A Family of Exponential Attractors and Inertial Manifolds for a Class of Higher Order Kirchhoff Equations

Guoguang Lin, Yingguo Wang

Department of Mathematics, Yunnan University, Kunming, China.

DOI: 10.4236/jamp.2022.103062                

Abstract

In this paper, the global dynamics of a class of higher order nonlinear Kirchhoff equations under n-dimensional conditions is studied. Firstly, the Lipschitz property and squeezing property of the nonlinear semigroup associated with the initial boundary value problem are proved, and the existence of a family of exponential attractors is obtained. Then, by constructing the corresponding graph norm, the condition of a spectral interval is established when N is sufficiently large. Finally, the existence of the family of inertial manifolds is obtained.

Keywords

Kirchhoff Equation, Lipschitz Property, Squeezing Property, a Family of the Exponential Attractors, a Family of Inertial Manifolds

1. Introduction

In the 1990s, C. Foiao proposed the concept of an exponential attractor. The exponential attractor has a compact positive invariant set with finite fractal dimension and is exponentially attractive to the solution orbit. The exponential attractor has deeper and more practical properties. Compared with the global attractor, the exponential attractor has a uniform exponential convergence rate on the invariant absorption set of its solution. The exponential attractor is more robust under numerical approximation and perturbation. The family of inertial manifolds is concerned with the long-time behavior of the solution of a dissipative evolution equation, which is a finite-dimensional invariant Lipschitz manifold and attracts all solution orbits in the phase space with an exponential rate. The family of inertial manifolds is an important link between finite-dimensional and infinite-dimensional dynamical systems. In this paper, we study the family of exponential attractors and inertial manifolds of the following nonlinear Kirchhoff equations

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

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

where $m>1$, and $m\in N^{+}$, $\Omega \in R^n\left(n\ge 1\right)$ is a bounded domain, $\partial \Omega$ denotes the boundary of $\Omega$, $g\left(u_t\right)$ is a nonlinear source term, $\beta {\left(-\Delta \right)}^{2m}{u}_t$ is a strongly dissipative term, $\beta >0$, $f\left(x\right)$ is an external force term.

In reference [1], Fan Xiaoming constructed the spatial discretization based on the wave equation on $R^{+}$, and studied the exponential attractor of the second-order lattice dynamical system with nonlinear damping:

$u_{tt}-\beta \Delta u_t+h\left(u_t\right)-\Delta u+\lambda u+\bar{g}u=\bar{q}$

In reference [2], Yang Zhijian et al. studied the exponential attractor of Kirchhoff equation with strong damping of nonlinear term and supercritical nonlinear term:

$\begin{array}{l}{u}_{tt}-\sigma \left({\Vert \nabla u\Vert }^2\right)\Delta u_t-\varphi \left({\Vert \nabla u\Vert }^2\right)\Delta u+f\left(u\right)=h\left(x\right),\\ {u|}_{\partial \Omega }=0,u\left(x,0\right)=u_0\left(x\right),u_t\left(x,0\right)=u_1\left(x\right),x\in \Omega .\end{array}$

They prove the existence of the exponential attractor by using the weak quasistability estimate; in reference [3], Xu Guigui, Wang Libo and Lin Guoguang studied a class of second-order nonlinear wave equations with time delay under the assumption that the delay time is small:

$\frac{{\partial }^{2}u}{{\partial }^{2}t}+\alpha \frac{\partial u}{\partial t}-\beta \Delta \frac{\partial u}{\partial t}-\Delta u+g\left(u\right)=f\left(x\right)+h\left(t,u_t\right),$

$t>0,\alpha >0,\beta >0.$

The existence of inertial manifolds. Need to know more references [4] - [17].

2. Basic Assumption

For convenience, space and symbols are defined as follows:

$H=L^2\left(\Omega \right)$, $H_0^{2m}\left(\Omega \right)=H^{2m}\left(\Omega \right)\cap H_0^1\left(\Omega \right)$, $H_0^{2m+k}\left(\Omega \right)=H^{2m+k}\left(\Omega \right)\cap H_0^1\left(\Omega \right)$, $E_0=H^{2m}\left(\Omega \right)\times L^2\left(\Omega \right)$,

$E_k=H_0^{2m+k}\left(\Omega \right)\times H_0^k\left(\Omega \right),k=1,2,\cdots ,2m$. $\left(\cdot \mathrm{,}\cdot \right)$ and $\Vert \cdot \Vert$ represent the inner product and the norm on H space, then $\left(u,v\right)={\displaystyle \underset{\Omega }{\int }}u\left(x\right)v\left(x\right)\text{d}x$, $\left(u,u\right)={{\displaystyle \Vert u\Vert }}^2$.

Nonlinear function $g\left(u_t\right)$ meets the following conditions:

(H₁) $2<p\le \frac{2n}{n-2m},n>2m;p\ge 2,n<2m.$

(H₂) $g\in C^k,k=1,2,\cdots ,2m.$

(H₃) $M\left(s\right)\in C^2\left(\left[0,+\infty \right),R^{+}\right),1\le {\mu }_0<M\left(s\right)<{\mu }_1,\mu =\{\begin{array}{l}{\mu }_0,\frac{\text{d}}{\text{d}t}{\Vert {\nabla }^{2m}u\Vert }^2\ge 0,\hfill \\ {\mu }_1,\frac{\text{d}}{\text{d}t}{\Vert {\nabla }^{2m}u\Vert }^2<0.\hfill \end{array}$

(H₄) ${{\displaystyle \Vert g^{\prime }\left(s\right)\Vert }}_{\infty }\le C_2\mathrm{.}$

3. Exponential Attractors

In order to prove the need of later, so the inner product and norm of $E_k$ are defined

$\forall U_i=\left(u_i,v_i\right)\in E_k,i=1,2,$

${\left(U_1,U_2\right)}_{E_k}=\left({\nabla }^{2m+k}{u}_1,{\nabla }^{2m+k}{u}_2\right)+\left({\nabla }^k{v}_1,{\nabla }^k{v}_2\right)$ (3.1)

${\Vert U_1\Vert }_{E_k}^2={\left(U_1,U_1\right)}_{E_k}={\Vert {\nabla }^{2m+k}{u}_1\Vert }^2+{\Vert {\nabla }^k{v}_1\Vert }^2$ (3.2)

Let $U=\left(u,v\right)\in E_k$, $v=u_t+\epsilon u$,

$\frac{1-\sqrt{1-\beta {\lambda }_1^{-2m}}}{\beta }\le \epsilon \le \min \left\{\frac{\beta {\lambda }_1^{2m}}{4}\mathrm{,}\frac{1+\sqrt{1-\beta {\lambda }_1^{-2m}}}{\beta }\mathrm{,}\sqrt[4]{\frac{\beta }{2{\lambda }_1^{-2m}}}\right\}$

then the Equation (1.1) is equivalent to

$U_t+HU=F\left(U\right)$ (3.3)

where

$HU=\left(\begin{array}{c}\epsilon u-v\\ \beta {\left(-\Delta \right)}^{2m}v+\left(1-\beta \epsilon \right){\left(-\Delta \right)}^{2m}u-\epsilon v+{\epsilon }^{2}u\end{array}\right)$

$F\left(U\right)=\left(\begin{array}{c}0\\ \left(1-M\left({\Vert D^{m}u\Vert }_p^p\right)\right){\left(-\Delta \right)}^{2m}u+f\left(x\right)-g\left(u_t\right)\end{array}\right)$

By definition, we know that $E_0\mathrm{,}{E}_k$ are two Hilbert spaces, $E_k$ is dense and compact in $E_0$, let $S\left(t\right)$ is the mapping of $E_i$ to $E_i$, $i=0,k$.

Definition 3.1. [4] If there is a compact set $A_k\subset E_k$, $A_k$ attracts all bounded sets in $E_k$, and it is an invariant set under $S\left(t\right)$, $S\left(t\right)A_k=A_k$, $\forall t\ge 0$. Then we say that a semigroup $S\left(t\right)$ has a family of $\left(E_k\mathrm{,}{E}_0\right)$ -compact attractors $A_k$.

Definition 3.2. [5] If $A_k\subset M_k\subset B_k$ and

1) $S\left(t\right)M_k\subseteq M_k$, $\forall t\ge 0$ ;

2) $M_k$ has finite fractal dimension, $d_f\left(M_k\right)<+\infty$ ;

3) There exist universal constants $\eta >0,\delta >0$, such that

$dist\left(S\left(t\right),B_k,M_k\right)\le \eta {\text{e}}^{-\delta t},t>0,$

where, $dis{t}_{E_k}\left(A_k,B_k\right)=\underset{x\in A_k}{\sup }\underset{y\in B_k}{\inf }{\left|x-y\right|}_{E_k}$, $B_k$ is the positive invariant set of

$S\left(t\right)$ in $E_k$. The compact set $M_k\in E_k$ is called a family of $\left(E_k\mathrm{,}{E}_0\right)$ exponential attractors for the system $\left(S\left(t\right)\mathrm{,}{E}_k\right)$.

Definition 3.3. [5] if there exists limited function $l\left(t\right)$, such that

${\left|S\left(t\right)u-S\left(t\right)v\right|}_{E_k}\le l\left(t\right){\left|u-v\right|}_{E_k}\mathrm{,}\forall u\mathrm{,}v\in E_k\mathrm{.}$ (3.4)

Then the semigroup $S\left(t\right)$ is Lipschitz continuous in $E_k$.

Definition 3.4. [5] If $\delta \le \left(\mathrm{0,}\frac{1}{8}\right)$ and exists an orthogonal projection $P_N$ of rank N, such that for $\forall \left(u\mathrm{,}v\right)\in E_k$,

${\left|S\left(t_{*}\right)u-S\left(t_{*}\right)v\right|}_{E_k}\le \delta {\left|u-v\right|}_{E_k},$ (3.5)

or

${\left|Q_N\left(S\left(t_{*}\right)u-S\left(t_{*}\right)v\right)\right|}_{E_k}\le {\left|P_N\left(S\left(t_{*}\right)u-S\left(t_{*}\right)v\right)\right|}_{E_k}.$ (3.6)

Then $S\left(t\right)$ is said to satisfy the discrete squeezing property in $E_k$, where $Q_N=I-P_N$.

Theorem 3.1. [5] Assume that

1) $S\left(t\right)$ possesses a family of $\left(E_k\mathrm{,}{E}_0\right)$ -compact attractors $A_k$ ;

2) In $E_k$, there exists a compact set $B_k$ with positive invariance to the action of $S\left(t\right)$ ;

3) $S\left(t\right)$ is Lipschitz continuous and is squeezed in $B_k$. $S\left(t\right)B_k$ possesses a family of $\left(E_k\mathrm{,}{E}_0\right)$ -compact attractors $M_k$, $M_k={\displaystyle \underset{0\le t\le t_{\mathrm{<mo>\; *\; </mo>}}}{\cup }S\left(t\right)M_{\mathrm{<mo>\; *\; </mo>}}}$,

$M_{*}=A_k\cup \left({\displaystyle \underset{j=1}{\overset{\infty }{\cup }}{\displaystyle \underset{i=1}{\overset{\infty }{\cup }}S{\left(t_{*}\right)}^j\left(E^{\left(i\right)}\right)}}\right)$. The fractal dimension of $M_k$ satisfies

$d_f\left(M_k\right)\le c{N}_0+1$, where, $N_0$ is the smallest N which makes the discrete squeezing property established.

Proposition 3.2. [6] After making reasonable assumptions about $M\left(s\right)$ and $g\left(u_t\right)$, the initial boundary value problem (1.1)-(1.3) has a unique smooth solution and the solution has the following properties:

${\Vert \left(u,v\right)\Vert }_{E_0}^2={\Vert {\nabla }^{2m}u\Vert }^2+{\Vert v\Vert }^2\le C\left(R_0\right),{\Vert \left(u,v\right)\Vert }_{E_k}^2={\Vert {\nabla }^{2m+k}u\Vert }^2+{\Vert {\nabla }^{k}v\Vert }^2\le C\left(R_1\right).$

The solution $S\left(t\right)\left(u_0,v_0\right)=\left(u\left(t\right),v\left(t\right)\right)$ of the equation is expressed by Theorem 3.1, then $S\left(t\right)$ is a semigroup of continuous operators in $E_k$, we have the ball

$B_1=\left\{\left(u,v\right)\in E_0:{\Vert \left(u,v\right)\Vert }_{E_0}^2\le C\left(R_0\right)\right\}$ (3.7)

$B_2=\left\{\left(u,v\right)\in E_k:{\Vert \left(u,v\right)\Vert }_{E_k}^2\le C\left(R_1\right)\right\}$ (3.8)

are absorbing sets of $S\left(t\right)$ in $E_0$ and $E_k$ respectively.

Lemma 3.1. For any $U=\left(u,v\right)\in E_k$, there is $\left(HU,U\right)\ge k_1{\Vert U\Vert }_{E_k}^2+k_2{\Vert {\nabla }^{2m+k}v\Vert }^2$.

Proof. By

${\left(U_1,U_2\right)}_{E_k}=\left({\nabla }^{2m+k}{u}_1,{\nabla }^{2m+k}{u}_2\right)+\left({\nabla }^k{v}_1,{\nabla }^k{v}_2\right)$

and

${\Vert U_1\Vert }_{E_k}^2={\left(U_1,U_1\right)}_{E_k}={\Vert {\nabla }^{2m+k}{u}_1\Vert }^2+{\Vert {\nabla }^k{v}_1\Vert }^2,$

then

$\begin{array}{c}\left(HU,U\right)=\left({\nabla }^{2m+k}\left(\epsilon u-v\right),{\nabla }^{2m+k}u\right)+\left({\nabla }^{k}H,{\nabla }^{k}v\right)\\ =\left(\epsilon {\nabla }^{2m+k}u,{\nabla }^{2m+k}u\right)-\left({\nabla }^{2m+k}v,{\nabla }^{2m+k}u\right)+\left({\nabla }^{k}H,{\nabla }^{k}v\right).\end{array}$ (3.9)

where $H=\beta {\left(-\Delta \right)}^{2m}v+\left(1-\beta \epsilon \right){\left(-\Delta \right)}^{2m}u-\epsilon v+{\epsilon }^{2}u$.

From the Young’s inequality and the Poincare’s inequality, we can get

$\begin{array}{c}\left({\nabla }^{k}H,{\nabla }^{k}v\right)=\left({\nabla }^k\left(\beta {\left(-\Delta \right)}^{2m}v+\left(1-\beta \epsilon \right){\left(-\Delta \right)}^{2m}u-\epsilon v+{\epsilon }^{2}u\right),{\nabla }^{k}v\right)\\ \ge \frac{\beta }{2}{\Vert {\nabla }^{2m+k}v\Vert }^2+\frac{\beta {\lambda }_1^{2m}}{2}{\Vert {\nabla }^{k}v\Vert }^2-\frac{\beta \epsilon }{2}{\Vert {\nabla }^{2m+k}u\Vert }^2-\frac{\beta \epsilon }{2}{\Vert {\nabla }^{2m+k}v\Vert }^2\\ +\left({\nabla }^{2m+k}u,{\nabla }^{2m+k}v\right)-\epsilon {\Vert {\nabla }^{k}v\Vert }^2+{\epsilon }^2\left({\nabla }^{k}u,{\nabla }^{k}v\right),\end{array}$ (3.10)

$\begin{array}{c}{\epsilon }^2\left({\nabla }^{k}u,{\nabla }^{k}v\right)\ge -{\epsilon }^2{\lambda }_1^{-m}\Vert {\nabla }^{m+k}u\Vert \Vert {\nabla }^{m+k}v\Vert \\ \ge -{\epsilon }^2{\lambda }_1^{-2m}\left(\frac{1}{2{\epsilon }^2}{\Vert {\nabla }^{2m+k}u\Vert }^2+\frac{{\epsilon }^2}{2}{\Vert {\nabla }^{2m+k}v\Vert }^2\right)\\ =-\frac{{\lambda }_1^{-2m}}{2}{\Vert {\nabla }^{2m+k}u\Vert }^2-\frac{{\lambda }_1^{-2m}{\epsilon }^4}{2}{\Vert {\nabla }^{2m+k}v\Vert }^2,\end{array}$ (3.11)

where ${\lambda }_1\left(>0\right)$ is the first eigenvalue of the operator $-\Delta$.

Derived from Formula (3.9) to Formula (3.11)

$\begin{array}{c}\left(HU,U\right)\ge \left(\frac{\beta {\lambda }_1^{2m}}{2}-\epsilon \right){\Vert {\nabla }^{k}v\Vert }^2+\left(\epsilon -\frac{\beta \epsilon }{2}-\frac{{\lambda }_1^{-2m}}{2}\right){\Vert {\nabla }^{2m+k}u\Vert }^2.\\ +\left(\frac{\beta }{2}-\frac{\beta \epsilon }{2}-\frac{{\lambda }_1^{-2m}{\epsilon }^4}{2}\right){\Vert {\nabla }^{2m+k}v\Vert }^2.\end{array}$

Due to

$\frac{{\lambda }_1^{-2m}}{2-\beta }\le \epsilon \le \min \left\{\frac{\beta {\lambda }_1^{2m}}{2}\mathrm{,}\frac{1+\sqrt{1-\beta {\lambda }_1^{-2m}}}{\beta }\mathrm{,}\sqrt[4]{\frac{\beta }{2{\lambda }_1^{-2m}}}\right\}\mathrm{,}$

then

$\frac{\beta {\lambda }_1^{2m}}{2}-\epsilon \ge 0,\epsilon -\frac{\beta \epsilon }{2}-\frac{{\lambda }_1^{-2m}}{2}\ge 0,\frac{\beta }{2}-\frac{\beta \epsilon }{2}-\frac{{\lambda }_1^{-2m}{\epsilon }^4}{2}\ge 0.$

thus

$\left(HU\mathrm{,}U\right)\ge k_1{\Vert U\Vert }_{E_k}^2+k_2{\Vert {\nabla }^{2m+k}v\Vert }^2\mathrm{.}$

where

$k_1=\min \left\{\left(\frac{\beta {\lambda }_1^{2m}}{2}-\epsilon \right),\left(\epsilon -\frac{\beta \epsilon }{2}-\frac{{\lambda }_1^{-2m}}{2}\right)\right\},k_2=\left(\frac{\beta }{2}-\frac{\beta \epsilon }{2}-\frac{{\lambda }_1^{-2m}{\epsilon }^4}{2}\right)\ge 0.$

Let $S\left(t\right)U_0=U\left(t\right)={\left(u\left(t\right),v\left(t\right)\right)}^{\text{T}}$, where $v=u_t\left(t\right)+\epsilon u\left(t\right)$ ; $S\left(t\right)V_0=V\left(t\right)={\left(\tilde{u}\left(t\right),\tilde{v}\left(t\right)\right)}^{\text{T}}$, where $\tilde{v}={\tilde{u}}_t\left(t\right)+\epsilon \tilde{u}\left(t\right)$ ;

Let $W\left(t\right)=S\left(t\right)U_0-S\left(t\right)V_0=U\left(t\right)-V\left(t\right)={\left(w\left(t\right),z\left(t\right)\right)}^{\text{T}}$ ; where $z\left(t\right)=w_t\left(t\right)+\epsilon w\left(t\right)$, then $W\left(t\right)$ satisfies

$W_t\left(t\right)+HU-HV+F\left(V\right)-F\left(U\right)=0,$ (3.12)

$W\left(0\right)=U_0-V_0.$ (3.13)

To verify that the Formulas (1.1)-(1.3) has an exponential attractor, it is necessary to prove that the dynamical system $S\left(t\right)$ is Lipschitz continuous on $E_k$.

Lemma 3.2. (Lipschitz property). For any $U_0\mathrm{,}{V}_0\in E_k\mathrm{,}T\ge 0$, there is

${\Vert S\left(t\right)U_0-S\left(t\right)V_0\Vert }_{E_k}^2\le {\text{e}}^{bt}{\Vert U_0-V_0\Vert }_{E_k}^2.$

Proof. We can get the inner product of $W\left(t\right)$ and Formula (3.4) in $E_k$ space,

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert W\left(t\right)\Vert }^2+\left(HU-HV,W\left(t\right)\right)+\left({\left(-\Delta \right)}^{2m+\frac{k}{2}}w,{\nabla }^{k}z\left(t\right)\right)\\ +\left(M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m+\frac{k}{2}}u-M\left({\Vert {\nabla }^m\tilde{u}\Vert }_p^p\right){\left(-\Delta \right)}^{2m+\frac{k}{2}}\tilde{u},{\nabla }^{k}z\left(t\right)\right)\\ +\left({\nabla }^k\left(g\left(u_t\right)-g\left({\tilde{u}}_t\right)\right),{\nabla }^{k}z\left(t\right)\right)=0.\end{array}$ (3.14)

A proof similar to lemma 3.1, obtained

${\left(HU-HV\mathrm{,}W\left(t\right)\right)}_{E_k}\ge k_1{\Vert W\left(t\right)\Vert }_{E_k}^2+k_2{\Vert {\nabla }^{2m+k}z\left(t\right)\Vert }_{E_k}^2\mathrm{.}$ (3.15)

From Young’s inequality and Poincare’s inequality and the mean value Theorem, we can get

$\begin{array}{c}-\left({\left(-\Delta \right)}^{2m+\frac{k}{2}}w\mathrm{,}{\nabla }^{k}z\left(t\right)\right)\ge -\Vert {\nabla }^{2m+k}w\Vert \Vert {\nabla }^{2m+k}z\Vert \\ \ge -\frac{1}{2}{\Vert {\nabla }^{2m+k}w\Vert }^2-\frac{1}{2}{\Vert {\nabla }^{2m+k}z\Vert }^2\mathrm{.}\end{array}$ (3.16)

let

$s={\Vert {\nabla }^{m}u\Vert }_p^p,\tilde{s}={\Vert {\nabla }^m\tilde{u}\Vert }_p^p,$

then

$\begin{array}{l}\left(M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m+\frac{k}{2}}u-M\left({\Vert {\nabla }^m\tilde{u}\Vert }_p^p\right){\left(-\Delta \right)}^{2m+\frac{k}{2}}\tilde{u},{\nabla }^{k}z\left(t\right)\right)\\ \le \left|\left(M\left(\tilde{s}\right){\left(-\Delta \right)}^{2m+\frac{k}{2}}w,{\nabla }^{k}z\left(t\right)\right)\right|+\left|\left(M^{\prime }\left(\vartheta \right)\left(\tilde{s}-s\right){\left(-\Delta \right)}^{2m+\frac{k}{2}}u,{\nabla }^{k}z\left(t\right)\right)\right|\\ \le \frac{{\mu }_1}{2}{\Vert {\nabla }^{2m+k}w\Vert }^2+\frac{{\mu }_1}{2}{\Vert {\nabla }^{2m+k}z\Vert }^2+C_0\Vert {\nabla }^{2m+k}w\Vert \Vert {\nabla }^{2m+k}z\Vert \\ \le \frac{{\mu }_1+C_0}{2}{\Vert {\nabla }^{2m+k}w\Vert }^2+\frac{{\mu }_1+C_0}{2}{\Vert {\nabla }^{2m+k}z\Vert }^2.\end{array}$ (3.17)

$\begin{array}{l}\left|\left({\nabla }^k\left(g\left(u_t\right)-g\left({\tilde{u}}_t\right)\right),{\nabla }^{k}z\left(t\right)\right)\right|=\left|\left({\nabla }^k{g}^{\prime }\left(s\right)w_t,{\nabla }^{k}z\left(t\right)\right)\right|\\ =g^{\prime }\left(s\right)\left|\left({\nabla }^k{w}_t\left(t\right),{\nabla }^{k}z\left(t\right)\right)\right|\le C_2\left|\left({\nabla }^k\left(z\left(t\right)-\epsilon w\left(t\right)\right),{\nabla }^{k}z\left(t\right)\right)\right|\\ \le C_2\left|\left({\nabla }^{k}z\left(t\right),{\nabla }^{k}z\left(t\right)\right)\right|-\epsilon C_2\left|\left({\nabla }^{k}w\left(t\right),{\nabla }^{k}z\left(t\right)\right)\right|\\ \le C_2{\Vert {\nabla }^{k}z\left(t\right)\Vert }^2+\frac{\epsilon C_2{\lambda }_1^{-2m}}{2}{\Vert {\nabla }^{2m+k}w\left(t\right)\Vert }^2+\frac{\epsilon C_2}{2}{\Vert {\nabla }^{k}z\left(t\right)\Vert }^2.\end{array}$ (3.18)

then

$\begin{array}{l}\frac{\text{d}}{\text{d}t}{\Vert W\left(t\right)\Vert }^2+2k_1{\Vert W\left(t\right)\Vert }^2+\left(2k_2-1-{\mu }_1-C_0\right)\Vert {\nabla }^{2m+k}z\left(t\right)\Vert \\ \le \left(1+{\mu }_1+C_0+\epsilon C_2{\lambda }_1^{-2m}\right){\Vert {\nabla }^{2m+k}w\left(t\right)\Vert }^2+\left(2C_2+\epsilon C_2\right){\Vert {\nabla }^{k}z\left(t\right)\Vert }^2\\ \le C_5{\Vert W\left(t\right)\Vert }^2\mathrm{.}\end{array}$ (3.19)

where

$C_5=\max \left\{1+{\mu }_1+C_0+\epsilon C_2{\lambda }_1^{-2m},2C_2+\epsilon C_2\right\}.$

By Gronwall’s inequality, we can get

${\Vert W\left(t\right)\Vert }^2\le {\text{e}}^{2C_{5}t}{\Vert W\left(0\right)\Vert }_{E_k}^2={\text{e}}^{bt}{\Vert W\left(0\right)\Vert }_{E_k}^2\mathrm{,}$ (3.20)

where $b=2C_5$.

Then

${\Vert S\left(t\right)U_0-S\left(t\right)V_0\Vert }_{E_k}^2\le {\text{e}}^{bt}{\Vert U_0-V_0\Vert }_{E_k}^2.$

Apparently, $-\Delta$ is an unbounded self-adjoint closed positive operator, and ${\left(-\Delta \right)}^{-1}$ is compact, we know that there is an Orthonormal basis of H through the basic theory of spectral spacing, it is made up of the eigenvector $w_j$ of $-\Delta$, so that

$-\Delta w_j={\lambda }_j{w}_j,0<{\lambda }_1\le {\lambda }_2\le \cdots \le {\lambda }_j\to +\infty ,j\to \infty .$

$P_N$ is an orthogonal projection in $E_k$. $Q_N=I-P_N$.

Next, were going to use

$\Vert \left({\left(-\Delta \right)}^{2m}u\right)\Vert \ge {\lambda }_{n+1}^{2m}\Vert u\Vert ,\forall u\in Q_n\left(H^{4m}\left(\Omega \right)\cap H_0^1\left(\Omega \right)\right),\Vert Q_{n}u\Vert \le \Vert u\Vert ,u\in H.$

Lemma 3.3. For any $U_0,V_0\in E_k$, $Q_{n_0}\left(t\right)=Q_{n_0}\left(U\left(t\right)-V\left(t\right)\right)=Q_{n_0}W\left(t\right)={\left(w_{n_0},z_{n_0}\right)}^{\text{T}}$, then

${\Vert W_{n_0}\left(t\right)\Vert }_{E_k}^2\le \left({\text{e}}^{-2k_{1}t}+\frac{C_3{\lambda }_{1+n_0}^{-m}}{2k_1+b}{\text{e}}^{bt}\right){\Vert W\left(0\right)\Vert }^2$

Proof. Apply $Q_{n_0}\left(t\right)$ to Equation (3.4), we get

$W_{n_0}\left(t\right)+Q_{n_0}\left(HU-HV\right)+Q_{n_0}\left(F\left(V\right)-F\left(U\right)\right)=0.$ (3.21)

The sum of (3.11) and $W_{n_0}\left(t\right)$ is the inner product of $E_k$, we get

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert W_{n_0}\left(t\right)\Vert }^2+k_1{\Vert W_{n_0}\left(t\right)\Vert }^2+k_2{\Vert {\nabla }^{2m+k}{z}_{n_0}\left(t\right)\Vert }^2\\ +\left(Q_{n_0}\left(F\left(V\right)-F\left(U\right)\right),{\nabla }^k{z}_{n_0}\left(t\right)\right)=0.\end{array}$ (3.22)

Known by Young’s inequality and hypothesis condition

$\begin{array}{c}-\left(Q_{n_0}{\left(-\Delta \right)}^{2m+\frac{k}{2}}w,{\nabla }^k{z}_{n_0}\left(t\right)\right)\ge -\Vert {\nabla }^{2m+k}{w}_{n_0}\Vert \Vert {\nabla }^{2m+k}{z}_{n_0}\Vert \\ \ge -\frac{1}{2}{\Vert {\nabla }^{2m+k}{w}_{n_0}\Vert }^2-\frac{1}{2}{\Vert {\nabla }^{2m+k}{z}_{n_0}\Vert }^2.\end{array}$ (3.23)

$\begin{array}{l}\left(Q_{n_0}\left(M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m+\frac{k}{2}}u-M\left({\Vert {\nabla }^m\tilde{u}\Vert }_p^p\right){\left(-\Delta \right)}^{2m+\frac{k}{2}}\tilde{u}\right),{\nabla }^k{z}_{n_0}\left(t\right)\right)\\ \le \left|\left(M\left(\tilde{s}\right){\left(-\Delta \right)}^{2m+\frac{k}{2}}{w}_{n_0},{\nabla }^k{z}_{n_0}\left(t\right)\right)\right|+\left|\left(M^{\prime }\left(\vartheta \right)\left(\tilde{s}-s\right){\left(-\Delta \right)}^{2m+\frac{k}{2}}u,{\nabla }^k{z}_{n_0}\left(t\right)\right)\right|\\ \le \frac{{\mu }_1}{2}{\Vert {\nabla }^{2m+k}{w}_{n_0}\Vert }^2+\frac{{\mu }_1}{2}{\Vert {\nabla }^{2m+k}{z}_{n_0}\Vert }^2+C_0\Vert {\nabla }^{2m+k}{w}_{n_0}\Vert \Vert {\nabla }^{2m+k}{z}_{n_0}\Vert \\ \le \frac{{\mu }_1+C_0}{2}{\Vert {\nabla }^{2m+k}{w}_{n_0}\Vert }^2+\frac{{\mu }_1+C_0}{2}{\Vert {\nabla }^{2m+k}{z}_{n_0}\Vert }^2.\end{array}$ (3.24)

$\begin{array}{l}\left|Q_{n_0}{\nabla }^k\left(g\left(u_t\right)-g\left({\tilde{u}}_t\right)\right),{\nabla }^k{z}_{n_0}\left(t\right)\right|=\left|\left({\nabla }^k{g}^{\prime }\left(s\right)w_{t{n}_0},{\nabla }^k{z}_{n_0}\left(t\right)\right)\right|\\ =g^{\prime }\left(s\right)\left|\left({\nabla }^k{w}_{t{n}_0}\left(t\right),{\nabla }^k{z}_{n_0}\left(t\right)\right)\right|\le C_3\left|\left({\nabla }^k\left(z_{n_0}\left(t\right)-\epsilon w_{n_0}\left(t\right)\right),{\nabla }^k{z}_{n_0}\left(t\right)\right)\right|\\ \le C_3\left|\left({\nabla }^k{z}_{n_0}\left(t\right),{\nabla }^k{z}_{n_0}\left(t\right)\right)\right|-\epsilon C_3\left|\left({\nabla }^k{w}_{n_0}\left(t\right),{\nabla }^k{z}_{n_0}\left(t\right)\right)\right|\\ \le C_3{\Vert {\nabla }^k{z}_{n_0}\left(t\right)\Vert }^2+\frac{\epsilon C_3{\lambda }_{1+n_0}^{-2m}}{2}{\Vert {\nabla }^{2m+k}{w}_{n_0}\left(t\right)\Vert }^2+\frac{\epsilon C_3}{2}{\Vert {\nabla }^k{z}_{n_0}\left(t\right)\Vert }^2\\ \le \left(1+{\mu }_1+C_0+\epsilon C_3{\lambda }_{1+n_0}^{-2m}\right){\Vert {\nabla }^{2m+k}{w}_{n_0}\left(t\right)\Vert }^2+\left(2C_3+\epsilon C_3\right){\Vert {\nabla }^k{z}_{n_0}\left(t\right)\Vert }^2.\end{array}$ (3.25)

Replace (3.13) with (3.12) to get

$\begin{array}{l}\frac{\text{d}}{\text{d}t}{\Vert W_{n_0}\left(t\right)\Vert }^2+2k_1{\Vert W_{n_0}\left(t\right)\Vert }^2+\left(2k_2-1-{\mu }_1-C_0\right){\Vert {\nabla }^{2m+k}{z}_{n_0}\left(t\right)\Vert }^2\\ \le \left(1+{\mu }_1+C_0+\epsilon C_3{\lambda }_{1+n_0}^{-2m}\right){\Vert {\nabla }^{2m+k}{w}_{n_0}\left(t\right)\Vert }^2+\left(2C_3+\epsilon C_3\right){\Vert {\nabla }^k{z}_{n_0}\left(t\right)\Vert }^2.\end{array}$ (3.26)

By Gronwall’s inequality, we can get

$\begin{array}{c}{\Vert W_{n_0}\left(t\right)\Vert }^2\le {\Vert W\left(0\right)\Vert }^2{\text{e}}^{-2k_{1}t}+\frac{C_3{\lambda }_{1+n_0}^{-2m}}{2k_1+b}{\text{e}}^{bt}{\Vert W\left(0\right)\Vert }^2\\ =\left({\text{e}}^{-2k_{1}t}+\frac{C_3{\lambda }_{1+n_0}^{-2m}}{2k_1+b}{\text{e}}^{bt}\right){\Vert W\left(0\right)\Vert }^2\end{array}$ (3.27)

Thus lemma 3.3 is proved.

Lemma 3.4. (Discrete squeezing property). For any $U_0\mathrm{,}{V}_0\in E_k$, if

${\Vert P_{n_0}\left(S\left(T^{\mathrm{<mo>\; *\; </mo>}}\right)U_0-S\left(T^{\mathrm{<mo>\; *\; </mo>}}\right)V_0\right)\Vert }_{E_k}\le {\Vert \left(I-P_{n_0}\right)\left(S\left(T^{\mathrm{<mo>\; *\; </mo>}}\right)U_0-S\left(T^{\mathrm{<mo>\; *\; </mo>}}\right)V_0\right)\Vert }_{E_k}\mathrm{,}$

then

${\Vert S\left(T^{\mathrm{<mo>\; *\; </mo>}}\right)U_0-S\left(T^{\mathrm{<mo>\; *\; </mo>}}\right)V_0\Vert }_{E_k}\le \frac{1}{8}{\Vert U_0-V_0\Vert }_{E_k}$

Proof. If

${\Vert P_{n_0}\left(S\left(T^{\mathrm{<mo>\; *\; </mo>}}\right)U_0-S\left(T^{\mathrm{<mo>\; *\; </mo>}}\right)V_0\right)\Vert }_{E_k}\le {\Vert \left(I-P_{n_0}\right)\left(S\left(T^{\mathrm{<mo>\; *\; </mo>}}\right)U_0-S\left(T^{\mathrm{<mo>\; *\; </mo>}}\right)V_0\right)\Vert }_{E_k}\mathrm{,}$

then

$\begin{array}{l}{\Vert S\left(T^{*}\right)U_0-S\left(T^{*}\right)V_0\Vert }_{E_k}^2\\ \le {\Vert \left(I-P_{n_0}\right)\left(S\left(T^{*}\right)U_0-S\left(T^{*}\right)V_0\right)\Vert }_{E_k}^2+{\Vert P_{n_0}\left(S\left(T^{*}\right)U_0-S\left(T^{*}\right)V_0\right)\Vert }_{E_k}^2\\ \le 2{\Vert \left(I-P_{n_0}\right)\left(S\left(T^{*}\right)U_0-S\left(T^{*}\right)V_0\right)\Vert }_{E_k}^2\\ \le 2\left({\text{e}}^{-2k_1{T}^{*}}+\frac{C_4{\lambda }_{1+n_0}^{-2m}}{2k_1+b}{\text{e}}^{b{T}^{*}}\right){\Vert U_0-V_0\Vert }_{E_k}^2.\end{array}$ (3.28)

Let $T^{\mathrm{<mo>\; *\; </mo>}}$ be big enough

${\text{e}}^{-2k_1{T}^{\mathrm{<mo>\; *\; </mo>}}}\le \frac{1}{256}\mathrm{.}$ (3.29)

and let $n_0$ be big enough

$\frac{C_4{\lambda }_{1+n_0}^{-2m}}{2k_1+b}{\text{e}}^{b{T}^{\mathrm{<mo>\; *\; </mo>}}}\le \frac{1}{256}\mathrm{.}$ (3.30)

Replace the Formulas (3.17) and (3.18) into the Formula (3.16), we get

${\Vert S\left(T^{\mathrm{<mo>\; *\; </mo>}}\right)U_0-S\left(T^{\mathrm{<mo>\; *\; </mo>}}\right)V_0\Vert }_{E_k}\le \frac{1}{8}{\Vert U_0-V_0\Vert }_{E_k}\mathrm{.}$ (3.31)

Theorem 3.3. Under the appropriate assumptions above, $\left(u_0,v_0\right)\in E_k$, $k=1,2,\cdots ,2m$, $f\in H$, $v=u_t+\epsilon u$,

$\frac{{\lambda }_1^{-2m}}{2-\beta }\le \epsilon \le \min \left\{\frac{\beta {\lambda }_1^{2m}}{2},\frac{1+\sqrt{1-\beta {\lambda }_1^{-2m}}}{\beta },\sqrt[4]{\frac{\beta }{2{\lambda }_1^{-2m}}}\right\},$

then the solution semigroup of the initial boundary value problem (1.1)-(1.3) has a family of $\left(E_k\mathrm{,}{E}_0\right)$ exponential attractors on $E_k$,

$M_k={\displaystyle \underset{0\le t\le T^{*}}{\cup }S\left(t\right)}\left(A_k\cup \left({\displaystyle \underset{j=1}{\overset{\infty }{\cup }}{\displaystyle \underset{i=1}{\overset{\infty }{\cup }}S{\left(T^{*}\right)}^j\left(E^{\left(i\right)}\right)}}\right)\right),$

and the fractal dimension is satisfied $d_f\left(M_k\right)\le c{N}_0+1$.

Proof: According to Theorem 3.1, Lemma 3.2, Lemma 3.4, Theorem 3.3 is easy to prove.

4. Inertial Manifolds

Definition 4.1. [18] Assum $S=S{\left(t\right)}_{t\ge 0}$ is a solution semigroup of Banach space $E_k=H_0^{2m+k}\left(\Omega \right)\times H_0^k\left(\Omega \right)$, $k=1,2,\cdots ,2m$, a subset ${\mu }_k\subset E_k$ satisfies the following three properties:

1) ${\mu }_k$ is finite dimensional Lipschitz manifold;

2) ${\mu }_k$ is positively invariant, $S\left(t\right){\mu }_0\subset {\mu }_k,\forall t\ge 0$ ;

3) ${\mu }_k$ attracts exponentially all the orbits of the solution, and $u\in E_k$, there are constants $\eta >0,\gamma >0$, then

$dist\left(S\left(t\right)u\mathrm{,}{\mu }_k\right)\le \gamma {\text{e}}^{-\eta t}\mathrm{,}t\ge 0.$

It is said that ${\mu }_k$ is an inertial manifold of ${\left\{S\left(t\right)\right\}}_{t\ge 0}$.

Definition 4.2. [19] Let $\Lambda \mathrm{<mo>\; :\; </mo>}{E}_k\to E_k$ be an operator and assume that $F\in C_b\left(E_k\mathrm{,}{E}_k\right)$ satisfies the Lipschitz condition

${\Vert F\left(U\right)-F\left(V\right)\Vert }_{E_k}\le l_F{\Vert U-V\Vert }_{E_k}\mathrm{,}U\mathrm{,}V\in E_k\mathrm{.}$ (4.1)

If the point spectrum of the operator $\Lambda$ can be divided into two parts ${\sigma }_1$ and ${\sigma }_2$, where ${\sigma }_1$ is finite,

${\Lambda }_1=\sup \left\{Re\lambda \mathrm{<mo>\; |\; </mo>}\lambda \in {\sigma }_1\right\}\mathrm{,}{\Lambda }_2=\inf \left\{Re\lambda \mathrm{<mo>\; |\; </mo>}\lambda \in {\sigma }_2\right\}\mathrm{,}$ (4.2)

$E_{k_i}=span\left\{{\omega }_j\mathrm{<mo>\; |\; </mo>}{\lambda }_j\in {\sigma }_i\mathrm{,}i=\mathrm{1,2}\right\}\mathrm{.}$ (4.3)

and satisfies the condition

${\Lambda }_2-{\Lambda }_1>4l_F,$ (4.4)

and the orthogonal decomposition

$E_k=E_{k_1}\oplus E_{k_2},$ (4.5)

set $P_1\mathrm{<mo>\; :\; </mo>}{E}_k\to E_{k_1}$ and $P_2\mathrm{<mo>\; :\; </mo>}{E}_k\to E_{k_2}$ are both continuous orthogonal projections, then the operator $\Lambda$ is said to satisfy the spectral interval condition.

Lemma 4.1. [18] Let the eigenvalues ${\mu }_j^{\pm }\left(j\ge 1\right)$ be non-decreasing, and $m\in N^{\mathrm{<mo>\; *\; </mo>}}$, there exists $N\ge m$, such that ${\mu }_{N+1}^{-}$ and ${\mu }_N^{-}$ are consecutive adjacent values.

Equation (1.1) is equivalent to the following first order evolution equation

$U_t+\Lambda U=F\left(U\right),$ (4.6)

where

$U\in E_k,U={\left(u,v\right)}^{\text{T}}={\left(u,u_t\right)}^{\text{T}},$

$\Lambda =\left(\begin{array}{cc}0& -I\\ M\left({{\displaystyle \Vert {{\displaystyle \nabla }}^{m}u\Vert }}_p^p\right){\left(-\Delta \right)}^{2m}& \beta {\left(-\Delta \right)}^{2m}\end{array}\right),$

$F\left(U\right)=\left(\begin{array}{c}0\\ f\left(x\right)-g\left(u_t\right)\end{array}\right)$

a graph defined on $E_k$ by the quantity product:

${\langle U,V\rangle }_{E_k}=\left(M\cdot {\nabla }^{2m+k}u,{\nabla }^{2m+k}\bar{y}\right)+\left(v,\bar{z}\right).$ (4.7)

where $U={\left(u,v\right)}^{\text{T}},V={\left(y,z\right)}^{\text{T}}\in E_k$, $\bar{y}\mathrm{,}\bar{z}$ respectively represents the conjugation of y and z $v\mathrm{,}z\in H_0^{2m+k}\left(\Omega \right)$, $u\mathrm{,}y\in H_0^{2m+k}\left(\Omega \right)$. $\forall U\in E_k$, there is

$\begin{array}{c}{\langle \Lambda U,U\rangle }_{E_k}=-\left(M\cdot {\nabla }^{2m+k}{u}_t,{\nabla }^{2m+k}\bar{u}\right)+\left(M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m}u+\beta {\left(-\Delta \right)}^{2m}{u}_t,\bar{v}\right)\\ \ge -\left(M\cdot {\nabla }^{2m+k}{u}_t,{\nabla }^{2m+k}\bar{u}\right)+M\left({\nabla }^{2m+k}u,{\nabla }^{2m+k}v\right)+\beta \left(-{\Delta }^{m}v,-{\Delta }^m\bar{v}\right)\\ \ge \beta {\Vert {\nabla }^{2m}v\Vert }^2>0.\end{array}$ (4.8)

Therefore, the operator $\Lambda$ is monotonically increasing, and ${\langle \Lambda U\mathrm{,}U\rangle }_{E_k}$ is a nonnegative real number.

The characteristic equation $\Lambda U=\lambda U$, $U={\left(u,v\right)}^{\text{T}}\in E_k$ is equivalent to

$-v=\lambda u$ (4.9)

$M\left({{\displaystyle \Vert {{\displaystyle \nabla }}^{m}u\Vert }}_p^p\right){\left(-\Delta \right)}^{2m}u+\beta {\left(-\Delta \right)}^{2m}v=\lambda v.$ (4.10)

Therefore, $\lambda$ satisfies the following eigenvalue problem

$\{\begin{array}{l}{\lambda }^{2}u+M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right){\left(-\Delta \right)}^{2m}u-\beta \lambda {\left(-\Delta \right)}^{2m}u=0,\\ {u|}_{\partial \Omega }={{\left(-\Delta \right)}^{2m}u|}_{\partial \Omega }=0.\end{array}$ (4.11)

Given by Formulas (4.11) and (4.12), the corresponding eigenvectors take the form

$U_j^{\pm }=\left(u_j,-{\lambda }_j^{\pm }{u}_j\right),{\mu }_j={\lambda }_1{j}^{\frac{2m}{n}}.$ (4.12)

where ${\mu }_j\left(j\ge 1\right)$ is the eigenvecroot of ${\left(-\Delta \right)}^{2m}$ in $H_0^{2m}\left(\Omega \right)$.

For $\forall j\ge 1$, there is

$\Vert {\nabla }^{2m+k}{u}_j\Vert =\sqrt{{\mu }_j},\Vert {\nabla }^k{u}_j\Vert =1,\Vert {\nabla }^{-2m-k}{u}_j\Vert =\frac{1}{\sqrt{{\mu }_j}},k=1,2,\cdots ,2m.$ (4.13)

Take the position of (4.12) u in $u_j$ and use ${\left(-\Delta \right)}^{k}u$ to get the inner product

${\lambda }^2{\Vert {\nabla }^{k}u\Vert }^2+M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right){\Vert {\nabla }^{2m+k}u\Vert }^2-\beta \lambda {\Vert {\nabla }^{2m+k}u\Vert }^2=0.$ (4.14)

Consider the Formula (4.16) as the quadratic equation of $\lambda$, as follows:

${\lambda }_j^{\pm }=\frac{\beta {\mu }_j\pm \sqrt{{\beta }^2{\mu }_j^2-4M{\mu }_j}}{2}\mathrm{.}$ (4.15)

Theorem 4.1. If $N_1\in N$ is large enough, when $N\ge N_1$, the following inequality holds

$\frac{1}{8}\left(\left({\mu }_{N+1}-{\mu }_N\right)\left(\beta -\sqrt{{\beta }^2{\mu }_j^2-4M\left(s\right)}\right)-1\right)\ge l_F\mathrm{.}$ (4.16)

then the operator $\Lambda$ is said to satisfy the spectral interval condition.

Proof. Because all the eigenvalues of $\Lambda$ are positive real numbers, and the known sequence ${\left\{{\lambda }_N^{-}\right\}}_{N\ge 1}$ and ${\left\{{\lambda }_N^{+}\right\}}_{N\ge 1}$ is incremented.

This theorem is then proved in four steps.

Step 1: Known non-subtractive sequence of ${\lambda }_N^{\pm }$, according to lemma 4.1, for $\forall m\in N$, $\exists N\ge m$ makes ${\lambda }_N^{-}$ and ${\lambda }_{N+1}^{-}$ adjacent, the eigenvalues of the operator $\Lambda$ can be decomposed into

${\sigma }_1=\left\{{\lambda }_j^{-},{\lambda }_k^{+}|\max \left\{{\lambda }_j^{-},{\lambda }_k^{+}\right\}\le {\lambda }_N^{-}\right\},$ (4.17)

${\sigma }_2=\left\{{\lambda }_j^{+},{\lambda }_k^{\pm }|{\lambda }_j^{-}\le {\lambda }_N^{-}\le \min \left\{{\lambda }_j^{+},{\lambda }_k^{\pm }\right\}\right\}.$ (4.18)

Step 2: The corresponding $E_k$ can be decomposed into

$E_{k_1}=span\left\{U_j^{-},U_k^{+}|{\lambda }_j^{-},{\lambda }_k^{+}\in {\sigma }_1\right\},$ (4.19)

$E_{k_2}=span\left\{U_j^{-},U_k^{\pm }|{\lambda }_j^{-},{\lambda }_k^{\pm }\in {\sigma }_2\right\}.$ (4.20)

In order to make the two subspace orthogonal and satisfy the interspectral Formula (4.4).

${\Lambda }_1={\lambda }_N^{-},{\Lambda }_2={\lambda }_{N+1}^{-}$. further decomposition $E_{k_2}=E_C+E_R$, i.e.

$E_C=span\left\{U_j^{-}|{\lambda }_j^{-}\le {\lambda }_N^{-}\le {\lambda }_j^{+}\right\},$ (4.21)

$E_R=span\left\{U_R^{\pm }|{\lambda }_N^{-}\le {\lambda }_k^{\pm }\right\}.$ (4.22)

and let $E_N=E_{k_1}\oplus E_C$.

Next, we specify the quantity product of the eigenvalues over $E_k$, makes $E_{k_1}$ and $E_{k_2}$ orthogonal, here are two functions

$\Phi \mathrm{<mo>\; :\; </mo>}{E}_N\to R$, $\psi \mathrm{<mo>\; :\; </mo>}{E}_R\to R$.

$\begin{array}{c}\Phi \left(U\mathrm{,}V\right)=\beta \left({\nabla }^{2m+k}u\mathrm{,}{\nabla }^{2m+k}\bar{y}\right)+2\beta \left({\nabla }^{-2m-k}\bar{z}\mathrm{,}{\nabla }^{2m}u\right)\\ +2\beta \left({\nabla }^{-2m-k}v\mathrm{,}{\nabla }^{2m}\bar{y}\right)+4\left({\nabla }^{-2m-k}v\mathrm{,}{\nabla }^{-2m-k}z\right)\\ -4M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right)\left({\nabla }^k\bar{u}\mathrm{,}{\nabla }^{k}y\right)+\left(2{\beta }^2-\beta \right)\left({\nabla }^{2m+k}\bar{u}\mathrm{,}{\nabla }^{2m+k}y\right)\mathrm{.}\end{array}$ (4.23)

$\begin{array}{c}\Psi \left(U,V\right)=\left({\nabla }^{2m+k}u,{\nabla }^{2m+k}\bar{y}\right)+\left({\nabla }^{-2m-k}\bar{z},{\nabla }^{2m+k}u\right)-\left({\nabla }^{-2m-k}v,{\nabla }^{2m+k}\bar{y}\right)\\ -4M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right)\left({\nabla }^k\bar{u},{\nabla }^{k}y\right)+\left({\beta }^2-1\right)\left({\nabla }^{2m+k}\bar{u},{\nabla }^{2m+k}y\right).\end{array}$ (4.24)

where $U={\left(u,v\right)}^{\text{T}},V={\left(y,z\right)}^{\text{T}}\in E_N$, $\bar{y}\mathrm{,}\bar{z}$ respectively represents the conjugation of y and z.

Set $\forall U=\left(u\mathrm{,}v\right)\in E_N$, then

$\begin{array}{c}\Phi \left(U,U\right)=\beta \left({\nabla }^{2m+k}u,{\nabla }^{2m+k}\bar{u}\right)+2\beta \left({\nabla }^{-2m-k}\bar{v},{\nabla }^{2m}u\right)\\ +2\beta \left({\nabla }^{-2m-k}v,{\nabla }^{2m}\bar{u}\right)+4\left({\nabla }^{-2m-k}v,{\nabla }^{-2m-k}v\right)\\ -4M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right)\left({\nabla }^k\bar{u},{\nabla }^{k}u\right)+\left(2{\beta }^2-\beta \right)\left({\nabla }^{2m+k}\bar{u},{\nabla }^{2m+k}u\right)\\ \ge \beta {\Vert {\nabla }^{2m+k}u\Vert }^2-4{\Vert {\nabla }^{-2m-k}\bar{v}\Vert }^2-{\beta }^2{\Vert {\nabla }^{2m+k}u\Vert }^2-4M\left(s\right){\Vert {\nabla }^{m+k}u\Vert }^2\\ +\left(2{\beta }^2-\beta \right){\Vert {\nabla }^{2m+k}u\Vert }^2+4{\Vert {\nabla }^{-2m-k}\bar{v}\Vert }^2\\ ={\beta }^2{\Vert {\nabla }^{2m+k}u\Vert }^2-4{\mu }_1{\Vert {\nabla }^{k}u\Vert }^2\ge \left({\beta }^2{\mu }_j^2-4M\left(s\right)\right){\Vert {\nabla }^{k}u\Vert }^2.\end{array}$ (4.25)

Since $\beta$ is sufficiently large, can Be obtained $\Phi \left(U\mathrm{,}U\right)\ge 0$, $\forall U\in E_N$, thus $\Phi$ is positive definite.

In the same way, $\forall U=\left(u\mathrm{,}v\right)\in E_R$, there is

$\begin{array}{c}\Psi \left(U\mathrm{,}U\right)=\left({\nabla }^{2m+k}u\mathrm{,}{\nabla }^{2m+k}\bar{u}\right)+\left({\nabla }^{-2m-k}\bar{v}\mathrm{,}{\nabla }^{2m+k}u\right)-\left({\nabla }^{-2m-k}v\mathrm{,}{\nabla }^{2m+k}\bar{u}\right)\\ -4M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right)\left({\nabla }^k\bar{u}\mathrm{,}{\nabla }^{k}u\right)+\left({\beta }^2-1\right)\left({\nabla }^{2m+k}\bar{u}\mathrm{,}{\nabla }^{2m+k}u\right)\\ \ge {\Vert {\nabla }^{2m+k}u\Vert }^2-4M\left(s\right){\Vert {\nabla }^{m+k}u\Vert }^2+\left({\beta }^2-1\right){\Vert {\nabla }^{2m+k}u\Vert }^2\\ \ge \left({\beta }^2{\mu }_j^2-4M\left(s\right)\right){\Vert {\nabla }^{k}u\Vert }^2\mathrm{.}\end{array}$ (4.26)

$\Psi \left(U\mathrm{,}U\right)\ge 0$, $\Psi$ are positive definite.

Specifies the inner product of $E_k$ :

${\langle \langle U\mathrm{,}V\rangle \rangle }_{E_k}=\Phi \left(P_{N}U\mathrm{,}{P}_{N}V\right)+\Psi \left(P_{R}U\mathrm{,}{P}_{R}V\right)\mathrm{.}$ (4.27)

where $P_N$ and $P_R$ are maps of $E_k\to E_N$ and $E_k\to E_R$ respectively.

The above formula can be abbreviated to

${\langle \langle U\mathrm{,}V\rangle \rangle }_{E_k}=\Phi \left(U\mathrm{,}V\right)+\Psi \left(U\mathrm{,}V\right)\mathrm{.}$ (4.28)

In the inner product of $E_k$, $E_{k_1}$ and $E_{k_2}$ orthogonal. If $E_N$ and $E_C$ orthogonal, just have to prove it

$\begin{array}{l}{\langle \langle U_j^{+},U_j^{-}\rangle \rangle }_{E_k}=\Phi \left(U_j^{+},U_j^{-}\right)\\ =\beta \left({\nabla }^{2m+k}{u}_j,{\nabla }^{2m+k}{\bar{u}}_j\right)+2\beta \left(-{\lambda }_j^{-}{\nabla }^{-2m-k}{\bar{u}}_j,{\nabla }^{2m}{u}_j\right)\\ +2\beta \left(-{\lambda }_j^{+}{\nabla }^{-2m-k}{u}_j,{\nabla }^{2m}{\bar{u}}_j\right)+4\left(-{\lambda }_j^{+}{\nabla }^{-2m-k}{u}_j,-{\lambda }_j^{-}{\nabla }^{-2m-k}{u}_j\right)\\ -4M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right)\left({\nabla }^k{\bar{u}}_j,{\nabla }^k{u}_j\right)+\left(2{\beta }^2-\beta \right)\left({\nabla }^{2m+k}{\bar{u}}_j,{\nabla }^{2m+k}{u}_j\right)\end{array}$

$\begin{array}{l}=\beta {\Vert {\nabla }^{2m+k}{u}_j\Vert }^2-2\beta \left({\lambda }_j^{-}+{\lambda }_j^{+}\right){\Vert {\nabla }^{-k}{u}_j\Vert }^2+4{\lambda }_j^{-}{\lambda }_j^{+}{\Vert {\nabla }^{-2m-k}{u}_j\Vert }^2\\ -4M\left(s\right){\Vert {\nabla }^k{u}_j\Vert }^2+\left(2{\beta }^2-\beta \right){\Vert {\nabla }^{2m+k}{u}_j\Vert }^2\\ =2{\beta }^2{\mu }_j-2\beta \left({\lambda }_j^{-}+{\lambda }_j^{+}\right)+4{\lambda }_j^{-}{\lambda }_j^{+}\frac{1}{{\mu }_j}-4M\left(s\right)=0.\end{array}$ (4.29)

According to (4.13), and

${\lambda }_j^{-}+{\lambda }_j^{+}=\beta {\mu }_j.$ (4.30)

${\lambda }_j^{-}{\lambda }_j^{+}=M{\mu }_j.$ (4.31)

Step 3: Further estimating the Lipschitz constant $l_F$ of F, where $F\left(U\right)={\left(0,f\left(x\right)-g\left(u_t\right)\right)}^{\text{T}}$, $g\mathrm{<mo>\; :\; </mo>}{H}_0^{2m}\left(\Omega \right)\to L^2\left(\Omega \right)$. If $\forall U{\left(u\mathrm{,}v\right)}^{\text{T}}\in E_k$, $U={\left(u,v\right)}^{\text{T}}$, $V={\left(\tilde{u},\tilde{v}\right)}^{\text{T}}={\left(y,z\right)}^{\text{T}}\in E_k$, then

$\begin{array}{c}{\Vert F\left(U\right)-F\left(V\right)\Vert }_{E_k}\le \Vert D^k\left(g\left(u_t\right)-g\left(v_t\right)\right)\Vert \\ \le C_0{\Vert g^{\infty }\left(u_t\right)\Vert }_{\infty }\Vert D^k\left(u_t-v_t\right)\Vert \\ \le C_0{\Vert g^{\left(k\right)}\left(\xi \right)\Vert }_{\infty }{\Vert u_t-v_t\Vert }_k\\ \le l_F{\Vert U-V\Vert }_{E_k}.\end{array}$ (4.32)

Step 4: Now verify that the spectral interval condition ${\Lambda }_2-{\Lambda }_1>4l_F$ holds.

Then

${\Lambda }_2-{\Lambda }_1={\lambda }_{N+1}^{-}-{\lambda }_N^{-}=\frac{\beta }{2}\left({\mu }_{N+1}-{\mu }_N\right)+\frac{1}{2}\left(\sqrt{R\left(N\right)}-\sqrt{R\left(N+1\right)}\right).$ (4.33)

where $R\left(N\right)={\beta }^2{\mu }_N^2-4M{\mu }_N^2$.

There exists $N_1\ge 0$, such that for $\forall N\ge N_1$, $R_1\left(N\right)=1-\sqrt{\frac{{\beta }^2}{{\beta }^2{\mu }_j^2-4M\left(s\right)}-\frac{4M}{{\beta }^2{\mu }_j^2-4M\left(s\right)}}$. We can get

$\begin{array}{l}\sqrt{R\left(N\right)}-\sqrt{R\left(N+1\right)}+\sqrt{{\beta }^2{\mu }_j^2-4M\left(s\right)}\left({\mu }_{N+1}-{\mu }_N\right)\\ =\sqrt{{\beta }^2{\mu }_j^2-4M\left(s\right)}\left({\mu }_{N+1}{R}_1\left(N+1\right)-{\mu }_N{R}_1\left(N\right)\right)\mathrm{,}\end{array}$ (4.34)

According to assumption (H₃), we can easily see that

$\underset{N\to +\infty }{\lim }\left(\sqrt{R\left(N\right)}-\sqrt{R\left(N+1\right)}+\sqrt{{\beta }^2{\mu }_j^2-4M\left(s\right)}\left({\mu }_{N+1}-{\mu }_N\right)\right)=\mathrm{0,}$ (4.35)

Then according to (4.16) and (4.32)-(4.35), we have

${\Lambda }_2-{\Lambda }_1\ge \frac{1}{2}\left(\left({\mu }_{N+1}-{\mu }_N\right)\left(\beta -\sqrt{{\beta }^2{\mu }_j^2-4M\left(s\right)}\right)-1\right)\ge 4l_F\mathrm{.}$ (4.36)

So the spectral interval condition holds.

Theorem 4.2. Under the assumption of Theorem 4.1, the initial boundary value problem (1.1)-(1.3) has the family of inertial manifolds $h_k$ on the space $E_k$, and the form is

$h_k=graph\left(m\right):=\left\{{\zeta }_k+m\left({\zeta }_k\right):{\zeta }_k\in E_{k_1}\right\}$ (4.37)

where $E_{k_1}\mathrm{,}{E}_{k_2}$ is defined in Formulas (4.19)-(4.20), $m\mathrm{<mo>\; :\; </mo>}{E}_{k_1}\to E_{k_2}$ is a Lipschitz continuous function.

Conflicts of Interest

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

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