The Dynamic Behavior of a Class of Kirchhoff Equations with High Order Strong Damping

Guoguang Lin, Chunmeng Zhou^*

Department of Mathematics, Yunnan University, Kunming, China.

DOI: 10.4236/jamp.2021.95071                

Abstract

In this paper, we study the long time behavior of a class of Kirchhoff equations with high order strong dissipative terms. On the basis of the proper hypothesis of rigid term and nonlinear term of Kirchhoff equation, firstly, we evaluate the equation via prior estimate in the space E₀ and E_k, and verify the existence and uniqueness of the solution of the equation by using Galerkin’s method. Then, we obtain the bounded absorptive set B_0k on the basis of the prior estimate. Moreover, by using the Rellich-Kondrachov Compact Embedding theorem, we prove that the solution semigroup S(t) of the equation has the family of the global attractor A_k in space E_k. Finally, we prove that the solution semigroup S(t) is Frechet differentiable on E_k via linearizing the equation. Furthermore, we can obtain the finite Hausdorff dimension and Fractal dimension of the family of the global attractor A_k.

Keywords

Kirchhoff Equation, Prior Estimate, The Existence and Uniqueness of the Solution, Family of Global Attractor, Dimension Estimation

1. Introduction

In this paper, we mainly study the initial boundary value problem of the Kirchhoff equation with high order strong damping:

$\begin{array}{l}{u}_{tt}+M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m}u+\beta {\left(-\Delta \right)}^{2m}{u}_t+{\left|u\right|}^{\rho }u=f\left(x\right),\\ u\left(x,t\right)=\frac{{\partial }^{j}u}{\partial v^j}=0,j=1,2,\cdots ,2m-1,x\in \partial \Omega ,\\ u\left(x,0\right)=u_0\left(x\right),u_t\left(x,0\right)=u_1\left(x\right),x\in \partial \Omega ,t>0,\end{array}$ (1.1)

where $m>1,p\ge 2$, $\Omega \subseteq R^n\left(n\ge 1\right)$ is a bounded region with a smooth boundary $\partial \Omega$, $Q=\Omega \times \left[0,\infty \right)$ stands for the cylinder in $R_x^n\times R_t$, the rigid term $M\left(s\right)\in C^1\left[\mathrm{0,}\infty \right)$ is a general function, $\beta {\left(-\Delta \right)}^{2m}{u}_t\left(\beta >0\right)$ is the strong dissipative term, ${\left|u\right|}^{\rho }u$ is the nonlinear term and $\rho \ge -1$, $f\left(x\right)$ denotes the external force.

Kirchhoff equation is a kind of important nonlinear wave equation, which is widely used in engineering physics, especially provides a strong support for measuring bridge vibration. Kirchhoff equation originates from a physical model, which is obtained by German physicist Gustav Robert Kirchhoff [1] when he studied the transverse vibration of elastic string:

$\rho h\frac{{\partial }^{2}u}{\partial t^2}-\left[P_0+\frac{Eh}{2L}{\displaystyle {\int }_0^L}{\left(\frac{\partial u}{\partial x}\right)}^2\text{d}x\right]\frac{{\partial }^{2}u}{\partial x^2}=f,0<x<L,t\ge 0,$

As $m=1,p=2,\beta =1$, Masanori et al. studied Kirchhoff equations with nonlinear dissipative terms in the literature [2]:

$u_{tt}-M\left({\Vert \nabla u\Vert }^2\right)\Delta u+\delta {\left|u\right|}^{\rho }u+r{u}_t=f,$

and they had discussed the existence and attenuation of the global solution of the initial boundary value problem by using Galerkin’s method.

Then Igor Chueshov studied the Kirchhoff equation with strong dissipative term for $m=1,p=2,\beta =\sigma \left({\Vert \nabla u\Vert }^2\right)$ in reference [3]:

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

They proved that its weak solution exists and is unique. Particularly, the equation has strong exponential attractor when the nonlinear term $g\left(u\right)$ is in a non-supercritical state. Further, Guoguang Lin, Yuhang Chen [4] et al. extended the equation to a higher order Kirchhoff-type equation based on the study of Igor Chueshov, and added a structural dissipation term ${\Delta }^{2m}u$ :

$u_{tt}+N\left({\Vert {\nabla }^{m}u\Vert }_q^q\right){\left(-\Delta \right)}^{m}u+a\left(t\right){\left(-\Delta \right)}^m{u}_t+{\Delta }^{2m}u=I\left(x\right),$

They discussed the relationship between the order m and q, made reasonable assumptions about the relevant terms, proved the existence and uniqueness of the comprehensions, and made finite-dimensional estimates of the global attractors. When the coefficient of structural dissipation term is not equal to 1, they can’t get the relevant conclusion. More research results can be referred to [5] - [16].

On the basis of previous studies, we improve the order of strong dissipation term further in this paper. At the same time, summing up previous experience, we discuss the difficult problem of the relationship between order m and p in the rigid term and $\rho$ in the nonlinear term, and get some theoretical results about the long time behavior of the equation.

2. The Basic Assumptions

For the convenience of later narration, the space and sign mentioned in the article are defined as follows:

$H=L^2\left(\Omega \right)$, $H_0^m\left(\Omega \right)=H^m\left(\Omega \right)\cap H_0^1\left(\Omega \right)$, $E_k=H_0^{2m+k}\left(\Omega \right)\times H_0^k\left(\Omega \right)$, $A^k$ is the global attractor family from $E_0$ to $E_k$, $B_{0k}$ is a bounded absorption set in $E_0$, where $k=1,2,\cdots ,2m$, $C_i>0\left(i=0,1,2,\cdots \right)$ are constants. The inner product $\left(\cdot \mathrm{,}\cdot \right)$ and norm $\Vert \cdot \Vert$ of H are given by $\left(u,v\right)={\displaystyle {\int }_{\Omega }}u\left(x\right)v\left(x\right)\text{d}x$, $\Vert \cdot \Vert ={\Vert \cdot \Vert }_{L^2\left(\Omega \right)}$, ${\Vert \cdot \Vert }_p={\Vert \cdot \Vert }_{L^p\left(\Omega \right)}$.

$M\left(s\right)\mathrm{,}\rho$ and p satisfy the following conditions:

(A1) For $\forall s\ge 0$, we have $\epsilon +1\le {\mu }_0\le M\left(s\right)\le {\mu }_1$, where ${\mu }_0\mathrm{,}{\mu }_1$ are constants, and

$\mu =\{\begin{array}{cc}{\mu }_0,& \frac{\text{d}}{\text{d}t}{\Vert {\Delta }^{m}u\Vert }^2\ge 0\\ {\mu }_1,& \frac{\text{d}}{\text{d}t}{\Vert {\Delta }^{m}u\Vert }^2\le 0\end{array}$

(A2) $\{\begin{array}{cc}1\le \rho \le \frac{8m}{n-4m},& n>4m\\ \rho \ge -1,& n\le 4m\end{array}$ ;

(A3) $\{\begin{array}{cc}2\le p\le \frac{2n}{n-2m},& n>2m\\ p\ge 2,& n\le 2m\end{array}$ ;

(A4) $0\le \epsilon \le \min \left\{\sqrt{1+\frac{\beta {\lambda }_1^{2m}}{4}-1},\frac{2\mu }{\beta },\frac{2\mu {\lambda }_1^{2m}}{\beta {\lambda }_1^{2m}+1}\right\}$ ; ${\lambda }_1$ denotes the first eigenvalue of $-\Delta$ with the homogeneous Dirichlet boundary on $\Omega$.

3. Existence and Uniqueness of Solutions

Lemma 1. Assume (A1)-(A4) are valid. Let $\left(u_0\mathrm{,}{u}_1\right)\in E_0$, $f\left(x\right)\in L^2\left(\Omega \right)$, then the initial boundary value problem (1.1) has a global solution $\left(u\mathrm{,}v\right)$ that satisfies $u\in L^{\infty }\left(\mathrm{0,}+\infty \mathrm{<mo>\; ;\; </mo>}{H}_0^{2m}\left(\Omega \right)\right)$,

$v\in L^{\infty }\left(\mathrm{0,}+\infty \mathrm{<mo>\; ;\; </mo>}{L}^2\left(\Omega \right)\right)\cap L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}_0^{2m}\left(\Omega \right)\right)$, and

${\Vert \left(u,v\right)\Vert }_{E_0}^2={\Vert {\Delta }^{m}u\Vert }^2+{\Vert v\Vert }^2\le y_1\left(0\right){\text{e}}^{-{\alpha }_{1}t}+\frac{C_1}{{\alpha }_1},$ (3.1)

$\frac{\beta }{2}{\displaystyle {\int }_0^T}{\Vert {\Delta }^{m}v\Vert }^2\text{d}t\le C_{1}T+y_1\left(0\right)\mathrm{,}$ (3.2)

where $v=u_t+\epsilon u$, ${\alpha }_1=\min \left\{\frac{\beta {\lambda }_1^{2m}}{2}-2\epsilon -{\epsilon }^2,2\epsilon ,\frac{2\epsilon \mu -\beta {\epsilon }^2}{\mu },\epsilon \left(\rho +2\right)\right\}$, $y_1\left(0\right)={\Vert v_0\Vert }^2+{\epsilon }^2{\Vert u_0\Vert }^2+\mu {\Vert {\Delta }^m{u}_0\Vert }^2+\frac{2}{\rho +2}{\Vert u_0\Vert }_{\rho +2}^{\rho +2}$.

So there exists a nonnegative real number $R_1$ and $t_1=t_1\left(\Omega \right)>0$ such that

${\Vert \left(u,v\right)\Vert }_{E_0}^2={\Vert {\Delta }^{m}u\Vert }^2+{\Vert v\Vert }^2\le R_1^2\left(t>t_1\right).$ (3.3)

Proof. Let $v=u_t+\epsilon u$, take the inner product of Equation (1.1) with v in H, and we get that

$\left(u_{tt}+M\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m}u+\beta {\left(-\Delta \right)}^{2m}{u}_t+{\left|u\right|}^{\rho }u,v\right)=\left(f\left(x\right),v\right).$ (3.4)

We process the terms in Equation (3.4) by using Young’s inequality, Holder’s inequality, Poincare’s inequality and differential mean value theorem, then we obtain that

$\left(u_{tt},v\right)=\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert v\Vert }^2+\frac{{\epsilon }^2}{2}{\Vert u\Vert }^2-\epsilon {\Vert v\Vert }^2+{\epsilon }^3{\Vert u\Vert }^2.$ (3.5)

$\left(M\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m}u\mathrm{,}v\right)\ge \frac{\mu }{2}\frac{\text{d}}{\text{d}t}{\Vert {\Delta }^{m}u\Vert }^2+\epsilon \mu {\Vert {\Delta }^{m}u\Vert }^2\mathrm{,}$ (3.6)

$\left(\beta {\left(-\Delta \right)}^{2m}{u}_t\mathrm{,}v\right)\ge \frac{\beta }{4}{\Vert {\Delta }^{m}v\Vert }^2+\frac{\beta {\lambda }_1^{2m}}{4}{\Vert v\Vert }^2-\frac{\beta {\epsilon }^2}{2}{\Vert {\Delta }^{m}u\Vert }^2\mathrm{,}$ (3.7)

$\left({\left|u\right|}^{\rho }u,v\right)=\frac{1}{\rho +2}\frac{\text{d}}{\text{d}t}{\Vert u\Vert }_{\rho +2}^{\rho +2}+\epsilon {\Vert u\Vert }_{\rho +2}^{\rho +2},$ (3.8)

$\left(f\left(x\right)\mathrm{,}v\right)\le \frac{1}{2{\epsilon }^2}{\Vert f\Vert }^2+\frac{{\epsilon }^2}{2}{\Vert v\Vert }^2\mathrm{.}$ (3.9)

Substitute (3.5)-(3.9) into (3.4), we have

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left({\Vert v\Vert }^2+{\epsilon }^2{\Vert u\Vert }^2+\mu {\Vert {\Delta }^{m}u\Vert }^2+\frac{2}{\rho +2}{\Vert u\Vert }_{\rho +2}^{\rho +2}\right)+\left(\frac{\beta {\lambda }_1^{2m}}{2}-2\epsilon -{\epsilon }^2\right){\Vert v\Vert }^2\\ +2{\epsilon }^3{\Vert u\Vert }^2+\left(2\epsilon \mu -\beta {\epsilon }^2\right){\Vert {\Delta }^{m}u\Vert }^2+2\epsilon {\Vert u\Vert }_{\rho +2}^{\rho +2}+\frac{\beta }{2}{\Vert {\Delta }^{m}v\Vert }^2\le \frac{1}{{\epsilon }^2}{\Vert f\Vert }^2=C_1\mathrm{.}\end{array}$

Let ${\alpha }_1=\min \left\{\frac{\beta {\lambda }_1^{2m}}{2}-2\epsilon -{\epsilon }^2,2\epsilon ,\frac{2\epsilon \mu -\beta {\epsilon }^2}{\mu },\epsilon \left(\rho +2\right)\right\}$, then

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

where $y_1\left(t\right)={\Vert v\Vert }^2+{\epsilon }^2{\Vert u\Vert }^2+\mu {\Vert {\Delta }^{m}u\Vert }^2+\frac{2}{\rho +2}{\Vert u\Vert }_{\rho +2}^{\rho +2}$.

By using the differential Gronwall’s inequality, we obtain

$y_1\left(t\right)\le y_1\left(0\right){\text{e}}^{-{\alpha }_{1}t}+\frac{C_1}{{\alpha }_1},$

$\frac{\beta }{2}{\displaystyle {\int }_0^T}{\Vert {\Delta }^{m}v\Vert }^2\text{d}t\le C_{1}T+y_1\left(0\right)\mathrm{,}$

thus ${\Vert \left(u,v\right)\Vert }_{E_0}^2={\Vert {\Delta }^{m}u\Vert }^2+{\Vert v\Vert }^2\le y_1\left(t\right)$, moreover

$\overline{\underset{t\to +\infty }{lim}}{\Vert \left(u,v\right)\Vert }_{E_0}^2⩽\frac{C_1}{{\alpha }_1}=R_1^2.$

Lemma 1 is proved. $■$

Lemma 2. Assume (A1)-(A4) are valid. If $\left(u_0,u_1\right)\in E_k,k=1,2,\cdots ,2m$, $f\left(x\right)\in L^2\left(\Omega \right)$, then the initial boundary value problem (1) has a global solution $\left(u\mathrm{,}v\right)$ that satisfies $u\in L^{\infty }\left(\mathrm{0,}+\infty \mathrm{<mo>\; ;\; </mo>}{H}_0^{2m+k}\left(\Omega \right)\right)$, $v\in L^{\infty }\left(\mathrm{0,}+\infty \mathrm{<mo>\; ;\; </mo>}{H}_0^k\left(\Omega \right)\right)\cap L^2\left(\mathrm{0,}T\mathrm{<mo>\; ;\; </mo>}{H}_0^{2m+k}\left(\Omega \right)\right)$, and

${\Vert \left(u,v\right)\Vert }_{E_k}^2={\Vert {\nabla }^{2m+k}u\Vert }^2+{\Vert {\nabla }^{k}v\Vert }^2\le y_2\left(0\right)e^{-{\alpha }_{2}t}+\frac{C_3}{{\alpha }_2},$

$\frac{\beta }{4}{\displaystyle {\int }_0^T}{\Vert {\nabla }^{2m+k}v\Vert }^2\text{d}t\le C_{3}T+y_2\left(0\right)\mathrm{,}$ (3.10)

where $v=u_t+\epsilon u$, ${\alpha }_2=\min \left\{\frac{\beta {\lambda }_1^{2m}}{4}-2\epsilon -{\epsilon }^2,2\epsilon -\frac{{\epsilon }^2}{{\lambda }_1^{2m}\mu }-\frac{\beta {\epsilon }^2}{\mu }\right\}$, $y_2\left(0\right)={\Vert {\nabla }^k{v}_0\Vert }^2+\mu {\Vert {\nabla }^{2m+k}{u}_0\Vert }^2$.

Thus there exists a nonnegative real number $R_k$ and $t_2=t_2\left(\Omega \right)>0$ such that

${\Vert \left(u,v\right)\Vert }_{E_k}^2={\Vert {\nabla }^{2m+k}u\Vert }^2+{\Vert {\nabla }^{k}v\Vert }^2\le R_k^2,\left(t>t_2\right).$ (3.11)

Proof. Taking the inner product of Equation (1.1) with ${\left(-\Delta \right)}^{k}v$ in H, and we get that

$\left(u_{tt}+M\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m}u+\beta {\left(-\Delta \right)}^{2m}{u}_t+{\left|u\right|}^{\rho }u,{\left(-\Delta \right)}^{k}v\right)=\left(f\left(x\right),{\left(-\Delta \right)}^{k}v\right).$ (3.12)

Similar to Lemma 1, each item of Equation (3.12) can be obtained

$\left(u_{tt}\mathrm{,}{\left(-\Delta \right)}^{k}v\right)\ge \frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert {\nabla }^{k}v\Vert }^2-\frac{2\epsilon +{\epsilon }^2}{2}{\Vert {\nabla }^{k}v\Vert }^2-\frac{{\epsilon }^2}{2{\lambda }_1^{2m}}{\Vert {\nabla }^{2m+k}u\Vert }^2\mathrm{,}$ (3.13)

$\left(M\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m}u\mathrm{,}{\left(-\Delta \right)}^{k}v\right)\ge \frac{\mu }{2}\frac{\text{d}}{\text{d}t}{\Vert {\nabla }^{2m+k}u\Vert }^2+\epsilon \mu {\Vert {\nabla }^{2m+k}u\Vert }^2\mathrm{,}$ (3.14)

$\begin{array}{l}\left(\beta {\left(-\Delta \right)}^{2m}{u}_t\mathrm{,}{\left(-\Delta \right)}^{k}v\right)\\ \ge \frac{\beta }{8}{\Vert {\nabla }^{2m+k}v\Vert }^2-\frac{\beta {\epsilon }^2}{2}{\Vert {\nabla }^{2m+k}u\Vert }^2+\frac{\beta {\lambda }_1^{2m-k}}{4}{\Vert {\left(-\Delta \right)}^{k}v\Vert }^2+\frac{\beta {\lambda }_1^{2m}}{8}{\Vert {\nabla }^{k}v\Vert }^2\mathrm{,}\end{array}$ (3.15)

$\left|\left({\left|u\right|}^{\rho }u,{\left(-\Delta \right)}^{k}v\right)\right|\ge -{\Vert u\Vert }_{\infty }^{\rho +1}\Vert {\left(-\Delta \right)}^{k}v\Vert \ge -\frac{2C_2}{\beta {\lambda }_1^{2m-k}}-\frac{\beta {\lambda }_1^{2m-k}}{8}{\Vert {\left(-\Delta \right)}^{k}v\Vert }^2,$ (3.16)

$\left(f\left(x\right)\mathrm{,}{\left(-\Delta \right)}^{k}v\right)\le \frac{2}{\beta {\lambda }_1^{2m-k}}{\Vert f\left(x\right)\Vert }^2+\frac{\beta {\lambda }_1^{2m-k}}{8}{\Vert {\left(-\Delta \right)}^{k}v\Vert }^2\mathrm{.}$ (3.17)

Substitute inequality (3.13)-(3.17) into (3.12), we have

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left({\Vert {\nabla }^{k}v\Vert }^2+\mu {\Vert {\nabla }^{2m+k}u\Vert }^2\right)+\left(\frac{\beta {\lambda }_1^{2m}}{4}-2\epsilon -{\epsilon }^2\right){\Vert {\nabla }^{k}v\Vert }^2\\ +\left(2\epsilon \mu -\frac{{\epsilon }^2}{{\lambda }_1^{2m}}-\beta {\epsilon }^2\right){\Vert {\nabla }^{2m+k}u\Vert }^2+\frac{\beta }{4}{\Vert {\nabla }^{2m+k}v\Vert }^2\\ \le \frac{4}{\beta {\lambda }_1^{2m-k}}{\Vert f\left(x\right)\Vert }^2+\frac{4C_2}{\beta {\lambda }_1^{2m-k}}=C_3\mathrm{.}\end{array}$ (18)

Let ${\alpha }_2=\min \left\{\frac{\beta {\lambda }_1^{2m}}{4}-2\epsilon -{\epsilon }^2,2\epsilon -\frac{{\epsilon }^2}{{\lambda }_1^{2m}\mu }-\frac{\beta {\epsilon }^2}{\mu }\right\}$, then

$\frac{\text{d}{y}_2\left(x\right)}{\text{d}t}+{\alpha }_2{y}_2\left(x\right)+\frac{\beta }{4}{\Vert {\nabla }^{2m+k}v\Vert }^2\le C_3\mathrm{,}$

where $y_2\left(x\right)={\Vert {\nabla }^{k}v\Vert }^2+\mu {\Vert {\nabla }^{2m+k}u\Vert }^2$.

By using the differential Gronwall’s inequality, we obtain

$y_2\left(x\right)\le y_2\left(0\right){\text{e}}^{-{\alpha }_{2}t}+\frac{C_3}{{\alpha }_2},$

$\frac{\beta }{4}{\displaystyle {\int }_0^T}{\Vert {\nabla }^{2m+k}v\Vert }^2\text{d}t\le y_2\left(0\right)+C_{3}T\mathrm{.}$

So ${\Vert \left(u,v\right)\Vert }_{E_k}^2={\Vert {\nabla }^{2m+k}u\Vert }^2+{\Vert {\nabla }^{k}v\Vert }^2\le y_2\left(t\right)$, moreover

$\overline{\underset{t\to +\infty }{\lim }}{\Vert \left(u,v\right)\Vert }_{E_k}^2\le \frac{C_3}{{\alpha }_2}=R_k^2.$

Lemma 2 is proved. $■$

Lemma 3. (Daprato G. [9] ) Let D be the bounded region of $R_x^n\times R_t$, $g_{\mu }$ and g are functions of $L^q\left(D\right)\left(1<q<\infty \right)$, ${\Vert g_{\mu }\Vert }_{L^q\left(D\right)}<C,g_{\mu }\to g$ a.e. in D, then $g_{\mu }\to g$ is weak convergence in $L^q\left(D\right)$.

Theorem 1. Assume (A1)-(A4) are valid. If $f\in H,\left(u_0,u_1\right)\in E_k$, then the initial boundary value problem (1.1) exists a unique smooth solution $\left(u\mathrm{,}v\right)\in L^{\infty }\left(\left[\mathrm{0,}+\infty \right)\mathrm{<mo>\; ;\; </mo>}{E}_k\right)$.

Proof. First of all, we prove the existence of solution.

Let ${\left(-\Delta \right)}^{2m+k}{\omega }_j={\lambda }_j^{2m+k}{\omega }_j\left(k=1,2,\cdots ,2m\right)$, where ${\lambda }_j$ represents the eigenvalue of $-\Delta$ with the homogeneous Dirichlet boundary on $\Omega$, ${\omega }_j$ represents the eigenfunction determined by the corresponding eigenvalue ${\lambda }_j$, then ${\omega }_1\mathrm{,}{\omega }_2\mathrm{,}\cdots \mathrm{,}{\omega }_h$ constitute the standard orthonormal basis of H. Suppose $u_h=u_h\left(t\right)={\displaystyle {\sum }_{j=1}^h}{g}_{jh}\left(t\right){\omega }_j$ is the approximate solution of the initial boundary value problem (1.1), where $g_{jh}\left(t\right)\left(1\le j\le h\right)$ is determined by

$\begin{array}{l}\left({u^{″}}_h+M\left({\Vert D^m{u}_h\Vert }_p^p\right){\left(-\Delta \right)}^{2m}{u}_h+\beta {\left(-\Delta \right)}^{2m}{u^{\prime }}_h+{\left|u_h\right|}^{\rho }{u}_h,{\left(-\Delta \right)}^k{\omega }_j\right)\\ =\left(f\left(x\right),{\left(-\Delta \right)}^k{\omega }_j\right)\left(j=1,2,\cdots ,h\right)\end{array}$ (3.19)

Equation (3.19) satisfies the initial conditions

$u_h\left(0\right)=u_{0h}={\displaystyle \underset{j=1}{\overset{h}{\sum }}}{\alpha }_{jh}\left(t\right){\omega }_j\to u_0\left(h\to +\infty \right)\text{in}{H}_0^{2m+k}\mathrm{,}$

${u^{\prime }}_h\left(0\right)=u_{1h}={\displaystyle \underset{j=1}{\overset{h}{\sum }}}{\beta }_{jh}\left(t\right){\omega }_j\to u_1\left(h\to +\infty \right)\text{in}{H}_0^k\mathrm{,}$

namely as $h\to +\infty$, $\left(u_{0h}\mathrm{,}{u}_{1h}\right)\to \left(u_0\mathrm{,}{u}_1\right)$ in $E_k$. According to the basic theory of solutions of ordinary differential equations, the approximate solution $u_h\left(t\right)$ exists on $\left(\mathrm{0,}{t}_h\right)$.

Multiply both sides of (3.19) by ${g^{\prime }}_{jh}+\epsilon g_{jh}\left(t\right)$ and sum over j. Let $v_h\left(t\right)={u^{\prime }}_h\left(t\right)+\epsilon u_h\left(t\right)$, according to Lemma 1 and Lemma 2, estimation (3.3), (3.10), (3.11) still hold for $u_h\in H_0^{2m+k}\left(\Omega \right)\mathrm{,}{v}_h\in H_0^k\left(\Omega \right)$, namely we have

${\Vert \left(u_h,v_h\right)\Vert }_{E_k}^2={\Vert {\nabla }^{2m+k}{u}_h\Vert }^2+{\Vert {\nabla }^k{v}_h\Vert }^2\le R_k^2,$ (3.20)

$\frac{\beta }{4}{\displaystyle {\int }_0^T}{\Vert {\nabla }^{2m+k}{v}_h\Vert }^2\text{d}t\le y_2\left(0\right)+{\displaystyle {\int }_0^T}{C}_3\text{d}t\mathrm{,}$ (3.21)

then we obtain $\left(u_h\left(t\right)\mathrm{,}{u^{\prime }}_h\left(t\right)\right)$ is uniformly bounded on $E_k$, and (3.20), (3.21) hold the priori estimate of the solution in the Lemma 2. Moreover, we know from (3.20) that $\left(u_h\mathrm{,}{v}_h\right)$ is bounded in $L^{\infty }\left(\left[\mathrm{0,}+\infty \right)\mathrm{<mo>\; ;\; </mo>}{E}_k\right)$, from (3.21) that $v_h$ is bounded in $L^2\left(\left(\mathrm{0,}T\right)\mathrm{<mo>\; ;\; </mo>}{H}_0^{2m+k}\right)$.

In space $E_k$, select the subsequence $\left\{u_s\right\}$ from the sequence $\left\{u_h\right\}$ such that $\left(u_s\mathrm{,}{v}_s\right)\to \left(u\mathrm{,}v\right)$ is weak * convergence in $L^{\infty }\left(\left[\mathrm{0,}+\infty \right)\mathrm{<mo>\; ;\; </mo>}{E}_k\right)$, and we obtain that $v_s$ is bounded in $L^2\left(\left(\mathrm{0,}T\right)\mathrm{<mo>\; ;\; </mo>}{H}_0^{2m+k}\right)$ via (3.21).

Due to the Rellich-Kohdrachov Compact Embedding theorem, $E_k$ ↪ $E_0$, $\left(u_s\mathrm{,}{v}_s\right)\to \left(u\mathrm{,}v\right)$ is strong convergence a.e. in $E_0$.

According to Lemma 3, taking $g_s={\left|u_s\right|}^{\rho }{u}_s$, $q=\frac{\rho +2}{\rho +1}$, $g_s\to {\left|u\right|}^{\rho }u$ a.e. in $L^q\left(Q\right)$, where $Q=\left[0,+\infty \right)\times \Omega$, then

${\left|u_s\right|}^{\rho }{u}_s\to {\left|u\right|}^{\rho }u$ is weak * convergence in $L^{\infty }\left(\left[\mathrm{0,}+\infty \right)\mathrm{<mo>\; ;\; </mo>}{L}^q\left(\Omega \right)\right)$. (3.22)

Then in Equation (3.19), let $h=s$ and take the limit. For fixed j and $s\ge j$, we obtain

$\begin{array}{l}\left({u^{″}}_s+M\left({\Vert D^m{u}_s\Vert }_p^p\right){\left(-\Delta \right)}^{2m}{u}_s+\beta {\left(-\Delta \right)}^{2m}{u^{\prime }}_s+{\left|u_s\right|}^{\rho }{u}_s,{\left(-\Delta \right)}^k{\omega }_j\right)\\ =\left(f\left(x\right),{\left(-\Delta \right)}^k{\omega }_j\right).\end{array}$

Due to $u_h\to u$ is weak * convergence in $L^{\infty }\left(\left[\mathrm{0,}+\infty \right)\mathrm{<mo>\; ;\; </mo>}{H}_0^{2m+k}\left(\Omega \right)\right)$, we have

$\left(u_s\left(t\right)\mathrm{,}{\left(-\Delta \right)}^k{\omega }_j\right)\to \left(u\left(t\right)\mathrm{,}{\lambda }_j^k{\omega }_j\right)$ is weak * convergence in $L^{\infty }\left[\mathrm{0,}+\infty \right)$,

$\left({u^{\prime }}_s\left(t\right)\mathrm{,}{\left(-\Delta \right)}^k{\omega }_j\right)\to \left(u^{\prime }\left(t\right)\mathrm{,}{\lambda }_j^k{\omega }_j\right)$ is weak * convergence in $L^{\infty }\left[\mathrm{0,}+\infty \right)$.

Then, we obtain $\left({u^{″}}_s\left(t\right),{\left(-\Delta \right)}^k{\omega }_j\right)=\frac{\text{d}}{\text{d}t}\left({u^{\prime }}_s\left(t\right),{\left(-\Delta \right)}^k{\omega }_j\right)\to \left(u^{″}\left(t\right),{\lambda }_j^k{\omega }_j\right)$ is convergence in $D^{\prime }\left[\mathrm{0,}+\infty \right)$, where $D^{\prime }\left[\mathrm{0,}+\infty \right)$ is the conjugate space of the infinitely differentiable space $D\left[\mathrm{0,}+\infty \right)$.

$\left(M\left({\Vert D^m{u}_s\Vert }_p^p\right){\left(-\Delta \right)}^{2m}{u}_s,{\left(-\Delta \right)}^k{\omega }_j\right)\to \left(M\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{\frac{2m+k}{2}}u,{\lambda }_j^{\frac{2m+k}{2}}{\left(-\Delta \right)}^k{\omega }_j\right)$ is weak * convergence in $L^{\infty }\left[\mathrm{0,}+\infty \right)$.

Due to

$\begin{array}{l}\left(\beta {\left(-\Delta \right)}^{2m}{u^{\prime }}_s,{\left(-\Delta \right)}^k{\omega }_j\right)\\ =\left(\beta {\left(-\Delta \right)}^{\frac{k}{2}}{v}_s,{\left(-\Delta \right)}^{\frac{2m+k}{2}}{\omega }_j\right)-\beta \epsilon \left({\left(-\Delta \right)}^{\frac{2m+k}{2}}{u}_s,{\left(-\Delta \right)}^{\frac{2m+k}{2}}{\omega }_j\right)\end{array}$,

$\begin{array}{l}\left(\beta {\left(-\Delta \right)}^{2m}{u^{\prime }}_s,{\left(-\Delta \right)}^k{\omega }_j\right)\\ \to \left(\beta {\left(-\Delta \right)}^{\frac{k}{2}}v,{\lambda }_j{\left(-\Delta \right)}^{\frac{2m+k}{2}}{\omega }_j\right)-\beta \epsilon \left({\left(-\Delta \right)}^{\frac{2m+k}{2}}u,{\lambda }_j^{\frac{2m+k}{2}}{\omega }_j\right)\end{array}$ is

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

According to Equation (3.22), we have

$\left({\left|u_s\right|}^{\rho }{u}_s\mathrm{,}{\left(-\Delta \right)}^k{\omega }_j\right)\to \left({\left|u\right|}^{\rho }u\mathrm{,}{\lambda }_j^k{\omega }_j\right)$ is weak * convergence in $L^{\infty }\left[\mathrm{0,}+\infty \right)$.

Because ${\omega }_j$ is arbitrary, for $\forall v\in H^k\left(\Omega \right)\cap H_0^1\left(\Omega \right)$ we have

$\begin{array}{l}\left(u_{tt}+M\left({\Vert D^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m}u+\beta {\left(-\Delta \right)}^{2m}{u}_t+{\left|u\right|}^{\rho }u,{\left(-\Delta \right)}^k{\omega }_j\right)\\ =\left(f\left(x\right),{\left(-\Delta \right)}^k{\omega }_j\right).\end{array}$ (3.23)

Hence, Equation (3.23) is established for all j. The existence is proved.

Then we prove the solution of the initial and boundary value problem is unique.

Assume $u_1\mathrm{,}{u}_2$ are the solution of the initial and boundary value problem (1.1). Let $w=u_1-u_2$, we have

$\begin{array}{l}{w}_{tt}+M\left({\Vert D^m{u}_1\Vert }_p^p\right){\left(-\Delta \right)}^{2m}{u}_1-M\left({\Vert D^m{u}_2\Vert }_p^p\right){\left(-\Delta \right)}^{2m}{u}_2\\ +\beta {\left(-\Delta \right)}^{2m}{w}_t+{\left|u_1\right|}^{\rho }{u}_1-{\left|u_2\right|}^{\rho }{u}_2=0,\\ w\left(0\right)=0,w_t\left(0\right)=0,\\ w\in L^{\infty }\left(\left[0,+\infty \right);H_0^{2m}\left(\Omega \right)\right),w_t\in L^{\infty }\left(\left[0,+\infty \right);L^2\left(\Omega \right)\right).\end{array}$ (3.24)

Taking the inner product of Equation (3.24) with $w_t$ in H, and we get that

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert w_t\Vert }^2+\beta {\Vert {\Delta }^m{w}_t\Vert }^2+\left(M\left({\Vert {\nabla }^m{u}_1\Vert }_p^p\right){\left(-\Delta \right)}^{2m}{u}_1-M\left({\Vert {\nabla }^m{u}_2\Vert }_p^p\right){\left(-\Delta \right)}^{2m}{u}_2\mathrm{,}{w}_t\right)\\ +\left({\left|u_1\right|}^{\rho }{u}_1-{\left|u_2\right|}^{\rho }{u}_2\mathrm{,}{w}_t\right)=0.\end{array}$ (3.25)

According to Lemma 1 and Lemma 2, the Equation (3.25) is processed as follows

$\begin{array}{l}\left(M\left({\Vert {\nabla }^m{u}_1\Vert }_p^p\right){\left(-\Delta \right)}^{2m}{u}_1-M\left({\Vert {\nabla }^m{u}_2\Vert }_p^p\right){\left(-\Delta \right)}^{2m}{u}_2\mathrm{,}{w}_t\right)\\ \ge \frac{1}{2}M\left({\Vert {\nabla }^m{u}_1\Vert }_p^p\right)\frac{\text{d}}{\text{d}t}{\Vert {\Delta }^{m}w\Vert }^2-M^{\prime }\left(\xi \right)\left({\Vert {\nabla }^m{u}_1\Vert }_p^p-{\Vert {\nabla }^m{u}_2\Vert }_p^p\right)\left({\left(-\Delta \right)}^{2m}{u}_2\mathrm{,}{w}_t\right)\\ \ge \frac{\mu }{2}\frac{\text{d}}{\text{d}t}{\Vert {\Delta }^{m}w\Vert }^2-\frac{\beta }{2}{\Vert {\Delta }^m{w}_t\Vert }^2-\frac{C_4^2}{2\beta {\lambda }_1^m}{\Vert {\Delta }^{m}w\Vert }^2\mathrm{,}\end{array}$ (3.26)

$\begin{array}{c}\left|\left({\left|u_1\right|}^{\rho }{u}_1-{\left|u_2\right|}^{\rho }{u}_2\mathrm{,}{w}_t\right)\right|\ge -C_5\left|{\displaystyle {\int }_{\Omega }}\left({\left|u_1\right|}^{\rho }+{\left|u_2\right|}^{\rho }\right)\left|w\right|\left|w_t\right|\text{d}x\right|\\ \ge -\frac{C_6}{2{\lambda }_1^{2m}}{\Vert {\Delta }^{m}w\Vert }^2-\frac{C_6}{2}{\Vert w_t\Vert }^2\mathrm{.}\end{array}$ (3.27)

Substituting (3.26)-(4.1) into (3.25), we obtain

$\frac{\text{d}}{\text{d}t}\left({\Vert w_t\Vert }^2+\mu {\Vert {\Delta }^{m}w\Vert }^2\right)\le \left|\beta {\lambda }_1^{2m}-C_6\right|{\Vert w_t\Vert }^2+\left(\frac{C_4^2}{\beta {\lambda }_1^m}+\frac{C_6}{{\lambda }_1^{2m}}\right){\Vert {\Delta }^{m}w\Vert }^2\mathrm{.}$

Let ${\alpha }_3=\max \left\{\left|\beta {\lambda }_1^{2m}-C_6\right|,\frac{C_4^2}{\beta \mu {\lambda }_1^m}+\frac{C_6}{\mu {\lambda }_1^{2m}}\right\}$, by using the integral Gronwall’s inequality we receive that

${\Vert w_t\left(t\right)\Vert }^2+\mu {\Vert {\Delta }^{m}w\left(t\right)\Vert }^2\le \left({\Vert w_t\left(0\right)\Vert }^2+\mu {\Vert {\Delta }^{m}w\left(0\right)\Vert }^2\right){\text{e}}^{{\alpha }_{3}t}=0.$

Hence $w_t\left(t\right)=0,{\Delta }^{m}w\left(t\right)=0$, a.e. $w\left(t\right)=0,u_1=u_2$. The uniqueness of solution is proved. $■$

4. Global Attractor and Dimension Estimation

Theorem 2. [9] Assume E is a Banach space, and ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ are the semigroup operators on E. $S\left(t\right):E\to E$, $S\left(t+s\right)=S\left(t\right)S\left(s\right)\left(\forall t,s\ge 0\right)$, $S\left(0\right)=I$, where I is the unit operator. Suppose $S\left(t\right)$ satisfies the following conditions:

1) $S\left(t\right)$ is uniformly bounded, i.e. $\forall R>0,{\Vert u\Vert }_E\le R$, there is a constant $C\left(R\right)$, such that

${\Vert S\left(t\right)u\Vert }_E\le C\left(R\right)\mathrm{,}\left(\forall t\in \left[\mathrm{0,}+\infty \right)\right)\mathrm{<mo>\; ;\; </mo>}$

2) There is a bounded absorbing set $B_0\subset E$, i.e. $\forall B\subset E$, there exists a constant $t_0$, so that

$S\left(t\right)B\subset B_0\mathrm{,}\left(\forall t\ge t_0\right)\mathrm{<mo>\; ;\; </mo>}$

3) As $t>0$, $S\left(t\right)$ is a completely continuous operator.

It is said that the semigroup operator $S\left(t\right)$ exists a compact global attractor A.

If $S\left(t\right)$ is the solution semigroup that generated by the initial boundary value problem (1.1), i.e. $u\left(t\right)=S\left(t\right)u_0$, according to Lemma 1 and Lemma 2 we obtain the existence theorem of the following family of global attractors.

Theorem 3. Under the assumption of Theorem 1, let $S\left(t\right)$ be the solution semigroup that generated by the initial boundary value problem (1.1), then the initial boundary value problem (1.1) exists a family of global attractors, i.e. exists compact set $A_k\subset E_k\subset E_0$, and

$A_k=\omega \left(B_{0k}\right)={\displaystyle \underset{s\ge 0}{\cap }}\overline{{\displaystyle \underset{t\ge s}{\cup }}S\left(t\right)B_{0k}},$

where $B_{0k}=\left\{\left(u,v\right)\in E_k:{\Vert \left(u,v\right)\Vert }_{E_k}^2={\Vert u\Vert }_{H_0^{2m+k}\left(\Omega \right)}^2+{\Vert v\Vert }_{H_0^k\left(\Omega \right)}^2\le R_k^2\right\}$, $B_{0k}$ is the bounded absorbing set in $E_k$ and satisfies:

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

2) ${\lim }_{t\to +\infty }dist\left(S\left(t\right)B_{0k},A_k\right)=0$, where for arbitrary bounded set $B_{0k}\subset E_k$, we have

$dist\left(S\left(t\right)B_{0k},A_k\right)=\underset{x\in B_{0k}}{\sup }\underset{y\in A_k}{\inf }{\Vert S\left(t\right)x-y\Vert }_{E_k}.$

Proof. The proof can be obtained by verifying the three conditions in Theorem 2. Under the conditions in Theorem 1, suppose that the Equation (1.1) has a solution semigroup $S\left(t\right)\mathrm{<mo>\; :\; </mo>}{E}_k\to E_k$.

(1)According to Lemma 2, for arbitrary $B_{0k}\subset E_k$, it contains the bounded set of the sphere $\left\{{\Vert \left(u\mathrm{,}v\right)\Vert }_{E_k}\le R\right\}$ such that there exists a constant C, then we have

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

where $t\ge 0$, $\left(u_0,v_0\right)\in B_{0k}$, ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ is uniformly bounded in $E_k$.

2) Further, $\forall \left(u_0,v_0\right)\in E_k,t\ge \max \left(t_1,t_2\right)$, we have

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

so $B_{0k}$ is a bounded absorption set of the semigroup $S\left(t\right)$.

3) Due to $E_k$ ↪ $E_0$, we obtain that the bounded set in $E_k$ is the compact set in $E_0$. Hence, $S\left(t\right)$ is completely continuous operator.

Theorem 3 is proved. $■$

In order to estimate the Hausdorff dimension and Fractal dimension of the family of global attractor, we linearize the problem (1.1):

$\begin{array}{l}{U}_{tt}+M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m}U+M^{\prime }\left({\Vert {\nabla }^{m}u\Vert }_p^p\right){\left({\Vert {\nabla }^{m}u\Vert }_p^p\right)}^{\prime }{\nabla }^{m}U{\left(-\Delta \right)}^{2m}u\\ +\beta {\left(-\Delta \right)}^{2m}{U}_t+\left(\rho +1\right){\left|u\right|}^{\rho }U=0,\\ U\left(x,t\right)=0,x\in \partial \Omega ,t>0,\\ U\left(x,0\right)=\xi ,U_t\left(x,0\right)=\eta ,\end{array}$ (4.1)

where $\left(\xi \mathrm{,}\eta \right)\in E_k$, $\left(u\mathrm{,}{u}_t\right)\in S\left(t\right)\left(u_0\mathrm{,}{u}_1\right)$ is the solution of question (1.1) obtained by $\left(u_0\mathrm{,}{u}_1\right)\in A_k$. Then for a given $\left(u_0\mathrm{,}{u}_1\right)$, $S\left(t\right)\mathrm{<mo>\; :\; </mo>}{E}_k\to E_k$, it can be proved that for arbitrary $\left(\xi \mathrm{,}\eta \right)\in E_k$, there is a unique solution $\left(U\left(t\right)\mathrm{,}{U}_t\left(t\right)\right)\in L^{\infty }\left(\left[\mathrm{0,}+\infty \right)\mathrm{<mo>\; ;\; </mo>}{E}_k\right)$ to the linear problem (4.1).

Lemma 4. For $\forall t>0,R>0$, the mapping $S\left(t\right)\mathrm{<mo>\; :\; </mo>}{E}_k\to E_k$ is differentiable on $E_k$. Then the derivative of $\phi =\left(u_0,u_1\right)$ is a linear operator of $G\left(t\right)\mathrm{<mo>\; :\; </mo>}\left(\xi \mathrm{,}\eta \right)\to \left(U\left(t\right)\mathrm{,}{U}_t\left(t\right)\right)$, where $\left(U\left(t\right)\mathrm{,}{U}_t\left(t\right)\right)$ is the solution of the linear initial boundary value problem (4.1).

Proof. Let ${\phi }_0=\left(u_0,v_0\right)\in E_k$, $\widetilde{{\phi }_0}=\left(u_0+\xi ,v_0+\eta \right)\in E_k$, and ${\Vert {\phi }_0\Vert }_{E_k}\le R$, ${\Vert \widetilde{{\phi }_0}\Vert }_{E_k}\le R$, define $\left(u,v\right)=S\left(t\right){\phi }_0$, $\left(\tilde{u},\tilde{v}\right)=S\left(t\right)\widetilde{{\phi }_0}$, then $S\left(t\right)$ has Lipschitz property on the bounded set $E_k$ :

${\Vert S\left(t\right){\phi }_0-S\left(t\right)\widetilde{{\phi }_0}\Vert }_{E_k}^2\le {\text{e}}^{ct}{\Vert \left(\xi \mathrm{,}\eta \right)\Vert }_{E_k}^2\mathrm{.}$ (4.2)

Let $\Theta =\tilde{u}-u-U$ is the solution of the linear initial boundary value problem (4.1), then we have

$\begin{array}{l}{\Theta }_{tt}+M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m}\Theta +\beta {\left(-\Delta \right)}^{2m}{\Theta }_t=h_1+h_2,\\ \Theta \left(0\right)={\Theta }_t\left(0\right)=0.\end{array}$ (4.3)

$\begin{array}{l}{h}_1=\left(M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right)-M\left({\Vert {\nabla }^m\tilde{u}\Vert }_p^p\right)\right){\left(-\Delta \right)}^{2m}\tilde{u}\\ +M^{\prime }\left({\Vert {\nabla }^{m}u\Vert }_p^p\right){\left({\Vert {\nabla }^{m}u\Vert }_p^p\right)}^{\prime }{\nabla }^{m}U{\left(-\Delta \right)}^{2m}u,\end{array}$

$h_2={\left|u\right|}^{\rho }u-{\left|\tilde{u}\right|}^{\rho }\tilde{u}+\left(\rho +1\right){\left|u\right|}^{\rho }U,$

Let $s={\nabla }^{m}u$, $\tilde{s}={\nabla }^m\tilde{u}$, $\bar{u}=\tilde{u}-u$, ${\xi }_1=\left(1-{\alpha }_4\right)\tilde{s}+{\alpha }_{4}s$, ${\xi }_2=\left(1-{\alpha }_5\right)\tilde{s}+{\alpha }_5{\xi }_1$, $N\left(\xi \right)=M^{\prime }\left({\Vert s\Vert }_p^p\right){\left({\Vert s\Vert }_p^p\right)}^{\prime }$, ${\alpha }_4,{\alpha }_5\in \left(0,1\right)$.

Then

$\begin{array}{c}{h}_1=-N\left({\xi }_1\right){\nabla }^m\bar{u}{\left(-\Delta \right)}^{2m}\tilde{u}+N\left(s\right){\nabla }^m\bar{u}{\left(-\Delta \right)}^{2m}u-N\left(s\right){\nabla }^m\Theta {\left(-\Delta \right)}^{2m}u\\ =-N\left({\xi }_1\right){\nabla }^m\bar{u}{\left(-\Delta \right)}^{2m}\bar{u}-N^{\prime }\left({\xi }_2\right)\left(1-{\alpha }_4\right){\left({\nabla }^m\bar{u}\right)}^2{\left(-\Delta \right)}^{2m}u\\ -N\left(s\right){\nabla }^m\Theta {\left(-\Delta \right)}^{2m}u,\end{array}$

$\begin{array}{c}{h}_2=-\left(\rho +1\right){\left|{\xi }_3\right|}^{\rho }\bar{u}+\left(\rho +1\right){\left|u\right|}^{\rho }\bar{u}-\left(\rho +1\right){\left|u\right|}^{\rho }\Theta \\ =\rho \left(\rho +1\right){\left|{\xi }_4\right|}^{\rho -1}{\bar{u}}^2-\left(\rho +1\right){\left|u\right|}^{\rho }\Theta .\end{array}$

Taking the inner product of Equation (4.3) with $s={\nabla }^{m}u$, we receive that

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert {\nabla }^k{\Theta }_t\Vert }^2+\frac{1}{2}M\left({\Vert {\nabla }^{k}u\Vert }_p^p\right)\frac{\text{d}}{\text{d}t}{\Vert {\nabla }^{2m+k}\Theta \Vert }^2+\beta {\Vert {\nabla }^{2m+k}{\Theta }_t\Vert }^2\\ =\left(h_1+h_2,{\left(-\Delta \right)}^k{\Theta }_t\right),\end{array}$ (4.4)

$\begin{array}{c}\left|\left(h_1\mathrm{,}{\left(-\Delta \right)}^k{\Theta }_t\right)\right|\le {\Vert N\left({\xi }_1\right)\Vert }_{\infty }\Vert {\nabla }^{2m+k}\bar{u}\Vert \Vert {\Delta }^m\bar{u}\Vert \Vert {\nabla }^{m+k}{\Theta }_t\Vert \\ +C_7{\Vert N^{\prime }\left({\xi }_2\right)\Vert }_{\infty }{\Vert {\nabla }^{2m+k}\bar{u}\Vert }^2\Vert {\left(-\Delta \right)}^{2}u\Vert \Vert {\Theta }_t\Vert \\ +{\Vert N\left(s\right)\Vert }_{\infty }\Vert {\left(-\Delta \right)}^{m}u\Vert \Vert {\nabla }^{m+k}\Theta \Vert \Vert {\nabla }^k{\Theta }_t\Vert \\ \le \left(\frac{C_8}{2{\lambda }_1^k}+\frac{C_{10}}{2}\right){\Vert {\nabla }^k{\Theta }_t\Vert }^2+\frac{C_{10}}{2{\lambda }_1^m}{\Vert {\nabla }^{2m+k}\Theta \Vert }^2\\ +\frac{\beta }{2}{\Vert {\nabla }^{2m+k}{\Theta }_t\Vert }^2+\left(\frac{C_8}{2}+\frac{C_9^2}{2\beta {\lambda }_1^{m+k}}\right){\Vert {\nabla }^{2m+k}\bar{u}\Vert }^4\mathrm{,}\end{array}$ (4.5)

$\begin{array}{l}\left|\left(h_2\mathrm{,}{\left(-\Delta \right)}^k{\Theta }_t\right)\right|\\ \le \rho \left(\rho +1\right){\Vert {\xi }_4\Vert }_{\infty }^{\rho -1}{\Vert {\nabla }^k\bar{u}\Vert }^2\Vert {\Theta }_t\Vert +\left(\rho +1\right){\Vert \left|u\right|\Vert }_{\infty }^{\rho }\Vert {\nabla }^k\Theta \Vert \Vert {\nabla }^k{\Theta }_t\Vert \\ =\frac{C_{11}}{2{\lambda }_1^{4m}}{\Vert {\nabla }^{2m+k}\bar{u}\Vert }^4+\frac{C_{12}}{2{\lambda }_1^{2m}}{\Vert {\nabla }^{2m+k}\Theta \Vert }^2+\left(\frac{C_{11}}{2{\lambda }_1^k}+\frac{C_{12}}{2}\right){\Vert {\nabla }^k{\Theta }_t\Vert }^2\mathrm{.}\end{array}$ (4.6)

According to (4.4)-(4.6), we obtain

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left({\Vert {\nabla }^k{\Theta }_t\Vert }^2+\mu {\Vert {\nabla }^{2m+k}\Theta \Vert }^2\right)+\beta {\Vert {\nabla }^{2m+k}{\Theta }_t\Vert }^2\\ \le \left(\frac{C_8+C_{11}}{{\lambda }_1^k}+C_{10}+C_{12}\right){\Vert {\nabla }^k{\Theta }_t\Vert }^2+\left(\frac{C_{10}}{{\lambda }_1^m}+\frac{C_{12}}{{\lambda }_1^{2m}}\right){\Vert {\nabla }^{2m+k}\Theta \Vert }^2\\ +\left(C_8+\frac{C_9^2}{\beta {\lambda }_1^{m+k}}+\frac{C_{11}}{{\lambda }_1^{4m}}\right){\Vert {\nabla }^{2m+k}\bar{u}\Vert }^4\mathrm{.}\end{array}$

Let ${\alpha }_4=\max \left\{\frac{C_8+C_{11}}{{\lambda }_1^k}+C_{10}+C_{12},\frac{C_{10}}{\mu {\lambda }_1^m}+\frac{C_{12}}{\mu {\lambda }_1^{2m}}\right\}$, we receive

$\frac{\text{d}}{\text{d}t}\left({\Vert {\nabla }^k{\Theta }_t\Vert }^2+\mu {\Vert {\nabla }^{2m+k}\Theta \Vert }^2\right)\le {\alpha }_4\left({\Vert {\nabla }^k{\Theta }_t\Vert }^2+\mu {\Vert {\nabla }^{2m+k}\Theta \Vert }^2\right)+C_{13}{\Vert {\nabla }^{2m+k}\bar{u}\Vert }^4.$ (4.7)

By using the differential Gronwall’s inequality, we have

${\Vert {\nabla }^k{\Theta }_t\Vert }^2+\mu {\Vert {\nabla }^{2m+k}\Theta \Vert }^2\le C_{14}{\text{e}}^{C_{15}t}{\Vert {\left(\xi ,\eta \right)}^{\text{T}}\Vert }_{E_k}^4.$

According to the Lipschitz property of $S\left(t\right)$, as ${\Vert {\left(\xi \mathrm{,}\eta \right)}^{\text{T}}\Vert }_{E_k}^2\to 0$, we obtain

$\frac{{\Vert S\left(t\right)\widetilde{{\phi }_0}-S\left(t\right){\phi }_0-G\left(t\right)\left(\xi ,\eta \right)\Vert }_{E_k}^2}{{\Vert {\left(\xi ,\eta \right)}^{\text{T}}\Vert }_{E_k}^2}\le C_{14}{\text{e}}^{C_{15}t}{\Vert {\left(\xi ,\eta \right)}^{\text{T}}\Vert }_{E_k}^2\to 0.$

That means $S\left(t\right)$ is uniformly differentiable on $E_k$. $■$

Theorem 4. Under the conditions of Theorem 3, the family of global attractor $A_k$ of the initial boundary value problem (1.1) have the finite Hausdorff dimension and Fractal dimension, and $d_H\left(A_k\right)<\frac{N}{2},d_F\left(A_k\right)<\frac{3N}{2}$.

Proof. In order to estimate the dimension of the family of global attractor $A_k$, we rewrite the initial boundary value problem as a first-order evolution equation

${\Psi }_t+{\Lambda }_{\epsilon }\Psi +g\left(\Psi \right)=F\left(\Psi \right),$

where $\Psi ={\left(u,v\right)}^{\text{T}}={\left(u,u_t+\epsilon u\right)}^{\text{T}}$, $g\left(\Psi \right)={\left(0,{\left|u\right|}^{\rho }u\right)}^{\text{T}}$, $F\left(\Psi \right)={\left(0,f\left(x\right)\right)}^{\text{T}}$,

${\Lambda }_{\epsilon }=\left(\begin{array}{cc}\epsilon I& -I\\ \left(M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right)-\epsilon \beta \right){\left(-\Delta \right)}^{2m}+{\epsilon }^{2}I& \beta {\left(-\Delta \right)}^{2m}-\epsilon I\end{array}\right).$

Further, let $L\left(\Psi \right)={\Psi }_t=F\left(\Psi \right)-{\Lambda }_{\epsilon }\Psi -g\left(\Psi \right)$, where $L\mathrm{<mo>\; :\; </mo>}{E}_k\to E_k$ is Frechet differential.

Similarly, we rewrite the linear Equation (4.1) as

$P_t+\overline{{\Lambda }_{\epsilon }}P+g^{\prime }\left(\varphi \right)P=0,$ (4.8)

where $P={\left(U,U_t+\epsilon U\right)}^{\text{T}}$, $P_t=L_t\left(\Psi \right)$, $g^{\prime }\left(\varphi \right)P={\left(0,\left(\rho +1\right){\left|u\right|}^{\rho }U\right)}^{\text{T}}$, U is the solution of Equation (4.1),

$\overline{{\Lambda }_{\epsilon }}=\left(\begin{array}{cc}\epsilon I& -I\\ \left(M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right)-\epsilon \beta \right){\left(-\Delta \right)}^{2m}+{\epsilon }^{2}I+D& \beta {\left(-\Delta \right)}^{2m}-\epsilon I\end{array}\right),$

$D\cdot U=M^{\prime }\left({\Vert {\nabla }^{m}u\Vert }_p^p\right){\left({\Vert {\nabla }^{m}u\Vert }_p^p\right)}^{\prime }{\nabla }^{m}U{\left(-\Delta \right)}^{2m}u.$

For every fixed $\left(u_0\mathrm{,}{v}_0\right)\in E_k$, assume that $U_1\left(t\right)\mathrm{,}{U}_2\left(t\right)\mathrm{,}\cdots \mathrm{,}{U}_N\left(t\right)$ are N solutions to Equation (4.8). The initial value $U_1\left(0\right)={\xi }_1,U_2\left(0\right)={\xi }_2,\cdots ,U_N\left(0\right)={\xi }_N$, where ${\xi }_i\in E_k\left(i=1,2,\cdots ,N\right)$.

Then

$\begin{array}{l}\frac{\text{d}}{\text{d}t}{\Vert U_1\left(t\right)\wedge U_2\left(t\right)\wedge \cdots \wedge U_N\left(t\right)\Vert }_{\wedge E_k}^2\\ -2tr\left(L_t\left(\Psi \right)\cdot Q_N\left(t\right)\right){\Vert {\xi }_1\wedge {\xi }_2\wedge \cdots \wedge {\xi }_N\Vert }_{\wedge E_k}^2=0,\end{array}$

where $\wedge$ represents the cross product, tr represents the trace of the operator, $Q_N$ represents the orthogonal projection from $E_k$ to $span\left\{U_1\left(t\right)\mathrm{,}{U}_2\left(t\right)\mathrm{,}\cdots \mathrm{,}{U}_N\left(t\right)\right\}$.

According to uniform Gronwall’s inequality, we obtain

$\begin{array}{l}{\Vert U_1\left(t\right)\wedge U_2\left(t\right)\wedge \cdots \wedge U_N\left(t\right)\Vert }_{\wedge E_k}^2\\ \le {\Vert {\xi }_1\wedge {\xi }_2\wedge \cdots \wedge {\xi }_N\Vert }_{\wedge E_k}^2\exp \left({\displaystyle {\int }_0^t}tr{L}_t\left(\Psi \left(\tau \right)\right)\cdot Q_N\left(\tau \right)\text{d}\tau \right).\end{array}$ (4.9)

For arbitrarily given time $\tau$, let ${\omega }_j\left(\tau \right)={\left({\xi }_j\left(\tau \right),{\eta }_j\left(\tau \right)\right)}^{\text{T}}\left(j=1,2,\cdots ,N\right)$ are the orthogonal projection of $span\left\{U_1\left(t\right)\mathrm{,}{U}_2\left(t\right)\mathrm{,}\cdots \mathrm{,}{U}_N\left(t\right)\right\}$. Then, we have

${\Vert {\nabla }^{2m+k}{\xi }_j\Vert }^2+{\Vert {\nabla }^k{\eta }_j\Vert }^2=1.$ (4.10)

Define the inner production on $E_k$ as $\left(\left(\varsigma ,\eta \right),\left(\bar{\varsigma },\bar{\eta }\right)\right)=\left({\nabla }^{2m+k}\varsigma ,{\nabla }^{2m+k}\bar{\varsigma }\right)+\left({\nabla }^k\eta ,{\nabla }^k\bar{\eta }\right)$.

According to the above conditions, we receive

$\begin{array}{c}tr{L}_t\left(\Psi \left(\tau \right)\cdot Q_N\left(\tau \right)\right)={\displaystyle \underset{j=1}{\overset{N}{\sum }}}{\left(L_t\left(\Psi \left(\tau \right)\right)\cdot Q_N\left(\tau \right){\omega }_j\left(\tau \right),{\omega }_j\left(\tau \right)\right)}_{E_k}\\ ={\displaystyle \underset{j=1}{\overset{N}{\sum }}}{\left(L_t\left(\Psi \left(\tau \right)\right){\omega }_j\left(\tau \right),{\omega }_j\left(\tau \right)\right)}_{E_k},\end{array}$ (4.11)

where $\begin{array}{c}{\left(L_t\left(\Psi \left(\tau \right)\right){\omega }_j\left(\tau \right),{\omega }_j\left(\tau \right)\right)}_{E_k}={\left(L_t\left(\Psi \left(\tau \right)\right){\omega }_j\left(\tau \right),{\omega }_j\left(\tau \right)\right)}_{E_k}\\ =-\left(\overline{{\wedge }_{\epsilon }}{\omega }_j,{\omega }_j\right)-\left(g^{\prime }\left(\varphi \right){\omega }_j,{\omega }_j\right)\end{array}$.

$\begin{array}{c}\left(\overline{{\wedge }_{\epsilon }}{\omega }_j,{\omega }_j\right)=((\epsilon {\xi }_j-{\eta }_j,M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right)-\epsilon \beta {\left(-\Delta \right)}^{2m}{\xi }_j+{\epsilon }^2{\xi }_j+D\cdot {\xi }_j\\ {}_{{}_{}{}^{}}^{}+\beta {\left(-\Delta \right)}^{2m}{\eta }_j-\epsilon {\eta }_j),\left({\xi }_j,{\eta }_j\right))\\ =\epsilon {\Vert {\nabla }^{2m+k}{\xi }_j\Vert }^2+\left(M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right)-\epsilon \beta -1\right)\left({\nabla }^{2m+k}{\xi }_j,{\nabla }^{2m+k}{\eta }_j\right)\\ +{\epsilon }^2\left({\nabla }^k{\xi }_j,{\nabla }^k{\eta }_j\right)+\beta {\Vert {\nabla }^{2m+k}{\eta }_j\Vert }^2-\epsilon {\Vert {\nabla }^k{\xi }_j\Vert }^2+\left({\nabla }^{k}D\cdot {\xi }_j,{\nabla }^k{\eta }_j\right)\\ \ge \left(\epsilon -\frac{{\epsilon }^2\left(\beta -1\right)}{2}\right){\Vert {\nabla }^{2m+k}{\xi }_j\Vert }^2+\frac{\left(\beta +1\right){\lambda }_1^{2m}-{\epsilon }^2-C_{16}}{2}{\Vert {\nabla }^k{\eta }_j\Vert }^2\\ -\frac{{\epsilon }^2+2\epsilon +C_{16}}{2}{\Vert {\nabla }^k{\xi }_j\Vert }^2,\end{array}$ (4.12)

$\begin{array}{c}\left(g^{\prime }\left(\varphi \right){\omega }_j\mathrm{,}{\omega }_j\right)=\left(\left(\mathrm{0,}\rho {\left|u\right|}^{\rho }{\eta }_j\right)\mathrm{,}\left({\xi }_j\mathrm{,}{\eta }_j\right)\right)\\ \ge -\rho {\Vert u\Vert }_{\infty }^{\rho }{\Vert {\nabla }^k{\eta }_j\Vert }^2\ge -C_{17}{\Vert {\nabla }^k{\eta }_j\Vert }^2\mathrm{.}\end{array}$ (4.13)

According to (4.10)-(4.13), let $\delta =\min \left\{\epsilon -\frac{{\epsilon }^2\left(\beta -1\right)}{2},\frac{\left(\beta +1\right){\lambda }_1^{2m}-{\epsilon }^2-C_{16}}{2}-C_{17}\right\}$, $\gamma =\frac{{\epsilon }^2+2\epsilon +C_{16}}{2}$, we obtain

${\displaystyle \underset{j=1}{\overset{N}{\sum }}}{\left(L_t\left(\Psi \left(\tau \right)\right){\omega }_j\left(\tau \right),{\omega }_j\left(\tau \right)\right)}_{E_k}\le -N\delta +\gamma {\displaystyle \underset{j=1}{\overset{N}{\sum }}}{\Vert {\nabla }^k{\xi }_j\Vert }^2.$ (4.14)

For almost all t, we have

${\displaystyle \underset{j=1}{\overset{N}{\sum }}}{\Vert {\nabla }^k{\xi }_j\Vert }^2\le {\displaystyle \underset{j=1}{\overset{N}{\sum }}}{\lambda }_j^{-\frac{2m-1}{2m}},$

where ${\lambda }_j$ is the eigenvalue of ${\left(-\Delta \right)}^{2m}$. Thus

$tr\left(L_t\left(\Psi \right)\left(\tau \right)\cdot Q_N\left(\tau \right)\right)\le -N\delta +\gamma {\displaystyle \underset{j=1}{\overset{N}{\sum }}}{\lambda }_j^{-\frac{2m-1}{2m}}.$

Suppose

$q_N\left(t\right)=\underset{{\Psi }_0\in A}{\sup }\underset{{\xi }_j\in E_k}{\inf }\left(\frac{1}{t}{\displaystyle {\int }_0^t}tr{L}_t\left(S\left(\tau \right){\Psi }_0\right)\cdot Q_N\left(\tau \right)\text{d}\tau \right),$

$q_N=\underset{t\to \infty }{\lim }{q}_N\left(t\right).$

Due to (4.14), we have

$q_N\le -N\delta +\gamma {\displaystyle \underset{j=1}{\overset{N}{\sum }}}{\lambda }_j^{-\frac{2m-1}{2m}}.$

Therefore, the Lyapunov exponents ${\sigma }_1\mathrm{,}{\sigma }_2\mathrm{,}\cdots \mathrm{,}{\sigma }_N$ of $B_{0k}$ are uniformly bounded, moreover

${\sigma }_1+{\sigma }_2+\cdots +{\sigma }_N\le -N\delta +\gamma {\displaystyle \underset{j=1}{\overset{n}{\sum }}}{\lambda }_j^{-\frac{2m-1}{2m}},$

such that

${\left(q_j\right)}_{+}\le -N\delta +\gamma {\displaystyle \underset{j=1}{\overset{n}{\sum }}}{\lambda }_j^{\frac{2m-1}{2m}}\le \gamma {\displaystyle \underset{j=1}{\overset{N}{\sum }}}{\lambda }_j^{\frac{2m-1}{2m}}\le -\frac{N\delta }{3},$

$q_N\le -N\delta \left(1-\frac{\gamma }{N\delta }{\displaystyle \underset{j=1}{\overset{N}{\sum }}}{\lambda }_j^{\frac{2m-1}{2m}}\right)\le -\frac{2N\delta }{3}.$

Further,

$\underset{1\le j\le N}{\max }\frac{{\left(q_j\right)}_{+}}{\left|q_N\right|}\le \frac{1}{2}\mathrm{.}$

Thus, we obtain $d_H\left(A_k\right)<\frac{N}{2},d_F\left(A_k\right)<\frac{3N}{2}$. $■$

Conflicts of Interest

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

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