A Family of Global Attractors for a Class of Generalized Kirchhoff-Beam Equations

Yuhuai Liao¹, Guoguang Lin², Jie Liu²

¹Wenshan Universiry, Wenshan, China.
²School of Mathematics and Statistics, Yunnan University, Kunming, China.

DOI: 10.4236/jamp.2022.103064                

Abstract

The initial boundary value problem for a class of high-order Beam equations with quasilinear and strongly damped terms is studied. Firstly, the existence and uniqueness of the global solution of the equation are proved by prior estimation and Galerkin finite element method. Then the bounded absorption set is obtained by prior estimation, and the family of global attractors for the high-order Kirchhoff-Beam equation is obtained. The Frechet differentiability of the solution semigroup is proved after the linearization of the equation, and the decay of the volume element of the linearization problem is further proved. Finally, the Hausdorff dimension and Fractal dimension of the family of global attractors are proved to be finite.

Keywords

High-Order Kirchhoff-Beam Equation, Galerkin’s Method, Family of Global Attractors, The Hausdorff Dimension

1. Introduction

In order to study the global stability of a wide norm model for vertical beam vibration, the initial boundary value problems of the following Kirchhoff-Beam equations are studied:

$u_{tt}+\beta {\left(-\Delta \right)}^{2m}{u}_t+M\left({\Vert D^{m}u\Vert }_p^p\right)u_t+\alpha {\Delta }^{2m}u+N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}u=f\left(x\right),$ (1)

$u\left(x,t\right)=0,\frac{{\partial }^{i}u}{\partial v^i}=0,i=1,2,\cdots ,m-1,x\in \partial \Omega ,t>0,$ (2)

$u\left(x,0\right)=u_0\left(x\right),u_t\left(x,0\right)=u_1\left(x\right),x\in \Omega \subset R^n.$ (3)

where $m>1$ is a positive integer, $\Omega$ is the bounded region in $R^n$ with smooth boundary $\partial \Omega$. $f\left(x\right)$ is the external force term. $\beta {\left(-\Delta \right)}^{2m}{u}_t$ is the strongly damped term, $\alpha ,\beta$ are positive constants, $\alpha \ge \frac{3}{2}$,

$M\left({\Vert D^{m}u\Vert }_p^p\right),N\left({\Vert D^{m}u\Vert }_p^p\right)$ are the general non-negative real-valued functions, ${\Vert D^{m}u\Vert }_p^p={\displaystyle \underset{\Omega }{\int }{\left|D^{m}u\right|}^p\text{d}x}$, and the relevant assumptions will be given later. In this

paper, we study the existence and uniqueness of global solutions for problems (1) - (3), prove the existence of a family global attractor and estimate its Hausdorff dimension and Fractal dimension.

In 1883, Kirchhoff [1] proposed the following model when studying the free vibration of elastic strings:

$u_{tt}-\sigma \left({\Vert Du\Vert }^2\right)\Delta u_t-\phi \left({\Vert Du\Vert }^2\right)\Delta u+g\left(u\right)=h\left(x\right).$

This model more accurately describes the motion of the elastic rod and reveals the physical significance of elastic vibration, from which the Kirchhoff equation model becomes a kind of classical problems in infinite-dimensional dynamic systems. Many scholars have also achieved some good results in their research on the global attractor of Kirchhoff equation and the estimation of Hausdorff dimension. For details, please refer to references [1] [2] [3].

Igor Chueshov [4]: the long time behavior of the following Kirchhoff wave equations with strong nonlinear damping is studied

${\partial }_{tt}u-\sigma \left({\Vert \nabla u\Vert }^2\right)\Delta {\partial }_{t}u-\varphi \left({\Vert \nabla u\Vert }^2\right)\Delta u+f\left(u\right)=h\left(x\right).$

Tokio Matsuyama and Ryo Ikehata [5]: the attenuation of global solution of Kirchhoff type wave equation with nonlinear damping is proved:

$u_{tt}+{\left(-\Delta \right)}^m{u}_t+\left({\Vert D^{m}u\Vert }^2\right){\left(-\Delta \right)}^{m}u+g\left(u\right)=f\left(x\right).$

${u\left(x,t\right)|}_{\partial \Omega }=0,t\ge 0.$

where $M\left(s\right)\in C^1\left[0,\infty \right),M\left(s\right)\ge m_0>0;\delta >0,\mu \in R$ are constants.

Guoguang Lin, Yunlong Gao [6]: the initial boundary value problem of Kirchhoff type equation with strongly damped term is studied

$u_{tt}+M\left({\Vert \nabla u\left(t\right)\Vert }_2^2\right)\Delta u+\delta {\left|u_t\right|}^{p-1}{u}_t=\mu {\left|u\right|}^{q-1}u.$

The existence and uniqueness of knowledge are proved by prior estimation and Galerkin’s method, and then the existence of global attractor is obtained. The Hausdorff dimension and Fractal dimension of global attractor are estimated.

Lin Chen, Guoguang Lin [7]: study the well-posedness and long-time behavior of solutions for a class of nonlinear higher-order Kirchhoff equations:

$u_{tt}+{\left(-\Delta \right)}^m{u}_t+\varphi \left({\Vert {\nabla }^{m}u\Vert }^2\right){\left(-\Delta \right)}^{m}u+g\left(u\right)=f\left(x\right).$

The existence and uniqueness of global solutions are proved by prior estimation and Galerkin’s method, and the Hausdorff dimension and Fractal dimension of global attractors are estimated. More studies of wave equations can be found in reference [8] - [19].

For the convenience of statement, the following Spaces and notations are defined:

$H=L^2\left(\Omega \right)$, $D=\nabla$, $H_0^m\left(\Omega \right)=H^m\left(\Omega \right)\cap H_0^1\left(\Omega \right)$, $H_0^{2m+k}\left(\Omega \right)=H^{2m+k}\left(\Omega \right)\cap H_0^1\left(\Omega \right)$. $C_i\left(i=1,2,\cdots \right)$ are the different positive constants. $E_k=H_0^{2m+k}\left(\Omega \right)\times H_0^k\left(\Omega \right),\left(k=1,2,\cdots ,2m\right)$, $k=0$, $E_0=H_0^{2m}\times L^2\left(\Omega \right)$.

Defines $\left(\cdot ,\cdot \right)$ and $\Vert \cdot \Vert$ represents H the inner product and norm respectively: $\left(u,v\right)={\displaystyle \underset{\Omega }{\int }u\left(x\right)v\left(x\right)\text{d}x}$, $\Vert \cdot \Vert ={\Vert \cdot \Vert }_{L^2}$, $\left(u,u\right)={\Vert u\Vert }^2$.

$A_k$ is $\left(E_0,E_k\right)$ -global attractors, $B_{0k}$ is the bounded absorption set in $E_k$, $k=1,2,\cdots ,2m$.

Kirchhoff stress $M\left(s\right),N\left(s\right)$ meet the conditions:

1) $M\left(s\right)\in C^2\left(\left[0,+\infty \right),R\right)$, $2\epsilon \le {\sigma }_0\le M\left(s\right)\le {\sigma }_1$, ${\sigma }_0,{\sigma }_1$ are the positive constants;

2) $N\left(s\right)\in C^2\left(\left[0,+\infty \right),R\right)$, ${\mu }_0\le N\left(s\right)\le {\mu }_1$, ${\mu }_0,{\mu }_1$ are the positive constants;

3) $\frac{2n}{n+2m}\le p<\{\begin{array}{l}\frac{2n}{n-2m},n>2m\\ \infty ,n\le 2m\end{array}$, $0<\epsilon <\min \{\frac{4{\lambda }_1^m-1}{4\beta {\lambda }_1^m},\frac{\alpha }{3\beta },2{\lambda }_1^m{\mu }_0\}$,

$\begin{array}{l}\beta \ge \max \{\frac{4{\mu }_1^2{\lambda }_1^m}{\left(4\alpha -1\right)\epsilon {\lambda }_1^{2m}+2\epsilon \left(\epsilon -{\sigma }_1\right)},\frac{\epsilon {\sigma }_1-2{\sigma }_0+2\epsilon +2{\epsilon }^2}{2{\lambda }_1^{2m}},\frac{2\epsilon {\sigma }_1}{{\lambda }_1^m},\frac{2C_4}{{\lambda }_1^m},\\ \frac{1}{2{\lambda }_1^m-1},\frac{3{\mu }_0}{2{\lambda }_1^m}+\frac{{\sigma }_1}{{\lambda }_1^{2m}},\frac{\alpha -1}{\epsilon +1},\frac{{\epsilon }^2+\varphi +\kappa -\epsilon {\sigma }_0}{\epsilon {\lambda }_1^{2m}}+\frac{{\mu }_0+\alpha -1}{\epsilon }-\epsilon ,\\ \frac{{\mu }_0+{\epsilon }^2+\varphi +\kappa -\left(\epsilon +2\right){\sigma }_0}{{\lambda }_1^{2m}}\}.\end{array}$

2. The Existence of the Family of Globals

For the initial boundary value problem (1) - (3), the existence of global solutions is proved by prior estimation and Galerkin’s finite element method, and then the uniqueness of global solutions is proved. Finally, the operator semigroup theory is used to prove that the solution semigroup of this problem has a family of global attractors.

Lemma 1. Assumes that (a), (b), (c) are held, $f\left(x\right)\in H$, $\left(u_0,u_1\right)\in E_0$, then the initial boundary value problem (1) - (3) has global smooth solutions $\left(u,v\right)\in E_0$,

${\Vert \left(u,v\right)\Vert }_{E_0}^2={\Vert D^{2m}u\Vert }^2+{\Vert v\Vert }^2\le \left({\Vert D^{2m}{u}_0\Vert }^2+{\Vert v_0\Vert }^2\right){\text{e}}^{-{\alpha }_{1}t}+\frac{C_1}{{\alpha }_1}\left(1-{\text{e}}^{-{\alpha }_{1}t}\right),$ (4)

${\Vert \left(u,v\right)\Vert }_{E_0}^2={\Vert D^{2m}u\Vert }^2+{\Vert v\Vert }^2\le R_0^2,\left(t>t_1\right)$ (5)

where $v=u_t+\epsilon u$, $C_1=\frac{1}{\epsilon }{\Vert f\left(x\right)\Vert }^2$, ${\alpha }_1=\min \left\{{\theta }_1,\frac{{\theta }_2}{\alpha }\right\}$, a non-negative real number $R_0$ and $t_1=t_1\left(\Omega \right)>0$.

Proof. Take the inner product of $v=u_t+\epsilon u$ and Equation (1) both sides,

$\begin{array}{l}\left(u_{tt}+\beta {\left(-\Delta \right)}^{2m}{u}_t+M\left({\Vert D^{m}u\Vert }_p^p\right)u_t+\alpha {\Delta }^{2m}u+N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}u,v\right)\\ =\left(f\left(x\right),v\right).\end{array}$ (6)

By using Holder’s Inequality, Young’s Inequality, and Poincare’s Inequality, The items in (6) can be obtained by successive processing:

$\left(u_{tt},v\right)=\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert v\Vert }^2-\epsilon {\Vert v\Vert }^2+{\epsilon }^2\left(u,v\right).$ (7)

$\begin{array}{c}\left(\beta {\left(-\Delta \right)}^{2m}{u}_t,v\right)=\beta {\Vert D^{2m}v\Vert }^2-\epsilon \beta \left(D^{2m}u,D^{2m}v\right)\\ \ge \beta {\Vert D^{2m}v\Vert }^2-\frac{\epsilon }{4}{\Vert D^{2m}u\Vert }^2-\epsilon {\beta }^2{\Vert D^{2m}v\Vert }^2,\end{array}$ (8)

$\begin{array}{c}\left(M\left({\Vert D^{m}u\Vert }_p^p\right)u_t,v\right)=M\left({\Vert D^{m}u\Vert }_p^p\right){\Vert v\Vert }^2-\epsilon M\left({\Vert D^{m}u\Vert }_p^p\right)\left(u,v\right).\\ \ge {\sigma }_0{\Vert v\Vert }^2-\epsilon M\left({\Vert D^{m}u\Vert }_p^p\right)\left(u,v\right).\end{array}$ (9)

$\begin{array}{c}\left(N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}u,v\right)\ge -N\left({\Vert D^{m}u\Vert }_p^p\right)\Vert D^{m}u\Vert \Vert D^{m}v\Vert \\ \ge -\frac{\beta }{4{\lambda }_1{}^m}{\Vert D^{2m}v\Vert }^2-\frac{{\mu }_1{}^2}{\beta }{\Vert D^{m}u\Vert }^2,\end{array}$ (10)

$\left(\alpha {\Delta }^{2m}u,v\right)=\frac{\alpha }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m}u\Vert }^2+\alpha \epsilon {\Vert D^{2m}u\Vert }^2.$ (11)

$\left(h\left(x\right),v\right)\le \frac{1}{2\epsilon }{\Vert f\left(x\right)\Vert }^2+\frac{\epsilon }{2}{\Vert v\Vert }^2.$ (12)

$\left({\epsilon }^2-\epsilon M\left({\Vert D^{m}u\Vert }_p^p\right)\right)\left(u,v\right)\ge \frac{{\epsilon }^2-\epsilon {\sigma }_1}{2}{\Vert u\Vert }^2+\frac{{\epsilon }^2-\epsilon {\sigma }_1}{2}{\Vert v\Vert }^2.$ (13)

where ${\lambda }_1$ be the first eigenvalue of $-\Delta$ with a homogeneous Dirichlet boundary.

Substitute Formulas (7) - (13) into Equation (6) to obtain

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}\left({\Vert v\Vert }^2+\alpha {\Vert D^{2m}u\Vert }^2\right)+\left({\sigma }_0-\frac{3\epsilon }{2}+\frac{{\epsilon }^2-\epsilon {\sigma }_1}{2}\right){\Vert v\Vert }^2\\ +\left(\alpha \epsilon -\frac{\epsilon }{4}+\frac{{\epsilon }^2-\epsilon {\sigma }_1}{2{\lambda }_1^{2m}}-\frac{{\mu }_1^2}{\beta {\lambda }_1^m}\right){\Vert D^{2m}u\Vert }^2+\left(\beta -\frac{\beta }{4{\lambda }_1^m}-\epsilon {\beta }^2\right){\Vert D^{2m}v\Vert }^2\\ \le \frac{{\Vert f\left(x\right)\Vert }^2}{2{\epsilon }^2},\end{array}$ (14)

According to the hypothesis (a), (c),

${\sigma }_0+\frac{{\epsilon }^2-\epsilon {\sigma }_1}{2}-\frac{3\epsilon }{2}\ge 0,\beta -\frac{\beta }{4{\lambda }_1^m}-\epsilon {\beta }^2\ge 0,\alpha \epsilon +\frac{{\epsilon }^2-\epsilon {\sigma }_1}{2{\lambda }_1^{2m}}\ge \frac{\epsilon }{4}+\frac{{\mu }_1^2}{\beta {\lambda }_1^m}.$ (15)

${\theta }_1=2{\sigma }_0-3\epsilon +{\epsilon }^2-\epsilon {\sigma }_1,{\theta }_2=2\left(\alpha \epsilon +\frac{{\epsilon }^2-\epsilon {\sigma }_1}{2{\lambda }_1^{2m}}-\frac{\epsilon }{4}-\frac{{\mu }_1^2}{\beta {\lambda }_1^m}\right).$ (16)

Thus

$\frac{\text{d}}{\text{d}t}\left({\Vert v\Vert }^2+\alpha {\Vert {\left(-\Delta \right)}^{m}u\Vert }^2\right)+{\alpha }_1\left({\Vert v\Vert }^2+\alpha {\Vert {\left(-\Delta \right)}^{m}u\Vert }^2\right)\le C_1,$ (17)

Using Gronwall’s inequation,

${\Vert v\Vert }^2+\alpha {\Vert {\left(-\Delta \right)}^{m}u\Vert }^2\le \left({\Vert v\Vert }^2+\alpha {\Vert {\left(-\Delta \right)}^{m}u\Vert }^2\right){\text{e}}^{-{\alpha }_{1}t}+\frac{C_1}{{\alpha }_1}\left(1-{\text{e}}^{-{\alpha }_{1}t}\right),$ (18)

$\overline{\underset{t\to \infty }{\lim }}{\Vert \left(u,v\right)\Vert }_{E_0}^2\le \frac{C_1}{{\alpha }_1}.$ (19)

So there’s a positive constant $R_0$, $t_1=t_1\left(\Omega \right)>0$, such that

${\Vert \left(u,v\right)\Vert }_{E_0}^2={\Vert D^{2m}u\Vert }^2+{\Vert v\Vert }^2\le R_0^2,\left(t>t_1\right).$ (20)

Lemma 1 is proved.

Lemma 2. Assumes that (a), (b), (c) are held, $h\left(x\right)\in H_1^0$, $\left(u_0,u_1\right)\in E_k$, then the initial boundary value problem (1) - (3) has global smooth solutions $\left(u,v\right)\in E_k$,

$\begin{array}{c}{\Vert \left(u,v\right)\Vert }_{E_k}^2={\Vert D^{2m+k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\\ \le \left({\Vert D^{2m+k}{u}_0\Vert }^2+{\Vert D^k{v}_0\Vert }^2\right){\text{e}}^{-{\alpha }_{2}t}+\frac{C_2}{{\alpha }_2}\left(1-{\text{e}}^{-{\alpha }_{2}t}\right),\end{array}$ (21)

${\Vert \left(u,v\right)\Vert }_{E_k}^2={\Vert D^{2m+k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\le R_1^2,\left(t>t_2\right).$ (22)

where $v=u_t+\epsilon u$, $C_2=\frac{1}{{\epsilon }^2}{\Vert D^{k}f\left(x\right)\Vert }^2$, ${\alpha }_2=\min \left\{{\theta }_3,\frac{{\theta }_4}{\sigma }\right\}$, $t_2=t_2\left(\Omega \right)>0$.

Proof. Take the inner product of ${\left(-\Delta \right)}^{k}v={\left(-\Delta \right)}^k{u}_t+{\left(-\Delta \right)}^k\epsilon u$ and Equation (1) both sides,

$\begin{array}{l}\left(u_{tt}+\beta {\left(-\Delta \right)}^m{}^{2m}{u}_t+M\left({\Vert D^{m}u\Vert }_p^p\right)u_t+\alpha {\Delta }^{2m}u+N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}u,{\left(-\Delta \right)}^m{}^{k}v\right)\\ =\left(f\left(x\right),{\left(-\Delta \right)}^m{}^{k}v\right),\end{array}$ (23)

By using Holder’s Inequality, Young’s Inequality, and Poincare’s Inequality, The items in (23) can be obtained by successive processing:

$\begin{array}{c}\left(u_{tt},{\left(-\Delta \right)}^{k}v\right)=\left(v_t-\epsilon u_t,{\left(-\Delta \right)}^{k}v\right)=\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{k}v\Vert }^2-\epsilon {\Vert D^{k}v\Vert }^2+{\epsilon }^2\left(u,{\left(-\Delta \right)}^{k}v\right)\\ \ge \frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{k}v\Vert }^2-\left(\epsilon +\frac{{\epsilon }^2}{2}\right){\Vert D^{k}v\Vert }^2-\frac{{\epsilon }^2}{2{\lambda }_1^m}{\Vert D^{m+k}u\Vert }^2,\end{array}$ (24)

$\begin{array}{l}\left(M\left({\Vert D^{m}u\Vert }_p^p\right)u_t,{\left(-\Delta \right)}^{k}v\right)\\ =M\left({\Vert D^{m}u\Vert }_p^p\right){\Vert D^{k}v\Vert }^2-\epsilon M\left({\Vert D^{m}u\Vert }_p^p\right)\Vert D^{k}u\Vert \Vert D^{k}v\Vert \\ \ge M\left({\Vert D^{m}u\Vert }_p^p\right){\Vert D^{k}v\Vert }^2-\frac{\epsilon M\left({\Vert D^{m}u\Vert }_p^p\right)}{2}{\Vert D^{k}v\Vert }^2-\frac{\epsilon M\left({\Vert D^{m}u\Vert }_p^p\right)}{2{\lambda }_1^{2m}}{\Vert D^{2m+k}u\Vert }^2\\ \ge M\left({\Vert D^{m}u\Vert }_p^p\right){\Vert D^{k}v\Vert }^2-\frac{\epsilon M\left({\Vert D^{m}u\Vert }_p^p\right)}{2}{\Vert D^{k}v\Vert }^2-\frac{\alpha \epsilon }{2}{\Vert D^{2m+k}u\Vert }^2\\ \ge {\sigma }_0{\Vert D^{k}v\Vert }^2-\frac{\epsilon {\sigma }_1}{2}{\Vert D^{k}v\Vert }^2-\frac{\alpha \epsilon }{2}{\Vert D^{2m+k}u\Vert }^2,\end{array}$ (25)

with $\alpha \ge \frac{{\sigma }_1}{{\lambda }_1^{2m}}$.

$\begin{array}{c}\left(\beta {\left(-\Delta \right)}^{2m}{u}_t,{\left(-\Delta \right)}^{k}v\right)\ge \beta {\lambda }_1^{2m}{\Vert D^{k}v\Vert }^2-\frac{\beta \epsilon }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m+k}u\Vert }^2-\beta {\epsilon }^2{\Vert D^{2m+k}u\Vert }^2\\ \ge \beta {\lambda }_1^{2m}{\Vert D^{k}v\Vert }^2-\frac{\beta \epsilon }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m+k}u\Vert }^2-\frac{\alpha \epsilon }{3}{\Vert D^{2m+k}u\Vert }^2,\end{array}$ (26)

$\begin{array}{l}\left(N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}u,{\left(-\Delta \right)}^{k}v\right)\\ =\frac{N\left({\Vert D^{m}u\Vert }_p^p\right)}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{m+k}u\Vert }^2+\epsilon N\left({\Vert D^{m}u\Vert }_p^p\right){\Vert D^{m+k}u\Vert }^2.\end{array}$ (27)

The following two cases are used for prior estimation of Equation (25):

1) As $\frac{\text{d}}{\text{d}t}{\Vert D^{m+k}u\Vert }^2\ge 0$, By assuming (b),

$\begin{array}{l}\frac{N\left({\Vert D^{m+k}u\Vert }_p^p\right)}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{m}u\Vert }^2+\epsilon N\left({\Vert D^{m+k}u\Vert }_p^p\right){\Vert D^{m}u\Vert }^2\\ \ge \frac{{\mu }_0}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{m}u\Vert }^2+\epsilon {\mu }_0{\Vert D^{m}u\Vert }^2,\end{array}$ (28)

let $\mu ={\mu }_0$,

$\left(N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}u,v\right)\ge \frac{\mu }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{m+k}u\Vert }^2+\epsilon {\mu }_0{\Vert D^{m+k}u\Vert }^2.$ (29)

2) As $\frac{\text{d}}{\text{d}t}{\Vert D^{m+k}u\Vert }^2<0$, By assuming (b),

$\begin{array}{l}\frac{N\left({\Vert D^{m+k}u\Vert }_p^p\right)}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{m}u\Vert }^2+\epsilon N\left({\Vert D^{m+k}u\Vert }_p^p\right){\Vert D^{m}u\Vert }^2\\ \ge \frac{{\mu }_1}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{m}u\Vert }^2+\epsilon {\mu }_0{\Vert D^{m}u\Vert }^2,\end{array}$ (30)

let $\mu ={\mu }_1$,

$\left(N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}u,v\right)\ge \frac{\mu }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{m+k}u\Vert }^2+\epsilon {\mu }_0{\Vert D^{m+k}u\Vert }^2.$ (31)

Similarly discussion (27) can be obtained

$\left(N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}u,{\left(-\Delta \right)}^{k}v\right)\ge \frac{\mu }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{m+k}u\Vert }^2+\epsilon {\mu }_0{\Vert D^{m+k}u\Vert }^2.$ (32)

$\begin{array}{c}\left(\alpha {\Delta }^{2m}u,{\left(-\Delta \right)}^{k}v\right)=\left(\alpha {\Delta }^{2m}u,{\left(-\Delta \right)}^k{u}_t+{\left(-\Delta \right)}^k\epsilon u\right)\\ =\frac{\alpha }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m+k}u\Vert }^2+\alpha \epsilon {\Vert D^{2m+k}u\Vert }^2,\end{array}$ (33)

$\left(f\left(x\right),{\left(-\Delta \right)}^{k}v\right)\le \frac{{\epsilon }^2}{2}{\Vert D^{k}v\Vert }^2+\frac{1}{2{\epsilon }^2}{\Vert D^{k}f\left(x\right)\Vert }^2.$ (34)

The comprehensive Equations (24) - (34) and (23) can be written as

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}\left({\Vert D^{k}v\Vert }^2+\left(\alpha -\epsilon \beta \right){\Vert D^{2m+k}u\Vert }^2+\mu {\Vert D^{m+k}u\Vert }^2\right)\\ +\left(\beta {\lambda }_1{}^{2m}+{\sigma }_0-\frac{\epsilon {\sigma }_1}{2}-\epsilon -{\epsilon }^2\right){\Vert D^{k}v\Vert }^2+\left(\epsilon {\mu }_0-\frac{{\epsilon }^2}{2{\lambda }_1^m}\right){\Vert D^{m+k}u\Vert }^2\\ +\frac{\alpha \epsilon }{6}{\Vert D^{2m+k}u\Vert }^2\le \frac{{\Vert D^{k}f\left(x\right)\Vert }^2}{2{\epsilon }^2}.\end{array}$ (35)

According to the hypothesis (c),

$\alpha -\beta \epsilon >0,\epsilon {\mu }_0-\frac{{\epsilon }^2}{2{\lambda }_1^m}\ge 0,\beta {\lambda }_1^{2m}+{\sigma }_0-\frac{\epsilon {\sigma }_1}{2}-\epsilon -{\epsilon }^2\ge 0.$ (36)

${\theta }_3=2\left(\beta {\lambda }_1^{2m}+{\sigma }_0-\frac{\epsilon {\sigma }_1}{2}-\epsilon -{\epsilon }^2\right),{\theta }_4=2\epsilon {\mu }_0-\frac{{\epsilon }^2}{{\lambda }_1^m},{\theta }_5=\frac{1}{3}\alpha \epsilon >0.$

Let ${\alpha }_2=\min \left\{{\theta }_3,\frac{{\theta }_4}{\mu },\frac{{\theta }_5}{\alpha -\beta \epsilon }\right\}$,

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left({\Vert D^{k}v\Vert }^2+\left(\alpha -\beta \epsilon \right){\Vert D^{2m+k}u\Vert }^2+\mu {\Vert D^{m+k}u\Vert }^2\right)\\ +{\alpha }_2\left({\Vert D^{k}v\Vert }^2+\left(\alpha -\beta \epsilon \right){\Vert D^{2m+k}u\Vert }^2+\mu {\Vert D^{m+k}u\Vert }^2\right)\le C_2.\end{array}$ (37)

By Gronwall’s inequality, we can get

$Y_1\le Y_1{\text{e}}^{-{\alpha }_{2}t}+\frac{C_2}{{\alpha }_2}\left(1-{\text{e}}^{-{\alpha }_{2}t}\right).$ (38)

where $Y_1={\Vert D^{k}v\Vert }^2+\left(\alpha -\beta \epsilon \right){\Vert D^{2m+k}u\Vert }^2+\mu {\Vert D^{m+k}u\Vert }^2$.

$\begin{array}{c}{\Vert \left(u,v\right)\Vert }_{E_k}^2={\Vert D^{2m+k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\\ \le \left({\Vert D^{2m+k}{u}_0\Vert }^2+{\Vert D^k{v}_0\Vert }^2\right){\text{e}}^{-{\alpha }_{2}t}+\frac{C_2}{{\alpha }_2}\left(1-{\text{e}}^{-{\alpha }_{2}t}\right),\end{array}$ (39)

$\underset{t\to \infty }{\bar{\lim }}{\Vert \left(u,v\right)\Vert }_{E_k}^2\le \frac{C_2}{{\alpha }_2}.$ (40)

So there’s a positive constant $R_1$, $t_2=t_2\left(\Omega \right)>0$, such that

${\Vert \left(u,v\right)\Vert }_{E_k}^2={\Vert D^{2m+k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\le R_1^2,\left(t>t_2\right).$ (41)

Lemma 2 is proved.

Theorem 1. (existence and uniqueness of solutions) Under the hypothesis of Lemma 1 and Lemma 2, $f\in H$, $\left(u_0,u_1\right)\in E_k$. Then there exists a unique global solution to the initial boundary value problem (1) - (3), $\left(u,v\right)\in L^{\infty }\left(\left[0,+\infty \right);E_k\right)$.

Proof. Existence: The existence of global solution is proved by Galerkin’s finite element method.

The first step: To construct approximate solutions

Let ${\left(-\Delta \right)}^{2m+k}{w}_j={\lambda }_j^{2m+k}{w}_j,k=1,2,\cdots ,2m$, where ${\lambda }_j$ is the eigenvalues of $-\Delta$ with homogeneous Dirichlet boundary on $\Omega$, $w_j$ is the eigenfunctions determined by the corresponding eigenvalues and $w_1,w_2,\cdots ,w_l$ are the orthonormal basis formed by the eigenvalue theory.

Let the approximate solution of problems (1) - (3) be $u_l\left(t\right)={\displaystyle \underset{j=1}{\overset{l}{\sum }}{g}_{jl}\left(t\right)w_j}$, and $g_{jl}\left(t\right)$ is determined by the following nonlinear ordinary differential equations

$\begin{array}{l}\left(u_{tt}+\beta {\left(-\Delta \right)}^{2m}{u}_t+M\left({\Vert D^{m}u\Vert }_p^p\right)u_t+\alpha {\Delta }^{2m}u+N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}u,w_j\right)\\ =\left(h\left(x\right),w_j\right),j=1,2,\cdots ,l.\end{array}$ (42)

It satisfies the initial conditions $u_{l0}\left(0\right)=u_{l0},u_{lt}\left(0\right)=u_{l1}$. As $l\to +\infty$, In $E_k$, $\left(u_{l0},u_{l1}\right)\to \left(u_0,u_1\right)$ in $E_k$. It is known from the basic theory of ordinary differential that approximate solutions exist on $\left(0,t_l\right)$.

The second step: A prior estimation

Because we want to prove the existence of weak solutions in space $E_k\left(k=1,2,\cdots ,2m-1\right)$.

So we multiply both sides of Equation (42) with ${\lambda }_j^k\left({g^{\prime }}_{jl}\left(t\right)+\epsilon g_{jl}\left(t\right)\right)$, and the sum of j, let $v_l\left(t\right)=u_{lt}\left(t\right)+\epsilon u_l\left(t\right)$.

As $k=0$, Get a priori estimate of the solution in space $E_0$ :

${\Vert \left(u_l,v_l\right)\Vert }_{E_0}^2={\Vert D^{2m}{u}_l\Vert }^2+{\Vert v_l\Vert }^2\le R_0^2,$ (43)

As $k=1,2,\cdots ,2m-1$, Get a priori estimate of the solution in space $E_k$ :

${\Vert \left(u_l,v_l\right)\Vert }_{E_k}^2={\Vert D^{2m+k}{u}_l\Vert }^2+{\Vert v_l\Vert }^2\le R_1^2.$ (44)

It can be known that the prior estimates of lemma 1 and lemma 2 of Equations (43) and (44) are valid respectively. According to Equations (43) and (44), $\left(u_l,v_l\right)$ be bounded in $L^{\infty }\left(\left[0,+\infty \right];E_0\right)$, $\left(u_l,v_l\right)$ be bounded in $L^{\infty }\left(\left[0,+\infty \right];E_k\right)$.

The third step: Limit process

In the space $E_k\left(k=1,2,\cdots ,2m-1\right)$, Selecting subcolumns $u_{\mu }$ from the sequence $u_l$,

$\left(u_{\mu },v_{\mu }\right)\to \left(u,v\right)$ Weak*-convergence, in $L^{\infty }\left(\left[0,+\infty \right];E_k\right)$ (45)

By Rellich-Kohdrachov Compact embedding theorem, $E_k$ compact embedded $E_0$, $\left(u_{\mu },v_{\mu }\right)\to \left(u,v\right)$

Strong convergence almost everywhere in $E_0$. (46)

Let $l=\mu \to +\infty$, From (45),

$\left(u_{\mu t},{\left(-\Delta \right)}^k{w}_j\right)\to \left(v,{\lambda }_j^k{w}_j\right)-\left(\epsilon u,{\lambda }_j^k{w}_j\right)$

Weak*-convergence in $L^{\infty }\left[0,+\infty \right)$.

$\left(u_{\mu tt},{\left(-\Delta \right)}^k{w}_j\right)=\frac{\text{d}}{\text{d}t}\left(u_{\mu t},{\left(-\Delta \right)}^k{w}_j\right)$

Thus, $\left(u_{\mu tt},{\left(-\Delta \right)}^k{w}_j\right)\to \left(u_t,{\left(-\Delta \right)}^k{w}_j\right)$ convergence in $D^{\prime }\left[0,+\infty \right)$, here $D^{\prime }\left[0,+\infty \right)$ is $D\left[0,+\infty \right)$ Conjugate space of infinitely differentiable space $D\left[0,+\infty \right)$.

$\left(\beta {\left(-\Delta \right)}^{2m}{u}_{\mu t},{\left(-\Delta \right)}^k{w}_j\right)\to \beta \left({\left(-\Delta \right)}^{\frac{2m+k}{2}}v-\epsilon {\left(-\Delta \right)}^{\frac{2m+k}{2}}u,{\lambda }_j^{\frac{2m+k}{2}}{w}_j\right)$

Weak*-convergence in $L^{\infty }\left[0,+\infty \right)$.

$\left(M\left({\Vert D^{m}u\Vert }_p^p\right)u_{\mu t},{\left(-\Delta \right)}^k{w}_j\right)\to M\left({\Vert D^{m}u\Vert }_p^p\right)\left({\left(-\Delta \right)}^{\frac{k}{2}}v-\epsilon {\left(-\Delta \right)}^{\frac{k}{2}}{u}_t,{\lambda }_j^{\frac{k}{2}}{w}_j\right)$

Weak*-convergence in $L^{\infty }\left[0,+\infty \right)$.

$\left(N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^m{u}_{\mu },{\left(-\Delta \right)}^k{w}_j\right)\to \left(N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{\frac{m+k}{2}}u,{\lambda }_j^{\frac{m+k}{2}}{w}_j\right)$

Weak*-convergence in $L^{\infty }\left[0,+\infty \right)$.

$\left(\alpha {\Delta }^{2m}{u}_{\mu },{\left(-\Delta \right)}^k{w}_j\right)\to \left(\alpha {\Delta }^{\frac{2m+k}{2}}u,{\lambda }_j^{\frac{2m+k}{2}}{w}_j\right)$.

Weak*-convergence in $L^{\infty }\left[0,+\infty \right)$.

In particular $u_{\mu 0}\to u_0$ weak convergence in $E_k$, $u_{\mu t}\to u_t=u_1$ weak convergence in $E_k$. For all j and $\mu \to +\infty$, It follows that

$\begin{array}{l}\left(u_{tt}+\beta {\left(-\Delta \right)}^{2m}{u}_t+M\left({\Vert D^{m}u\Vert }_p^p\right)u_t+\alpha {\Delta }^{2m}u+N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}u,w_j\right)\\ =\left(f\left(x\right),w_j\right),j=1,2,\cdots ,l.\end{array}$ (47)

Therefore, the existence of the weak solution of the problem (1) - (3) is obtained, and the existence is proved.

The uniqueness of the solution of the following proof.

Let $u,v$ be two solutions of Equation (1), set $w=u-v$ and take the inner product of $w_t+\epsilon w$ in H,

$\begin{array}{l}(w_{tt}+\beta {\left(-\Delta \right)}^{2m}{w}_t+M\left({\Vert D^{m}u\Vert }_p^p\right)u_t-M\left({\Vert D^{m}v\Vert }_p^p\right)v_t+\alpha {\Delta }^{2m}w\\ +N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}u-N\left({\Vert D^{m}v\Vert }_p^p\right){\left(-\Delta \right)}^{m}v,w_t+\epsilon w)=0\end{array}$ (48)

Similar to Lemma 1, can be obtained

$\left(w_{tt},w_t+\epsilon w\right)=\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert w_t\Vert }^2+\epsilon \frac{\text{d}}{\text{d}t}\left(w_t,w\right)-\epsilon {\Vert w_t\Vert }^2,$ (49)

$\left(\beta {\left(-\Delta \right)}^{2m}{w}_t,w_t+\epsilon w\right)=\beta {\Vert D^{2m}{w}_t\Vert }^2+\frac{\beta \epsilon }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m}w\Vert }^2,$ (50)

$\begin{array}{l}\left(M\left({\Vert D^{m}u\Vert }_p^p\right)u_t-M\left({\Vert D^{m}v\Vert }_p^p\right)v_t,w_t+\epsilon w\right)\\ =M\left({\Vert D^{m}u\Vert }_p^p\right)\left(w_t,w_t\right)+\epsilon M\left({\Vert D^{m}u\Vert }_p^p\right)\left(w_t,w\right)\\ +M^{\prime }\left({\Vert D^m\xi \Vert }_p^p\right){\left({\Vert D^m\xi \Vert }_p^p\right)}^{\prime }\left(D^{m}w{v}_t,w_t+\epsilon w\right)\\ \ge M\left({\Vert D^{m}u\Vert }_p^p\right){\Vert w_t\Vert }^2-\epsilon M\left({\Vert D^{m}u\Vert }_p^p\right)\left(w_t,w\right)-C_3{v}_t\left(D^{m}w,\epsilon w+w_t\right)\end{array}$

$\begin{array}{l}\ge M\left({\Vert D^{m}u\Vert }_p^p\right){\Vert w_t\Vert }^2-\frac{\epsilon M\left({\Vert D^{m}u\Vert }_p^p\right)}{2{\lambda }_1^m}{\Vert D^m{w}_t\Vert }^2-\frac{\epsilon M\left({\Vert D^{m}u\Vert }_p^p\right)}{2}{\Vert w\Vert }^2\\ -C_4\Vert D^{m}w\Vert \cdot \Vert w_t\Vert -\epsilon C_4\Vert D^{m}w\Vert \cdot \Vert w\Vert \\ \ge M\left({\Vert D^{m}u\Vert }_p^p\right){\Vert w_t\Vert }^2-\frac{\epsilon M\left({\Vert D^{m}u\Vert }_p^p\right)}{2{\lambda }_1^m}{\Vert D^m{w}_t\Vert }^2-\frac{\epsilon M\left({\Vert D^{m}u\Vert }_p^p\right)}{2}{\Vert w\Vert }^2\\ -\frac{C_4}{2}{\Vert D^{m}w\Vert }^2-\frac{C_4}{2{\lambda }_1^m}{\Vert D^m{w}_t\Vert }^2-\frac{\epsilon C_4}{2}{\Vert D^{m}w\Vert }^2-\frac{\epsilon C_4}{2}{\Vert w\Vert }^2\\ \ge {\sigma }_0{\Vert w_t\Vert }^2-\frac{\beta }{2}{\Vert D^m{w}_t\Vert }^2-\left(\frac{\epsilon {\sigma }_1}{2}+\frac{\epsilon C_4}{2}\right){\Vert w\Vert }^2-\frac{C_4\left(1+\epsilon \right)}{2{\lambda }_1^m}{\Vert D^{2m}w\Vert }^2,\end{array}$ (51)

$C_3={\Vert M^{\prime }\left({\Vert D^m\xi \Vert }_p^p\right){\left({\Vert D^m\xi \Vert }_p^p\right)}^{\prime }\Vert }_{\infty },C_4={\Vert M^{\prime }\left({\Vert D^m\xi \Vert }_p^p\right){\left({\Vert D^m\xi \Vert }_p^p\right)}^{\prime }\Vert }_{\infty }\Vert v_t\Vert .$

$\xi =s{D}^{m}u+\left(1-s\right)D^{m}v,s\in \left(0,1\right).$

$\left(\alpha {\Delta }^{2m}w,w_t+\epsilon w\right)\ge \frac{\alpha }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m}w\Vert }^2+\frac{\alpha \epsilon }{2}{\Vert D^{2m}w\Vert }^2+\frac{\alpha \epsilon {\lambda }_1^{2m}}{2}{\Vert w\Vert }^2,$ (52)

$\begin{array}{l}\left(N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}u-N\left({\Vert D^{m}v\Vert }_p^p\right){\left(-\Delta \right)}^{m}v,w_t+\epsilon w\right)\\ =\left(N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}w,w_t\right)+\left(N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}v-N\left({\Vert D^{m}v\Vert }_p^p\right){\left(-\Delta \right)}^{m}v,w_t\right)\\ +\epsilon \left(N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}w,w\right)\\ +\epsilon \left(N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}v-N\left({\Vert D^{m}v\Vert }_p^p\right){\left(-\Delta \right)}^{m}v,w\right)\\ \ge \frac{N\left({\Vert D^{m}u\Vert }_p^p\right)}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{m}w\Vert }^2-C_5\Vert D^{m}w\Vert \Vert D^{m}v\Vert \Vert D^m{w}_t\Vert \end{array}$

$\begin{array}{l}+\epsilon N\left({\Vert D^{m}u\Vert }_p^p\right){\Vert D^{m}w\Vert }^2-\epsilon C_5\Vert D^{m}w\Vert \Vert D^{m}v\Vert \Vert D^{m}w\Vert \\ \ge \frac{N\left({\Vert D^{m}u\Vert }_p^p\right)}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{m}w\Vert }^2-\frac{C_6^2}{2}{\Vert D^{m}w\Vert }^2-\frac{1}{2}{\Vert D^m{w}_t\Vert }^2\\ +\epsilon N\left({\Vert D^{m}u\Vert }_p^p\right){\Vert D^{m}w\Vert }^2-\epsilon C_6{\Vert D^{m}w\Vert }^2\\ \ge \frac{N\left({\Vert D^{m}u\Vert }_p^p\right)}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{m}w\Vert }^2+\left(\epsilon N\left({\Vert D^{m}u\Vert }_p^p\right)-\frac{C_6^2}{2}-\epsilon C_6\right){\Vert D^{m}w\Vert }^2-\frac{1}{2}{\Vert D^m{w}_t\Vert }^2\\ \ge \frac{\mu }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{m}w\Vert }^2+\left(\epsilon {\mu }_0-\frac{C_6^2}{2}-\epsilon C_6\right){\Vert D^{m}w\Vert }^2-\frac{1}{2}{\Vert D^m{w}_t\Vert }^2,\end{array}$ (53)

where $C_5={\Vert N^{\prime }\left({\Vert D^m\xi \Vert }_p^p\right){\left({\Vert D^m\xi \Vert }_p^p\right)}^{\prime }\Vert }_{\infty }$.

Substitute (49) - (53) into Equation (48)

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left[{\Vert w_t\Vert }^2+2\epsilon \left(w_t,w\right)+\mu {\Vert D^{m}w\Vert }^2+\left(\epsilon \beta +\alpha \right){\Vert {\left(-\Delta \right)}^{m}w\Vert }^2\right]\\ +2\left({\sigma }_0-\epsilon \right){\Vert w_t\Vert }^2+\left(\alpha \epsilon -\frac{C_4+\epsilon C_4}{{\lambda }_1^m}\right){\Vert {\left(-\Delta \right)}^{m}w\Vert }^2+\left(2\beta {\lambda }_1^m-\beta -1\right){\Vert D^m{w}_t\Vert }^2\\ +2\left(\epsilon {\mu }_0-\epsilon C_6-\frac{C_6^2}{2}\right){\Vert D^{m}w\Vert }^2+\left(\alpha \epsilon {\lambda }_1^{2m}-\epsilon {\sigma }_1-\epsilon C_4\right){\Vert w\Vert }^2\le 0,\end{array}$ (54)

$\epsilon \left(w_t,w\right)\ge -\frac{\epsilon }{2}{\Vert w_t\Vert }^2-\frac{\epsilon }{2}{\Vert w\Vert }^2\ge -\frac{\epsilon }{2}{\Vert w_t\Vert }^2-\frac{\epsilon }{2{\lambda }_1^m}{\Vert D^{m}w\Vert }^2,$

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left[\left(1-\epsilon \right){\Vert w_t\Vert }^2+\left(\mu -\frac{\epsilon }{{\lambda }_1^m}\right){\Vert D^{m}w\Vert }^2+\left(\beta \epsilon +\alpha \right){\Vert {\left(-\Delta \right)}^{m}w\Vert }^2\right]\\ +2\left({\sigma }_0-\epsilon \right){\Vert w_t\Vert }^2+\left(\alpha \epsilon -\frac{C_4+\epsilon C_4}{{\lambda }_1^m}\right){\Vert {\left(-\Delta \right)}^{m}w\Vert }^2\\ +2\left(\epsilon {\mu }_0-\epsilon C_3-\frac{C_3{}^2}{2}\right){\Vert D^{m}w\Vert }^2\le 0,\end{array}$ (55)

$\alpha \ge \max \{\frac{{\sigma }_1+C_4}{{\lambda }_1^{2m}},\frac{C_4+\epsilon C_4}{{\lambda }_1^m}\}$, ${\mu }_0\ge \max \{\frac{\epsilon }{{\lambda }_1^m},\frac{C_6^2}{2\epsilon }+C_6\}$

and (c),

$\begin{array}{l}\mu -\frac{\epsilon }{{\lambda }_1^m}\ge 0,2\beta {\lambda }_1{}^m-1-\beta \ge 0,\alpha {\lambda }_1^{2m}-{\sigma }_1-\epsilon C_4\ge 0,{\theta }_5={\sigma }_0-\epsilon ,\\ {\theta }_6=\alpha \epsilon -\frac{C_4+\epsilon C_4}{{\lambda }_1^m},{\theta }_7=\epsilon {\mu }_0-\epsilon C_6-\frac{C_6^2}{2}\ge 0.\end{array}$

Take ${\alpha }_3=\max \left\{\frac{2{\theta }_5}{1-\epsilon },\frac{{\theta }_6}{\alpha +\beta \epsilon },\frac{2{\theta }_7}{\mu -\frac{\epsilon }{{\lambda }_1^m}}\right\}$

Then, according to Gronwall’s inequality, get

$\begin{array}{l}\left(1-\epsilon \right){\Vert w_t\Vert }^2+\left(\beta \epsilon +\alpha \right){\Vert D^{2m}w\Vert }^2\\ \le {\alpha }_3\left({\Vert w_t\left(0\right)\Vert }^2+\left(\beta \epsilon +\alpha \right){\Vert D^{2m}w\left(0\right)\Vert }^2\right){\text{e}}^{-{\alpha }_{3}t}=0,\end{array}$ (56)

Then ${\Vert w_t\Vert }^2={\Vert D^{2m}w\Vert }^2=0$, that is $w\left(t\right)=0,u=v$, So the uniqueness is proved. Theorem 1 is proved.

Theorem 2. According to lemma 1 and theorem 1, then the initial boundary value problem (1) - (3) has a family of global attractors

$A_k=\omega \left(B_{0k}\right)={\displaystyle \underset{\tau \ge 0}{\cap }\overline{{\displaystyle \underset{t\ge \tau }{\cup }S\left(t_k\right)B_{0k}}}},\left(k=1,2,3,\cdots ,2m\right)$,

where $B_{0k}=\left\{\left(u,v\right)\in E_k:{\Vert \left(u,v\right)\Vert }_{E_k}^2={\Vert D^{2m+k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\le R_k^2+R_0^2\right\}$ is a bounded absorbing set in $E_k$ and satisfies the following conditions:

1) $S\left(t\right)A_k=A_k,t>0$ ;

2) $\underset{t\to \infty }{\lim }dist\left(S\left(t\right)B_k,A_k\right)=0\left(\forall B_k\subset E_k\right)$ $B_k$ is a bounded set;

Where $dist\left(S\left(t\right)B_k,A_k\right)=\underset{x\in B_k}{\sup }\underset{y\in A_k}{\inf }{\Vert S\left(t\right)x-y\Vert }_{E_k}$, $S\left(t\right)$ is the solution semigroup generated by the initial boundary value problem (1) - (3).

Proof. It is necessary to verify the conditions (I), (II) and (III) for the existence of attractors in reference [8]. Under the condition of Theorem 1, there exists a solution semigroup $S\left(t\right):E_k\to E_k$ of the initial boundary value problem (1) - (3).

From lemma 1, we can obtain that $\forall B_k\subset E_k$ is a bounded set that includes in the ball $\left\{{\Vert \left(u,v\right)\Vert }_{E_k}\le R_k\right\}$.

$\begin{array}{c}{\Vert S\left(t\right)\left(u_0,v_0\right)\Vert }_{E_k}^2={\Vert u\Vert }_{H_0^{2m+k}\left(\Omega \right)}^2+{\Vert v\Vert }_{H_0^k\left(\Omega \right)}^2\\ \le {\Vert u_0\Vert }_{H_0^{2m+k}\left(\Omega \right)}^2+{\Vert v_0\Vert }_{H_0^k\left(\Omega \right)}^2+C\\ \le R_k^2+C,\end{array}$ (57)

where $t\ge 0$, $\left(u_0,v_0\right)\in B_k$, this shows that $\left\{S\left(t\right)\right\}\left(t\ge 0\right)$ is uniformly bounded in $E_k$.

Furthermore, for any $\left(u_0,v_0\right)\in E_k$, when $t\ge \max \left\{t_1,t_k\right\}$, we have

${\Vert S\left(t\right)\left(u_0,v_0\right)\Vert }_{E_k}^2={\Vert u\Vert }_{H_0^{2m+k}\left(\Omega \right)}^2+{\Vert v\Vert }_{H_0^k\left(\Omega \right)}^2\le R_k^2+R_0^2$, (58)

Therefore, $B_{0k}=\left\{\left(u,v\right)\in E_k:{\Vert \left(u,v\right)\Vert }_{E_k}^2={\Vert D^{2m+k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\le R_k^2+R_0^2\right\}$ is a bounded absorbing set in semigroup $S\left(t\right)$.

According to the rellich kondrachov’s compact embedding theorem, if $E_k$ is compactly embedded in $E_0$, then the bounded set in $E_k$ is the compact set in $E_0$. Therefore, the solution semigroup $S\left(t\right)$ is a completely continuous operator, thus the family of global attractors $A_k$ of solution semigroup $S\left(t\right)$ is

obtained. Where $A_k=\omega \left(B_{0k}\right)={\displaystyle \underset{\tau \ge 0}{\cap }\overline{{\displaystyle \underset{t\ge \tau }{\cup }S\left(t\right)B_{0k}}}}$.

The proof is completed.

3. The Dimension Estimation for the Family of Global Attractors

In this part, we first linearize the equation into a first-order variational equation and prove that the solution semigroup $s\left(t\right)$ is Fréchet differentiable on $E_k$. furthermore, we prove the decay of the volume element of the linearization problem. Finally, we estimate the upper bound of the Hausdorff dimension and fractal dimension of $A_k$

Linearize problems (1) - (3),

$\begin{array}{l}{U}_{tt}+M\left({\Vert D^{m}u\Vert }_p^p\right)U_t+M^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{u}_t+\beta {\left(-\Delta \right)}^{2m}{U}_t+\alpha {\Delta }^{2m}U\\ +N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}U+N^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{\left(-\Delta \right)}^{m}u=0,\end{array}$ (59)

$U\left(x,t\right)=0,x\in \partial \Omega ,t>0,$ (60)

$U\left(x,0\right)=\xi ,U_t\left(x,0\right)=\eta .$ (61)

where $\left(\xi ,\eta \right)\in E_k$, $\left(u,u_t\right)=S\left(t\right)\left(u_0,u_1\right)$ is the solution of problems (1) - (3) with $\left(u_0,u_1\right)\in A_k$.

Given $\left(u_0,u_1\right)\in A_k$, $S\left(t\right):E_k\to E_k$, Verifiable to any $\left(\xi ,\eta \right)\in E_k$, Linearization of initial boundary value problems (59) - (61) has a unique solution:

$\left(U\left(t\right),U_t\left(t\right)\right)\in L^{\infty }\left(\left[0,+\infty \right);E_k\right).$ (62)

Lemma 3. If $S\left(t\right):E_k\to E_k$, the Frechet differential on ${\varphi }_0={\left(u_0,u_1\right)}^{\text{T}}$ is a linear operator $F:{\left(\xi ,\eta \right)}^{\text{T}}\to {\left(U\left(t\right),U_t\left(t\right)\right)}^{\text{T}}$, then $\forall t>0$, $R>0$, the mapping $S\left(t\right):E_k\to E_k$ is Fréchet differentiable on $E_k$, where $U\left(t\right),U_t\left(t\right)$ is the solution of the linearized initial boundary value problem (59) - (61).

Proof. Set ${\varphi }_0={\left(u_0,u_1\right)}^{\text{T}}\in E_k,{\bar{\varphi }}_0={\left(u_0+\xi ,u_1+\eta \right)}^{\text{T}}\in E_k$, ${\varphi }_0={\left(u_0,u_1\right)}^{\text{T}}\in E_k,{\bar{\varphi }}_0={\left(u_0+\xi ,u_1+\eta \right)}^{\text{T}}\in E_k$ and ${\Vert {\varphi }_0\Vert }_{E_k}\le R,{\Vert {\bar{\varphi }}_0\Vert }_{E_k}\le R$,

the semigroup $S\left(t\right)$ is Lipschitz continuous on the bounded set of $E_k$, i.e.

${\Vert S\left(t\right){\varphi }_0-S\left(t\right){\bar{\varphi }}_0\Vert }_{E_k}^2\le {\text{e}}^{ct}{\Vert {\left(\xi ,\eta \right)}^{\text{T}}\Vert }_{E_k}^2.$ (63)

Let $\theta =U_t+\epsilon U$, then

$\begin{array}{l}{\theta }_t-\epsilon \theta +{\epsilon }^{2}u+M\left({\Vert D^{m}u\Vert }_p^p\right)\theta -\epsilon M\left({\Vert D^{m}u\Vert }_p^p\right)U\\ +M^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{u}_t+\beta {\left(-\Delta \right)}^{2m}\left(\theta -\epsilon u\right)\\ +N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}U+N^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{\left(-\Delta \right)}^{m}u+\alpha {\Delta }^{2m}U=0,\end{array}$ (64)

$\theta \left(0\right)=0,{\theta }_t\left(0\right)=0.$ (65)

Set $\left(\psi ,\varphi \right)=\bar{\eta }-\eta -\omega =\left(\bar{u}-u-U,\bar{v}-v-\theta \right)$,

$\{\begin{array}{l}{\bar{u}}_{tt}+\beta {\left(-\Delta \right)}^{2m}{\bar{u}}_t+M\left({\Vert D^m\bar{u}\Vert }_p^p\right){\bar{u}}_t+\alpha {\Delta }^{2m}\bar{u}+N\left({\Vert D^m\bar{u}\Vert }_p^p\right){\left(-\Delta \right)}^m\bar{u}=f\left(x\right),\\ u_{tt}+\beta {\left(-\Delta \right)}^{2m}{u}_t+M\left({\Vert D^{m}u\Vert }_p^p\right)u_t+\alpha {\Delta }^{2m}u+N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}u=f\left(x\right),\\ U_{tt}+\beta {\left(-\Delta \right)}^{2m}{\bar{U}}_t+M\left({\Vert D^{m}u\Vert }_p^p\right)U_t+M^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^{m}U\cdot u_t\\ +\alpha {\Delta }^{2m}U+N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}U+N^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^{m}U\cdot {\left(-\Delta \right)}^{m}u=0.\end{array}$

Then the three equations can be subtracted

$\begin{array}{l}{\psi }_{tt}+\beta {\left(-\Delta \right)}^{2m}{\psi }_t+\alpha {\Delta }^{2m}\psi +M\left({\Vert D^m\bar{u}\Vert }_p^p\right){\bar{u}}_t-M\left({\Vert D^{m}u\Vert }_p^p\right)u_t\\ -M\left({\Vert D^{m}u\Vert }_p^p\right)U_t-M^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^{m}U\cdot u_t\\ +N\left({\Vert D^m\bar{u}\Vert }_p^p\right){\left(-\Delta \right)}^m\bar{u}-N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}u-N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}U\\ -N^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^{m}U\cdot {\left(-\Delta \right)}^{m}u=0,\end{array}$ (66)

$\begin{array}{c}H=M\left({\Vert D^m\bar{u}\Vert }_p^p\right){\bar{u}}_t-M\left({\Vert D^{m}u\Vert }_p^p\right)u_t-M\left({\Vert D^{m}u\Vert }_p^p\right)U_t\\ -M^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^{m}U{u}_t+N\left({\Vert D^m\bar{u}\Vert }_p^p\right){\left(-\Delta \right)}^m\bar{u}\\ -N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}u-N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}U\\ -N^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^{m}U{\left(-\Delta \right)}^{m}u.\end{array}$

then $H=h_1+h_2$,

$\begin{array}{c}{h}_1=M\left({\Vert D^m\bar{u}\Vert }_p^p\right){\bar{u}}_t-M\left({\Vert D^{m}u\Vert }_p^p\right)u_t-M\left({\Vert D^{m}u\Vert }_p^p\right)U_t\\ -M^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^{m}U{u}_t,\end{array}$ (67)

$\begin{array}{c}{h}_2=N\left({\Vert D^m\bar{u}\Vert }_p^p\right){\left(-\Delta \right)}^m\bar{u}-N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}u-N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}U\\ -N^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^{m}U{\left(-\Delta \right)}^{m}u,\end{array}$ (68)

$\begin{array}{c}{h}_1=M\left({\Vert D^m\bar{u}\Vert }_p^p\right){\bar{u}}_t-M\left({\Vert D^{m}u\Vert }_p^p\right){\bar{u}}_t+M\left({\Vert D^{m}u\Vert }_p^p\right){\psi }_t\\ -M^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^m\left(\bar{u}-u-\psi \right)u_t\\ =M^{\prime }\left({\Vert D^m\bar{\xi }\Vert }_p^p\right){\left({\Vert D^m\bar{\xi }\Vert }_p^p\right)}^{\prime }{D}^m\left(\bar{u}-u\right){\bar{u}}_t\\ -M^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^m\left(\bar{u}-u\right)u_t+M^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }\end{array}$

$\begin{array}{l}{D}^m\psi u_t+M\left({\Vert D^{m}u\Vert }_p^p\right){\psi }_t\\ =M^{″}\left({\Vert D^m\varsigma \Vert }_p^p\right){\left({\Vert D^m\varsigma \Vert }_p^p\right)}^{\prime }\left(1-s\right){\left(D^m\left(\bar{u}-u\right)\right)}^2{\bar{u}}_t\\ +M^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^m\left(\bar{u}-u\right)\left({\bar{u}}_t-u_t\right)+M^{\prime }\end{array}$

$\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^m\psi u_t+M\left({\Vert D^{m}u\Vert }_p^p\right){\psi }_t,$ (69)

$\bar{\xi }=s{D}^m\bar{u}+\left(1-s\right)D^{m}u,s\in \left(0,1\right),$

$\varsigma =s{D}^m\bar{\xi }+\left(1-s\right)D^{m}u,s\in \left(0,1\right).$

$\begin{array}{l}\left|\left(M^{″}\left({\Vert D^m\varsigma \Vert }_p^p\right){\left({\Vert D^m\varsigma \Vert }_p^p\right)}^{\prime }\left(1-s\right){\left(D^m\left(\bar{u}-u\right)\right)}^2{\bar{u}}_t,{\left(-\Delta \right)}^k\varphi \right)\right|\\ \le C_7\left|{\displaystyle \underset{\Omega }{\int }{\left(D^m\left(\bar{u}-u\right)\right)}^2{D}^k{\bar{u}}_t{D}^k\varphi \text{d}x}\right|\\ \le C_8{\Vert \bar{u}-u\Vert }_{H^m}^2\Vert D^k\varphi \Vert \\ \le \frac{C_8^2{\lambda }_1^{-3m}}{2{\mu }_0}{\Vert \bar{u}-u\Vert }_{E_k}^4+\frac{{\mu }_0}{2}{\Vert D^{m+k}\varphi \Vert }^2,\end{array}$ (70)

$\begin{array}{l}{C}_7={\Vert M^{″}\left({\Vert D^m\varsigma \Vert }_p^p\right){\left({\Vert D^m\varsigma \Vert }_p^p\right)}^{\prime }\left(1-s\right)\Vert }_{\infty },\\ C_8={\Vert M^{″}\left({\Vert D^m\varsigma \Vert }_p^p\right){\left({\Vert D^m\varsigma \Vert }_p^p\right)}^{\prime }\left(1-s\right)\Vert }_{\infty }\Vert D^k{\bar{u}}_t\Vert .\end{array}$

$\begin{array}{l}\left|\left(M^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^m\left(\bar{u}-u\right)\left({\bar{u}}_t-u_t\right),{\left(-\Delta \right)}^k\varphi \right)\right|\\ \le C_3\left|{\displaystyle \underset{\Omega }{\int }{D}^m\left(\bar{u}-u\right)D^k\left({\bar{u}}_t-u_t\right)D^k\varphi \text{d}x}\right|\\ \le \frac{C_3}{{\lambda }_1^{\frac{m}{2}}}\Vert D^m\left(\bar{u}-u\right)\Vert \cdot \Vert D^k\left({\bar{u}}_t-u_t\right)\Vert \cdot \Vert D^{m+k}\varphi \Vert \\ \le \frac{C_3}{2{\lambda }_1^{\frac{2m+k}{2}}}\left({\Vert D^{2m+k}\left(\bar{u}-u\right)\Vert }^2+{\Vert D^k\left({\bar{u}}_t-u_t\right)\Vert }^2\right)\Vert D^{m+k}\varphi \Vert \end{array}$

$\begin{array}{l}\le \frac{C_3}{2{\lambda }_1^{\frac{2m+k}{2}}}({\Vert D^{2m+k}\left(\bar{u}-u\right)\Vert }^2+{\Vert D^k\left(\bar{v}-v\right)\Vert }^2\\ +\epsilon {\lambda }_1^{-2m}{\Vert D^{2m+k}\left(\bar{u}-u\right)\Vert }^2)\Vert D^{m+k}\varphi \Vert \\ \le C_9{\Vert \bar{u}-u\Vert }_{E_k}^2\cdot \Vert D^{m+k}\varphi \Vert \\ \le \frac{{\mu }_0}{2}{\Vert D^{m+k}\varphi \Vert }^2+\frac{C_9^2}{2{\mu }_0}{\Vert \bar{u}-u\Vert }_{E_k}^4,\end{array}$ (71)

$\begin{array}{l}\left|\left(M^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^m\psi u_t,{\left(-\Delta \right)}^k\varphi \right)\right|\\ \le C_3\left|{\displaystyle \underset{\Omega }{\int }{D}^m\psi D^k{u}_t{D}^k\varphi \text{d}x}\right|\\ \le C_3{\Vert D^m\psi \Vert }_{\infty }\cdot \Vert D^k{u}_t\Vert \cdot \Vert D^k\varphi \Vert \\ \le C_3\Vert D^k{u}_t\Vert \cdot {\lambda }_1^{-m-\frac{k}{2}}\Vert D^{2m+k}\psi \Vert \cdot \Vert D^{m+k}\varphi \Vert \\ \le \frac{C_{10}^2}{2{\mu }_0}{\Vert D^{2m+k}\psi \Vert }^2+\frac{{\mu }_0}{2}{\Vert D^{m+k}\varphi \Vert }^2,\end{array}$ (72)

$\begin{array}{l}\left|\left(M\left({\Vert D^{m}u\Vert }_p^p\right){\psi }_t,{\left(-\Delta \right)}^k\varphi \right)\right|\\ =\left|\left(M\left({\Vert D^{m}u\Vert }_p^p\right)\left(\varphi -\epsilon \psi \right),{\left(-\Delta \right)}^k\varphi \right)\right|\\ \le {\lambda }_1^{-m}{\sigma }_1{\Vert D^{m+k}\varphi \Vert }^2+\frac{{\sigma }_1}{4}{\Vert D^k\varphi \Vert }^2+{\epsilon }^2{\sigma }_1{\Vert D^k\psi \Vert }^2,\end{array}$ (73)

we obtain from Equations (69) - (73),

$\begin{array}{c}\left(h_1,{\left(-\Delta \right)}^k\varphi \right)\le \left(\frac{3{\mu }_0}{2}+\frac{{\sigma }_1}{{\lambda }_1^m}\right){\Vert D^{m+k}\varphi \Vert }^2+\frac{C_8^2{\lambda }_1^{-3m}+C_9^2}{2{\mu }_0}{\Vert \bar{u}-u\Vert }_{E_k}^4\\ +C_{11}\left({\Vert D^k\psi \Vert }^2+{\Vert D^{2m+k}\psi \Vert }^2+{\Vert D^k\varphi \Vert }^2\right),\end{array}$

with $C_{11}=\max \{\frac{C_{10}^2}{2{\mu }_0},{\epsilon }^2{\sigma }_1,\frac{{\sigma }_1}{4}\}$.

Do the same thing,

$\begin{array}{c}{h}_2=N\left({\Vert D^m\bar{u}\Vert }_p^p\right){\left(-\Delta \right)}^m\bar{u}-N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}u-N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{m}U\\ -N^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^{m}U{\left(-\Delta \right)}^{m}u\\ =N\left({\Vert D^m\bar{u}\Vert }_p^p\right){\left(-\Delta \right)}^m\bar{u}-N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^m\bar{u}\\ -N^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^m\left(\bar{u}-u\right){\left(-\Delta \right)}^{m}u\\ +N^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^m\psi {\left(-\Delta \right)}^{m}u+N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^m\psi \end{array}$

$\begin{array}{l}=N^{″}\left({\Vert \bar{\varsigma }\Vert }_p^p\right){\left({\Vert \bar{\varsigma }\Vert }_p^p\right)}^{\prime }\left(1-s\right){\left(D^m\left(\bar{u}-u\right)\right)}^2{\left(-\Delta \right)}^m\bar{u}\\ +N^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^m\left(\bar{u}-u\right){\left(-\Delta \right)}^m\left(\bar{u}-u\right)\\ +N^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^m\psi {\left(-\Delta \right)}^{m}u+N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^m\psi \end{array}$ (74)

Take inner product $h_2$ with ${\left(-\Delta \right)}^k\varphi$, by usingYoung’s inequality, Poincare’s inequality,

$\begin{array}{l}\left|\left(N^{″}\left({\Vert \bar{\varsigma }\Vert }_p^p\right){\left({\Vert \bar{\varsigma }\Vert }_p^p\right)}^{\prime }\left(1-s\right){\left(D^m\left(\bar{u}-u\right)\right)}^2{\left(-\Delta \right)}^m\bar{u},{\left(-\Delta \right)}^k\varphi \right)\right|\\ \le C_{12}{\Vert D^m\left(\bar{u}-u\right)\Vert }^2\cdot \Vert D^{2m+k}\bar{u}\Vert \cdot \Vert D^k\varphi \Vert \\ \le C_{12}\Vert D^{2m+k}\bar{u}\Vert {\lambda }_1^{-m-k}\cdot {\Vert D^{2m+k}\left(\bar{u}-u\right)\Vert }^2\cdot \Vert D^k\varphi \Vert \\ \le \frac{C_{13}^2}{2{\mu }_0}{\Vert D^{2m+k}\left(\bar{u}-u\right)\Vert }^4+\frac{{\mu }_0}{2}{\Vert D^k\varphi \Vert }^2,\end{array}$ (75)

where

$\begin{array}{l}{C}_{12}={\Vert N^{″}\left({\Vert D^m\varsigma \Vert }_p^p\right){\left({\Vert D^m\varsigma \Vert }_p^p\right)}^{\prime }\left(1-s\right)\Vert }_{\infty },\\ C_{13}={\Vert N^{″}\left({\Vert D^m\varsigma \Vert }_p^p\right){\left({\Vert D^m\varsigma \Vert }_p^p\right)}^{\prime }\left(1-s\right)\Vert }_{\infty }\Vert D^{2m+k}\bar{u}\Vert {\lambda }_1^{-m-k}.\end{array}$

$\begin{array}{l}\left(N^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^m\left(\bar{u}-u\right){\left(-\Delta \right)}^m\left(\bar{u}-u\right),{\left(-\Delta \right)}^k\varphi \right)\\ \le C_5{\lambda }_1^{\frac{m+k}{-2}}{\Vert D^{2m+k}\left(\bar{u}-u\right)\Vert }^2\Vert D^k\varphi \Vert \\ \le \frac{{\mu }_0}{2}{\Vert D^k\varphi \Vert }^2+\frac{C_5^2\cdot {\lambda }_1^{-m-k}}{2{\mu }_0}{\Vert D^{2m+k}\left(\bar{u}-u\right)\Vert }^4.\end{array}$ (76)

$\begin{array}{l}\left(N^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^m\psi {\left(-\Delta \right)}^{m}u,{\left(-\Delta \right)}^k\varphi \right)\\ \le C_5{\lambda }_1^{\frac{-k}{2}}\Vert D^{m+k}\psi \Vert \cdot \Vert D^{2m+k}u\Vert \cdot \Vert D^k\varphi \Vert \\ \le \frac{{\mu }_0}{2}{\Vert D^k\varphi \Vert }^2+\frac{C_{14}^2}{2{\mu }_0}{\Vert D^{m+k}\psi \Vert }^2,\end{array}$ (77)

$\left(N\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^m\psi ,{\left(-\Delta \right)}^k\varphi \right)\le \frac{{\mu }_0}{2}{\Vert D^k\varphi \Vert }^2+\frac{{\mu }_1^2}{2{\mu }_0}{\Vert D^{2m+k}\psi \Vert }^2,$ (78)

From Equation (75) - (78), it follows

$\left|\left(h_2,{\left(-\Delta \right)}^k\varphi \right)\right|\le C_{15}\left({\Vert D^{2m+k}\psi \Vert }^2+{\Vert D^k\varphi \Vert }^2\right)+\frac{C_{13}^2+C_5{\lambda }_1^{-m-k}}{2{\mu }_0}{\Vert D^{2m+k}\left(\bar{u}-u\right)\Vert }^4,$ (79)

where $C_{15}=\max \{2{\mu }_0,\frac{C_{14}^2{\lambda }_1^{-m}+{\mu }_1^2}{2{\mu }_0}\}$.

Equation (73) and Equation (79) deduce that,

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}\left({\Vert D^k\varphi \Vert }^2+{\epsilon }^2{\Vert D^k\psi \Vert }^2+\left(\alpha -\beta \epsilon \right){\Vert D^{2m+k}\psi \Vert }^2\right)+{\epsilon }^3{\Vert D^k\psi \Vert }^2\\ +\left(\alpha \epsilon -\beta {\epsilon }^2\right){\Vert D^{2m+k}\psi \Vert }^2+\left(\beta {\lambda }_1^m-\frac{3}{2}{\mu }_0-\frac{{\sigma }_1}{{\lambda }_1^m}\right){\Vert D^{m+k}\varphi \Vert }^2\\ \le \left(C_{11}+C_{15}\right)\left({\Vert D^k\varphi \Vert }^2+{\Vert D^{2m+k}\psi \Vert }^2\right)+C_{11}{\Vert D^k\psi \Vert }^2\\ +\frac{C_8^2{\lambda }_1^{-3m}+C_9^2+C_{13}^2+C_5{\lambda }_1^{-m-k}}{2{\mu }_0}{\Vert \bar{u}-u\Vert }_{E_k}^4.\end{array}$ (80)

According to the hypothesis (c), $\beta {\lambda }_1^m-\frac{3}{2}{\mu }_0-\frac{{\sigma }_1}{{\lambda }_1^m}\ge 0$,

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left({\Vert D^k\varphi \Vert }^2+{\epsilon }^2{\Vert D^k\psi \Vert }^2+\left(\alpha -\beta \epsilon \right){\Vert D^{2m+k}\psi \Vert }^2\right)\\ \le C_{16}\left({\Vert D^k\varphi \Vert }^2+{\epsilon }^2{\Vert D^k\psi \Vert }^2+\left(\alpha -\beta \epsilon \right){\Vert D^{2m+k}\psi \Vert }^2\right)\\ +\frac{C_8^2{\lambda }_1^{-3m}+C_9^2+C_{13}^2+C_5{\lambda }_1^{-m-k}}{2{\mu }_0}{\Vert \bar{u}-u\Vert }_{E_k}^4,\end{array}$ (81)

with $C_{16}=2\max \{C_{11}+C_{15},C_{11}\}=2\left(C_{11}+C_{15}\right)$,

Obtained by Gronwall inequality

${\Vert D^k\varphi \Vert }^2+{\epsilon }^2{\Vert D^k\psi \Vert }^2+\left(\alpha -\beta \epsilon \right){\Vert D^{2m+k}\psi \Vert }^2\le C_{17}{\text{e}}^{C_{18}t}{\Vert {\left(\xi ,\zeta \right)}^{\text{T}}\Vert }_{E_k}^4$.

${\Vert D^k\varphi \Vert }^2+\left(\alpha -\beta \epsilon \right){\Vert D^{2m+k}\psi \Vert }^2\le C_{17}{\text{e}}^{C_{18}t}{\Vert {\left(\xi ,\zeta \right)}^{\text{T}}\Vert }_{E_k}^4$ (82)

From (82), as ${\Vert {\left(\xi ,\eta \right)}^{\text{T}}\Vert }_{E_k}^2\to 0$,

$\frac{{\Vert S\left(t\right)\bar{\eta }-S\left(t\right)\eta -DS\left(t\right)\left(\xi ,\zeta \right)\Vert }_{E_k}}{{\Vert \left(\xi ,\zeta \right)\Vert }_{E_k}^2}\le C_{19}{\text{e}}^{C_{20}t}{\Vert \left(\xi ,\zeta \right)\Vert }_{E_k}^2\to 0.$ (83)

Lemma 3 is proved.

Theorem 3. Under the assumptions and conditions of theorem 2, then a family of global attractors $A_k$ of initial boundary value problem (1) - (3) has Hausdorff dimension and fractal dimension, and

$d_H\left(A_k\right)<\frac{2}{7}n\text{,}{d}_F\left(A_k\right)<\frac{4}{7}n$.

Proof. Let $\psi =R_{\epsilon }\phi ={\left(u,v\right)}^{\text{T}},\phi ={\left(u,u_t\right)}^{\text{T}},v=u_t+\epsilon u$, then $R_{\epsilon }:\left\{u,u_t\right\}\to \left\{u,u_t+\epsilon u\right\}$ is an isomorphic mapping. If $A_i\left(i=1,2,\cdots ,2m\right)$ is the global attractor of $\left\{S\left(t\right)\right\}$, then $A_{\epsilon }{}_i$ is the global attractor of $\left\{S_{\epsilon }\left(t\right)\right\}$, and they have the same dimension.

From Lemma 3, we can get that $S\left(t\right):E_k\to E_k$ is Fréchet differentiable, then the linearized first order variational Equation (59) can be rewritten as

$P_t+\Lambda \left(\psi \right)P=0$, (84)

${\psi }_t+{\Lambda }_{\epsilon }\psi =0$. (85)

${\Lambda }_{\epsilon }=\left(\begin{array}{cc}\epsilon I& -I\\ \left(N\left({\Vert A^{\frac{m}{2}}u\Vert }_p^P\right)A^m+\left(\alpha -\beta \epsilon \right)A^{2m}+{\epsilon }^2-\epsilon M+\varphi +\kappa \right)I& \left(\beta A^{2m}+M-\epsilon \right)I\end{array}\right)$ (86)

where

$\varphi =M^{\prime }{\displaystyle \underset{\Omega }{\int }p{\left|A^{\frac{m}{2}}u\right|}_p^{p-2}{A}^{\frac{m}{2}}u{A}^{\frac{m}{2}}U\text{d}x{u}_t}=M^{\prime }{\displaystyle \underset{\Omega }{\int }p{\left|A^{\frac{m}{2}}u\right|}_p^{p-2}{A}^{\frac{m}{2}}u{A}^{\frac{m}{2}}\cdot \text{d}x{u}_t}U,$

$\kappa =N^{\prime }{\displaystyle \underset{\Omega }{\int }p{\left|A^{\frac{m}{2}}u\right|}_p^{p-2}{A}^{\frac{m}{2}}u{A}^{\frac{m}{2}}U\text{d}x{A}^{m}u}=N^{\prime }{\displaystyle \underset{\Omega }{\int }p{\left|A^{\frac{m}{2}}u\right|}_p^{p-2}{A}^{\frac{m}{2}}u{A}^{\frac{m}{2}}\cdot \text{d}x{A}^{m}uU},$

$-\Delta =A$.

For a fixed $\left(u_0,v_0\right)\in E_k$, let ${\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_n$ be n elements of $E_k$, and $U_1\left(t\right),U_2\left(t\right),\cdots ,U_n\left(t\right)$ be n solutions of linear Equation (84), whose initial value is $U_1\left(0\right)={\gamma }_1,U_2\left(0\right)={\gamma }_2,\cdots ,U_n\left(0\right)={\gamma }_n$.

Therefore,

$\begin{array}{l}{\Vert U_1\left(t\right)\Lambda U_2\left(t\right)\Lambda \cdots \Lambda U_n\left(t\right)\Vert }_{\Lambda E_k}^2\\ ={\Vert {\gamma }_1\Lambda {\gamma }_1\Lambda \cdots \Lambda {\gamma }_n\Vert }_{\Lambda E_k}\exp \left({\displaystyle {\int }_0^{t}tr{F}^{\prime }\left(\psi \left(\tau \right)\right)\cdot Q_n\left(\tau \right)\text{d}\tau }\right).\end{array}$ (87)

where $\Lambda$ is the outer product, $tr$ is the trace, $Q_n\left(\tau \right)$ is an orthogonal projection from space $E_k$ to $span\left\{U_1\left(t\right),U_2\left(t\right),\cdots ,U_n\left(t\right)\right\}$.

For a given time $\tau$, let ${\omega }_j\left(\tau \right)={\left({\xi }_j\left(\tau \right),{\eta }_j\left(\tau \right)\right)}^{\text{T}}\left(j=1,2,\cdots ,n\right)$ be the standard orthogonal basis of spance $span\left\{w_1\left(t\right),w_2\left(t\right),\cdots ,w_n\left(t\right)\right\}$.

We define the inner product of $E_k$ as

$\left(\left(\xi ,\eta \right),\left(\bar{\xi },\bar{\eta }\right)\right)=\left(\left(D^{2m+k}\xi ,D^{2m+k}\bar{\xi }\right)+\left(D^k\eta ,D^k\bar{\eta }\right)\right).$ (88)

To sum up, it can be concluded that

$\begin{array}{c}tr{F}^{\prime }\left(\psi \left(\tau \right)\right)\cdot Q_n\left(\tau \right)={\displaystyle \underset{j=1}{\overset{n}{\sum }}{\left(F^{\prime }\left(\psi \left(\tau \right)\right)\cdot Q_n\left(\tau \right){\omega }_j\left(\tau \right),{\omega }_j\left(\tau \right)\right)}_{E_k}}\\ ={\displaystyle \underset{j=1}{\overset{n}{\sum }}{\left(F^{\prime }\left(\psi \left(\tau \right)\right){\omega }_j\left(\tau \right),{\omega }_j\left(\tau \right)\right)}_{E_k}},\end{array}$ (89)

where

${\left(F^{\prime }\left(\psi \left(\tau \right)\right){\omega }_j\left(\tau \right),{\omega }_j\left(\tau \right)\right)}_{E_k}=-\left({\Lambda }_{\epsilon }{\omega }_j,{\omega }_j\right).$

$\begin{array}{l}-\left({\Lambda }_{\epsilon }{\omega }_j,{\omega }_j\right)=-((\epsilon {\xi }_j-{\eta }_j,N\left({\Vert A^{\frac{m}{2}}u\Vert }_p^p\right)A^m{\xi }_j+\left(\alpha -\beta \epsilon \right)A^{2m}{\xi }_j\\ +\left({\epsilon }^2-\epsilon M\left({\Vert A^{\frac{m}{2}}u\Vert }_p^p\right)+\varphi +\kappa \right){\xi }_j+\beta A^{2m}{\eta }_j+M\left({\Vert A^{\frac{m}{2}}u\Vert }_p^p\right){\eta }_j-\epsilon {\eta }_j),\left({\xi }_j,{\eta }_j\right))\\ +\frac{{\epsilon }^2}{2}{\Vert D^k{\eta }_j\Vert }^2-{\lambda }_1^m{\mu }_0{\Vert D^k{\eta }_j\Vert }^2+\epsilon {\Vert D^k{\eta }_j\Vert }^2+\frac{\epsilon }{2}{\Vert D^{2m+k}{\xi }_j\Vert }^2+\frac{C_{14}^2{\lambda }_1^{-m-k}}{2\epsilon }{\Vert D^k{\eta }_j\Vert }^2\\ =-\epsilon {\Vert D^{2m+k}{\xi }_j\Vert }^2-\left(\alpha -1-\beta \epsilon \right)\left(D^{2m+k}{\xi }_j,D^{2m+k}{\eta }_j\right)\\ -N\left({\Vert A^{\frac{m}{2}}u\Vert }_p^p\right)\left(D^{2m+k}{\xi }_j,D^k{\eta }_j\right)-\left({\epsilon }^2+\varphi +\kappa \right)\left(D^k{\xi }_j,D^k{\eta }_j\right)\end{array}$

$\begin{array}{l}-\beta {\Vert D^{2m+k}{\eta }_j\Vert }^2-M\left({\Vert A^{\frac{m}{2}}u\Vert }_p^p-\epsilon \right){\Vert D^k{\eta }_j\Vert }^2\\ \le -\epsilon {\Vert D^{2m+k}{\xi }_j\Vert }^2+\frac{\alpha -1-\beta \epsilon }{2}{\Vert D^{2m+k}{\xi }_j\Vert }^2+\frac{\alpha -1-\beta \epsilon }{2}{\Vert D^{2m+k}{\eta }_j\Vert }^2\\ +\frac{{\mu }_0}{2}{\Vert D^{2m+k}{\xi }_j\Vert }^2+\frac{{\mu }_0}{2}{\Vert D^k{\eta }_j\Vert }^2+\frac{{\epsilon }^2+\varphi +\kappa -\epsilon {\sigma }_0}{2{\lambda }_1^{2m}}{\Vert D^{2m+k}{\xi }_j\Vert }^2\\ +\frac{{\epsilon }^2+\varphi +\kappa -\epsilon {\sigma }_0}{2}{\Vert D^k{\eta }_j\Vert }^2-\beta {\Vert D^{2m+k}{\eta }_j\Vert }^2-\left({\sigma }_0-\epsilon \right){\Vert D^k{\eta }_j\Vert }^2\\ =\left(\frac{\alpha -1-\beta \epsilon }{2}-\epsilon +\frac{{\mu }_0}{2}+\frac{{\epsilon }^2+\varphi +\kappa -\epsilon {\sigma }_0}{2{\lambda }_1^{2m}}\right){\Vert D^{2m+k}{\xi }_j\Vert }^2\end{array}$

$\begin{array}{l}+\left(\frac{\alpha -1-\beta \epsilon -\beta }{2}\right){\Vert D^{2m+k}{\eta }_j\Vert }^2\\ +\frac{1}{2}\left({\mu }_1+{\epsilon }^2+\varphi +\kappa -\epsilon {\sigma }_0-\beta {\lambda }_1^{2m}-2{\sigma }_0\right){\Vert D^k{\eta }_j\Vert }^2+\epsilon {\Vert D^k{\eta }_j\Vert }^2\\ \le \left(\frac{\alpha -1-\beta \epsilon }{2}-\epsilon +\frac{{\mu }_1}{2}+\frac{{\epsilon }^2+\varphi +\kappa -\epsilon {\sigma }_0}{2{\lambda }_1^{2m}}\right){\Vert D^{2m+k}{\xi }_j\Vert }^2\\ +\frac{1}{2}\left(\left(\alpha -1-\beta \epsilon -2\beta \right){\lambda }_1^{2m}+{\mu }_0+{\epsilon }^2+\varphi +\kappa -\epsilon {\sigma }_0-2{\sigma }_0\right){\Vert D^k{\eta }_j\Vert }^2+\epsilon {\Vert D^k{\eta }_j\Vert }^2\\ \le -C_{21}\left({\Vert D^{2m+k}{\xi }_j\Vert }^2+{\Vert D^k{\eta }_j\Vert }^2\right)+\epsilon {\Vert D^k{\eta }_j\Vert }^2,\end{array}$ (90)

$\begin{array}{l}{C}_{21}=\min \{\frac{\beta \epsilon +1+\alpha }{2}+\epsilon -\frac{{\mu }_0}{2}-\frac{{\epsilon }^2+\varphi +\kappa -\epsilon {\sigma }_0}{2{\lambda }_1^{2m}},\frac{-1}{2}(\left(\alpha -1-\beta \epsilon -2\beta \right){\lambda }_1^{2m}\\ \stackrel{}{}+{\mu }_0+{\epsilon }^2+\varphi +\kappa -\epsilon {\sigma }_0-\beta {\lambda }_1^{2m}-2{\sigma }_0){\Vert D^k{\eta }_j\Vert }^2+\epsilon {\Vert D^k{\eta }_j\Vert }^2\}.\end{array}$ (91)

According to the hypothesis (c),

$\begin{array}{l}\alpha -\beta \epsilon -1\ge 0,\alpha -\beta \epsilon -1-\beta \le 0,\\ \frac{\alpha -1-\beta \epsilon }{2}-\epsilon +\frac{{\mu }_1}{2}+\frac{{\epsilon }^2+\varphi +\kappa -\epsilon {\sigma }_0}{2{\lambda }_1^{2m}}\le 0,\\ {\mu }_0+{\epsilon }^2+\varphi +\kappa -\epsilon {\sigma }_0-2{\sigma }_0\le 0.\end{array}$

From (84) and (85),

$\begin{array}{c}{\left(F^{\prime }\left(\psi \left(\tau \right)\right){\omega }_j\left(\tau \right),{\omega }_j\left(\tau \right)\right)}_{E_k}\le -C_{21}\left({\Vert D^{2m+k}{\xi }_j\Vert }^2+{\Vert D^k{\eta }_j\Vert }^2\right)+\epsilon {\Vert D^k{\eta }_j\Vert }^2\\ \le -C_{22}\left({\Vert {\xi }_j\Vert }^2+{\Vert {\eta }_j\Vert }^2\right)+r{\Vert D^k{\eta }_j\Vert }^2.\end{array}$ (92)

Owing to the ${\omega }_j\left(\tau \right)={\left({\xi }_j\left(\tau \right),{\eta }_j\left(\tau \right)\right)}^{\text{T}},j=1,2,\cdots ,n$ is the standard orthonormal basis of $span\left\{U_1\left(t\right),U_2\left(t\right),\cdots ,U_n\left(t\right)\right\}$, so

${\Vert {\xi }_j\Vert }^2+{\Vert {\eta }_j\Vert }^2=1,$ (93)

${\displaystyle \underset{j=1}{\overset{n}{\sum }}{\left(F^{\prime }\left(\psi \left(\tau \right)\right){\omega }_j\left(\tau \right),{\omega }_j\left(\tau \right)\right)}_{E_k}\le -n{C}_{22}+r{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\Vert D^k{\eta }_j\Vert }^2}},$ (94)

${\displaystyle \underset{j=1}{\overset{n}{\sum }}{\Vert D^k{\eta }_j\Vert }^2}\le {\displaystyle \underset{j=1}{\overset{n}{\sum }}{\lambda }_j^{s-1}},$ (95)

Therefore

$Tr{F}^{\prime }\left(\psi \left(\tau \right)\right)\cdot Q_n\left(\tau \right)\le -n{C}_{22}+r{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\lambda }_j^{s-1}}.$ (96)

$q_n\left(t\right)=\underset{{\psi }_0\in B_{0k}}{\sup }\underset{\begin{array}{l}{\eta }_j\in E_k\\ \Vert {\eta }_j\Vert \le 1\end{array}}{\sup }\left(\frac{1}{t}{\displaystyle {\int }_0^{t}tr{F}^{\prime }\left(S\left(\tau \right){\psi }_0\right)}\cdot Q_n\left(\tau \right)\text{d}\tau \right),q_n=\underset{t\to \infty }{\lim }{q}_n\left(t\right),$ (97)

And

$q_n\le -n{C}_{22}+r{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\lambda }_j^{s-1}}.$ (98)

Thus, the Lyapunov exponent ${\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_n\left(n>1\right)$ of $B_{0k}$ is uniformly bounded

${\kappa }_1+{\kappa }_2+\cdot \cdot \cdot +{\kappa }_n\le -n{C}_{22}+r{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\lambda }_j^{s-1}}.$ (99)

then, there is a $s\in \left[0,1\right]$, such that

${\left(q_j\right)}_{+}\le -n{C}_{22}+r{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\lambda }_j^{s-1}}\le r{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\lambda }_j^{s-1}}\le \frac{n{C}_{22}}{9},$ (100)

where ${\lambda }_j$ is the eigenvalue of $A_k^m$, ${\lambda }_1<{\lambda }_2<\cdots <{\lambda }_n$.

$q_n\le -\frac{n{C}_{22}}{2}\left(1-\frac{2r}{n{C}_{22}}{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\lambda }_j^{s-1}}\right)\le -\frac{7}{18}n{C}_{22}.$ (101)

$\underset{1\le j\le n}{\max }\frac{{\left(q_j\right)}_{+}}{\left|q_n\right|}\le \frac{2}{7}.$ (102)

Then

$d_H\left(A_k\right)<\frac{2}{7}n,d_F\left(A_k\right)<\frac{4}{7}n$,

the Hausdorff dimension and Fractal dimension of the family of global attractors are finite. Theorem 3 is proved.

4. Conclusion

We have shown the existence and uniqueness of global solutions, the existence of the family of global attractors and the upper bound estimates of Hausdoff and Fractal dimensions for the wide norm model of vertical beam vibration. The global stability of the problem is obtained.

Conflicts of Interest

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

References

[1]	Kirchhoff, G. (1883) Vorlesungen Uber Mechanik, Teubner, Leipzig. Stutgaety, 56-70.
[2]	Teman, R. (1988) Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer-Verlag, New York. https://doi.org/10.1007/978-1-4684-0313-8
[3]	Guo, B. and Jing, Z. (1996) Infinite-Dimensional Dynamical Systems. China Academic Journal, 2, 290-296.
[4]	Chueshov, I. (2012) Long-Time Dynamics of Kirchhoff Wave Models with Strong Nonlinear Damping. Journal of Differential Equations, 252, 1229-1262. https://doi.org/10.1016/j.jde.2011.08.022
[5]	Matsuyama, T. and Ikehata, R. (1996) On Global Solutions and Energy Decay for the Wave Equations of Kirchhoff Type with Nonlinear Damping Terms. Journal of Mathematical Analysis and Applications, 204, 729-753. https://doi.org/10.1006/jmaa.1996.0464
[6]	Gao, Y., Sun, Y. and Lin, G. (2016) The Global Attractors and Their Hausdorff and Fractal Dimensions Estimation for the Higer-Order Nonlinear Kirchhoff-Type Equation with Strong Linear Damping. International Journal of Modern Nonlinear Theory and Application, 5, 185-202. https://doi.org/10.4236/ijmnta.2016.54018
[7]	Chen, L., Wang, W. and Lin, G. (2016) The Global Attractors and the Hausdorff and Fractal Demensions Estimation for the Higher-Order Nonlinear Kirchhoff-Type Equation. Journal of Advances in Mathematics, 12, 6608-6621. https://doi.org/10.24297/jam.v12i9.133
[8]	Lin, G. (2011) Nonlinear Evolution Equation. Yunnan University Press, Kunming, 123-184.
[9]	Zhou, S. (1999) Global Attractor for Strongly Damped Nonlinear Wave Equations. Functional Differential Equations, 6, 451-470.
[10]	Xu, G. and Lin, G. (2011) Global Attractors and Their Dimensions Estimation for the Generalized Boussinesq Equation. China Science and Technology Information, 10, 54-61.
[11]	Lyu, P. (2016) Long-Time Behavior of a Class of Generalized Nonlinear Kirchhoff-Boussinesq Equations. Yunnan University, Kunming.
[12]	Lin, G. and Yang, S. (2017) Hausdorff Dimension and Fractal Dimension of the Global Attractor for the Higher-Order Coupled Kirchhoff-Type Equations. Journal of Applied Mathematics and Physics, 5, 2411-2424.
[13]	Lyu, P., Lu, J. and Lin, G. (2016) Global Attractor for a Class of Nonlinear Generalized Kirchhoff Models. Journal of Advances in Mathematics, 12, 6452-6462. https://doi.org/10.24297/jam.v12i8.135
[14]	Lyu, P., Lou, R. and Lin, G. (2016) Global Attractor for a Class of Nonlinear Generalized Kirchhoff-Boussinesq Model. International Journal of Modern Nonlinear Theory and Application, 5, 82-92. https://doi.org/10.4236/ijmnta.2016.51009
[15]	Sun, Y., Gao, Y. and Lin, G. (2016) The Global Attractors for the Higher-Order Kirchhoff-Type Equation with Nonlinear Strongly Damped Term. International Journal of Modern Nonlinear Theory and Application, 5, 203-217. https://doi.org/10.4236/ijmnta.2016.54019
[16]	Li, Q.X. and Zhong, T. (2002) Existence of Global Solutions for Kirchhoff Type Equations with Dissipation and Damping Terms. Journal of Xiamen University: Natural Science Edition, 41, 419-422.
[17]	Silva, M.A.J. and Ma, T.F. (2013) Long-Time Dynamics for a Class of Kirchhoff Models with Memory. Journal of Mathematical Physics, 54, Article ID: 021505. https://doi.org/10.1063/1.4792606
[18]	Alves, C.O. and Figueiredo, G.M. (2016) Nonlinear Perturbations of a Periodic Kirchhoff Equation in RN. Journal of Mathematical Analysis and Applications, 435, 1826-1851.
[19]	Lin, G. and Li, Z. (2019) A Class of Attractor Families and Their Dimensions for Higher Order Nonlinear Kirchhoff Equations. Journal of Shandong University (Natural Science), 54, 1-11.