Long Time Behavior of a Class of Generalized Beam-Kirchhoff Equations

Abstract

In this paper, we study the long time behavior of a class of generalized Beam-Kirchhoff equation , and prove the existence and uniqueness of the global solution of this class of equation by Galerkin method by making some assumptions about the nonlinear function term . The existence of the family of global attractor and its Hausdorff dimension and Fractal dimension estimation are proved.

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Lin, G. and Peng, K. (2023) Long Time Behavior of a Class of Generalized Beam-Kirchhoff Equations. Journal of Applied Mathematics and Physics, 11, 2963-2981. doi: 10.4236/jamp.2023.1110196.

1. Introduction

We study the initial boundary value problem of the following higher order Beam-Kirchhoff equation:

u t t + β Δ 2 m u t + δ Ψ ( u t 2 ) Δ 2 m u t + α Δ 2 m u + γ Φ ( D m u p p ) Δ 2 m u = g ( x ) , (1.1)

u ( x , t ) = 0 , i u v i = 0 , i = 1 , 2 , , 2 m 1 , x Ω , t > 0 , (1.2)

u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x Ω R n . (1.3)

where m > 1 is a positive integer, Ω is a bounded region in R n with a smooth homogeneous Dirichlet boundary, Ω represents the boundary of Ω , and g ( x ) is the external force term. β ( Δ ) 2 m u t is a strongly damped term, α , β , δ , γ is a constant greater than 0, Ψ ( u t 2 ) , Φ ( D m u p p ) is a given non-negative function, and D m u p p = Ω | D m u | p d x .

Kirchhoff type equation is a hot topic in the research of mathematical physics equations in recent years, and many scholars have done research on this kind of equation. Igor Chueshov [1] studied the long-term behavior of the solution of the following nonlinear strongly damped Kirchhoff wave equation:

u t t σ ( u 2 ) Δ u t ϕ ( u 2 ) Δ u + g 1 ( u ) = h ( x )

In the energy space H ( Ω ) = H 0 1 L p + 1 ( Ω ) × L 2 ( Ω ) , the author can find that the growth of exponential p of strong nonlinear g ( u ) is supercritical, that is, when p N + 2 ( N 2 ) + , p = N + 4 ( N 4 ) + , there is p p p , where when p < p , the growth of exponential p for H 1 ( Ω ) L p + 1 is supercritical. In H ( Ω ) , under the locally strong topology, a global attractor is established that is finite-dimensional. Especially in the non-supercritical case: 1) The partial strong topology will become a strong topology; 2) In H ( Ω ) , the exponential attractor can be obtained due to the feature of strong stability estimation. In addition, Chueshov [1] also considers the well-fitting solution of the Kirchhoff equation with structural damping term σ ( u 2 ) θ Δ u t at the abstract level and the long-term dynamic system, where 1 2 θ 1 .

Guoguang Lin, Yunlong Gao [2] studied the long time behavior of the solutions of a class of higher-order Kirchhoff type equation

u t t + ( α + β D m u 2 ) q ( Δ ) m + ( Δ ) m u t + g ( u ) = f ( x ) .

For Kirchhoff dissipation term, Newton binomial theorem is used to process analysis. In dimension estimation, a novel method is used to deal with the variational problem. The lemma and theorem are obtained by adding α q ( Δ ) m u to both sides of the variational equation at the same time, and the existence of global attractor and exponential attractor and Hausdorff dimension estimation are proved.

Guoguang Lin and Zhuoqian Li [3] studied the family of global attractor and its dimension estimation of a class of high-order nonlinear Kirchhoff equation

u t t + M ( D m u 2 ) ( Δ ) m u + β ( Δ ) m u t + g ( x , t ) = f ( x ) .

They use Sobolev embedding theorem H 0 m ( Ω ) L 2 p ( Ω ) to deal with source term g ( x , t ) , analyzing Kirchhoff stress term by case treatment, and prove that compacting family of global attractor exists in solution semigroup by means of the family of global attractor correlation theory. Of course, there are many studies on the family of global attractor higher order Kirchhoff equations.

Zhijian Yang and Zhiming Liu [4] explored the long-term behavior of the solution of the Kirchhoff equation

u t t Δ u t M ( u 2 ) Δ u + u t + g ( x , u ) = f ( x ) , ( x , u ) R N × R + ,

u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x R N

with nonlinear strong damping and critical term. Under the premise of critical nonlinearity, the adaptability of the solution, the existence of the global attractor and exponential attractor of the solution exist in the energy space H = H 1 ( R N ) × L 2 ( R N ) . Their innovation is that the obtained results improve Yang’s research results.

Guigui Xu, Libo Wang and Guoguang Lin [5] explored the inertial manifold of the strongly damped wave equation:

u t t α Δ u + β Δ 2 u γ Δ u t + g ( u ) = f ( x , t ) , ( x , t ) Ω × R + ,

u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x Ω ,

u | Ω = 0 , Δ u | Ω = 0 , ( x , t ) Ω × R + ,

The Fadeo-Galerkin method and the uniformly compact method are used to prove the existence and uniqueness of the strong solution to this problem. The existence of the inertial manifold is proved under the assumption of a certain spectral interval and a sufficiently small delay.

Naimen [6] studied the Kirchhoff type elliptic boundary value problem with Sobolev critical growth:

( a + b Ω | u | 2 d x ) Δ u = μ g ( x , u ) + u 5 .

When u 0 + , g ( x , u ) = o ( u ) and u + , g ( x , u ) = o ( u 5 ) , x Ω , The Sobolev embedding theorem H 0 1 ( Ω ) L p ( Ω ) is used to deal with the nonlinear source term μ g ( x , u ) , and the corresponding energy functional is reduced to the critical value level, so the equation satisfies the compactness condition.

Masamro [7] studied the initial boundary value problem of a class of Kirchhoff type equation

u t t M ( u 2 ) Δ u + δ | u | p u + γ u t = f ( x ) , x Ω , t > 0 ,

u ( x , t ) = 0 , x Ω , t 0 ,

u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( t ) , x Ω .

containing dissipative terms and damping terms, and proved the existence of the global solution of the equation under the initial boundary value condition by using Galerkin’s method, where Ω R n is a bounded region with a smooth boundary Ω , and

M ( γ ) C 1 [ 0 , ) , M ( γ ) m 0 > 0.

when δ > 0 , α > 0 and γ 0 are present.

Matsugama and Ikehata [8] used the potential well method to discuss the existence of the global solution of the Kirchhoff equation

u t t M ( u 2 ) Δ u + δ | u t | p 1 u t = μ | u | q 1 u ,

u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x Ω ,

u ( x , t ) = 0 , x Ω , t > 0.

with damping term and the attenuation estimation of the global solution, where

Ω R n , M ( s ) = a + b s r C 1 [ 0 , ) , a , b 0 , a + b > 0 , r 1 , M ( s ) m 0 > 0.

Nakao and Zhijian Yang [9] studied the long time behavior of the solution of the Kirchhoff type equation

u t t M ( u 2 ) Δ u Δ u t + g ( x , u ) = f ( x ) ,

u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x Ω ,

u ( x , t ) = 0 , x Ω , t > 0.

with strong damping term. When g ( x , u ) satisfies the local Lipschitz condition, they prove the existence of the global attractor by proving the existence of the bounded absorption set and the asymptotic compactness of the corresponding continuous semigroup. Where g ( x , u ) C 1 ( R n , R ) and g ( x , u ) have order p : 1 p < 4 ( N 4 ) + with respect to u.

2. Background Knowledge and Assumptions

In this paper, we need the following mathematical notation:

f = f ( x ) , D = , H = L 2 ( Ω ) , H 0 2 m ( Ω ) = H 2 m ( Ω ) H 0 1 ( Ω ) .

H 0 m + k ( Ω ) = H m + k ( Ω ) H 0 1 ( Ω ) , ( k = 0 , 1 , 2 , , 2 m )

H 0 m + k ( Ω ) = D ( Ω ) , Closure of infinitely differentiable Spaces in

H m + k ( Ω ) .

E k = H 0 2 m + k ( Ω ) × H 0 k ( Ω ) and E 0 = H 0 2 m ( Ω ) × H , where C i 0 ( i = 1 , 2 , ) is constant.

λ 1 is the first eigenvalue of Δ on Ω .

The inner product and norm expressed by ( , ) and respectively, that is, ( u , v ) = Ω u ( x ) v ( x ) d x , ( u , u ) = u 2 .

Note the global attractor from E 0 to E k as A k , while B 0 k is the bounded absorption set in E k , where k = 1 , 2 , , 2 m .

Lemma 2.1. [10] (Holder inequality) set

1 p + 1 q = 1 , ( p 1 , q 1 ) , f ( x ) L p ( Ω ) , g ( x ) L q ( Ω )

We have Ω | f ( x ) g ( x ) d x | ( Ω | f ( x ) | p d x ) 1 p ( Ω | g ( x ) | q d x ) 1 q .

Lemma 2.2. [10] (Poincare inequality) If Ω R n is a bounded open subset, then

u L 2 ( Ω ) λ 1 1 2 u L 2 ( Ω ) , u H 0 1 ( Ω )

where λ 1 is the first eigenvalue of Δ on Ω .

Lemma 2.3. [10] (Young Inequality) For any real number a , b 0 , then

a b ε p p a p + 1 q ε q b q , ( 1 p + 1 q = 1 , p > 1 , q > 1 ) ,

Let function Ψ ( u t 2 ) , Φ ( D m u p p ) satisfy the condition:

(A1) Φ ( D m u p p ) C 2 ( [ 0 , + ) , R ) , 0 σ 0 Φ ( D m u p p ) σ 1 , δ 2 = { σ 0 , d d t D 2 m u 2 0 σ 1 , d d t D 2 m u 2 0

Where σ 0 , σ 1 is the positive constant;

(A2) Ψ ( u t 2 ) C 2 ( [ 0 , + ) , R ) , 0 B 0 Ψ ( u t 2 ) B 1 ,

Where B 0 , B 1 is the positive constant;

(A3) 2 n n + 2 m p { < 2 n n 2 m , n > 2 m < , n 2 m

3. The Existence and Uniqueness of the Global Solution

Lemma 3.1. Assuming that condition (A1), (A2), (A3) is true, g ( x ) H , ( u 0 , u 1 ) E 0 , then the initial boundary value problem (1.1)-(1.3) has a global smooth solution ( u , v ) E 0 and is satisfied

( u , v ) E 0 2 = D 2 m u 2 + v 2 ( D 2 m u 0 2 + v 0 2 ) e a 1 t + C 1 a 1 ( 1 e a 1 t ) ,

where v = u t + ε u ,

Ream a 1 = min { β λ 1 2 m + δ B 0 λ 1 2 m 2 ε 2 ε 2 , 2 ε ( α + r δ 0 ) ε 2 ( λ 1 2 m + β + δ B 1 ) α + r δ }

So there is a non-negative real number R 0 and t 0 that makes

( u , v ) E 0 2 = D 2 m u 2 + v 2 R 0 2 , ( t > t 0 )

Prove we set v = u t + ε u and take the inner product of both sides of Equation (1.1) and v, we get

( u t t + β Δ 2 m u t + δ Ψ ( u t 2 ) Δ 2 m u t + α Δ 2 m u + γ Φ ( D m u p p ) Δ 2 m u , v ) = ( g ( x ) , v ) (3.1)

This is obtained by using the Holder inequality, the Young inequality, the Poincare inequality and the terms of condition (A1), (A2), (A3) in successive processing (3.1)

( u t t , v ) = 1 2 d d t v 2 ε v 2 + ε 2 ( u , v ) 1 2 d d t v 2 ε v 2 ε 2 2 v 2 ε 2 2 u 2 1 2 d d t v 2 2 ε + ε 2 2 v 2 ε 2 λ 1 2 m 2 D 2 m u 2 (3.2)

( α ( Δ ) 2 m u , v ) = ( α ( Δ ) 2 m u , u t + ε u ) = α 2 d d t D 2 m u 2 + α ε D 2 m u 2 (3.3)

( β ( Δ ) 2 m u t , v ) = ( β ( Δ ) 2 m ( v ε u ) , v ) = β D 2 m v 2 β ε ( D 2 m u , D 2 m v ) β D 2 m v 2 β ε 2 2 D 2 m u 2 β 2 D 2 m v 2 β λ 1 2 m 2 v 2 β ε 2 2 D 2 m u 2 (3.4)

( δ ψ ( u t 2 ) ( Δ ) 2 m u t , v ) = δ ψ ( u t 2 ) D 2 m v 2 ε δ ψ ( u t 2 ) ( D 2 m u , D 2 m v ) δ B 0 D 2 m v 2 ε δ B 1 ( ε D 2 m u 2 + 1 ε D 2 m v 2 2 ) δ B 0 λ 1 2 m 2 v 2 ε 2 δ B 1 2 D 2 m u 2 (3.5)

( r ϕ ( D m u p D ) ( Δ ) 2 m u , u t + ε u ) = r ϕ ( D m u p D ) 2 d d t D 2 m u 2 + ε r ϕ ( D m u p D ) D 2 m u 2 r δ 2 d d t D 2 m u 2 + ε r δ 0 D 2 m u 2 (3.6)

( g ( x ) , v ) ε 2 2 v 2 + 1 2 ε 2 g ( x ) 2 (3.7)

Synthetically available

1 2 d d t v 2 + 1 2 ( α + r δ ) d d t D 2 m u 2 + ( β λ 1 2 m 2 ε 2 ε 2 + δ B 0 λ 1 2 m 2 ) v 2 + ( α ε ε 2 λ 1 2 m 2 β ε 2 2 ε 2 δ B 1 2 + ε r δ 0 ) D 2 m u 2 1 2 ε 2 g ( x ) 2

d d t ( v 2 + ( α + r δ ) D 2 m u 2 ) + ( β λ 1 2 m 2 ε 2 ε 2 + δ B 0 λ 1 2 m ) v 2 + ( ε 2 ( λ 1 2 m + β + δ B 1 ) + 2 ε ( α + r B 0 ) α + r δ ) ( α + r δ ) D 2 m u 2 1 ε 2 g ( x ) 2 (3.8)

Ream a 1 = min { β λ 1 2 m + δ B 0 λ 1 2 m 2 ε 2 ε 2 , 2 ε ( α + r δ 0 ) ε 2 ( λ 1 2 m + β + δ B 1 ) α + r δ }

In a result d d t ( v 2 + ( α + r δ ) D 2 m u 2 ) + a 1 ( v 2 + ( α + r δ ) D 2 m u 2 ) C 1 (3.9)

It’s derived from the Gronwall inequality

v 2 + ( α + r δ ) D 2 m u 2 ( v 0 2 + ( α + r δ ) D 2 m u 0 2 ) e a 1 t + C 1 a ( 1 e a 1 t ) (3.10)

So there exist R 0 and t 0 to make ( u , v ) E 0 2 = D 2 m u + v 2 R 0 2 ( t > t 0 ) .

Lemma 3.2. Assuming that condition (A1), (A2), (A3) is true and g ( x ) H k , ( u 0 , u 1 ) E k is true, then the global solution ( u , v ) E k of the initial boundary value problem (1.1)-(1.3) is satisfied

( u , v ) E k 2 = D 2 m + k u 2 + D k v 2 ( D 2 m + k u 0 2 + D k v 0 2 ) e a 1 t + C 2 a 1 ( 1 e a 1 t ) ,

where C 2 = 1 ε 2 D k g ( x ) 2 ,

a 1 = min { β λ 1 2 m + δ B 0 λ 1 2 m 2 ε 2 ε 2 , 2 ε ( α + r δ 0 ) ε 2 ( λ 1 2 m + β + δ B 1 ) α + r δ }

Then there are positive constants R k and t k that make

( u , v ) E k 2 = D 2 m + k u 2 + D k v 2 R k 2 , ( t > t k )

Prove set ( Δ ) k v = ( Δ ) k u t + ε ( Δ ) k u , and take the inner product of both sides of Equation (1.1) with ( Δ ) k v , i.e.

( u t t + β Δ 2 m u t + δ Ψ ( u t 2 ) Δ 2 m u t + α Δ 2 m u + γ Φ ( D m u p p ) Δ 2 m u , ( Δ ) k v ) = ( g ( x ) , ( Δ ) k v ) . (3.11)

By using Holder inequality, Young inequality, Poincare inequality and conditions (A1), (A2), (A3), the terms in Equation (3.11) are processed successively

( u t t , ( Δ ) k v ) = ( v t ε u t , ( Δ ) k v ) = ( D k v t , D k v ) ε ( D k u t , D k v ) = 1 2 d d t D k v 2 ε D k v 2 + ε 2 ( D k u , D k v ) 1 2 d d t D k v 2 ε + ε 2 2 D k v 2 ε 2 λ 1 2 m 2 D 2 m + k u 2 (3.12)

( α ( Δ ) 2 m u , ( Δ ) k v ) = α ( D 2 m + k u , D 2 m + k u t ) + α ε D 2 m + k u 2 = α 2 d d t D 2 m + k u 2 + α ε D 2 m + k u 2 (3.13)

( β ( Δ ) 2 m u t , ( Δ ) k v ) = β ( ( Δ ) 2 m ( v ε u ) , ( Δ ) k v ) = β D 2 m + k v 2 ε β ( D 2 m + k u , D 2 m + k v ) β D 2 m + k v 2 ε β 2 ( ε D 2 m + k u 2 + D 2 m + k v 2 ε ) β λ 1 2 m 2 D k v 2 ε 2 β 2 D 2 m + k u 2 (3.14)

( δ ψ ( u t 2 ) ( Δ ) 2 m u t , ( Δ ) k v ) = δ ψ ( u t 2 ) D 2 m + k v 2 ε δ ψ ( u t 2 ) ( D 2 m + k u , D 2 m + k v ) δ B 0 D 2 m + k v 2 ε δ B 1 2 ( ε D 2 m + k u 2 + D 2 m + k v 2 ε ) δ B 0 λ 1 2 m 2 D k v 2 ε 2 δ B 1 2 D 2 m + k u 2 (3.15)

( r ϕ ( D m u p D ) ( Δ ) 2 m u , ( Δ ) k v ) = r ϕ ( D m u p D ) 2 d d t D 2 m + k u 2 + ε r ϕ ( D m u p D ) D 2 m + k u 2 r δ 2 d d t D 2 m + k u 2 + ε r δ 0 D 2 m + k u 2 (3.16)

( g ( x ) , ( Δ ) k v ) = ( D k g ( x ) , D k v ) ε 2 2 D k v 2 + 1 2 ε 2 D k g ( x ) 2 (3.17)

In summary can be obtained

1 2 d d t D k v 2 + α + r δ 2 d d t D 2 m + k u 2 + β λ 1 2 m + δ B 0 λ 1 2 m 2 ε 2 2 ε 2 D k v 2 + 2 ε ( α + r δ 0 ) ε 2 ( λ 1 2 m + β + δ B 1 ) 2 D 2 m + k u 2 C 2

From the Gronwall inequality:

d d t ( D k v 2 + ( α + r δ ) D 2 m + k u 2 ) + a 1 ( D k v 2 + ( α + r δ ) D 2 m + k u 2 ) C 2

D k v 2 + ( α + r δ ) D 2 m + k u 2 v 0 2 + ( α + r δ ) D 2 m + k u 0 2 e a 1 t + C 2 a 1 ( 1 e a 1 t ) (3.18)

So there exist R k and t k to make ( u , v ) E k 2 = D 2 m + k u 2 + D k v 2 R k 2 ( t > t k )

Theorem 3.3. (existence and uniqueness of solutions) Under the conditions of lemma 3.1 and Lemma 3.2, and assuming ψ < 0 , g H k , ( u 0 , u 1 ) E k , then the initial boundary value problem (1.1)-(1.3) has a unique global solution ( u , v ) L ( [ 0 , + ) ; E k )

Proof: Galerkin’s finite element method is used to prove the existence of the global solution.

The first step is to construct an approximate solution.

Let ( Δ ) 2 m + k w j = λ j 2 m + k w j , k = 1 , 2 , , 2 m , where λ j is the eigenvalue of Δ with a homogeneous Dirichlet boundary on Ω , and w j is the eigenfunction determined by the corresponding eigenvalue λ j , and the orthonormal basis of H is formed by the eigenvalue theory known w 1 , w 2 , , w l .

Construct the approximate solution u l ( t ) = j = 1 l g j l ( t ) w j of problem where g j l ( t ) is determined by the following nonlinear system of ordinary differential equations

( u t t + β Δ 2 m u t + δ Ψ ( u t 2 ) Δ 2 m u t + α Δ 2 m u + γ Φ ( D m u p p ) Δ 2 m u , w j ) = ( g ( x ) , w j ) , j = 1 , 2 , , l . (3.19)

When the initial condition u l 0 ( 0 ) = u l 0 , u l t ( 0 ) = u l 1 is satisfied and l + is satisfied, ( u l 0 , u l 1 ) ( u 0 , u 1 ) , in E k is known from the basic theory of ordinary differentiation that the approximate solution u l t exists on ( 0 , t l ) .

The second step is prior estimation.

Recording v l ( t ) = u l t ( t ) + ε u l ( t ) , multiply both sides of the equation v l ( t ) = u l t ( t ) + ε u l ( t ) , by λ j k ( g j l ( t ) + ε g j l ( t ) ) , and sum over j to get

1) k = 0 , have

( u l , v l ) E 0 2 = D 2 m u l 2 + v l 2 R 0 2 .

2) k = 1 , 2 , , 2 m , have

( u l , v l ) E k 2 = D 2 m + k u l 2 + v l 2 R k 2 .

then ( u l , v l ) in L ( [ 0 , + ] ; E k ) Middle bounded.

The third step is the limiting process.

In space E k ( k = 1 , 2 , , 2 m ) , choosing a subcolumn u μ from sequence u l causes ( u μ , v μ ) ( u , v ) to converge weakly * in L .

From the space compact embedding theorem we know that E k is compactly embedded in E 0 , and ( u μ , v μ ) ( u , v ) is strongly convergent almost everywhere in E 0 .

Let l = μ and take the limit, which can be obtained from the above equation

( u μ t , ( Δ ) k w j ) = ( v μ , λ j k w j ) ( ε u μ , λ j k w j ) ( v , λ j k w j ) ( ε u , λ j k w j ) (3.20)

Weakly convergent in L [ 0 , + ) . Due to

( u μ t t , ( Δ ) k w j ) = d d t ( u μ t , ( Δ ) k w j ) ,

Thus ( u μ t t , ( Δ ) k w j ) ( u t t , ( Δ ) k w j ) converges in D [ 0 , + ) , and D [ 0 , + ) is the conjugate space of the infinitely differentiable space of D [ 0 , + ) .

( β Δ 2 m u t , ( Δ ) k w j ) β ( Δ 2 m + k 2 v ε Δ 2 m + k 2 u , λ j 2 m + k 2 w j )

is weakly convergent in L [ 0 , + ) .

( δ Ψ ( u t 2 ) Δ 2 m u μ t , ( Δ ) k w j ) δ Ψ ( u t 2 ) ( Δ 2 m + k 2 v ε Δ 2 m + k 2 u , λ j 2 m + k 2 w j )

is weakly convergent in L [ 0 , + ) .

( γ Φ ( D m u p p ) Δ 2 m u μ , ( Δ ) k w j ) ( γ Φ ( D m u p p ) ( Δ ) 2 m + k 2 u , λ j 2 m + k 2 w j )

is weakly convergent in L [ 0 , + ) .

( α Δ 2 m u μ , ( Δ ) k w j ) ( α Δ 2 m + k 2 u , λ j 2 m + k 2 w j )

is weakly convergent in L [ 0 , + ) .

In particular, u μ 0 u 0 is weakly convergent in E k , and u μ t u t = u 1 is weakly convergent in E k . For all j and μ + ,

( u t t + β Δ 2 m u t + δ Ψ ( u t 2 ) Δ 2 m u t + α Δ 2 m u + γ Φ ( D m u p p ) Δ 2 m u , w j ) = ( g ( x ) , w j ) , j = 1 , 2 , , l (3.21)

can be derived.

Therefore, the existence of the weak solution of the problem (1.1)-(1.3) is obtained, the existence is proved, and the uniqueness of the solution below is obtained.

Let u , v be two solutions to the problem (1.1)-(1.3), and let w = u v have

( w t t , w t + ε w ) = ( w t t , w t ) + ε ( w t t , w ) = 1 2 d d t w t 2 + ε d d t ( w t , w ) ε 2 w t 2 (3.22)

( β ( Δ ) 2 m w t , w t + ε w ) = β ε 2 d d t D 2 m w 2 + β D 2 m w t 2 (3.23)

( α ( Δ ) 2 m w , w t + ε w ) = α 2 d d t D 2 m w 2 + α ε D 2 m w 2 (3.24)

( δ ψ ( u t 2 ) ( Δ ) 2 m u t δ ψ ( u t 2 ) ( Δ ) 2 m v t , w t + ε w ) = ( δ ψ ( u t 2 ) ( Δ ) 2 m w t , w t + ε w ) + ( [ δ ψ ( u t 2 ) δ ψ ( v t 2 ) ] ( Δ ) 2 m v t , w t + ε w ) = ( δ ψ ( u t 2 ) ( Δ ) 2 m w t , w t + ε w ) + ( δ ψ ( ξ ) ( u t + v t ) ( u t v t ) ( Δ ) 2 m u t , w t + ε w )

By ψ < 0 , then the original formula

δ B 0 ( D 2 m w t , D 2 m w t + ε D 2 m w ) + C 3 ( w t , w t + ε w ) δ B 0 D 2 m w t 2 + ε δ B 2 d d t D 2 m w 2 + C 3 ( 2 + ε 2 ) w t 2 + C 3 ε 2 w 2 (3.25)

( r ϕ ( D m u p D ) ( Δ ) 2 m u r ϕ ( D m v p D ) ( Δ ) 2 m v , w t + ε w ) = ( r ϕ ( D m u p D ) ( Δ ) 2 m u r ϕ ( D m u p D ) ( Δ ) 2 m v + r ϕ ( D m v p D ) ( Δ ) 2 m v , w t + ε w ) = r ϕ ( D m u p D ) ( ( Δ ) 2 m w , w t + ε w ) + r ϕ ( D m ξ p p ) ( D m ξ p p ) ( D m w ( Δ ) 2 m w , w t + ε w )

r δ d d t D 2 m w 2 + ε r δ 0 D 2 m w 2 + C 4 ( D m w ( Δ ) 2 m w , w t + ε w ) r δ d d t D 2 m w 2 + ε r δ 0 D 2 m w 2 C 4 2 ( D m w ( Δ ) 2 m w 2 + w t 2 )

C 4 ε ( D m w ( Δ ) 2 m w 2 + w t 2 ) r δ d d t D 2 m w 2 + ε r δ 0 D 2 m w 2 C 4 ( 1 + 2 ε 2 ) ( Δ ) 2 m w D m w 2 C 4 2 w t 2 > r δ d d t D 2 m w 2 + ε r δ 0 D 2 m w 2 C 5 D m w 2 C 4 2 w t 2 (3.26)

d d t ( 1 2 w t 2 + ε ( w t , w ) + α + β ε + ε δ B + 2 r δ 2 D 2 m w 2 ) + ( 2 ε ε C 3 2 C 3 + C 4 2 ) w t 2 + ε C 3 2 w 2 + ( α ε + ε r δ C 5 λ 1 m ) D 2 m w 2 0

So there exist a 2 to make that

d d t ( 1 + ε 2 w t 2 + ε 2 w 2 + α + β ε + ε δ B + 2 r δ 2 D 2 m w 2 ) + a 2 ( 1 + ε 2 w t 2 + ε 2 w 2 + α + β ε + ε δ B + 2 r δ 2 D 2 m w 2 ) 0 (3.27)

From Gronwall’s inequality:

1 + ε 2 w t 2 + ε 2 w 2 + α + β ε + ε δ B + 2 r δ 2 D 2 m w 1 2 1 + ε 2 w 0 t 2 + ε 2 w 0 2 + α + β ε + ε δ B + 2 r δ 2 D 2 m w 0 2 = 0 (3.28)

In a result, we have u = v and the Uniqueness obtained.

4. The Existence and Dimension Estimation of the Family of Global Attractor

Having studied the existence and uniqueness of global solutions for problems (1.1)-(1.3), the following proves the existence of the family of global attractor and estimates the hausdorff dimension and fractal dimension.

Theorem 4.1. [10] [11] Let E be a Banach space and semigroup S ( t ) : E E satisfy the following conditions:

1) The semigroup S ( t ) is uniformly bounded in E, that is r > 0 , when u E r , there is a constant C ( r ) , such that

S ( t ) u E C ( r ) , t [ 0 , + )

is true;

2) There exists a bounded absorption set B 0 in E;

3) S ( t ) is a completely continuous operator, and the semigroup S ( t ) has a compact complete attractor A 0 .

By changing the Banach space E from the above theorem to the Hilbert space E k , the existence theorem of the family of global attractor is obtained.

Theorem 4.2. If the global smooth solution of the problem (1.1)-(1.3) satisfies the conditions of Lemma 3.1, Lemma 3.2 and theorem 3.3 in the existence and uniqueness of the solution, then the problem (1.1)-(1.3) has a family of global attractor

A k = ω ( K 0 k ) = τ 0 t τ S ( t ) K 0 k ¯ , k = 1 , 2 , , 2 m ,

where K 0 k = { ( u , v ) E k : ( u , v ) E k 2 = D 2 m + k u 2 + D k v 2 R k 2 } is the bounded absorption set in E k and satisfies:

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

2) lim t d i s t ( S ( t ) B k , A k ) = 0 ( B k E k is a bounded set), where

lim t d i s t ( S ( t ) B k , A k ) = sup x B k inf y A k S ( t ) x y E k ,

where S ( t ) is the solution semigroup generated by problem (1.1)-(1.3).

Proof: It is necessary to verify the condition (1), (2), (3) of theorem 4.1, under the condition of theorem, the equation has a solution semigroup S ( t ) : E k E k . The bounded set known as K 0 k E k and contained in ball { ( u , v ) E k 2 R k 2 } ,

S ( t ) ( u 0 , v 0 ) E k 2 = u H 0 2 m + k ( Ω ) 2 + v H 0 k ( Ω ) 2 u 0 H 0 2 m + k ( Ω ) 2 + v 0 H 0 k ( Ω ) 2 R k 2 (4.1)

Then { S ( t ) } ( t 0 ) is uniformly bounded within E k . For any ( u 0 , v 0 ) E k ,

S ( t ) ( u 0 , v 0 ) E k 2 = u H 2 m + k ( Ω ) 2 + v H k ( Ω ) 2 R k 2

It follows that K 0 k = { ( u , v ) E k : ( u , v ) E k 2 = D 2 m + k u 2 + D k v 2 R k 2 } is the bounded absorption set of a semigroup S ( t ) .

Since E k E 0 is compactly embedded, that is the bounded set in E k is the compactly set in E 0 , the solution semigroup S ( t ) is a completely continuous operator, so there exists a family of global attractor A k = ω ( K 0 k ) = τ 0 t τ S ( t ) K 0 k ¯ of the solution semigroup S ( t ) .

Theorem 4.2 is proved.

To linearize Equation (1.1)-(1.3), consider the following initial boundary value problem:

U t t + δ ψ ( u t 2 ) Δ 2 m U t + δ ψ ( u t 2 ) ( u t 2 ) U t Δ 2 m u t + β Δ 2 m U t + γ Φ ( D m u p p ) Δ 2 m U + γ Φ ( D m u p p ) ( D m u p p ) D m U Δ 2 m u + α Δ 2 m U = 0 (4.2)

U ( x , t ) = 0 , i U V i = 0 , i = 1 , 2 , , 2 m 1 , x Ω , t > 0 , (4.3)

U ( x , 0 ) = ξ , U t ( x , 0 ) = η . (4.4)

where ( u 0 , u 1 ) A k , S ( t ) : E k E k , ( ξ , η ) E k , ( u , u t ) = S ( t ) ( u 0 , u 1 ) is the solution of problem (1.1)-(1.3) obtained by ( u 0 , u 1 ) A k , it can be shown that for any ( ξ , η ) E k , the linearized initial boundary value problem (4.2)-(4.3) has a unique solution ( U ( t ) , U t ( t ) ) L ( [ 0 , + ) ; E k ) .

Lemma 4.3. t > 0 , R > 0 , The map E k E k is Frechet differentiable on E k , and the Frechet differential in ϕ 0 = ( u 0 , u 1 ) T is a linear operator F : ( ξ , η ) T ( U ( t ) , U t ( t ) ) T .

Proof: Let ϕ 0 = ( u 0 , u 1 ) T E k , ϕ ¯ 0 = ( u 0 + ξ , u 1 + η ) T E k and ϕ 0 E k , ϕ ¯ 0 E k R give the Lipchitz continuity of S ( t ) on E k , i.e.

S ( t ) ϕ 0 S ( t ) ϕ ¯ 0 E k 2 e c 16 t ( ξ , η ) T E k 2 .

If h = u ¯ u U , μ = h t + ε h , then

{ u ¯ t t + β Δ 2 m u ¯ t + δ Ψ ( u ¯ t 2 ) Δ 2 m u ¯ t + α Δ 2 m u ¯ + γ Φ ( D m u ¯ p p ) Δ 2 m u ¯ = g ( x ) u t t + β Δ 2 m u t + δ Ψ ( u t 2 ) Δ 2 m u t + α Δ 2 m u + γ Φ ( D m u p p ) Δ 2 m u = g ( x ) U t t + δ ψ ( u t 2 ) Δ 2 m U t + δ ψ ( u t 2 ) ( u t 2 ) U t Δ 2 m u t + β Δ 2 m U t + γ Φ ( D m u p p ) Δ 2 m U + γ Φ ( D m u p p ) ( D m u p p ) D m U Δ 2 m u + α Δ 2 m U = 0

Subtract from the three formulas to get

( u ¯ u U ) t t + β ( Δ ) 2 m ( u ¯ u U ) t + α ( Δ ) 2 m ( u ¯ u U ) + g 1 + g 2 = 0 (4.5)

where

g 1 = δ ψ ( u ¯ t 2 ) ( Δ ) 2 m u ¯ t δ ψ ( u t 2 ) ( Δ ) 2 m u t δ ψ ( u t 2 ) ( Δ ) 2 m U t δ ψ ( D m u t 2 ) ( D m u t 2 ) D m U t ( Δ ) 2 m u t (4.6)

g 2 = r ϕ ( D m u ¯ p p ) ( Δ ) 2 m u ¯ r ϕ ( D m u p p ) ( Δ ) 2 m u r ϕ ( D m u p p ) ( Δ ) 2 m U r ϕ ( D m u p p ) ( D m u p p ) D m U ( Δ ) 2 m u (4.7)

u ¯ u U = h , Use the mean value theorem to deal with

g 1 = δ ψ ( ξ t 2 ) ( ξ t 2 ) ( u ¯ t u t ) ( Δ ) 2 m u ¯ t + δ ψ ( u t 2 ) ( Δ ) 2 m h t δ ψ ( u t 2 ) ( u t 2 ) U t ( Δ ) 2 m u t = g 11 + g 12 (4.8)

where

g 11 = δ ψ ( ξ t 2 ) ( ξ t 2 ) ( Δ ) 2 m u ¯ t ( u ¯ t u t ) δ ψ ( u t 2 ) ( u t 2 ) ( Δ ) 2 m u t ( u ¯ t u t )

g 12 = δ ψ ( u t 2 ) ( Δ ) 2 m h t + δ ψ ( u t 2 ) ( u t 2 ) h t ( Δ ) 2 m u t

g 11 = δ ψ ( ξ t 2 ) ( ξ t 2 ) ( Δ ) 2 m u ¯ t δ ψ ( u t 2 ) ( Δ ) 2 m u t ( u ¯ t u t ) = [ ( δ ψ ( ξ t 2 ) ( ξ t 2 ) δ ψ ( u t 2 ) ( u t 2 ) ) ( Δ ) 2 m u ¯ t + δ ψ ( u t 2 ) ( u t 2 ) ( Δ ) 2 m ( u ¯ t u t ) ] ( u ¯ t u t )

Make f ( x ) = δ ψ ( x 2 ) ( x 2 ) and ξ t = λ 1 u t + ( 1 λ 1 ) u ¯ t , λ 1 ( 0 , 1 ) , Then the above formula is

g 1 = f ( s ) λ 1 ( Δ ) 2 m u ¯ t ( u ¯ t u t ) 2 + f ( u t ) ( Δ ) 2 m ( u ¯ t u t ) ( u ¯ t u t ) + δ ψ ( u t 2 ) ( Δ ) 2 m h t + f ( u t ) ( Δ ) 2 m u t h t (4.9)

Similarly deal with

g 2 = r ϕ ( D m ξ t p p ) ( D m ξ t p p ) ( D m u ¯ D m u ) ( Δ ) 2 m u ¯ r ϕ ( D m u p p ) ( D m u p p ) D m u ¯ ( Δ ) 2 m u + r ϕ ( D m u p p ) ( D m u p p ) D m u ( Δ ) 2 m u + r ϕ ( D m u p p ) ( Δ ) 2 m h + r ϕ ( D m u p p ) ( D m u p p ) D m h ( Δ ) 2 m u = g 21 + g 22 (4.10)

where

g 21 = r ϕ ( D m ξ t p p ) ( D m ξ t p p ) ( D m u ¯ D m u ) ( Δ ) 2 m u ¯ r ϕ ( D m u p p ) ( D m u p p ) ( D m u ¯ D m u ) ( Δ ) 2 m u

g 22 = r ϕ ( D m u p p ) ( Δ ) 2 m h + r ϕ ( D m u p p ) ( D m u p p ) D m h ( Δ ) 2 m u

Similarly deal with

g 21 = [ r ϕ ( D m p p ) ( D m p p ) ( Δ ) 2 m u ¯ r ϕ ( D m u p p ) ( D m u p p ) ( Δ ) 2 m u ] ( D m u ¯ D m u ) = [ r ϕ ( D m p p ) ( D m p p ) r ϕ ( D m u p p ) ( D m u p p ) ( Δ ) 2 m u ¯ + r ϕ ( D m u p p ) ( D m u p p ) ( Δ ) 2 m ( u ¯ u ) ] ( D m u ¯ D m u )

Recording m ( y ) = r ϕ ( b p p ) ( b p p ) , D m ξ = D m u + λ 2 ( D m u ¯ D m u ) , λ 2 ( 0 , 1 ) .

So we can obtain

g 21 = m ( s ) λ 2 ( Δ ) 2 m u ¯ ( D m u ¯ D m u ) 2 + r ϕ ( D m u p p ) ( D m u p p ) ( Δ ) 2 m ( u ¯ u ) ( D m u ¯ D m u )

g 2 = m ( s ) δ ( Δ ) 2 m u ¯ ( D m u ¯ D m u ) 2 + r ϕ ( D m u p p ) ( D m u p p ) ( Δ ) 2 m ( u ¯ u ) ( D m u ¯ D m u ) + r ϕ ( D m u p p ) ( Δ ) 2 m h + r ϕ ( D m u p p ) ( D m u p p ) ( Δ ) 2 m u D m h (4.11)

Let ( Δ ) k μ and each of them take the inner product: where μ = h t + ε h .

( h t t , ( Δ ) k μ ) = ( k t ε k + ε 2 h , ( Δ ) k μ ) = 1 2 d d t D k μ 2 ε D k μ 2 + ε 2 2 d d t D k h 2 + ε 3 D k h 2 (4.12)

( β + δ ψ ( u t 2 ) ( Δ ) 2 m h t , ( Δ ) k μ ) = β + δ ψ ( u t 2 ) ( D 2 m + k h t 2 + ε 2 d d t h 2 ) ( β + δ B ) ε 2 d d t h 2 (4.13)

( α + r ϕ ( D m u p p ) ( Δ ) 2 m h , ( Δ ) k μ ) = ( α + r ϕ ( D m u p p ) ) ( 1 2 d d t D 2 m + k h 2 + ε D 2 m + k h 2 ) α + r δ 2 d d t D 2 m + k h 2 + ε ( α + r δ 0 ) D 2 m + k h 2 (4.14)

( f ( s ) λ 1 ( Δ ) 2 m u ¯ t ( u ¯ t u t ) 2 , ( Δ ) k μ ) = f ( s ) λ 1 Ω D 2 m + k u ¯ ( u ¯ t u t ) 2 D 2 m + k k d x C 6 u ¯ t u t 2 D 2 m + k μ

Due to

u t v + ε u v + u 1 λ 1 k 2 D k v + 1 λ 1 2 m + k 2 D 2 m + k u λ 0 ( D k v + D 2 m + k u ) = λ 0 u E k

So there are

( f ( s ) λ 1 ( Δ ) 2 m u ¯ t ( u ¯ t u t ) 2 , ( Δ ) k μ ) C 6 λ 0 u ¯ t u t E k 2 D 2 m + k μ C 6 λ 0 2 u ¯ t u t E k 4 + C 6 λ 0 2 D 2 m + k μ 2 (4.15)

( f ( u t ) ( Δ ) 2 m u t h t , ( Δ ) k μ ) C 7 u ¯ t u t 2 D 2 m + k μ

( f ( u t ) ( Δ ) 2 m u t h t , ( Δ ) k μ ) C 8 ( D k h t 2 + ε 2 d d t D k h 2 )

( m ( s ) λ 2 ( Δ ) 2 m u ¯ ( D m u ¯ D m u ) 2 , ( Δ ) k μ ) C 9 D m u ¯ D m u 2 D 2 m + k μ C 9 λ 1 m + k D 2 m + k u ¯ D 2 m + k u 2 D 2 m + k w C 9 λ 1 m + k u ¯ t u t E k 2 D 2 m + k μ C 9 2 λ 1 m + k u ¯ t u t E k 4 + C 9 2 λ 1 m + k D 2 m + k μ 2 (4.16)

( r ϕ ( D m u p p ) ( D m u p p ) ( Δ ) 2 m ( u ¯ u ) ( D m u ¯ D m u ) , ( Δ ) k μ ) C 10 D 2 m + k u ¯ D 2 m + k u D m ( u ¯ u ) D 2 m + k μ C 10 λ 1 m + k 2 D 2 m + k ( u ¯ u ) 2 D 2 m + k μ C 10 2 λ 1 m + k 2 ( u ¯ u E k 4 + D 2 m + k μ 2 ) (4.17)

( r ϕ ( D m u p p ) ( D m u p p ) ( Δ ) 2 m u D m h , ( Δ ) k μ ) C 11 D m h D 2 m + k h C 11 2 D m h 2 + C 11 2 D 2 m + k h 2 ( C 11 2 + C 11 2 λ 1 m + k ) D 2 m + k h 2 (4.18)

So we obtain

d d t ( 1 2 D k μ 2 + 1 2 D 2 m + k μ 2 + ε ( C 8 + ε ) 2 D k h 2 + β + δ B ε 2 h 2 + 2 ε + 1 2 ( α + r δ ) D 2 m + k h 2 ) C 12 u ¯ u E k 4 + C 13 D 2 m + k μ 2 + C 14 D k μ 2 + C 11 2 ( 1 + 1 λ 1 m + k ) D 2 m + k h 2 + ( ε ε 2 ) D k h 2 (4.19)

1 2 D k μ 2 + 1 2 D 2 m + k μ 2 + ε ( C 8 + ε ) 2 D k h 2 + β + δ B ε 2 h 2 + 2 ε + 1 2 ( α + r δ ) D 2 m + k h 2 C 15 e C 16 t u ¯ u E k 4 (4.20)

when ( ξ , η ) E k 2 0 , 1 2 u ¯ u U E k 2 ( ξ , η ) E k 2 C 15 e C 16 t ( ξ , η ) E k 2 0 .

Theorem 4.4. Under the condition of Theorem 4.2, the family of global attractor A k of the problem (1.1)-(1.3) has the Hausdorff dimension and the Fractal dimension and d H ( A k ) < 2 3 n , d F ( A k ) < 4 3 n .

Proof: Let P ε : { u , u t } { u , u t + ε u } be an isomorphic mapping, then

Ψ = P ε φ ¯ = ( U , V ) T , where φ ¯ = ( U , U t ) T , V = U t + ε U ,

The Frechet differentiability of S ( t ) : E k E k is known from lemma 4.3 to estimate the Hausdorff dimension and Fractal dimension of problem (1.1)-(1.3). Consider the variational equation Y t + A ε Y = 0 of Equation (4.2) under initial conditions, where

Y = [ U V ]

A ε = [ ε I I α ε β r ϕ ( D m u p p ) ε δ ψ ( u t 2 ) A 2 m + ε 2 2 δ ε B + p r C β + δ ψ ( u t 2 ) A 2 m ε + 2 δ D ] (4.21)

where ream Δ = A , D V = ψ ( u t 2 ) Ω u t V d x ( Δ ) 2 m u t ,

B U = ψ ( u t 2 ) Ω u t U d x ( Δ ) 2 m u t , C U = ϕ ( D m u p p ) Ω | D m u | p 2 D m u D m U d x ( Δ ) 2 m u

For A fixed ( u 0 , v 0 ) E k , let γ 1 , γ 2 , , γ n be n elements of E k , and let U 1 ( t ) , U 2 ( t ) , , U n ( t ) be n solutions of the linear Equation (4.2) with corresponding initial values of U 1 ( 0 ) = γ 1 , U 2 ( 0 ) = γ 2 , , U n ( 0 ) = γ n . Get

U 1 ( t ) Λ U 2 ( t ) Λ Λ U n ( t ) Λ E k 2 = γ 1 Λ γ 1 Λ Λ γ n Λ E k 2 exp ( 0 t t r F ( Ψ ( τ ) ) Q n ( τ ) d τ ) , t [ 0 , 1 ]

where Λ represents the outer product, t r represents the trace of the operator, and Q N is the orthogonal projection from space E k to

s p a n { U 1 ( t ) , U 2 ( t ) , , U n ( t ) } .

Given τ , let ω j ( τ ) = ( ξ j ( τ ) , η j ( τ ) ) T , j = 1 , 2 , , n be the orthonormal basis of s p a n { U 1 ( t ) , U 2 ( t ) , , U n ( t ) } .

Define the inner product on E k as

( ( ξ , η ) , ( ξ ¯ , η ¯ ) ) = ( ( D 2 m + k ξ , D 2 m + k ξ ¯ ) + ( D k η , D k η ¯ ) ) , (4.22)

t r F ( Ψ ( τ ) ) Q n ( τ ) = j = 1 n ( Λ ε ( Ψ ( τ ) ) Q n ( τ ) ω j ( τ ) , ω j ( τ ) ) E k = j = 1 n ( Λ ε ( Ψ ( τ ) ) ω j ( τ ) , ω j ( τ ) ) E k (4.23)

( A ε w j , w j ) E k = [ ( ε ξ j η j , ( α ε β r ϕ ( D m u p p ) ε δ ψ ( u t 2 ) A 2 m + ε 2 2 δ ε B + p r C ) ξ j + ( β + δ ψ ( u t 2 ) A 2 m ε + 2 δ D ) μ j ) ( ξ j , μ j ) ] E k = ε D 2 m + k ξ j 2 + ( 1 α + ε β + r ϕ ( D m u p p ) ε δ ψ ( u t 2 ) ) ( D 2 m + k μ j , D 2 m + k ξ j ) ( ε 2 2 δ ε B + p r C ) ( D 2 m + k μ j , D 2 m + k ξ j ) ( β + δ ψ ) ( u t 2 ) D 2 m + k μ j 2

+ ( ε 2 δ | D | D k μ j 2 ) ε D 2 m + k ξ j 2 + 1 α + ε β + r δ 1 ε δ B 0 2 ( D 2 m + k μ j 2 + D 2 m + k ξ j 2 ) + ε 2 2 δ ε | B | + p r | C | 2 ( D k ξ j 2 + D k μ j 2 ) ( ε + 1 α + ε β + r δ 1 ε δ B 0 2 + ε 2 2 δ ε | B | + p r | C | 2 λ 1 2 m + k ) D 2 m + k ξ j 2 + ε 2 2 δ ε | B | + p r | C | 2 D k μ j 2 + ε D k μ j 2

Let

C 17 = max { ( ε + 1 α + ε β + r δ 1 ε δ B 0 2 + ε 2 2 δ ε | B | + p r | C | 2 λ 1 2 m ) , ε 2 2 δ ε | B | + p r | C | 2 }

ε = r , ( A ε w j , w j ) E k C 17 ( D k μ j 2 + D 2 m + k ξ j 2 ) + r D k μ j 2 . (4.24)

Since ω j ( τ ) = ( ξ j ( τ ) , η j ( τ ) ) T , j = 1 , 2 , , n , s p a n { U 1 ( t ) , U 2 ( t ) , , U n ( t ) } is an orthonormal basis,

j = 1 n ( F ( ψ ( τ ) ) ω j ( τ ) , ω j ( τ ) ) E k n C 17 + r j = 1 n D k ξ j 2 (4.25)

For any t, there’s a

j = 1 n D k ξ j 2 j = 1 n λ j s 1 , (4.26)

So there is

T r F ( Ψ ( τ ) ) Q n ( τ ) n C 17 + r j = 1 n λ j s 1 . (4.27)

Let

q n ( t ) = sup Ψ 0 K 0 k sup η j E k η j E k 1 ( 1 t 0 t t r F ( S ( τ ) Ψ 0 ) Q n ( τ ) d τ ) , q n = lim t sup q n ( t )

Then q n n C 17 + r j = 1 n λ j s 1 , Therefore, the Lyapunov exponent μ ˜ 1 , μ ˜ 2 , , μ ˜ j of K 0 k is uniformly bounded, and

μ ˜ 1 + μ ˜ 2 + + μ ˜ j n C 17 + r j = 1 n λ j s 1

s [ 0 , 1 ] make ( q j ) + n C 17 7 , q n = n C 17 7 ( 1 2 r n C 17 j = 1 n λ j s 1 ) 3 14 n C 17 (4.28)

max ( q j ) + | q n | 2 3 , d H ( A k ) 2 3 n , d F ( A k ) 4 3 n (4.29)

It can be concluded that N-dimensional volume elements decay exponentially in E k and d H ( A k ) < 2 3 n , d F ( A k ) < 4 3 n , then the Hausdorff dimension and Fractal dimension of the family of global attractor are finite, and theorem 4.4 can be proved.

5. Conclusion

In this paper, we studied a class of generalized Bean-Kirchhoff equations, proved the existence and uniqueness of the global solution of this class of equations by Galerkin method, and further proved the existence of the family of global attractor and its Hausdorff dimension and Fractal dimension estimation, which has certain scientific significance. And there’s more to study.

Conflicts of Interest

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

References

[1] Chueshov, I. (2012) Long-Time Dynamics of Kirchhoff Wave Models with Strong Nonlinear Damping. Journal of Differential Equations, 252, 1229-1262.
[2] Lin, G.G. and Gao, Y.L. (2018) Long Time Behavior of Solution for a Class of Higher-Order Kirchhoff Equation. Yunnan University, Kunming.
[3] Lin, G.G. and Li, Z.Q. (2019) A Class of Attractors of Higher Order Nonlinear Kirchhoff Equations and Their Dimensions. Journal of Shandong University (Science Edition), 54, 1-11.
[4] Yang, Z.J. and Liu, Z.M. (2015) Exponential Attractor for the Kirchhoff Equations with Strong Nonlinear Damping and Supercritical Nonlinearity. Applied Mathematics Letters, 46, 127-132.
https://doi.org/10.1016/j.aml.2015.02.019
[5] Xu, G.G., Wang, L.B. and Lin, G.G. (2014) Inertial Manifolds for a Class of the Retarded Nonlinear Wave Equations. Mathematica Applicata, 27, 887-891.
[6] Naimen, D. (2014) Positive Solutions of Kirchhoff Type Elliptic Equations Involving a Critical Sobolev Exponent. NoDEA Nonlinear Differential Equations and Applications, 21, 885-914.
https://doi.org/10.1007/s00030-014-0271-4
[7] Masamro, H. and Yoshio, Y. (1991) On Some Nonlinear Wave Equations 2: Global Existence and Energy Decay of Solutions. Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics, 38, 239-250.
[8] Matsuyama, T. and Ikehata, R. (1996) On Global Solution and Energy Decay for the Wave Equation of Kirchhoff Type with Nonlinear Damping Term. Journal of Mathematical Analysis and Applications, 204, 729-753.
https://doi.org/10.1006/jmaa.1996.0464
[9] Nakao, M. and Zhijian, Y. (2007) Global Attractors for Some Quasi-Linear Wave Equations with a Strong Dissipation. Advances in Mathematical Sciences and Applications, 17, 89.
[10] Lin, G.G. (2019) Dynamic Properties of Several Kinds of the Kirchhoff Equations. Chongqing University Press, Chongqing.
[11] Lin, G.G. (2011) Nonlinear Evolution Equation. Yunnan University Press, Kunming.

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