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

Guoguang Lin^*, Keshun Peng

Department of Mathematics, Yunnan University, Kunming, China.

DOI: 10.4236/jamp.2023.1110196                

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.

Keywords

Beam-Kirchhoff Equation, Galerkin’s Method, The Family of Global Attractor, Dimension Estimation

1. Introduction

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

$u_{tt}+\beta {\Delta }^{2m}{u}_t+\delta \Psi \left({\Vert u_t\Vert }^2\right){\Delta }^{2m}{u}_t+\alpha {\Delta }^{2m}u+\gamma \Phi \left({\Vert D^{m}u\Vert }_p^p\right){\Delta }^{2m}u=g\left(x\right),$ (1.1)

$u\left(x,t\right)=0,\frac{{\partial }^{i}u}{\partial v^i}=0,i=1,2,\cdots ,2m-1,x\in \partial \Omega ,t>0,$ (1.2)

$u\left(x,0\right)=u_0\left(x\right),u_t\left(x,0\right)=u_1\left(x\right),x\in \Omega \subset R^n.$ (1.3)

where $m>1$ is a positive integer, $\Omega$ is a bounded region in $R^n$ with a smooth homogeneous Dirichlet boundary, $\partial \Omega$ represents the boundary of $\Omega$ , and $g\left(x\right)$ is the external force term. $\beta {\left(-\Delta \right)}^{2m}{u}_t$ is a strongly damped term, $\alpha ,\beta ,\delta ,\gamma$ is a constant greater than 0, $\Psi \left({\Vert u_t\Vert }^2\right),\Phi \left({\Vert D^{m}u\Vert }_p^p\right)$ is a given non-negative function, and ${\Vert D^{m}u\Vert }_p^p={\displaystyle \underset{\Omega }{\int }{\left|D^{m}u\right|}^p\text{d}x}$ .

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:

$u_{tt}-\sigma \left({\Vert \nabla u\Vert }^2\right)\Delta u_t-\varphi \left({\Vert \nabla u\Vert }^2\right)\Delta u+g_1\left(u\right)=h(\; x\; )$

In the energy space $H\left(\Omega \right)=H_0^1\cap L^{p+1}\left(\Omega \right)\times L^2\left(\Omega \right)$ , the author can find that the growth of exponential p of strong nonlinear $g\left(u\right)$ is supercritical, that is, when $p^{\ast }\equiv \frac{N+2}{{\left(N-2\right)}^{+}}$ , $p^{\ast \ast }=\frac{N+4}{{\left(N-4\right)}^{+}}$ , there is $p^{\ast }\le p\le p^{\ast \ast }$ , where when $p<p^{\ast }$ , the growth of exponential p for $H^1\left(\Omega \right)\to L^{p+1}$ is supercritical. In $H\left(\Omega \right)$ , 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 $H\left(\Omega \right)$ , 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 $\sigma {\left({\Vert \nabla u\Vert }^2\right)}^{\theta }\Delta u_t$ at the abstract level and the long-term dynamic system, where $\frac{1}{2}\le \theta \le 1$ .

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

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

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 ${\alpha }^q{\left(-\Delta \right)}^{m}u$ 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

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

They use Sobolev embedding theorem $H_0^m\left(\Omega \right)\subset L^{2p}\left(\Omega \right)$ to deal with source term $g\left(x,t\right)$ , 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

$u_{tt}-\Delta u_t-M\left({\Vert \nabla u\Vert }^2\right)\Delta u+u_t+g\left(x,u\right)=f\left(x\right),\left(x,u\right)\in R^N\times R^{+},$

$u\left(x,0\right)=u_0\left(x\right),u_t\left(x,0\right)=u_1\left(x\right),x\in R^N$

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 $H=H^1\left(R^N\right)\times L^2\left(R^N\right)$ . 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:

$u_{tt}-\alpha \Delta u+\beta {\Delta }^{2}u-\gamma \Delta u_t+g\left(u\right)=f\left(x,t\right),\left(x,t\right)\in \Omega \times R^{+},$

$u\left(x,0\right)=u_0\left(x\right),u_t\left(x,0\right)=u_1\left(x\right),x\in \Omega ,$

${u|}_{\partial \Omega }=0,{\Delta u|}_{\partial \Omega }=0,\left(x,t\right)\in \partial \Omega \times R^{+},$

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:

$-\left(a+b{\displaystyle \underset{\Omega }{\int }{\left|\nabla u\right|}^2\text{d}x}\right)\Delta u=\mu g\left(x,u\right)+u^5.$

When $u\to 0^{+},g\left(x,u\right)=o\left(u\right)$ and $u\to +\infty ,g\left(x,u\right)=o\left(u^5\right),x\in \Omega$ , The Sobolev embedding theorem $H_0^1\left(\Omega \right)\subseteq L^p\left(\Omega \right)$ is used to deal with the nonlinear source term $\mu g\left(x,u\right)$ , 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

$u_{tt}-M\left({\Vert \nabla u\Vert }^2\right)\Delta u+\delta {\left|u\right|}^{p}u+\gamma u_t=f\left(x\right),x\in \Omega ,t>0,$

$u\left(x,t\right)=0,x\in \partial \Omega ,t\ge 0,$

$u\left(x,0\right)=u_0\left(x\right),u_t\left(x,0\right)=u_1\left(t\right),x\in \Omega .$

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 $\Omega \in R^n$ is a bounded region with a smooth boundary $\partial \Omega$ , and

$M\left(\gamma \right)\in C^1\left[0,\infty \right),M\left(\gamma \right)\ge m_0>0.$

when $\delta >0,\alpha >0$ and $\forall \gamma \ge 0$ are present.

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

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

$u\left(x,0\right)=u_0\left(x\right),u_t\left(x,0\right)=u_1\left(x\right),x\in \Omega ,$

$u\left(x,t\right)=0,x\in \partial \Omega ,t>0.$

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

$\Omega \subset R^n,M\left(s\right)=a+b{s}^r\in C^1\left[0,\infty \right),a,b\ge 0,a+b>0,r\ge 1,M\left(s\right)\ge m_0>0.$

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

$u_{tt}-M\left({\Vert \nabla u\Vert }^2\right)\Delta u-\Delta u_t+g\left(x,u\right)=f\left(x\right),$

$u\left(x,0\right)=u_0\left(x\right),u_t\left(x,0\right)=u_1\left(x\right),x\in \Omega ,$

$u\left(x,t\right)=0,x\in \partial \Omega ,t>0.$

with strong damping term. When $g\left(x,u\right)$ 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 $g\left(x,u\right)\in C^1\left(R^n,R\right)$ and $g\left(x,u\right)$ have order $p:1\le p<\frac{4}{{\left(N-4\right)}^{+}}$ with respect to u.

2. Background Knowledge and Assumptions

In this paper, we need the following mathematical notation:

$f=f\left(x\right)$ , $D=\nabla$ , $H=L^2\left(\Omega \right)$ , $H_0^{2m}\left(\Omega \right)=H^{2m}\left(\Omega \right)\cap H_0^1\left(\Omega \right)$ .

$H_0^{m+k}\left(\Omega \right)=H^{m+k}\left(\Omega \right)\cap H_0^1\left(\Omega \right)$ , $\left(k=0,1,2,\cdots ,2m\right)$

$H_0^{m+k}\left(\Omega \right)=D\left(\Omega \right)$ , Closure of infinitely differentiable Spaces in

$H^{m+k}\left(\Omega \right)$ .

$E_k=H_0^{2m+k}\left(\Omega \right)\times H_0^k\left(\Omega \right)$ and $E_0=H_0^{2m}\left(\Omega \right)\times H$ , where $C_i\ge 0\left(i=1,2,\cdots \right)$ is constant.

${\lambda }_1$ is the first eigenvalue of $-\Delta$ on $\Omega$ .

The inner product and norm expressed by $\left(•,•\right)$ and $\Vert •\Vert$ respectively, that is, $\left(u,v\right)={\displaystyle {\int }_{\Omega }u\left(x\right)v\left(x\right)\text{d}x}$ , $\left(u,u\right)={\Vert u\Vert }^2$ .

Note the global attractor from $E_0$ to $E_k$ as $A_k$ , while $B_{0k}$ is the bounded absorption set in $E_k$ , where $k=1,2,\cdots ,2m$ .

Lemma 2.1. [10] (Holder inequality) set

$\frac{1}{p}+\frac{1}{q}=1,\left(p\ge 1,q\ge 1\right),\forall f\left(x\right)\in L^p\left(\Omega \right),\forall g\left(x\right)\in L^q(\; \Omega \; )$

We have ${\displaystyle {\int }_{\Omega }\left|f\left(x\right)g\left(x\right)\text{d}x\right|}\le {\left({{\displaystyle {\int }_{\Omega }\left|f\left(x\right)\right|}}^p\text{d}x\right)}^{\frac{1}{p}}{\left({{\displaystyle {\int }_{\Omega }\left|g\left(x\right)\right|}}^q\text{d}x\right)}^{\frac{1}{q}}$ .

Lemma 2.2. [10] (Poincare inequality) If $\Omega \in R^n$ is a bounded open subset, then

${\Vert u\Vert }_{L^2\left(\Omega \right)}\le {\lambda }_1^{-\frac{1}{2}}{\Vert \nabla u\Vert }_{L^2\left(\Omega \right)},\forall u\in H_0^1(\; \Omega \; )$

where ${\lambda }_1$ is the first eigenvalue of $-\Delta$ on $\Omega$ .

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

$ab\le \frac{{\epsilon }^p}{p}{a}^p+\frac{1}{q{\epsilon }^q}{b}^q,\left(\frac{1}{p}+\frac{1}{q}=1,p>1,q>1\right)$ ,

Let function $\Psi \left({\Vert u_t\Vert }^2\right),\Phi \left({\Vert D^{m}u\Vert }_p^p\right)$ satisfy the condition:

(A₁) $\Phi \left({\Vert D^{m}u\Vert }_p^p\right)\in C^2\left(\left[0,+\infty \right),R\right)$ , $0\le {\sigma }_0\le \Phi \left({\Vert D^{m}u\Vert }_p^p\right)\le {\sigma }_1$ , ${\delta }_2=\{\begin{array}{l}{\sigma }_0,\frac{\text{d}}{\text{d}t}{\Vert D^{2m}u\Vert }^2\ge 0\\ {\sigma }_1,\frac{\text{d}}{\text{d}t}{\Vert D^{2m}u\Vert }^2\le 0\end{array}$

Where ${\sigma }_0,{\sigma }_1$ is the positive constant;

(A₂) $\Psi \left({\Vert u_t\Vert }^2\right)\in C^2\left(\left[0,+\infty \right),R\right)$ , $0\le B_0\le \Psi \left({\Vert u_t\Vert }^2\right)\le B_1$ ,

Where $B_0,B_1$ is the positive constant;

(A₃) $\frac{2n}{n+2m}\le p\{\begin{array}{l}<\frac{2n}{n-2m},n>2m\\ <\infty ,n\le 2m\end{array}$

3. The Existence and Uniqueness of the Global Solution

Lemma 3.1. Assuming that condition (A₁), (A₂), (A₃) is true, $g\left(x\right)\in H$ , $\left(u_0,u_1\right)\in E_0$ , then the initial boundary value problem (1.1)-(1.3) has a global smooth solution $\left(u,v\right)\in E_0$ and is satisfied

${\Vert \left(u,v\right)\Vert }_{E_0}^2={\Vert D^{2m}u\Vert }^2+{\Vert v\Vert }^2\le \left({\Vert D^{2m}{u}_0\Vert }^2+{\Vert v_0\Vert }^2\right){\text{e}}^{-a_{1}t}+\frac{C_1}{a_1}\left(1-{\text{e}}^{-a_{1}t}\right),$

where $v=u_t+\epsilon u$ ,

Ream $a_1=\min \left\{\beta {\lambda }_1^{2m}+\delta B_0{\lambda }_1^{2m}-2\epsilon -2{\epsilon }^2,\frac{2\epsilon \left(\alpha +r{\delta }_0\right)-{\epsilon }^2\left({\lambda }_1^{-2m}+\beta +\delta B_1\right)}{\alpha +r\delta }\right\}$

So there is a non-negative real number $R_0$ and $t_0$ that makes

${\Vert \left(u,v\right)\Vert }_{E_0}^2={\Vert D^{2m}u\Vert }^2+{\Vert v\Vert }^2\le R_0^2,\left(t>t_0\right)$

Prove we set $v=u_t+\epsilon u$ and take the inner product of both sides of Equation (1.1) and v, we get

$\left(u_{tt}+\beta {\Delta }^{2m}{u}_t+\delta \Psi \left({\Vert u_t\Vert }^2\right){\Delta }^{2m}{u}_t+\alpha {\Delta }^{2m}u+\gamma \Phi \left({\Vert D^{m}u\Vert }_p^p\right){\Delta }^{2m}u,v\right)=\left(g\left(x\right),v\right)$ (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)

$\begin{array}{c}\left(u_{tt},v\right)=\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert v\Vert }^2-\epsilon {\Vert v\Vert }^2+{\epsilon }^2\left(u,v\right)\ge \frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert v\Vert }^2-\epsilon {\Vert v\Vert }^2-\frac{{\epsilon }^2}{2}{\Vert v\Vert }^2-\frac{{\epsilon }^2}{2}{\Vert u\Vert }^2\\ \ge \frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert v\Vert }^2-\frac{2\epsilon +{\epsilon }^2}{2}{\Vert v\Vert }^2-\frac{{\epsilon }^2{\lambda }_1^{-2m}}{2}{\Vert D^{2m}u\Vert }^2\end{array}$ (3.2)

$\left(\alpha {\left(-\Delta \right)}^{2m}u,v\right)=\left(\alpha {\left(-\Delta \right)}^{2m}u,u_t+\epsilon u\right)=\frac{\alpha }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m}u\Vert }^2+\alpha \epsilon {\Vert D^{2m}u\Vert }^2$ (3.3)

$\begin{array}{c}\left(\beta {\left(-\Delta \right)}^{2m}{u}_t,v\right)=\left(\beta {\left(-\Delta \right)}^{2m}\left(v-\epsilon u\right),v\right)=\beta {\Vert D^{2m}v\Vert }^2-\beta \epsilon \left(D^{2m}u,D^{2m}v\right)\\ \ge \beta {\Vert D^{2m}v\Vert }^2-\frac{\beta {\epsilon }^2}{2}{\Vert D^{2m}u\Vert }^2-\frac{\beta }{2}{\Vert D^{2m}v\Vert }^2\\ \ge \frac{\beta {\lambda }_1^{2m}}{2}{\Vert v\Vert }^2-\frac{\beta {\epsilon }^2}{2}{\Vert D^{2m}u\Vert }^2\end{array}$ (3.4)

$\begin{array}{c}\left(\delta \psi \left({\Vert u_t\Vert }^2\right){\left(-\Delta \right)}^{2m}{u}_t,v\right)=\delta \psi \left({\Vert u_t\Vert }^2\right){\Vert D^{2m}v\Vert }^2-\epsilon \delta \psi \left({\Vert u_t\Vert }^2\right)\left(D^{2m}u,D^{2m}v\right)\\ \ge \delta B_0{\Vert D^{2m}v\Vert }^2-\epsilon \delta B_1\left(\frac{\epsilon {\Vert D^{2m}u\Vert }^2+\frac{1}{\epsilon }{\Vert D^{2m}v\Vert }^2}{2}\right)\\ \ge \frac{\delta B_0{\lambda }_1^{2m}}{2}{\Vert v\Vert }^2-\frac{{\epsilon }^2\delta B_1}{2}{\Vert D^{2m}u\Vert }^2\end{array}$ (3.5)

$\begin{array}{l}\left(r\varphi \left({\Vert D^{m}u\Vert }_p^D\right){\left(-\Delta \right)}^{2m}u,u_t+\epsilon u\right)\\ =\frac{r\varphi \left({\Vert D^{m}u\Vert }_p^D\right)}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m}u\Vert }^2+\epsilon r\varphi \left({\Vert D^{m}u\Vert }_p^D\right){\Vert D^{2m}u\Vert }^2\\ \ge \frac{r\delta }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m}u\Vert }^2+\epsilon r{\delta }_0{\Vert D^{2m}u\Vert }^2\end{array}$ (3.6)

$\left(g\left(x\right),v\right)\le \frac{{\epsilon }^2}{2}{\Vert v\Vert }^2+\frac{1}{2{\epsilon }^2}{\Vert g\left(x\right)\Vert }^2$ (3.7)

Synthetically available

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert v\Vert }^2+\frac{1}{2}\left(\alpha +r\delta \right)\frac{\text{d}}{\text{d}t}{\Vert D^{2m}u\Vert }^2+\left(\frac{\beta {\lambda }_1^{2m}-2\epsilon -2{\epsilon }^2+\delta B_0{\lambda }_1^{2m}}{2}\right){\Vert v\Vert }^2\\ +\left(\alpha \epsilon -\frac{{\epsilon }^2{\lambda }_1^{-2m}}{2}-\frac{\beta {\epsilon }^2}{2}-\frac{{\epsilon }^2\delta B_1}{2}+\epsilon r{\delta }_0\right){\Vert D^{2m}u\Vert }^2\le \frac{1}{2{\epsilon }^2}{\Vert g\left(x\right)\Vert }^2\end{array}$

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left({\Vert v\Vert }^2+\left(\alpha +r\delta \right){\Vert D^{2m}u\Vert }^2\right)+\left(\beta {\lambda }_1^{2m}-2\epsilon -2{\epsilon }^2+\delta B_0{\lambda }_1^{2m}\right){\Vert v\Vert }^2\\ +\left(\frac{-{\epsilon }^2\left({\lambda }_1^{-2m}+\beta +\delta B_1\right)+2\epsilon \left(\alpha +r{B}_0\right)}{\alpha +r\delta }\right)\left(\alpha +r\delta \right){\Vert D^{2m}u\Vert }^2\le \frac{1}{{\epsilon }^2}{\Vert g\left(x\right)\Vert }^2\end{array}$ (3.8)

Ream $a_1=\min \left\{\beta {\lambda }_1^{2m}+\delta B_0{\lambda }_1^{2m}-2\epsilon -2{\epsilon }^2,\frac{2\epsilon \left(\alpha +r{\delta }_0\right)-{\epsilon }^2\left({\lambda }_1^{-2m}+\beta +\delta B_1\right)}{\alpha +r\delta }\right\}$

In a result $\frac{\text{d}}{\text{d}t}\left({\Vert v\Vert }^2+\left(\alpha +r\delta \right){\Vert D^{2m}u\Vert }^2\right)+a_1\left({\Vert v\Vert }^2+\left(\alpha +r\delta \right){\Vert D^{2m}u\Vert }^2\right)\le C_1$ (3.9)

It’s derived from the Gronwall inequality

${\Vert v\Vert }^2+\left(\alpha +r\delta \right){\Vert D^{2m}u\Vert }^2\le \left({\Vert v_0\Vert }^2+\left(\alpha +r\delta \right){\Vert D^{2m}{u}_0\Vert }^2\right){\text{e}}^{-a_{1}t}+\frac{C_1}{a}\left(1-{\text{e}}^{-a_{1}t}\right)$ (3.10)

So there exist $R_0$ and $t_0$ to make ${\Vert \left(u,v\right)\Vert }_{E_0}^2=\Vert D^{2m}u\Vert +{\Vert v\Vert }^2\le R_0^2\left(t>t_0\right)$ .

Lemma 3.2. Assuming that condition (A₁), (A₂), (A₃) is true and $g\left(x\right)\in H^k$ , $\left(u_0,u_1\right)\in E_k$ is true, then the global solution $\left(u,v\right)\in E_k$ of the initial boundary value problem (1.1)-(1.3) is satisfied

${\Vert \left(u,v\right)\Vert }_{E_k}^2={\Vert D^{2m+k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\le \left({\Vert D^{2m+k}{u}_0\Vert }^2+{\Vert D^k{v}_0\Vert }^2\right){\text{e}}^{-a_{1}t}+\frac{C_2}{a_1}\left(1-{\text{e}}^{-a_{1}t}\right),$

where $C_2=\frac{1}{{\epsilon }^2}{\Vert D^{k}g\left(x\right)\Vert }^2$ ,

$a_1=\min \left\{\beta {\lambda }_1^{2m}+\delta B_0{\lambda }_1^{2m}-2\epsilon -2{\epsilon }^2,\frac{2\epsilon \left(\alpha +r{\delta }_0\right)-{\epsilon }^2\left({\lambda }_1^{-2m}+\beta +\delta B_1\right)}{\alpha +r\delta }\right\}$

Then there are positive constants $R_k$ and $t_k$ that make

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

Prove set ${\left(-\Delta \right)}^{k}v={\left(-\Delta \right)}^k{u}_t+\epsilon {\left(-\Delta \right)}^{k}u$ , and take the inner product of both sides of Equation (1.1) with ${\left(-\Delta \right)}^{k}v$ , i.e.

$\begin{array}{l}\left(u_{tt}+\beta {\Delta }^{2m}{u}_t+\delta \Psi \left({\Vert u_t\Vert }^2\right){\Delta }^{2m}{u}_t+\alpha {\Delta }^{2m}u+\gamma \Phi \left({\Vert D^{m}u\Vert }_p^p\right){\Delta }^{2m}u,{\left(-\Delta \right)}^{k}v\right)\\ =\left(g\left(x\right),{\left(-\Delta \right)}^{k}v\right).\end{array}$ (3.11)

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

$\begin{array}{c}\left(u_{tt},{\left(-\Delta \right)}^{k}v\right)=\left(v_t-\epsilon u_t,{\left(-\Delta \right)}^{k}v\right)=\left(D^k{v}_t,D^{k}v\right)-\epsilon \left(D^k{u}_t,D^{k}v\right)\\ =\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{k}v\Vert }^2-\epsilon {\Vert D^{k}v\Vert }^2+{\epsilon }^2\left(D^{k}u,D^{k}v\right)\\ \ge \frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{k}v\Vert }^2-\frac{\epsilon +{\epsilon }^2}{2}{\Vert D^{k}v\Vert }^2-\frac{{\epsilon }^2{\lambda }_1^{-2m}}{2}{\Vert D^{2m+k}u\Vert }^2\end{array}$ (3.12)

$\begin{array}{c}\left(\alpha {\left(-\Delta \right)}^{2m}u,{\left(-\Delta \right)}^{k}v\right)=\alpha \left(D^{2m+k}u,D^{2m+k}{u}_t\right)+\alpha \epsilon {\Vert D^{2m+k}u\Vert }^2\\ =\frac{\alpha }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m+k}u\Vert }^2+\alpha \epsilon {\Vert D^{2m+k}u\Vert }^2\end{array}$ (3.13)

$\begin{array}{c}\left(\beta {\left(-\Delta \right)}^{2m}{u}_t,{\left(-\Delta \right)}^{k}v\right)=\beta \left({\left(-\Delta \right)}^{2m}\left(v-\epsilon u\right),{\left(-\Delta \right)}^{k}v\right)\\ =\beta {\Vert D^{2m+k}v\Vert }^2-\epsilon \beta \left(D^{2m+k}u,D^{2m+k}v\right)\\ \ge \beta {\Vert D^{2m+k}v\Vert }^2-\frac{\epsilon \beta }{2}\left(\epsilon {\Vert D^{2m+k}u\Vert }^2+\frac{{\Vert D^{2m+k}v\Vert }^2}{\epsilon }\right)\\ \ge \frac{\beta {\lambda }_1^{2m}}{2}{\Vert D^{k}v\Vert }^2-\frac{{\epsilon }^2\beta }{2}{\Vert D^{2m+k}u\Vert }^2\end{array}$ (3.14)

$\begin{array}{l}\left(\delta \psi \left({\Vert u_t\Vert }^2\right){\left(-\Delta \right)}^{2m}{u}_t,{\left(-\Delta \right)}^{k}v\right)\\ =\delta \psi \left({\Vert u_t\Vert }^2\right){\Vert D^{2m+k}v\Vert }^2-\epsilon \delta \psi \left({\Vert u_t\Vert }^2\right)\left(D^{2m+k}u,D^{2m+k}v\right)\\ \ge \delta B_0{\Vert D^{2m+k}v\Vert }^2-\frac{\epsilon \delta B_1}{2}\left(\epsilon {\Vert D^{2m+k}u\Vert }^2+\frac{{\Vert D^{2m+k}v\Vert }^2}{\epsilon }\right)\\ \ge \frac{\delta B_0{\lambda }_1^{2m}}{2}{\Vert D^{k}v\Vert }^2-\frac{{\epsilon }^2\delta B_1}{2}{\Vert D^{2m+k}u\Vert }^2\end{array}$ (3.15)

$\begin{array}{l}\left(r\varphi \left({\Vert D^{m}u\Vert }_p^D\right){\left(-\Delta \right)}^{2m}u,{\left(-\Delta \right)}^{k}v\right)\\ =\frac{r\varphi \left({\Vert D^{m}u\Vert }_p^D\right)}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m+k}u\Vert }^2+\epsilon r\varphi \left({\Vert D^{m}u\Vert }_p^D\right){\Vert D^{2m+k}u\Vert }^2\\ \ge \frac{r\delta }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m+k}u\Vert }^2+\epsilon r{\delta }_0{\Vert D^{2m+k}u\Vert }^2\end{array}$ (3.16)

$\left(g\left(x\right),{\left(-\Delta \right)}^{k}v\right)=\left(D^{k}g\left(x\right),D^{k}v\right)\le \frac{{\epsilon }^2}{2}{\Vert D^{k}v\Vert }^2+\frac{1}{2{\epsilon }^2}{\Vert D^{k}g\left(x\right)\Vert }^2$ (3.17)

In summary can be obtained

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{k}v\Vert }^2+\frac{\alpha +r\delta }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m+k}u\Vert }^2+\frac{\beta {\lambda }_1^{2m}+\delta B_0{\lambda }_1^{2m}-2{\epsilon }^2-2\epsilon }{2}{\Vert D^{k}v\Vert }^2\\ +\frac{2\epsilon \left(\alpha +r{\delta }_0\right)-{\epsilon }^2\left({\lambda }_1^{2m}+\beta +\delta B_1\right)}{2}{\Vert D^{2m+k}u\Vert }^2\le C_2\end{array}$

From the Gronwall inequality:

$\frac{\text{d}}{\text{d}t}\left({\Vert D^{k}v\Vert }^2+\left(\alpha +r\delta \right){\Vert D^{2m+k}u\Vert }^2\right)+a_1\left({\Vert D^{k}v\Vert }^2+\left(\alpha +r\delta \right){\Vert D^{2m+k}u\Vert }^2\right)\le C_2$

${\Vert D^{k}v\Vert }^2+\left(\alpha +r\delta \right){\Vert D^{2m+k}u\Vert }^2\le {\Vert v_0\Vert }^2+\left(\alpha +r\delta \right){\Vert D^{2m+k}{u}_0\Vert }^2{\text{e}}^{-a_{1}t}+\frac{C_2}{a_1}\left(1-{\text{e}}^{-a_{1}t}\right)$ (3.18)

So there exist $R_k$ and $t_k$ to make ${\Vert \left(u,v\right)\Vert }_{E_k}^2={\Vert D^{2m+k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\le R_k^2\left(t>t_k\right)$

Theorem 3.3. (existence and uniqueness of solutions) Under the conditions of lemma 3.1 and Lemma 3.2, and assuming ${\psi }^{\prime }<0$ , $g\in H^k$ , $\left(u_0,u_1\right)\in E_k$ , then the initial boundary value problem (1.1)-(1.3) has a unique global solution $\left(u,v\right)\in L^{\infty }\left(\left[0,+\infty \right);E_k\right)$

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

Construct the approximate solution $u_l\left(t\right)={\displaystyle \underset{j=1}{\overset{l}{\sum }}{g}_{jl}\left(t\right)w_j}$ of problem where $g_{jl}\left(t\right)$ is determined by the following nonlinear system of ordinary differential equations

$\begin{array}{l}\left(u_{tt}+\beta {\Delta }^{2m}{u}_t+\delta \Psi \left({\Vert u_t\Vert }^2\right){\Delta }^{2m}{u}_t+\alpha {\Delta }^{2m}u+\gamma \Phi \left({\Vert D^{m}u\Vert }_p^p\right){\Delta }^{2m}u,w_j\right)\\ =\left(g\left(x\right),w_j\right),j=1,2,\cdots ,l.\end{array}$ (3.19)

When the initial condition $u_{l0}\left(0\right)=u_{l0},u_{lt}\left(0\right)=u_{l1}$ is satisfied and $l\to +\infty$ is satisfied, $\left(u_{l0},u_{l1}\right)\to \left(u_0,u_1\right)$ , in $E_k$ is known from the basic theory of ordinary differentiation that the approximate solution $u_{lt}$ exists on $\left(0,t_l\right)$ .

The second step is prior estimation.

Recording $v_l\left(t\right)=u_{lt}\left(t\right)+\epsilon u_l\left(t\right)$ , multiply both sides of the equation $v_l\left(t\right)=u_{lt}\left(t\right)+\epsilon u_l\left(t\right)$ , by ${\lambda }_j^k\left({g^{\prime }}_{jl}\left(t\right)+\epsilon g_{jl}\left(t\right)\right)$ , and sum over j to get

1) $k=0$ , have

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

2) $k=1,2,\cdots ,2m$ , have

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

then $\left(u_l,v_l\right)$ in $L^{\infty }\left(\left[0,+\infty \right];E_k\right)$ Middle bounded.

The third step is the limiting process.

In space $E_k\left(k=1,2,\cdots ,2m\right)$ , choosing a subcolumn $u_{\mu }$ from sequence $u_l$ causes $\left(u_{\mu },v_{\mu }\right)\to \left(u,v\right)$ to converge weakly * in $L^{\infty }$ .

From the space compact embedding theorem we know that $E_k$ is compactly embedded in $E_0$ , and $\left(u_{\mu },v_{\mu }\right)\to \left(u,v\right)$ is strongly convergent almost everywhere in $E_0$ .

Let $l=\mu$ and take the limit, which can be obtained from the above equation

$\left(u_{\mu t},{\left(-\Delta \right)}^k{w}_j\right)=\left(v_{\mu },{\lambda }_j^k{w}_j\right)-\left(\epsilon u_{\mu },{\lambda }_j^k{w}_j\right)\to \left(v,{\lambda }_j^k{w}_j\right)-\left(\epsilon u,{\lambda }_j^k{w}_j\right)$ (3.20)

Weakly convergent in $L^{\infty }\left[0,+\infty \right)$ . Due to

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

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

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

is weakly convergent in $L^{\infty }\left[0,+\infty \right)$ .

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

is weakly convergent in $L^{\infty }\left[0,+\infty \right)$ .

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

is weakly convergent in $L^{\infty }\left[0,+\infty \right)$ .

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

is weakly convergent in $L^{\infty }\left[0,+\infty \right)$ .

In particular, $u_{\mu 0}\to u_0$ is weakly convergent in $E_k$ , and $u_{\mu t}\to u_t=u_1$ is weakly convergent in $E_k$ . For all j and $\mu \to +\infty$ ,

$\begin{array}{l}\left(u_{tt}+\beta {\Delta }^{2m}{u}_t+\delta \Psi \left({\Vert u_t\Vert }^2\right){\Delta }^{2m}{u}_t+\alpha {\Delta }^{2m}u+\gamma \Phi \left({\Vert D^{m}u\Vert }_p^p\right){\Delta }^{2m}u,w_j\right)\\ =\left(g\left(x\right),w_j\right),j=1,2,\cdots ,l\end{array}$ (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

$\left(w_{tt},w_t+\epsilon w\right)=\left(w_{tt},w_t\right)+\epsilon \left(w_{tt},w\right)=\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert w_t\Vert }^2+\epsilon \frac{\text{d}}{\text{d}t}\left(w_t,w\right)-{\epsilon }^2{\Vert w_t\Vert }^2$ (3.22)

$\left(\beta {\left(-\Delta \right)}^{2m}{w}_t,w_t+\epsilon w\right)=\frac{\beta \epsilon }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m}w\Vert }^2+\beta {\Vert D^{2m}{w}_t\Vert }^2$ (3.23)

$\left(\alpha {\left(-\Delta \right)}^{2m}w,w_t+\epsilon w\right)=\frac{\alpha }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m}w\Vert }^2+\alpha \epsilon {\Vert D^{2m}w\Vert }^2$ (3.24)

$\begin{array}{l}\left(\delta \psi \left({\Vert u_t\Vert }^2\right){\left(-\Delta \right)}^{2m}{u}_t-\delta \psi \left({\Vert u_t\Vert }^2\right){\left(-\Delta \right)}^{2m}{v}_t,w_t+\epsilon w\right)\\ =\left(\delta \psi \left({\Vert u_t\Vert }^2\right){\left(-\Delta \right)}^{2m}{w}_t,w_t+\epsilon w\right)+\left(\left[\delta \psi \left({\Vert u_t\Vert }^2\right)-\delta \psi \left({\Vert v_t\Vert }^2\right)\right]{\left(-\Delta \right)}^{2m}{v}_t,w_t+\epsilon w\right)\\ =\left(\delta \psi \left({\Vert u_t\Vert }^2\right){\left(-\Delta \right)}^{2m}{w}_t,w_t+\epsilon w\right)\\ +\left(\delta {\psi }^{\prime }\left(\xi \right)\left(\Vert u_t\Vert +\Vert v_t\Vert \right)\left(\Vert u_t\Vert -\Vert v_t\Vert \right){\left(-\Delta \right)}^{2m}{u}_t,w_t+\epsilon w\right)\end{array}$

By ${\psi }^{\prime }<0$ , then the original formula

$\begin{array}{l}\ge \delta B_0\left(D^{2m}{w}_t,D^{2m}{w}_t+\epsilon D^{2m}w\right)+C_3\left(\Vert w_t\Vert ,w_t+\epsilon w\right)\\ \ge \delta B_0{\Vert D^{2m}{w}_t\Vert }^2+\frac{\epsilon \delta B}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m}w\Vert }^2+C_3\left(\frac{2+\epsilon }{2}\right){\Vert w_t\Vert }^2+\frac{C_3\epsilon }{2}{\Vert w\Vert }^2\end{array}$ (3.25)

$\begin{array}{l}\left(r\varphi \left({\Vert D^{m}u\Vert }_p^D\right){\left(-\Delta \right)}^{2m}u-r\varphi \left({\Vert D^{m}v\Vert }_p^D\right){\left(-\Delta \right)}^{2m}v,w_t+\epsilon w\right)\\ =\left(r\varphi \left({\Vert D^{m}u\Vert }_p^D\right){\left(-\Delta \right)}^{2m}u-r\varphi \left({\Vert D^{m}u\Vert }_p^D\right){\left(-\Delta \right)}^{2m}v+r\varphi \left({\Vert D^{m}v\Vert }_p^D\right){\left(-\Delta \right)}^{2m}v,w_t+\epsilon w\right)\\ =r\varphi \left({\Vert D^{m}u\Vert }_p^D\right)\left({\left(-\Delta \right)}^{2m}w,w_t+\epsilon w\right)\\ +r{\varphi }^{\prime }\left({\Vert D^m\xi \Vert }_p^p\right){\left({\Vert D^m\xi \Vert }_p^p\right)}^{\prime }\left(D^{m}w{\left(-\Delta \right)}^{2m}w,w_t+\epsilon w\right)\end{array}$

$\begin{array}{l}\ge r\delta \frac{\text{d}}{\text{d}t}{\Vert D^{2m}w\Vert }^2+\epsilon r{\delta }_0{\Vert D^{2m}w\Vert }^2+C_4\left(D^{m}w\cdot {\left(-\Delta \right)}^{2m}w,w_t+\epsilon w\right)\\ \ge r\delta \frac{\text{d}}{\text{d}t}{\Vert D^{2m}w\Vert }^2+\epsilon r{\delta }_0{\Vert D^{2m}w\Vert }^2-\frac{C_4}{2}\left({\Vert D^{m}w\cdot {\left(-\Delta \right)}^{2m}w\Vert }^2+{\Vert w_t\Vert }^2\right)\end{array}$

$\begin{array}{l}-C_4\epsilon \left({\Vert D^{m}w\cdot {\left(-\Delta \right)}^{2m}w\Vert }^2+{\Vert w_t\Vert }^2\right)\\ \ge r\delta \frac{\text{d}}{\text{d}t}{\Vert D^{2m}w\Vert }^2+\epsilon r{\delta }_0{\Vert D^{2m}w\Vert }^2-C_4\left(\frac{1+2\epsilon }{2}\right){\Vert {\left(-\Delta \right)}^{2m}w\Vert }_{\infty }{\Vert D^{m}w\Vert }^2-\frac{C_4}{2}{\Vert w_t\Vert }^2\\ >r\delta \frac{\text{d}}{\text{d}t}{\Vert D^{2m}w\Vert }^2+\epsilon r{\delta }_0{\Vert D^{2m}w\Vert }^2-C_5{\Vert D^{m}w\Vert }^2-\frac{C_4}{2}{\Vert w_t\Vert }^2\end{array}$ (3.26)

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left(\frac{1}{2}{\Vert w_t\Vert }^2+\epsilon \left(w_t,w\right)+\frac{\alpha +\beta \epsilon +\epsilon \delta B+2r\delta }{2}{\Vert D^{2m}w\Vert }^2\right)\\ +\left(-\frac{2\epsilon -\epsilon C_3-2C_3+C_4}{2}\right){\Vert w_t\Vert }^2+\frac{\epsilon C_3}{2}{\Vert w\Vert }^2+\left(\alpha \epsilon +\epsilon r\delta -\frac{C_5}{{\lambda }_1^m}\right){\Vert D^{2m}w\Vert }^2\le 0\end{array}$

So there exist $a_2$ to make that

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left(\frac{1+\epsilon }{2}{\Vert w_t\Vert }^2+\frac{\epsilon }{2}{\Vert w\Vert }^2+\frac{\alpha +\beta \epsilon +\epsilon \delta B+2r\delta }{2}{\Vert D^{2m}w\Vert }^2\right)\\ +a_2\left(\frac{1+\epsilon }{2}{\Vert w_t\Vert }^2+\frac{\epsilon }{2}{\Vert w\Vert }^2+\frac{\alpha +\beta \epsilon +\epsilon \delta B+2r\delta }{2}{\Vert D^{2m}w\Vert }^2\right)\le 0\end{array}$ (3.27)

From Gronwall’s inequality:

$\begin{array}{l}\frac{1+\epsilon }{2}{\Vert w_t\Vert }^2+\frac{\epsilon }{2}{\Vert w\Vert }^2+\frac{\alpha +\beta \epsilon +\epsilon \delta B+2r\delta }{2}{\Vert D^{2m}{w}_1\Vert }^2\\ \le \frac{1+\epsilon }{2}{\Vert w_{0t}\Vert }^2+\frac{\epsilon }{2}{\Vert w_0\Vert }^2+\frac{\alpha +\beta \epsilon +\epsilon \delta B+2r\delta }{2}{\Vert D^{2m}{w}_0\Vert }^2=0\end{array}$ (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 $S\left(t\right):E\to E$ satisfy the following conditions:

1) The semigroup $S\left(t\right)$ is uniformly bounded in E, that is $\forall r>0$ , when ${\Vert u\Vert }_E\le r$ , there is a constant $C\left(r\right)$ , such that

${\Vert S\left(t\right)u\Vert }_E\le C\left(r\right),\forall t\in \left[0,+\infty \right)$

is true;

2) There exists a bounded absorption set $B_0$ in E;

3) $S\left(t\right)$ is a completely continuous operator, and the semigroup $S\left(t\right)$ 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

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

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

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

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

$\underset{t\to \infty }{\lim }dist\left(S\left(t\right)B_k,A_k\right)=\underset{x\in B_k}{\sup }\underset{y\in A_k}{\inf }{\Vert S\left(t\right)x-y\Vert }_{E_k}$ ,

where $S\left(t\right)$ 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 $S\left(t\right):E_k\to E_k$ . The bounded set known as $\forall K_{0k}\subset E_k$ and contained in ball $\{{\Vert \left(u,v\right)\Vert }_{E_k}^2\le R_k^2\}$ ,

${\Vert S\left(t\right)\left(u_0,v_0\right)\Vert }_{E_k}^2={\Vert u\Vert }_{H_0^{2m+k}\left(\Omega \right)}^2+{\Vert v\Vert }_{H_0^k\left(\Omega \right)}^2\le {\Vert u_0\Vert }_{H_0^{2m+k}\left(\Omega \right)}^2+{\Vert v_0\Vert }_{H_0^k\left(\Omega \right)}^2\le R_k^2$ (4.1)

Then $\left\{S\left(t\right)\right\}\left(t\ge 0\right)$ is uniformly bounded within $E_k$ . For any $\left(u_0,v_0\right)\in E_k$ ,

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

It follows that $K_{0k}=\{\left(u,v\right)\in E_k:{\Vert \left(u,v\right)\Vert }_{E_k}^2={\Vert D^{2m+k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\le R_k^2\}$ is the bounded absorption set of a semigroup $S\left(t\right)$ .

Since $E_k\subset E_0$ is compactly embedded, that is the bounded set in $E_k$ is the compactly set in $E_0$ , the solution semigroup $S\left(t\right)$ is a completely continuous operator, so there exists a family of global attractor $A_k=\omega \left(K_{0k}\right)=\underset{\tau \ge 0}{\cap }\overline{\underset{t\ge \tau }{\cup }S\left(t\right)K_{0k}}$ of the solution semigroup $S\left(t\right)$ .

Theorem 4.2 is proved.

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

$\begin{array}{l}{U}_{tt}+\delta \psi \left({\Vert u_t\Vert }^2\right){\Delta }^{2m}{U}_t+\delta {\psi }^{\prime }\left({\Vert u_t\Vert }^2\right){\left({\Vert u_t\Vert }^2\right)}^{\prime }{U}_t\cdot {\Delta }^{2m}{u}_t+\beta {\Delta }^{2m}{U}_t\\ +\gamma \Phi \left({\Vert D^{m}u\Vert }_p^p\right){\Delta }^{2m}U+\gamma {\Phi }^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^{m}U\cdot {\Delta }^{2m}u+\alpha {\Delta }^{2m}U=0\end{array}$ (4.2)

$U\left(x,t\right)=0,\frac{{\partial }^{i}U}{\partial V^i}=0,i=1,2,\cdots ,2m-1,x\in \partial \Omega ,t>0,$ (4.3)

$U\left(x,0\right)=\xi ,U_t\left(x,0\right)=\eta .$ (4.4)

where $\left(u_0,u_1\right)\in A_k$ , $S\left(t\right):E_k\to E_k$ , $\left(\xi ,\eta \right)\in E_k$ , $\left(u,u_t\right)=S\left(t\right)\left(u_0,u_1\right)$ is the solution of problem (1.1)-(1.3) obtained by $\left(u_0,u_1\right)\in A_k$ , it can be shown that for any $\left(\xi ,\eta \right)\in E_k$ , the linearized initial boundary value problem (4.2)-(4.3) has a unique solution $\left(U\left(t\right),U_t\left(t\right)\right)\in L^{\infty }\left(\left[0,+\infty \right);E_k\right)$ .

Lemma 4.3. $\forall t>0,R>0$ , The map $E_k\to E_k$ is Frechet differentiable on $E_k$ , and the Frechet differential in ${\varphi }_0={\left(u_0,u_1\right)}^{\text{T}}$ is a linear operator $F:{\left(\xi ,\eta \right)}^{\text{T}}\to {\left(U\left(t\right),U_t\left(t\right)\right)}^{\text{T}}$ .

Proof: Let ${\varphi }_0={\left(u_0,u_1\right)}^{\text{T}}\in E_k$ , ${\bar{\varphi }}_0={\left(u_0+\xi ,u_1+\eta \right)}^{\text{T}}\in E_k$ and ${\Vert {\varphi }_0\Vert }_{E_k},{\Vert {\bar{\varphi }}_0\Vert }_{E_k}\le R$ give the Lipchitz continuity of $S\left(t\right)$ on $E_k$ , i.e.

${\Vert S\left(t\right){\varphi }_0-S\left(t\right){\bar{\varphi }}_0\Vert }_{E_k}^2\le {\text{e}}^{c_{16}t}{\Vert {\left(\xi ,\eta \right)}^{\text{T}}\Vert }_{E_k}^2.$

If $h=\bar{u}-u-U,\mu =h_t+\epsilon h$ , then

$\{\begin{array}{l}{\bar{u}}_{tt}+\beta {\Delta }^{2m}{\bar{u}}_t+\delta \Psi \left({\Vert {\bar{u}}_t\Vert }^2\right){\Delta }^{2m}{\bar{u}}_t+\alpha {\Delta }^{2m}\bar{u}+\gamma \Phi \left({\Vert D^m\bar{u}\Vert }_p^p\right){\Delta }^{2m}\bar{u}=g\left(x\right)\\ u_{tt}+\beta {\Delta }^{2m}{u}_t+\delta \Psi \left({\Vert u_t\Vert }^2\right){\Delta }^{2m}{u}_t+\alpha {\Delta }^{2m}u+\gamma \Phi \left({\Vert D^{m}u\Vert }_p^p\right){\Delta }^{2m}u=g\left(x\right)\\ U_{tt}+\delta \psi \left({\Vert u_t\Vert }^2\right){\Delta }^{2m}{U}_t+\delta {\psi }^{\prime }\left({\Vert u_t\Vert }^2\right){\left({\Vert u_t\Vert }^2\right)}^{\prime }{U}_t\cdot {\Delta }^{2m}{u}_t+\beta {\Delta }^{2m}{U}_t\\ +\gamma \Phi \left({\Vert D^{m}u\Vert }_p^p\right){\Delta }^{2m}U+\gamma {\Phi }^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^{m}U\cdot {\Delta }^{2m}u+\alpha {\Delta }^{2m}U=0\end{array}$

Subtract from the three formulas to get

${\left(\bar{u}-u-U\right)}_{tt}+\beta {\left(-\Delta \right)}^{2m}{\left(\bar{u}-u-U\right)}_t+\alpha {\left(-\Delta \right)}^{2m}\left(\bar{u}-u-U\right)+g_1+g_2=0$ (4.5)

where

$\begin{array}{c}{g}_1=\delta \psi \left({\Vert {\bar{u}}_t\Vert }^2\right){\left(-\Delta \right)}^{2m}{\bar{u}}_t-\delta \psi \left({\Vert u_t\Vert }^2\right){\left(-\Delta \right)}^{2m}{u}_t-\delta \psi \left({\Vert u_t\Vert }^2\right){\left(-\Delta \right)}^{2m}{U}_t\\ -\delta {\psi }^{\prime }\left({\Vert D^m{u}_t\Vert }^2\right){\left({\Vert D^m{u}_t\Vert }^2\right)}^{\prime }{D}^m{U}_t{\left(-\Delta \right)}^{2m}{u}_t\end{array}$ (4.6)

$\begin{array}{c}{g}_2=r\varphi \left({\Vert D^m\bar{u}\Vert }_p^p\right){\left(-\Delta \right)}^{2m}\bar{u}-r\varphi \left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m}u-r\varphi \left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m}U\\ -r{\varphi }^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^{m}U{\left(-\Delta \right)}^{2m}u\end{array}$ (4.7)

$\bar{u}-u-U=h$ , Use the mean value theorem to deal with

$\begin{array}{c}{g}_1=\delta {\psi }^{\prime }\left({\Vert {\xi }_t\Vert }^2\right){\left({\Vert {\xi }_t\Vert }^2\right)}^{\prime }\left({\bar{u}}_t-u_t\right){\left(-\Delta \right)}^{2m}{\bar{u}}_t+\delta \psi \left({\Vert u_t\Vert }^2\right){\left(-\Delta \right)}^{2m}ht\\ -\delta {\psi }^{\prime }\left({\Vert u_t\Vert }^2\right){\left({\Vert u_t\Vert }^2\right)}^{\prime }{U}_t{\left(-\Delta \right)}^{2m}{u}_t\\ =g_{11}+g_{12}\end{array}$ (4.8)

where

$g_{11}=\delta {\psi }^{\prime }\left({\Vert {\xi }_t\Vert }^2\right){\left({\Vert {\xi }_t\Vert }^2\right)}^{\prime }{\left(-\Delta \right)}^{2m}{\bar{u}}_t\left({\bar{u}}_t-u_t\right)-\delta {\psi }^{\prime }\left({\Vert u_t\Vert }^2\right){\left({\Vert u_t\Vert }^2\right)}^{\prime }{\left(-\Delta \right)}^{2m}{u}_t\left({\bar{u}}_t-u_t\right)$

$g_{12}=\delta \psi \left({\Vert u_t\Vert }^2\right){\left(-\Delta \right)}^{2m}ht+\delta {\psi }^{\prime }\left({\Vert u_t\Vert }^2\right){\left({\Vert u_t\Vert }^2\right)}^{\prime }ht{\left(-\Delta \right)}^{2m}{u}_t$

$\begin{array}{c}{g}_{11}=\delta {\psi }^{\prime }\left({\Vert {\xi }_t\Vert }^2\right){\left({\Vert {\xi }_t\Vert }^2\right)}^{\prime }{\left(-\Delta \right)}^{2m}{\bar{u}}_t-\delta {\psi }^{\prime }\left({\Vert u_t\Vert }^2\right){\left(-\Delta \right)}^{2m}{u}_t\left({\bar{u}}_t-u_t\right)\\ =[\left(\delta {\psi }^{\prime }\left({\Vert {\xi }_t\Vert }^2\right){\left({\Vert {\xi }_t\Vert }^2\right)}^{\prime }-\delta {\psi }^{\prime }\left({\Vert u_t\Vert }^2\right){\left({\Vert u_t\Vert }^2\right)}^{\prime }\right){\left(-\Delta \right)}^{2m}{\bar{u}}_t\\ +\delta {\psi }^{\prime }\left({\Vert u_t\Vert }^2\right){\left({\Vert u_t\Vert }^2\right)}^{\prime }{\left(-\Delta \right)}^{2m}\left({\bar{u}}_t-u_t\right)]\left({\bar{u}}_t-u_t\right)\end{array}$

Make $f\left(x\right)=\delta {\psi }^{\prime }\left({\Vert x\Vert }^2\right){\left({\Vert x\Vert }^2\right)}^{\prime }$ and ${\xi }_t={\lambda }_1{u}_t+\left(1-{\lambda }_1\right){\bar{u}}_t,{\lambda }_1\in \left(0,1\right)$ , Then the above formula is

$\begin{array}{c}{g}_1=f^{\prime }\left(s\right){\lambda }_1{\left(-\Delta \right)}^{2m}{\bar{u}}_t{\left({\bar{u}}_t-u_t\right)}^2+f\left(u_t\right){\left(-\Delta \right)}^{2m}\left({\bar{u}}_t-u_t\right)\cdot \left({\bar{u}}_t-u_t\right)\\ +\delta \psi \left({\Vert u_t\Vert }^2\right){\left(-\Delta \right)}^{2m}ht+f\left(u_t\right){\left(-\Delta \right)}^{2m}{u}_t\cdot ht\end{array}$ (4.9)

Similarly deal with

$\begin{array}{c}{g}_2=r{\varphi }^{\prime }\left({\Vert D^m{\xi }_t\Vert }_p^p\right){\left({\Vert D^m{\xi }_t\Vert }_p^p\right)}^{\prime }\left(D^m\bar{u}-D^{m}u\right){\left(-\Delta \right)}^{2m}\bar{u}\\ -r{\varphi }^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^m\bar{u}{\left(-\Delta \right)}^{2m}u\\ +r{\varphi }^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^{m}u\cdot {\left(-\Delta \right)}^{2m}u+r\varphi \left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m}h\\ +r{\varphi }^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^{m}h\cdot {\left(-\Delta \right)}^{2m}u\\ =g_{21}+g_{22}\end{array}$ (4.10)

where

$\begin{array}{c}{g}_{21}=r{\varphi }^{\prime }\left({\Vert D^m{\xi }_t\Vert }_p^p\right){\left({\Vert D^m{\xi }_t\Vert }_p^p\right)}^{\prime }\left(D^m\bar{u}-D^{m}u\right){\left(-\Delta \right)}^{2m}\bar{u}\\ -r{\varphi }^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }\left(D^m\bar{u}-D^{m}u\right){\left(-\Delta \right)}^{2m}u\end{array}$

$g_{22}=r\varphi \left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m}h+r{\varphi }^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{D}^{m}h\cdot {\left(-\Delta \right)}^{2m}u$

Similarly deal with

$\begin{array}{c}{g}_{21}=\left[r{\varphi }^{\prime }\left({\Vert D^m\Vert }_p^p\right){\left({\Vert D^m\Vert }_p^p\right)}^{\prime }{\left(-\Delta \right)}^{2m}\bar{u}-r{\varphi }^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{\left(-\Delta \right)}^{2m}u\right]\left(D^m\bar{u}-D^{m}u\right)\\ =[r{\varphi }^{\prime }\left({\Vert D^m\Vert }_p^p\right){\left({\Vert D^m\Vert }_p^p\right)}^{\prime }-r{\varphi }^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{\left(-\Delta \right)}^{2m}\bar{u}\\ +r{\varphi }^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }\cdot {\left(-\Delta \right)}^{2m}\left(\bar{u}-u\right)]\left(D^m\bar{u}-D^{m}u\right)\end{array}$

Recording $m\left(y\right)=r{\varphi }^{\prime }\left({\Vert b\Vert }_p^p\right){\left({\Vert b\Vert }_p^p\right)}^{\prime }$ , $D^m\xi =D^{m}u+{\lambda }_2\left(D^m\bar{u}-D^{m}u\right),{\lambda }_2\in \left(0,1\right)$ .

So we can obtain

$\begin{array}{c}{g}_{21}=m^{\prime }\left(s\right){\lambda }_2{\left(-\Delta \right)}^{2m}\bar{u}{\left(D^m\bar{u}-D^{m}u\right)}^2\\ +r{\varphi }^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{\left(-\Delta \right)}^{2m}\left(\bar{u}-u\right)\left(D^m\bar{u}-D^{m}u\right)\end{array}$

$\begin{array}{c}{g}_2=m^{\prime }\left(s\right)\delta {\left(-\Delta \right)}^{2m}\bar{u}{\left(D^m\bar{u}-D^{m}u\right)}^2\\ +r{\varphi }^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{\left(-\Delta \right)}^{2m}\left(\bar{u}-u\right)\left(D^m\bar{u}-D^{m}u\right)\\ +r\varphi \left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m}h+r{\varphi }^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{\left(-\Delta \right)}^{2m}u\cdot D^{m}h\end{array}$ (4.11)

Let ${\left(-\Delta \right)}^k\mu$ and each of them take the inner product: where $\mu =h_t+\epsilon h$ .

$\begin{array}{c}\left(h_{tt},{\left(-\Delta \right)}^k\mu \right)=\left(k_t-\epsilon k+{\epsilon }^{2}h,{\left(-\Delta \right)}^k\mu \right)\\ =\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert D^k\mu \Vert }^2-\epsilon {\Vert D^k\mu \Vert }^2+\frac{{\epsilon }^2}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{k}h\Vert }^2+{\epsilon }^3{\Vert D^{k}h\Vert }^2\end{array}$ (4.12)

$\begin{array}{l}\left(\beta +\delta \psi \left({\Vert u_t\Vert }^2\right){\left(-\Delta \right)}^{2m}{h}_t,{\left(-\Delta \right)}^k\mu \right)\\ =\beta +\delta \psi \left({\Vert u_t\Vert }^2\right)\left({\Vert D^{2m+k}{h}_t\Vert }^2+\frac{\epsilon }{2}\frac{\text{d}}{\text{d}t}{\Vert h\Vert }^2\right)\ge \frac{\left(\beta +\delta B\right)\epsilon }{2}\frac{\text{d}}{\text{d}t}{\Vert h\Vert }^2\end{array}$ (4.13)

$\begin{array}{l}\left(\alpha +r\varphi \left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m}h,{\left(-\Delta \right)}^k\mu \right)\\ =\left(\alpha +r\varphi \left({\Vert D^{m}u\Vert }_p^p\right)\right)\left(\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m+k}h\Vert }^2+\epsilon {\Vert D^{2m+k}h\Vert }^2\right)\\ \ge \frac{\alpha +r\delta }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{2m+k}h\Vert }^2+\epsilon \left(\alpha +r{\delta }_0\right){\Vert D^{2m+k}h\Vert }^2\end{array}$ (4.14)

$\begin{array}{l}\left(f^{\prime }\left(s\right){\lambda }_1{\left(-\Delta \right)}^{2m}{\bar{u}}_t{\left({\bar{u}}_t-u_t\right)}^2,{\left(-\Delta \right)}^k\mu \right)\\ =f^{\prime }\left(s\right){\lambda }_1{\displaystyle {\int }_{\Omega }{D}^{2m+k}}\bar{u}{\left({\bar{u}}_t-u_t\right)}^2{D}^{2m+k}k\text{d}x\le C_6{\Vert {\bar{u}}_t-u_t\Vert }^2\cdot \Vert D^{2m+k}\mu \Vert \end{array}$

Due to

$\begin{array}{c}\Vert u_t\Vert \le \Vert v\Vert +\epsilon \Vert u\Vert \le \Vert v\Vert +\Vert u\Vert \le \frac{1}{{\lambda }_1^{\frac{k}{2}}}\Vert D^{k}v\Vert +\frac{1}{{\lambda }_1^{\frac{2m+k}{2}}}\Vert D^{2m+k}u\Vert \\ \le {\lambda }_0\left(\Vert D^{k}v\Vert +\Vert D^{2m+k}u\Vert \right)={\lambda }_0{\Vert u\Vert }_{E_k}\end{array}$

So there are

$\begin{array}{l}\left(f^{\prime }\left(s\right){\lambda }_1{\left(-\Delta \right)}^{2m}{\bar{u}}_t{\left({\bar{u}}_t-u_t\right)}^2,{\left(-\Delta \right)}^k\mu \right)\\ \le C_6{\lambda }_0{\Vert {\bar{u}}_t-u_t\Vert }_{{}_{E_k}}^2\cdot \Vert D^{2m+k}\mu \Vert \le \frac{C_6{\lambda }_0}{2}{\Vert {\bar{u}}_t-u_t\Vert }_{{}_{E_k}}^4+\frac{C_6{\lambda }_0}{2}{\Vert D^{2m+k}\mu \Vert }^2\end{array}$ (4.15)

$\left(f\left(u_t\right){\left(-\Delta \right)}^{2m}{u}_t{h}_t,{\left(-\Delta \right)}^k\mu \right)\le C_7{\Vert {\bar{u}}_t-u_t\Vert }^2\cdot \Vert D^{2m+k}\mu \Vert$

$\left(f\left(u_t\right){\left(-\Delta \right)}^{2m}{u}_t\cdot h_t,{\left(-\Delta \right)}^k\mu \right)\ge C_8\left({\Vert D^k{h}_t\Vert }^2+\frac{\epsilon }{2}\frac{\text{d}}{\text{d}t}{\Vert D^{k}h\Vert }^2\right)$

$\begin{array}{l}\left(m^{\prime }\left(s\right){\lambda }_2{\left(-\Delta \right)}^{2m}\bar{u}{\left(D^m\bar{u}-D^{m}u\right)}^2,{\left(-\Delta \right)}^k\mu \right)\\ \le C_9{\Vert D^m\bar{u}-D^{m}u\Vert }^2\cdot \Vert D^{2m+k}\mu \Vert \le \frac{C_9}{{\lambda }_1^{m+k}}{\Vert D^{2m+k}\bar{u}-D^{2m+k}u\Vert }^2-\Vert D^{2m+k}w\Vert \\ \le \frac{C_9}{{\lambda }_1^{m+k}}{\Vert {\bar{u}}_t-u_t\Vert }_{{}_{E_k}}^2\cdot \Vert D^{2m+k}\mu \Vert \le \frac{C_9}{2{\lambda }_1^{m+k}}{\Vert {\bar{u}}_t-u_t\Vert }_{{}_{E_k}}^4+\frac{C_9}{2{\lambda }_1^{m+k}}{\Vert D^{2m+k}\mu \Vert }^2\end{array}$ (4.16)

$\begin{array}{l}\left(r{\varphi }^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{\left(-\Delta \right)}^{2m}\left(\bar{u}-u\right)\left(D^m\bar{u}-D^{m}u\right),{\left(-\Delta \right)}^k\mu \right)\\ \le C_{10}\Vert D^{2m+k}\bar{u}-D^{2m+k}u\Vert \cdot \Vert D^m\left(\bar{u}-u\right)\Vert \cdot \Vert D^{2m+k}\mu \Vert \\ \le \frac{C_{10}}{{\lambda }_1^{\frac{m+k}{2}}}{\Vert D^{2m+k}\left(\bar{u}-u\right)\Vert }^2\cdot \Vert D^{2m+k}\mu \Vert \le \frac{C_{10}}{2{\lambda }_1^{\frac{m+k}{2}}}\left({\Vert \bar{u}-u\Vert }_{E_k}^4+{\Vert D^{2m+k}\mu \Vert }^2\right)\end{array}$ (4.17)

$\begin{array}{l}\left(r{\varphi }^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\left({\Vert D^{m}u\Vert }_p^p\right)}^{\prime }{\left(-\Delta \right)}^{2m}u{D}^{m}h,{\left(-\Delta \right)}^k\mu \right)\\ \le C_{11}\Vert D^{m}h\Vert \cdot \Vert D^{2m+k}h\Vert \le \frac{C_{11}}{2}{\Vert D^{m}h\Vert }^2+\frac{C_{11}}{2}{\Vert D^{2m+k}h\Vert }^2\\ \le \left(\frac{C_{11}}{2}+\frac{C_{11}}{2{\lambda }_1^{m+k}}\right){\Vert D^{2m+k}h\Vert }^2\end{array}$ (4.18)

So we obtain

$\begin{array}{l}\frac{\text{d}}{\text{d}t}(\frac{1}{2}{\Vert D^k\mu \Vert }^2+\frac{1}{2}{\Vert D^{2m+k}\mu \Vert }^2+\frac{\epsilon \left(C_8+\epsilon \right)}{2}{\Vert D^{k}h\Vert }^2+\frac{\beta +\delta B\epsilon }{2}{\Vert h\Vert }^2\\ +\frac{2\epsilon +1}{2}\left(\alpha +r\delta \right){\Vert D^{2m+k}h\Vert }^2)\\ \le C_{12}{\Vert \bar{u}-u\Vert }_{E_k}^4+C_{13}{\Vert D^{2m+k}\mu \Vert }^2+C_{14}{\Vert D^k\mu \Vert }^2\\ +\frac{C_{11}}{2}\left(1+\frac{1}{{\lambda }_1^{m+k}}\right){\Vert D^{2m+k}h\Vert }^2+\left(\epsilon -{\epsilon }^2\right){\Vert D^{k}h\Vert }^2\end{array}$ (4.19)

$\begin{array}{l}\frac{1}{2}{\Vert D^k\mu \Vert }^2+\frac{1}{2}{\Vert D^{2m+k}\mu \Vert }^2+\frac{\epsilon \left(C_8+\epsilon \right)}{2}{\Vert D^{k}h\Vert }^2+\frac{\beta +\delta B\epsilon }{2}{\Vert h\Vert }^2\\ +\frac{2\epsilon +1}{2}\left(\alpha +r\delta \right){\Vert D^{2m+k}h\Vert }^2\le C_{15}{\text{e}}^{C_{16}t}{\Vert \bar{u}-u\Vert }_{E_k}^4\end{array}$ (4.20)

when ${\Vert \left(\xi ,\eta \right)\Vert }_{E_k}^2\to 0$ , $\frac{1}{2}\frac{{\Vert \bar{u}-u-U\Vert }_{E_k}^2}{{\Vert \left(\xi ,\eta \right)\Vert }_{E_k}^2}\le C_{15}{\text{e}}^{C_{16}t}{\Vert \left(\xi ,\eta \right)\Vert }_{E_k}^2\to 0$ .

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 $d_H\left(A_k\right)<\frac{2}{3}n$ , $d_F\left(A_k\right)<\frac{4}{3}n$ .

Proof: Let $P_{\epsilon }:\{u,u_t\}\to \{u,u_t+\epsilon u\}$ be an isomorphic mapping, then

$\Psi =P_{\epsilon }\bar{\phi }={\left(U,V\right)}^{\text{T}}$ , where $\bar{\phi }={\left(U,U_t\right)}^{\text{T}}$ , $V=U_t+\epsilon U$ ,

The Frechet differentiability of $S\left(t\right):E_k\to E_k$ is known from lemma 4.3 to estimate the Hausdorff dimension and Fractal dimension of problem (1.1)-(1.3). Consider the variational equation $Y_t+A_{\epsilon }Y=0$ of Equation (4.2) under initial conditions, where

$Y=\left[\begin{array}{c}U\\ V\end{array}\right]$

$A_{\epsilon }=\left[\begin{array}{cc}\epsilon I& -I\\ \alpha -\epsilon \beta -r\varphi \left({\Vert D^{m}u\Vert }_p^p\right)-\epsilon \delta \psi \left({\Vert u_t\Vert }^2\right)A^{2m}+{\epsilon }^2-2\delta \epsilon B+prC& \beta +\delta \psi \left({\Vert u_t\Vert }^2\right)A^{2m}-\epsilon +2\delta D\end{array}\right]$ (4.21)

where ream $-\Delta =A$ , $DV={\psi }^{\prime }\left({\Vert u_t\Vert }^2\right){\displaystyle {\int }_{\Omega }{u}_{t}V\text{d}x}{\left(-\Delta \right)}^{2m}{u}_t$ ,

$\begin{array}{l}BU={\psi }^{\prime }\left({\Vert u_t\Vert }^2\right){\displaystyle {\int }_{\Omega }{u}_{t}U\text{d}x}{\left(-\Delta \right)}^{2m}{u}_t,\\ CU={\varphi }^{\prime }\left({\Vert D^{m}u\Vert }_p^p\right){\displaystyle {\int }_{\Omega }{\left|D^{m}u\right|}^{p-2}}{D}^{m}u{D}^{m}U\text{d}x{\left(-\Delta \right)}^{2m}u\end{array}$

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 let $U_1\left(t\right),U_2\left(t\right),\cdots ,U_n\left(t\right)$ be n solutions of the linear Equation (4.2) with corresponding initial values of $U_1\left(0\right)={\gamma }_1,U_2\left(0\right)={\gamma }_2,\cdots ,U_n\left(0\right)={\gamma }_n$ . Get

$\begin{array}{l}{\Vert U_1\left(t\right)\Lambda U_2\left(t\right)\Lambda \cdot \cdot \cdot \Lambda U_n\left(t\right)\Vert }_{\Lambda E_k}^2\\ ={\Vert {\gamma }_1\Lambda {\gamma }_1\Lambda \cdot \cdot \cdot \Lambda {\gamma }_n\Vert }_{\Lambda E_k}^2\exp \left({\displaystyle {\int }_0^{t}tr{F}^{\prime }\left(\Psi \left(\tau \right)\right)\cdot Q_n\left(\tau \right)\text{d}\tau }\right),t\in \left[0,1\right]\end{array}$

where $\Lambda$ represents the outer product, $tr$ represents the trace of the operator, and $Q_N$ is the orthogonal projection from space $E_k$ to

$span\left\{U_1\left(t\right),U_2\left(t\right),\cdots ,U_n\left(t\right)\right\}$ .

Given $\tau$ , let ${\omega }_j\left(\tau \right)={\left({\xi }_j\left(\tau \right),{\eta }_j\left(\tau \right)\right)}^{\text{T}},j=1,2,\cdots ,n$ be the orthonormal basis of $span\left\{U_1\left(t\right),U_2\left(t\right),\cdots ,U_n\left(t\right)\right\}$ .

Define the inner product on $E_k$ as

$\left(\left(\xi ,\eta \right),\left(\bar{\xi },\bar{\eta }\right)\right)=\left(\left(D^{2m+k}\xi ,D^{2m+k}\bar{\xi }\right)+\left(D^k\eta ,D^k\bar{\eta }\right)\right)$ , (4.22)

$\begin{array}{c}tr{F}^{\prime }\left(\Psi \left(\tau \right)\right)\cdot Q_n\left(\tau \right)=-{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\left({\Lambda }_{\epsilon }\left(\Psi \left(\tau \right)\right)\cdot Q_n\left(\tau \right){\omega }_j\left(\tau \right),{\omega }_j\left(\tau \right)\right)}_{E_k}}\\ =-{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\left({\Lambda }_{\epsilon }\left(\Psi \left(\tau \right)\right){\omega }_j\left(\tau \right),{\omega }_j\left(\tau \right)\right)}_{E_k}}\end{array}$ (4.23)

$\begin{array}{l}-{\left(A_{\epsilon }{w}_j,w_j\right)}_{E_k}\\ =-[(\epsilon {\xi }_j-{\eta }_j,\left(\alpha -\epsilon \beta -r\varphi \left({\Vert D^{m}u\Vert }_p^p\right)-\epsilon \delta \psi \left({\Vert u_t\Vert }^2\right)A^{2m}+{\epsilon }^2-2\delta \epsilon B+prC\right){\xi }_j\\ +{\left(\beta +\delta \psi \left({\Vert u_t\Vert }^2\right)A^{2m}-\epsilon +2\delta D\right){\mu }_j)\cdot \left({\xi }_j,{\mu }_j\right)]}_{E_k}\\ =-\epsilon {\Vert D^{2m+k}{\xi }_j\Vert }^2+\left(1-\alpha +\epsilon \beta +r\varphi \left({\Vert D^{m}u\Vert }_p^p\right)-\epsilon \delta \psi \left({\Vert u_t\Vert }^2\right)\right)\left(D^{2m+k}{\mu }_j,D^{2m+k}{\xi }_j\right)\\ -\left({\epsilon }^2-2\delta \epsilon B+prC\right)\left(D^{2m+k}{\mu }_j,D^{2m+k}{\xi }_j\right)-\left(\beta +\delta \psi \right)\left({\Vert u_t\Vert }^2\right){\Vert D^{2m+k}{\mu }_j\Vert }^2\end{array}$

$\begin{array}{l}+\left(\epsilon -2\delta \left|D\right|{\Vert D^k{\mu }_j\Vert }^2\right)\\ \le -\epsilon {\Vert D^{2m+k}{\xi }_j\Vert }^2+\frac{1-\alpha +\epsilon \beta +r{\delta }_1-\epsilon \delta B_0}{2}\left({\Vert D^{2m+k}{\mu }_j\Vert }^2+{\Vert D^{2m+k}{\xi }_j\Vert }^2\right)\\ +\frac{{\epsilon }^2-2\delta \epsilon \left|B\right|+pr\left|C\right|}{2}\left({\Vert D^k{\xi }_j\Vert }^2+{\Vert D^k{\mu }_j\Vert }^2\right)\\ \le \left(-\epsilon +\frac{1-\alpha +\epsilon \beta +r{\delta }_1-\epsilon \delta B_0}{2}+\frac{{\epsilon }^2-2\delta \epsilon \left|B\right|+pr\left|C\right|}{2{\lambda }_1^{2m+k}}\right){\Vert D^{2m+k}{\xi }_j\Vert }^2\\ +\frac{{\epsilon }^2-2\delta \epsilon \left|B\right|+pr\left|C\right|}{2}{\Vert D^k{\mu }_j\Vert }^2+\epsilon {\Vert D^k{\mu }_j\Vert }^2\end{array}$

Let

$C_{17}=\max \left\{\left(-\epsilon +\frac{1-\alpha +\epsilon \beta +r{\delta }_1-\epsilon \delta B_0}{2}+\frac{{\epsilon }^2-2\delta \epsilon \left|B\right|+pr\left|C\right|}{2{\lambda }_1^{2m}}\right),\frac{{\epsilon }^2-2\delta \epsilon \left|B\right|+pr\left|C\right|}{2}\right\}$

$\epsilon =r$ , $\therefore -{\left(A_{\epsilon }{w}_j,w_j\right)}_{E_k}\le C_{17}\left({\Vert D^k{\mu }_j\Vert }^2+{\Vert D^{2m+k}{\xi }_j\Vert }^2\right)+r{\Vert D^k{\mu }_j\Vert }^2$ . (4.24)

Since ${\omega }_j\left(\tau \right)={\left({\xi }_j\left(\tau \right),{\eta }_j\left(\tau \right)\right)}^{\text{T}},j=1,2,\cdots ,n$ , $span\left\{U_1\left(t\right),U_2\left(t\right),\cdots ,U_n\left(t\right)\right\}$ is an orthonormal basis,

${\displaystyle \underset{j=1}{\overset{n}{\sum }}{\left(F^{\prime }\left(\psi \left(\tau \right)\right){\omega }_j\left(\tau \right),{\omega }_j\left(\tau \right)\right)}_{E_k}}\le -n{C}_{17}+r{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\Vert D^k{\xi }_j\Vert }^2}$ (4.25)

For any t, there’s a

${\displaystyle \underset{j=1}{\overset{n}{\sum }}{\Vert D^k{\xi }_j\Vert }^2}\le {\displaystyle \underset{j=1}{\overset{n}{\sum }}{\lambda }_j^{s-1}}$ , (4.26)

So there is

$Tr{F}^{\prime }\left(\Psi \left(\tau \right)\right)\cdot Q_n\left(\tau \right)\le -n{C}_{17}+r{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\lambda }_j^{s-1}}$ . (4.27)

Let

$q_n\left(t\right)=\underset{{\Psi }_0\in K_{0k}}{\sup }\underset{{\Vert {\eta }_j\Vert }_{E_k}\le 1}{\underset{{\eta }_j\in E_k}{\sup }}\left(\frac{1}{t}{\displaystyle {\int }_0^{t}tr{F}^{\prime }\left(S\left(\tau \right){\Psi }_0\right)}\cdot Q_n\left(\tau \right)\text{d}\tau \right),q_n=\underset{t\to \infty }{\lim }\sup q_n(\; t\; )$

Then $q_n\le -n{C}_{17}+r{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\lambda }_j^{s-1}}$ , Therefore, the Lyapunov exponent ${\tilde{\mu }}_1,{\tilde{\mu }}_2,\cdots ,{\tilde{\mu }}_j$ of $K_{0k}$ is uniformly bounded, and

${\tilde{\mu }}_1+{\tilde{\mu }}_2+\cdots +{\tilde{\mu }}_j\le -n{C}_{17}+r{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\lambda }_j^{s-1}}$

$\exists s\in \left[0,1\right]$ make ${\left(q_j\right)}_{+}\le \frac{n{C}_{17}}{7}$ , $q_n=-\frac{n{C}_{17}}{7}\left(1-\frac{2r}{n{C}_{17}}{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\lambda }_j^{s-1}}\right)\le -\frac{3}{14}n{C}_{17}$ (4.28)

$\therefore \max \frac{{\left(q_j\right)}_{+}}{\left|q_n\right|}\le \frac{2}{3}$ , $\therefore d_H\left(A_k\right)\le \frac{2}{3}n$ , $d_F\left(A_k\right)\le \frac{4}{3}n$ (4.29)

It can be concluded that N-dimensional volume elements decay exponentially in $E_k$ and $d_H\left(A_k\right)<\frac{2}{3}n$ , $d_F\left(A_k\right)<\frac{4}{3}n$ , 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.

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