖ 2 ) ( − Δ ) m θ + β ( − Δ ) m θ t = h 1 , ω t t + M ( ‖ ∇ m u 1 ‖ 2 + ‖ ∇ m v 1 ‖ 2 ) ( − Δ ) 2 m ω + β ( − Δ ) 2 m ω t = h 2 , ω ( 0 ) = ω t ( 0 ) = 0 , θ ( 0 ) = θ t ( 0 ) = 0. (53)

where

h 1 = [ M ( s ) − M ( s ˜ ) ] ( − Δ ) m u 2     + 2 M ′ ( s ) [ ( ∇ m u 1 , ∇ m U ) + ( ∇ m v 1 , ∇ m V ) ] ( − Δ ) m u 1     + ∂ g ( u 1 t , v 1 ) ∂ u 1 t U t + ∂ g ( u 1 t , v 1 ) ∂ v 1 V + g ( u 1 t , v 1 ) − g ( u 2 t , v 2 ) , (54)

h 2 = [ M ( s ) − M ( s ˜ ) ] ( − Δ ) 2 m v 2     + 2 M ′ ( s ) [ ( ∇ m u 1 , ∇ m U ) + ( ∇ m v 1 , ∇ m V ) ] ( − Δ ) 2 m v 1     + ∂ g ( u 1 , v 1 t ) ∂ u 1 U + ∂ g ( u 1 , v 1 t ) ∂ v 1 t V t + g ( u 1 , v 1 t ) − g ( u 2 , v 2 t ) , (55)

where s = ‖ ∇ m u 1 ‖ 2 + ‖ ∇ m v 1 ‖ 2 , s ˜ = ‖ ∇ m u 2 ‖ 2 + ‖ ∇ m v 2 ‖ 2 .

Some items are treated as follows

[ M ( s ) − M ( s ˜ ) ] ( − Δ ) m u 2 + 2 M ′ ( s ) [ ( ∇ m U , ∇ m u 1 ) + ( ∇ m V , ∇ m v 1 ) ] ( − Δ ) m u 1 = ( M ( s ) − M ( s ˜ ) ) ( − Δ ) m u 2 + 2 M ′ ( s ) [ ( ∇ m u 1 , ∇ m U ) + ( ∇ m v 1 , ∇ m V ) ] ( − Δ ) m u 1 = − 2 M ′ ( s ) [ ( − ∇ m U , ∇ m u 1 ) + ( − ∇ m V , ∇ m v 1 ) ] ( − Δ ) m u 1 + M ′ ( α s + ( 1 − α ) s ˜ )     × [ ( ∇ m ( u 1 − u 2 ) , ∇ m ( u 2 + u 1 ) ) + ( ∇ m ( v 1 − v 2 ) , ∇ m ( v 2 + v 1 ) ) ] ( − Δ ) m u 2

≤ M ″ ( ξ ) [ ( ∇ m ( u 1 − u 2 ) , ∇ m ( u 2 + u 1 ) ) + ( ∇ m ( v 1 − v 2 ) , ∇ m ( v 2 + v 1 ) ) ] 2 ( − Δ ) m u 2     + M ′ ( s ) [ ( ∇ m ( u 1 − u 2 ) , ∇ m ( u 2 − u 1 ) ) + ( ∇ m ( v 1 − v 2 ) , ∇ m ( v 2 − v 1 ) ) ] ( − Δ ) m u 2     + 2 M ′ ( s ) [ ( ∇ m ( u 1 − u 2 ) , ∇ m u 1 ) + ( ∇ m ( v 1 − v 2 ) , ∇ m v 1 ) ] ( − Δ ) m ( u 2 − u 1 )     − 2 M ′ ( s ) [ ( ∇ m θ , ∇ m u 1 ) + ( ∇ m ω , ∇ m v 1 ) ] ( − Δ ) m u 1 = I 1 + I 2 + I 3 + I 4 . (56)

Let ( − Δ ) k θ t inner product with the first equation in (53), ( − Δ ) 2 k ω t and inner product with the second equation in (53), obtain

( I 1 , ( − Δ ) k θ t ) ≤ ( 2 M ″ ( ξ ) [ ( ∇ m ( u 1 − u 2 ) , ∇ m ( u 2 + u 1 ) ) 2 + ( ∇ m ( v 1 − v 2 ) , ∇ m ( v 2 + v 1 ) ) 2 ] ∇ m + k u 2 , ∇ m + k θ t ) ≤ 2 c 1 ( 4 ε 2 ‖ ∇ m ( u 2 − u 1 ) ‖ 4 + 4 ε 2 ‖ ∇ m ( v 2 − v 1 ) ‖ 4 + ε 2 ‖ ∇ m + k θ t ‖ 2 8 ) , ≤ 2 c 1 ( 4 ε 2 ( λ 1 − 2 k + λ 1 − 2 m − 4 k ) ( ‖ ∇ m + k ( u 2 − u 1 ) ‖ 4 + ‖ ∇ 2 m + 2 k ( v 2 − v 1 ) ‖ 4 ) + ε 2 ‖ ∇ m + k θ t ‖ 2 8 ) ,

( I 2 , ( − Δ ) k θ t ) = ( M ′ ( s ) [ ( ∇ m ( u 1 − u 2 ) , ∇ m ( u 2 − u 1 ) ) + ( ∇ m ( v 1 − v 2 ) , ∇ m ( v 2 − v 1 ) ) ] ( − Δ ) m + k u 2 , ∇ m + k θ t ) ≤ c 2 ( 4 ε 2 ‖ ∇ m ( u 2 − u 1 ) ‖ 4 + 4 ε 2 ‖ ∇ m ( v 2 − v 1 ) ‖ 4 + ε 2 ‖ ∇ m + k θ t ‖ 2 8 ) ≤ c 2 ( 4 ε 2 ( λ 1 − 2 k + λ 1 − 2 m − 4 k ) ( ‖ ∇ m + k ( u 2 − u 1 ) ‖ 4 + ‖ ∇ 2 m + 2 k ( v 2 − v 1 ) ‖ 4 + ) ε 2 ‖ ∇ m + k θ t ‖ 2 8 ) ,

( I 3 , ( − Δ ) k θ t ) ≤ ( 2 M ′ ( s ) [ ( ∇ m ( u 1 − u 2 ) , ∇ m u 1 ) + ( ∇ m ( v 1 − v 2 ) , ∇ m v 1 ) ] ∇ m + k ( u 2 − u 1 ) , ∇ m + k θ t ) ≤ 2 c 3 ( 4 ε 2 ‖ ∇ m ( u 2 − u 1 ) ‖ 4 + 4 ε 2 ‖ ∇ m ( v 2 − v 1 ) ‖ 4 + 4 ε 2 ‖ ∇ m + k ( u 2 − u 1 ) ‖ 4 + 3 ε 2 ‖ ∇ m + k θ t ‖ 2 16 ) ≤ 2 c 3 ( 4 ε 2 ( λ 1 − 2 k + λ 1 − 2 m − 4 k + 1 ) ( ‖ ∇ m + k ( u 2 − u 1 ) ‖ 4 + ‖ ∇ 2 m + 2 k ( v 2 − v 1 ) ‖ 4 ) + 3 ε 2 ‖ ∇ m + k θ t ‖ 2 16 ) ,

( I 4 , ( − Δ ) k θ t ) = ( − 2 M ′ ( s ) [ ( ∇ m θ , ∇ m u 1 ) + ( ∇ m ω , ∇ m v 1 ) ] ∇ m + k u 1 , ∇ m + k θ t ) ≤ 2 c 4 ( 4 ‖ ∇ m + k θ ‖ 2 ε 2 + 4 ‖ ∇ m + k ω ‖ 2 ε 2 + ε 2 ‖ ∇ m + k θ t ‖ 2 8 ) ≤ 2 c 4 ( 4 ε 2 ( λ 1 − 2 k + λ 1 − 2 m − 4 k ) ( ‖ ∇ m + k θ ‖ 2 + ‖ ∇ 2 m + 2 k ω ‖ 2 ) + ε 2 ‖ ∇ m + k θ t ‖ 2 8 ) ,

which implies that

| ( [ M ( s ) − M ( s ˜ ) ] ( − Δ ) m u 2 + 2 M ′ ( s ) [ ( ∇ m U , ∇ m u 1 ) + ( ∇ m V , ∇ m v 1 ) ] ( − Δ ) m u 1 , ( − Δ ) k θ t ) | ≤ c 5 ε 2 ( λ 1 − 2 k + λ 1 − 2 m − 4 k + 1 ) ( ‖ ∇ m + k ( u 2 − u 1 ) ‖ 4 + ‖ ∇ 2 m + 2 k ( v 2 − v 1 ) ‖ 4     + ‖ ∇ m + k θ ‖ 2 + ‖ ∇ 2 m + 2 k ω ‖ 2 ) + c 6 ε 2 ‖ ∇ m + k θ t ‖ 2 . (57)

Analogously,

| ( [ M ( s ) − M ( s ˜ ) ] ( − Δ ) 2 m v 2 + 2 M ′ ( s ) [ ( ∇ m u 1 , ∇ m U ) + ( ∇ m v 1 , ∇ m V ) ] ( − Δ ) 2 m v 1 , ( − Δ ) 2 k ω t ) | ≤ c 7 ε 2 ( λ 1 − 2 k + λ 1 − 2 m − 4 k + 1 ) ( ‖ ∇ m + k ( u 2 − u 1 ) ‖ 4 + ‖ ∇ 2 m + 2 k ( v 2 − v 1 ) ‖ 4     + ‖ ∇ m + k θ ‖ 2 + ‖ ∇ 2 m + 2 k ω ‖ 2 ) + c 8 ε 2 ‖ ∇ 2 m + 2 k ω t ‖ 2 . (58)

Further,

( ∂ g ( u 1 t , v 1 ) ∂ u 1 t U t + ∂ g ( u 1 t , v 1 ) ∂ v 1 V + g ( u 1 t , v 1 ) − g ( u 2 t , v 2 ) , ( − Δ ) k θ t ) = ( ∂ g ( ξ 1 , v 1 ) ∂ ξ 1 ( u 1 t − u 2 t ) + ∂ g ( u 2 t , η 1 ) ∂ η 1 ( v 1 − v 2 ) + ∂ g ( u 1 t , v 1 ) ∂ u 1 t U t + ∂ g ( u 1 t , v 1 ) ∂ v 1 V , ( − Δ ) k θ t ) ≤ ( ∂ 2 g ( ξ 2 , v 1 ) ∂ ξ 2 ( u 1 t − u 2 t ) ( ξ 1 − u 1 t ) + ∂ g ( u 1 t , v 1 ) ∂ u 1 t ( − θ t ) + ∂ 2 g ( u 2 t , η 2 ) ∂ , η 2 2 ( v 1 − v 2 ) ( η 1 − v 1 )     + ∂ 2 g ( ξ 3 , v 1 ) ∂ ξ 3 ∂ v ( v 1 − v 2 ) ( u 2 t − u 1 t ) + ∂ g ( u 1 t , v 1 ) ∂ v 1 ( − ω ) , ( − Δ ) k θ t )

≤ c 9 λ 1 − k 2 − m ‖ ∇ m + k ( u 1 t − u 2 t ) ‖ 2 ‖ ∇ k θ t ‖ + c 10 ‖ ∇ k θ t ‖ 2 + c 11 λ 1 − 3 k 2 − 2 m ‖ ∇ 2 m + 2 k ( v 1 − v 2 ) ‖ 2 ‖ ∇ k θ t ‖     + c 12 λ 1 − k − 3 m 2 ‖ ∇ m + k ( u 1 t − u 2 t ) ‖ ‖ ∇ 2 k + 2 m ( v 1 − v 2 ) ‖ ‖ ∇ k θ t ‖ + c 13 λ 1 − k 2 − m ‖ ∇ 2 m + 2 k ω ‖ ‖ ∇ k θ t ‖ ≤ c 14 2 ( 3 2 ε 2 ( ‖ ∇ m + k ( u 1 t − u 2 t ) ‖ 4 + ‖ ∇ 2 m + 2 k ( v 1 − v 2 ) ‖ 4 ) + ε 2 ( ‖ ∇ k θ t ‖ 2 + 5 ‖ ∇ 2 m + 2 k ω ‖ 2 ) ) , (59)

where,

ξ 1 = r 1 u 1 t − ( 1 − r 1 ) u 2 t , η 1 = r 2 v 1 − ( 1 − r 2 ) v 2 , ξ 2 = r 3 ξ 1 − ( 1 − r 3 ) u 1 t , η 2 = r 4 η 1 − ( 1 − r 4 ) v 1 , ξ 3 = r 5 u 2 t − ( 1 − r 5 ) u 1 t , c 14 = max { c 9 λ 1 − k 2 − m , 2 c 10 ε 2 , c 11 λ 1 − 3 k 2 − 2 m , c 12 λ 1 − k − 3 m 2 , c 13 λ 1 − k 2 − m } .

Similarly

( ∂ g ( u , v t ) ∂ u U + ∂ g ( u , v t ) ∂ v t V t + g ( u , v t ) − g ( u ˜ , v ˜ t ) , ω t ) ≤ c 15 2 ( 3 2 ε 2 ( ‖ ∇ m + k ( u 1 − u 2 ) ‖ 4 + ‖ ∇ 2 m + 2 k ( v 1 t − v 2 t ) ‖ 4 )         + ε 2 ( ‖ ∇ m + k θ ‖ 2 + 5 ‖ ∇ 2 k ω t ‖ 2 ) ) . (60)

Based on the above Equations (57)-(60), it is sorted out that

1 2 d d t { ‖ ∇ k θ t ‖ 2 + ‖ ∇ 2 k ω t ‖ 2 + μ ( ‖ ∇ m + k θ ‖ 2 + ‖ ∇ 2 m + 2 k ω ‖ 2 ) } + ( β − ( c 6 + c 8 ) ε 2 ) ( ‖ ∇ m + k θ t ‖ 2 + ‖ ∇ 2 m + 2 k ω t ‖ 2 ) ≤ c 16 ( ‖ ∇ m + k ( u 1 − u 2 ) ‖ 4 + ‖ ∇ m + k ( u 1 t − u 2 t ) ‖ 4 + ‖ ∇ 2 m + 2 k ( v 1 − v 2 ) ‖ 4     + ‖ ∇ 2 m + 2 k ( v 1 t − v 2 t ) ‖ 4 ) + c 17 ( ‖ ∇ k θ t ‖ 2 + ‖ ∇ 2 k ω t ‖ 2 + ‖ ∇ m + k θ ‖ 2 + ‖ ∇ 2 m + 2 k ω ‖ 2 ) , (61)

where β ≥ ( c 6 + c 8 ) ε 2 ,

by Gronwall inequality

‖ θ t ‖ 2 + ‖ ω t ‖ 2 + μ ( ‖ ∇ m θ ‖ 2 + ‖ ∇ 2 m ω ‖ 2 ) ≤ c 18 e C 19 t ∫ 0 t ( ‖ ∇ m + k ( u 1 − u 2 ) ‖ 4 + ‖ ∇ m + k ( u 1 t − u 2 t ) ‖ 4       + ‖ ∇ 2 m + 2 k ( v 1 − v 2 ) ‖ 4 + ‖ ∇ 2 m + 2 k ( v 1 t − v 2 t ) ‖ 4 ) d τ ≤ c 20 e c 21 t ‖ ( ξ , ζ , η , σ ) ‖ E k 4 , (62)

so as ‖ ( ξ , ζ , η , σ ) ‖ E k 2 → 0 in E k , there are

‖ S ( t ) ρ ˜ 0 − S ( t ) ρ 0 − ( D S ( t ) ρ 0 ) ( ξ 1 , ξ 2 , η 1 , η 2 ) ‖ E k 2 ‖ ( ξ , ζ , η , σ ) ‖ E k 2 ≤ c 20 e c 21 t ‖ ( ξ , ζ , η , σ ) ‖ E k 2 → 0.

The differentiability of S(t) is proved.

The next step will be used in demonstrating the process of dimension estimation. it seems obvious that the Equations (1)-(5) also can be written as

φ t + H ( φ ) = F ( φ ) , (63)

where φ = ( u , p , v , q ) T , p = u t + ε u , q = v t + ε v ,

H ( φ ) = ( ε u − p ( − Δ ) m u − ε ( β ( − Δ ) m − ε ) u + ( β ( − Δ ) m − ε ) p ε v − q ( − Δ ) 2 m v − ε ( β ( − Δ ) 2 m − ε ) v + ( β ( − Δ ) 2 m − ε ) q ) , (64)

F ( φ ) = ( 0 f 1 ( x ) − g ( u t , v ) + ( 1 − M ( ‖ ∇ m u ‖ 2 + ‖ ∇ m v ‖ 2 ) ) ( − Δ ) m u 0 f 2 ( x ) − g ( u , v t ) + ( 1 − M ( ‖ ∇ m u ‖ 2 + ‖ ∇ m v ‖ ) ) ( − Δ ) 2 m v ) . (65)

Consider the first variation equation of (63)

Ψ ′ + P ( φ ) Ψ = Γ 1 ( φ ) Ψ + Γ 2 ( φ ) Ψ , (66)

where Ψ = ( U , P , V , Q ) T , P = U t + ε U , Q = V t + ε V , and φ = ( u , p , v , q ) T is the solution to problem (63), and

P ( φ ) = ( ε I − I 0 0 ( 1 − β ε ) ( − Δ ) m + ε 2 I β ( − Δ ) m − ε I 0 0 0 0 ε I − I 0 0 ( 1 − β ε ) ( − Δ ) 2 m + ε 2 I β ( − Δ ) 2 m − ε I ) , (67)

Γ 1 ( φ ) = ( 0 0 0 0 ε ∂ g ( u t , v ) ∂ u t − ∂ g ( u t , v ) ∂ u t − ∂ g ( u t , v ) ∂ v 0 0 0 0 0 − ∂ g ( u , v t ) ∂ u 0 ε ∂ g ( u , v t ) ∂ v t − ∂ g ( u , v t ) ∂ v t ) , (68)

Γ 2 ( φ ) Ψ = ( 0 ( 1 − M ( s ) ) ( − Δ ) m U − 2 M ′ ( s ) [ ( ∇ m u , ∇ m U ) + ( ∇ m v , ∇ m V ) ] ( − Δ ) m u 0 ( 1 − M ( s ) ) ( − Δ ) 2 m V − 2 M ′ ( s ) [ ( ∇ m u , ∇ m U ) + ( ∇ m v , ∇ m V ) ] ( − Δ ) 2 m v ) . (69)

Theorem 4 Under the condition of Theorem 3, the global attractors of initial boundary value problems (1)-(5) have finite dimensional Hausdroff dimension

and fractal dimension, and then d H ( A k ) < 2 3 n 0 , d F ( A k ) < 4 3 n 0 .

Proof: For any fixed ( u 0 , p 0 , v 0 , q 0 ) ∈ E k , assume that χ 1 , χ 2 , ⋯ , χ n are n 0 elements in E k , and ψ 1 ( t ) , ψ 2 ( t ) , ⋯ , ψ n 0 ( t ) are n 0 solutions of the linearized Equation (66) with an initial value ψ 1 ( 0 ) = χ 1 , ψ 2 ( 0 ) = χ 2 , ⋯ , ψ n 0 ( 0 ) = χ n 0 , where n 0 is a natural number. It can be obtained by calculation

‖ ψ 1 ( t ) ∧ ⋯ ∧ ψ n 0 ( t ) ‖ ∧ n 0 E k ≤ ‖ χ 1 ∧ ⋯ ∧ χ n ‖ ∧ n 0 E k exp ∫ 0 t t r F ′ ( φ ( τ ) ) ∘ Q n ​ 0 ( τ ) d τ . (70)

where ∧ represents the outer product, tr represents the trace of the operator, Q n 0 ( τ ) = Q n 0 ( τ , φ 0 ; χ 1 , χ 2 , ⋯ , χ n 0 ) represents the orthogonal projection from E k to s p a n { ψ 1 ( t ) , ψ 2 ( t ) , ⋯ , ψ n 0 ( t ) } .

At a given time τ , let h j ( τ ) = ( ξ j ( τ ) , ζ j ( τ ) , η j ( τ ) , σ j ( τ ) ) T , j = 1 , 2 , ⋯ , n 0

are the standard orthogonal basis of space s p a n { ψ 1 ( t ) , ψ 2 ( t ) , ⋯ , ψ n 0 ( t ) } , then define the inner product on E k

( h j , h j ¯ ) = ( ∇ m + k ξ j , ∇ m + k ξ j ¯ ) + ( ∇ k ζ j , ∇ k ζ j ¯ )     + ( ∇ 2 m + 2 k η j , ∇ 2 m + 2 k η j ¯ ) + ( ∇ 2 k σ j , ∇ 2 k σ j ¯ ) ,

‖ h j ‖ E k 2 = ( h j , h j ) E k = ‖ ∇ m + k ξ j ‖ 2 + ‖ ∇ k ζ j ‖ 2 + ‖ ∇ 2 m + 2 k η j ‖ 2 + ‖ ∇ 2 k σ j ‖ 2 = 1.

Through the above conditions, can get

t r F ′ ( φ ( τ ) ) ∘ Q n 0 ( τ ) = ∑ j = 1 + ∞ ( F ′ ( φ ( τ ) ) ∘ Q n 0 ( τ ) h j ( τ ) , h j ( τ ) ) E k = ∑ j = 1 n 0 ( F ′ ( φ ( τ ) ) h j ( τ ) , h j ( τ ) ) E k ,

by the Holder inequality, Young and Poincare inequality

( P ( φ ) h j , h j ) = ε ( ∇ m + k ξ j , ∇ m + k ξ j ) − ( ∇ m + k ζ j , ∇ m + k ξ j ) + ( 1 − β ε ) ( ∇ m + k ξ j , ∇ m + k ζ j )       + ε 2 ( ∇ k ξ j , ∇ k ζ j ) + β ( ∇ m + k ζ j , ∇ m + k ζ j ) − ε ( ∇ k ζ j , ∇ k ζ j )       + ε ( ∇ 2 m + 2 k η j , ∇ 2 m + 2 k η j ) − ( ∇ 2 m + 2 k σ j , ∇ 2 m + 2 k η j ) + ε 2 ( ∇ 2 k η j , ∇ 2 k σ j )       + β ( ∇ 2 m + 2 k σ j , ∇ 2 m + 2 k σ j ) − ε ( ∇ 2 k σ j , ∇ 2 k σ j )

      + ( 1 − β ε ) ( ∇ 2 m + 2 k η j , ∇ 2 m + 2 k σ j ) ≥ ε ‖ ∇ m + k ξ j ‖ 2 − ( 2 − β ε ) ‖ ∇ m + k ζ j ‖ ‖ ∇ m + k ξ j ‖ − ε 2 ‖ ∇ k ξ j ‖ ‖ ∇ k ζ j ‖       + β ‖ ∇ m + k ζ j ‖ 2 − ε ‖ ∇ k ζ j ‖ 2 − ( 2 − β ε ) ‖ ∇ 2 m + 2 k σ j ‖ ‖ ∇ 2 m + 2 k η j ‖       − ε 2 ‖ ∇ 2 k η j ‖ ‖ ∇ 2 k σ j ‖ + β ‖ ∇ 2 m + 2 k σ j ‖ 2 − ε ‖ ∇ 2 k σ j ‖ 2 + ε ‖ ∇ 2 m + 2 k η j ‖ 2

≥ ( β λ 1 m − ε 2 + 2 ε 2 − c 21 ( 2 − β ε ) λ 1 m 2 ) ‖ ∇ k ζ j ‖ 2 − ε 2 2 ‖ ∇ k ξ j ‖ 2 − ε 2 2 ‖ ∇ 2 k η j ‖ 2     + ( ε − c 21 ( 2 − β ε ) 2 ) ‖ ∇ m + k ξ j ‖ 2 + ( ε − c 22 ( 2 − β ε ) 2 ) ‖ ∇ 2 m + 2 k η j ‖ 2     + ( β λ 1 2 m − ε 2 + 2 ε 2 − c 22 ( 2 − β ε ) λ 1 2 m 2 ) ‖ ∇ 2 k σ j ‖ 2 , (71)

( Γ 1 ( φ ) h j , h j ) = ε ( ∂ g ( u t , v ) ∂ u t ∇ k ξ j , ∇ k ζ j ) − ( ∂ g ( u t , v ) ∂ u t ∇ k ζ j , ∇ k ζ j )     − ( ∂ g ( u t , v ) ∂ v ∇ k η j , ∇ k ζ j ) − ( ∂ g ( u , v t ) ∂ u ∇ 2 k ξ j , ∇ 2 k σ j )     + ε ( ∂ g ( u , v t ) ∂ v t ∇ 2 k η j , ∇ 2 k σ j ) − ( ∂ g ( u , v t ) ∂ v t ∇ 2 k σ j , ∇ 2 k σ j )

≤ ε ‖ ∂ g ( u t , v ) ∂ u t ‖ ∞ ‖ ∇ k ξ j ‖ ‖ ∇ k ζ j ‖ − ‖ ∂ g ( u t , v ) ∂ u t ‖ ∞ ‖ ∇ k ζ j ‖ 2     − ‖ ∂ g ( u t , v ) ∂ v ‖ ∞ ‖ ∇ k η j ‖ ‖ ∇ k ζ j ‖ − ‖ ∂ g ( u , v t ) ∂ u ‖ ∞ ‖ ∇ 2 k ξ j ‖ ‖ ∇ 2 k σ j ‖     + ε ‖ ∂ g ( u , v t ) ∂ v t ‖ ∞ ‖ ∇ 2 k η j ‖ ‖ ∇ 2 k σ j ‖ − ‖ ∂ g ( u , v t ) ∂ v t ‖ ∞ ‖ ∇ 2 k σ j ‖ 2 ≤ ε 2 c 23 2 ‖ ∇ k ξ j ‖ 2 + c 23 − c 24 2 ‖ ∇ k ζ j ‖ 2 + ε 2 c 25 2 ‖ ∇ 2 k η j ‖ 2 + c 23 − c 26 2 ‖ ∇ 2 k σ j ‖ 2 , (72)

( Γ 2 ( φ ) h j , h j ) = ( 1 − M ( s ) ) ( ∇ m + k ξ j , ∇ m + k ζ j ) − 2 M ′ ( s ) ( ∇ m u , ∇ m ξ j ) ( ∇ 2 m + k u , ∇ k ζ j )       − 2 M ′ ( s ) ( ∇ m v , ∇ m η j ) ( ∇ 2 m + k u , ∇ k ζ j ) + ( 1 − M ( s ) ) ( ∇ 2 m + 2 k η j , ∇ 2 m + 2 k σ j )       − 2 M ′ ( s ) ( ∇ m u , ∇ m ξ j ) ( ∇ 4 m + 2 k v , ∇ 2 k σ j )       − 2 M ′ ( s ) ( ∇ m v , ∇ m η j ) ( ∇ 4 m + 2 k v , ∇ 2 k σ j )

≤ ( 1 − δ 0 ) λ 1 m 2 ( ‖ ∇ m + k ξ j ‖ ‖ ∇ k ζ j ‖ + ‖ ∇ 2 m + 2 k η j ‖ ‖ ∇ 2 k σ j ‖ )       + 2 c 26 ( λ 1 − k 2 + λ 1 − m 2 − k ) ( ‖ ∇ m + k ξ j ‖ ‖ ∇ k ζ j ‖ + ‖ ∇ 2 m + 2 k η j ‖ ‖ ∇ k ζ j ‖ )       + 2 c 27 ( λ 1 − k 2 + λ 1 − m 2 − k ) ( ‖ ∇ m + k ξ j ‖ ‖ ‖ ∇ 2 k σ j ‖ ‖ + ‖ ∇ 2 m + 2 k η j ‖ ‖ ∇ 2 k σ j ‖ )

≤ 1 − δ 0 2 λ 1 m 2 ( ‖ ∇ m + k ξ j ‖ 2 + ‖ ∇ k ζ j ‖ 2 + ‖ ∇ 2 m + 2 k η j ‖ 2 + ‖ ∇ 2 k σ j ‖ 2 )       + 2 c 28 ( λ 1 − k 2 + λ 1 − m 2 − k ) ( ‖ ∇ m + k ξ j ‖ 2 + ‖ ∇ k ζ j ‖ 2 + ‖ ∇ 2 m + 2 k η j ‖ 2 + ‖ ∇ 2 k σ j ‖ 2 ) ≤ 1 − δ 0 2 λ 1 m 2 + 2 c 28 ( λ 1 − k 2 + λ 1 − m 2 − k ) . (73)

Based on the above Equations (71)-(73), it is sorted out that

( F ′ ( φ ( τ ) ) h j ( τ ) , h j ( τ ) ) E k = ( ( − P ( φ ) + Γ 1 ( φ ) + Γ 2 ( φ ) ) h j , h j ) ≤ − ( β λ 1 m − ε 2 + 2 ε 2 − c 21 ( 2 − β ε ) λ 1 m 2 − c 23 − c 24 2 ) ‖ ∇ k ζ j ‖ 2       − ( β λ 1 2 m − ε 2 + 2 ε 2 − c 22 ( 2 − β ε ) λ 1 2 m 2 − c 25 − c 26 2 ) ‖ ∇ 2 k σ j ‖ 2       − ( ε − c 21 ( 2 − β ε ) 2 ) ‖ ∇ m + k ξ j ‖ 2 − ( ε − c 22 ( 2 − β ε ) 2 ) ‖ ∇ 2 m + 2 k η j ‖ 2       + ε 2 ( c 23 − 1 ) 2 ‖ ∇ k ξ j ‖ 2 + ε 2 ( c 25 − 1 ) 2 ‖ ∇ 2 k η j ‖ 2 + 1 − δ 0 2 λ 1 m 2       + 2 c 28 ( λ 1 − k 2 + λ 1 − m 2 − k ) , (74)

let

b = min { β λ 1 m − ε 2 + 2 ε 2 − c 21 ( 2 − β ε ) λ 1 m 2 − c 23 − c 24 2 , ε − c 21 ( 2 − β ε ) 2 ,                   ε − c 22 ( 2 − β ε ) 2 , β λ 1 2 m − ε 2 + 2 ε 2 − c 22 ( 2 − β ε ) λ 1 2 m 2 − c 25 − c 26 2 ,                   − 1 − δ 0 2 λ 1 m 2 − 2 c 28 ( λ 1 − k 2 + λ 1 − m 2 − k ) } ,

a = max { ε 2 ( c 23 − 1 ) 2 , ε 2 ( c 25 − 1 ) 2 } ,

we can obtain

∑ j = 1 n 0 ( F ′ ( φ ( τ ) ) h j ( τ ) , h j ( τ ) ) E k ≤ − n 0 b + a ∑ j = 1 n 0 ( ‖ ∇ k ξ j ‖ 2 + ‖ ∇ 2 k η j ‖ 2 )     , (75)

for almost all times t, there is ∑ i = 1 n 0 ‖ ∇ k ξ i ‖ 2 ≤ ∑ i = 1 n 0 λ i s ′ − 1 , ∑ j = 1 n 0 ‖ ∇ 2 k η j ‖ 2 ≤ ∑ j = 1 n 0 λ j s ″ − 1 , so

t r F ′ ( φ ( τ ) ) ∘ Q n 0 ( τ ) ≤ − n 0 b + a ( ∑ i = 1 n 0 λ i s ′ − 1 + ∑ j = 1 n 0 λ j s ″ − 1 ) , (76)

because of

q n 0 ( t ) = sup φ 0 ∈ B 0 k sup Ψ j ( 0 ) ∈ E k { 1 t ∫ 0 t t r F ′ ( φ ( τ ) ) ∘ Q n 0 ( τ ) d τ } , q n 0 = lim t → ∞ q n 0 ( t ) , (77)

q n 0 ( t ) ≤ − n 0 b + a ( ∑ i = 1 n 0 λ i s ′ − 1 + ∑ j = 1 n 0 λ j s ″ − 1 ) , q n 0 ≤ − n 0 b + a ( ∑ i = 1 n 0 λ i s ′ − 1 + ∑ j = 1 n 0 λ j s ″ − 1 ) , (78)

Therefore, the Lyapunov exponent K 1 , K 2 , ⋯ , K n 0 on set B 0 k is uniformly bounded, and

K 1 + K 2 + ⋯ + K n 0 ≤ − n 0 b + a ( ∑ i = 1 n 0 λ i s ′ − 1 + ∑ j = 1 n 0 λ j s ″ − 1 ) , (79)

so

( q i j ) + ≤ − n 0 b + a ( ∑ i = 1 n 0 λ i s ′ − 1 + ∑ j = 1 n 0 λ j s ″ − 1 ) ≤ a ( ∑ i = 1 n 0 λ i s ′ − 1 + ∑ j = 1 n 0 λ j s ″ − 1 ) ≤ 2 5 n 0 b , (80)

q n 0 ≤ − n 0 b ( 1 − a n 0 b ( ∑ i = 1 n 0 λ i s ′ − 1 + ∑ j = 1 n 0 λ j s ″ − 1 ) ) ≤ − 3 5 n 0 b , (81)

further

max 1 ≤ i , j ≤ n 0 ( q i j ) + | q n 0 | ≤ 2 3 . (82)

Thus, can obtain d H ( A k ) < 2 3 n 0 , d F ( A k ) < 4 3 n 0 .

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

1. Lin, G.G. and Gao, Y.L. (2017) The Global and Exponential Attractors for the Higher-Order Kirchhoff-Type Equation with Strong Linear Damping. Journal of Mathematics Research, 9, 145-167. https://doi.org/10.5539/jmr.v9n4p145

2. Lin, G.G. and Zhang, M. (2018) The Estimates of the Upper Bounds of Hausdorff Dimensions for the Global Attractor for a Class of Nonlinear Coupled Kirchhoff-Type Equations. Advances in Pure Mathematics, 8, 1-10.https://doi.org/10.4236/apm.2018.81001

3. Lin, G.G. and Yang, L.J. (2021) Global Attractors and Their Dimension Estimates for a Class of Generalized Kirchhoff Equations. Advances in Pure Mathematics, 11, 317-333. https://doi.org/10.4236/apm.2021.114020

4. Lin, G.G. and Chen, Y.H. (2020) A Family of Global Attractors for the Higher-Order Kirchhoff-Type Equations and Its Dimension Estimation. American Journal of Applied Mathematics, 8, 300-310. https://doi.org/10.11648/j.ajam.20200806.12

5. Lin, G.G. and Zhou, C.M. (2021) The Dynamic Behavior of a Class of Kirchhoff Equations with High Order Strong Damping. Journal of Applied Mathematics and Physics, 9, 1041-1055. https://doi.org/10.4236/jamp.2021.95071

6. Lin, G.G. and Hu, L.J. (2018) Estimate on the Dimension of Global Attractor for Nonlinear Higher-Order Coupled Kirchhoff Type Equations. Advances in Pure Mathematics, 8, 11-24.

7. Cavalcanti, M.M., Domingos Cavalcanti, V.N., Prates Filho, J.S. and Soriano, J.A. (1998) Existence and Exponential Decay for a Kirchhoff-Carrier Model with Viscosity. Journal of Mathematical Analysis and Applications, 226, 40-60.https://doi.org/10.1006/jmaa.1998.6057

8. Ono, K. (1997) Global Existence, Decay, and Blowup of Solutions for Some Mildly Degenerate Nonlinear Kirchhoff Strings. Journal of Differential Equations, 137, 273-301. https://doi.org/10.1006/jdeq.1997.3263

9. Lin, G.G. (2011) Nonlinear Evolution Equations. Yunnan University Press, Kunming.

10. Nakao, M. (2008) An Attractor for a Nonlinear Dissipative Wave Equation of Kirchhoff Type. Journal of Mathematical Analysis and Applications, 353, 652-659. https://doi.org/10.1016/j.jmaa.2008.09.010

11. Lin, G.G. (2019) Dynamic Properties of Several Kinds of the Kirchhoff Equations. Chongqing University Press, Chongqing.

12. Lin, G.G. and Guan, L.P. (2019) Random Attractor Family for a Class of Stochastic Higher-Order Kirchhoff Equations. Journal of Mathematics Research, 11, 23-33. https://doi.org/10.5539/jmr.v11n3p23

13. Wang, M.X., Tian, C.C. and Lin, G.G. (2014) Global Attractor and Dimension Estimation for a 2D Generalized Anisotropy Kuramoto-Sivashinsky Equation. International Journal of Modern Nonlinear Theory and Application, 3, 163-172. https://doi.org/10.4236/ijmnta.2014.34018

14. Tchaho, C.T.D., Omanda, H.M., Mbourou, G.N., Bogning, J.R. and Kofané, T.C. (2021) Higher Order Solitary Wave Solutions of the Standard KdV Equations. Open Journal of Applied Sciences, 11, 103-125. https://doi.org/10.4236/ojapps.2021.111008

15. Qin, C.L., Du, J.J. and Lin. G.G. (2017) Random Attractors for the Kirchhoff-Type Suspension Bridge Equations with Strong Damping and White Noises. International Journal of Modern Nonlinear Theory and Application, 6, 134-147.https://doi.org/10.4236/ijmnta.2017.64012

16. Liao, Y.H., Lin, G.G. and Liu, J. (2022) A Family of Global Attractors for a Class of Generalized Kirchhoff-Beam Equations. Journal of Applied Mathematics and Physics, 10, 930-951. https://doi.org/10.4236/jamp.2022.103064

17. Kirchhoff, G. (1883) Vorlesungen Fiber Mechanic. Tenbner, Stuttgart.

18. Hosoya, M. and Yamada, Y. (1991) On Some Nonlinear Wave Equations II: Global Existence and Energy Decay of Solutions. Journal of the Faculty of Science, the University of Tokyo. Section 1A, Mathematics, 38, 239-250.