A Family of the Inertial Manifolds for a Class of Generalized Kirchhoff-Type Coupled Equations

Guoguang Lin, Jiaying Zhou

Department of Mathematics, Yunnan University, Kunming, China.

DOI: 10.4236/ojapps.2022.127076                

Abstract

The paper considers the long-time behavior for a class of generalized high-order Kirchhoff-type coupled equations, under the corresponding hypothetical conditions, according to the Hadamard graph transformation method, obtain the equivalent norm in space , and we obtain the existence of a family of the inertial manifolds while such equations satisfy the spectral interval condition.

Keywords

Kirchhoff-Type Coupled Equations, Spectral Interval Condition, A Family of the Inertial Manifolds

1. Introduction

This paper investigates the following primal value problems of a system of generalized Kirchhoff-type coupled equations:

$\{\begin{array}{ll}{u}_{tt}+M\left({\Vert D^{m}u\Vert }_p^p+{\Vert D^{m}v\Vert }_p^p\right){\left(-\Delta \right)}^{2m}u+\beta {\left(-\Delta \right)}^{2m}{u}_t+g_1\left(u,v\right)=f_1\left(x\right),\hfill & \text{(1)}\hfill \\ v_{tt}+M\left({\Vert D^{m}u\Vert }_p^p+{\Vert D^{m}v\Vert }_p^p\right){\left(-\Delta \right)}^{2m}v+\beta {\left(-\Delta \right)}^{2m}{v}_t+g_2\left(u,v\right)=f_2\left(x\right),\hfill & \text{(2)}\hfill \\ u\left(x,0\right)=u_{\text{0}}\left(x\right),u_t\left(x,0\right)=u_{\text{1}}\left(x\right),x\in \Omega ,\hfill & \text{(3)}\hfill \\ v\left(x,0\right)=v_{\text{0}}\left(x\right),v_t\left(x,0\right)=v_{\text{1}}\left(x\right),x\in \Omega ,\hfill & \text{(4)}\hfill \\ \frac{{\partial }^{i}u}{\partial n^i}=0,\frac{{\partial }^{i}v}{\partial n^i}=0\left(i=0,1,2,\cdots ,2m\right).\hfill & \text{(5)}\hfill \end{array}$

where $\Omega$ is a bounded region with a smooth boundary in $R^n$, $\partial \Omega$ represents the boundary of $\Omega$, $u_0\left(x\right),u_1\left(x\right)$ and $v_0\left(x\right),v_1\left(x\right)$ are known functions, where $g_j\left(u,v\right),f_j\left(u,v\right)\left(j=1,2\right)$ are nonlinear terms and external interference terms, respectively, and are known functions on $\Omega \times \left(0,T\right)$, $\beta$ is the normal number, $M\left({\Vert D^{m}u\Vert }_p^p+{\Vert D^{m}v\Vert }_p^p\right)$ is a non-negative first-order continuous derivative function, and $m>1$ is the normal number, ${\Vert D^{m}u\Vert }_p^p={\displaystyle \underset{\Omega }{\int }{\left|D^{m}u\right|}^p\text{d}x}$.

In order to overcome the research difficulties, G. Foias, G. R. Sell and R. Temam [1] proposed the concept of inertial manifolds, which greatly promoted the study of infinite-dimensional dynamical systems. Where the inertial manifold is a positive, finite-dimensional Lipschitz manifold, and the existence of an inertial manifold depends on the establishment of a spectral interval condition. Therefore, the research on a family of inertial manifolds is of great significance from both theoretical and practical aspects, and the relevant theoretical achievements can be referred to [2] - [9].

Guoguang Lin, Lingjuan Hu [10] studied a system of coupled wave equations of higher-order Kirchhoff type with strong damping terms

$\{\begin{array}{l}{u}_{tt}+M\left({\Vert {\nabla }^{m}u\Vert }^2+{\Vert {\nabla }^{m}v\Vert }^2\right){\left(-\Delta \right)}^{m}u+\beta {\left(-\Delta \right)}^m{u}_t+g_1\left(u,v\right)=f_1\left(x\right),\\ v_{tt}+M\left({\Vert {\nabla }^{m}u\Vert }^2+{\Vert {\nabla }^{m}v\Vert }^2\right){\left(-\Delta \right)}^{m}v+\beta {\left(-\Delta \right)}^m{v}_t+g_2\left(u,v\right)=f_2\left(x\right),\\ u\left(x,0\right)=u_0\left(x\right),u_t\left(x,0\right)=u_1\left(x\right),x\in \Omega ,\\ v\left(x,0\right)=v_0\left(x\right),v_t\left(x,0\right)=v_1\left(x\right),x\in \Omega ,\\ \frac{{\partial }^{i}u}{\partial n^i}=0,\frac{{\partial }^{i}v}{\partial n^i}=0\left(i=0,1,2,\cdots ,2m-1\right)x\in \partial \Omega .\end{array}$

where $\Omega$ is a bounded region with a smooth boundary in $R^n$, $\partial \Omega$ represents the boundary of $\Omega$, $g_j\left(u,v\right)\left(j=1,2\right)$ is a nonlinear source term, $f_1\left(x\right),f_2\left(x\right)$ is an external force interference term, and $\beta {\left(-\Delta \right)}^{m}u$, $\beta {\left(-\Delta \right)}^{m}v$ $\left(\beta \ge 0\right)$ is a strong dissipation terms. Using the Hadamard graph transformation method, the Lipschitz constant $l_F$ of F is further estimated, and the inertial manifolds that satisfies the spectral interval condition is obtained.

Lin Guoguang, Liu Xiaomei [11] studied a family of inertial manifolds for a class of generalized higher-order Kirchhoff equations with strong dissipation terms

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

where $m\in N^{+}$, $\Omega \subset R^n\left(n\ge 1\right)$ is a bounded domain with a smooth boundary in $\partial \Omega$, $f\left(x\right)$ is an external force term, $M\left({\Vert {\nabla }^{m}u\Vert }_p^p\right)$ is the stress term of Kirchhoff equation, $\beta {\left(-\Delta \right)}^{2m}{u}_t$ is a strong dissipative term, ${\left|u\right|}^{\rho }\left(u_t+u\right)$ is a nonlinear source term. Based on appropriate assumptions and the Hadamard graph transformation method, the spectral interval condition is verified, and the existence of a family of the inertial manifolds of the equation is obtained.

On the basis of previous research, rigid term strengthening becomes

$M\left({\Vert D^{m}u\Vert }_p^p+{\Vert D^{m}v\Vert }_p^p\right){\left(-\Delta \right)}^{2m}u$ and $M\left({\Vert D^{m}u\Vert }_p^p+{\Vert D^{m}v\Vert }_p^p\right){\left(-\Delta \right)}^{2m}v$, and this paper seeks a family of inertial manifolds. When defining the equivalence norm in space $E_k$, by making reasonable assumptions, it is obtained that the equation satisfies the spectral interval condition so that there is a family of inertial manifolds.

2. Preliminaries

For narrative convenience, we introduce the following symbols and assumptions:

Set $\nabla =D$. Consider Hilbert space family $V_{\alpha }=D\left({\left(-\Delta \right)}^{\alpha /2}\right),\alpha \in R$, whose inner product and norm are ${\left(•,•\right)}_{V_{\alpha }}=\left({\left(-\Delta \right)}^{\alpha /2},{\left(-\Delta \right)}^{\alpha /2}\right)$ and ${\Vert •\Vert }_{V_{\alpha }}=\Vert {\left(-\Delta \right)}^{\alpha /2}\Vert$, respectively. Apparently there are

$\begin{array}{l}{V}_0=L^2\left(\Omega \right),V_{2m}=H^{2m}\left(\Omega \right)\cap H_0^1\left(\Omega \right),V_{2m+k}=H^{2m+k}\left(\Omega \right)\cap H_0^1\left(\Omega \right),\\ V_k=H^k\left(\Omega \right)\cap H_0^1\left(\Omega \right),E_0=V_{2m}\times V_0\times V_{2m}\times V_0,\\ E_k=V_{2m+k}\times V_k\times V_{2m+k}\times V_k,\left(k=1,2,\cdots ,2m\right).\end{array}$

The assumption is as follows:

Let $M\left(s\right)$ be a continuous function on interval $D_{\text{1}}\left(D_{\text{1}}\in \Omega \right)$, and $M\left(s\right)\in C^1\left(R^{+}\right)$ :

$1\le {\mu }_0\le M\left(s\right)\le {\mu }_1$, set $M\left(s\right)=M\left({\Vert D^{m}u\Vert }_p^p+{\Vert D^{m}v\Vert }_p^p\right)$.

3. A Family of Inertial Manifolds

Definition 1 [12] lets $S={\left\{S\left(t\right)\right\}}_{t\ge 0}$ be the solution semigroup on Banach space $E_k=H_0^{2m+k}\left(\Omega \right)\times H_0^k\left(\Omega \right)\left(k=1,2,\cdots ,2m\right)$, and a subset ${\mu }_k\subset E_k$ satisfies:

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

2) ${\mu }_k$ is positively unchanging, ${\left\{S\left(t\right)\right\}}_{t\ge 0}:\forall u_0\in {\mu }_k,S\left(t\right)u_0\subset {\mu }_k,t\ge 0$ ;

3) ${\mu }_k$ attracts the solution orbit exponentially, i.e. for any $u\in E_k$, the existence constant $\eta >0,c>0$ makes $dist\left(S\left(t\right)u,\mu \right)\le c{\text{e}}^{-\eta t},t\ge 0$.

Then ${\mu }_k$ is called $E_k$ is a family of inertial manifolds.

In order to describe the spectral interval condition, first consider that the nonlinear term $F:E_k\to E_k$ is integrally bounded and continuous, and has a positive Lipschitz constant $l_F$, and its operator A has several eigenvalues and eigenfunctions of the positive real part.

Definition 2 [12] Set operator $A:{\rm X}\to {\rm X}$ has several eigenvalues of positive real numbers, and $F\in C_b\left({\rm X},{\rm X}\right)$ satisfies the Lipschitz condition:

${\Vert F\left(u\right)-F\left(v\right)\Vert }_{{\rm X}}\le l_F{\Vert u-v\Vert }_{{\rm X}},u,v\in {\rm X},$

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

$\begin{array}{l}{\Lambda }_1=\sup \left\{\mathrm{Re}\lambda |\lambda \in {\sigma }_1\right\},\\ {\Lambda }_2=\sup \left\{\mathrm{Re}\lambda |\lambda \in {\sigma }_2\right\},\\ {{\rm X}}_i=span\left\{{\omega }_j|{\lambda }_j\in {\sigma }_i\right\},i=1,2.\end{array}$

and conditions

${\Lambda }_2-{\Lambda }_1>4l_F$ (6)

are satisfied.

Where the continuous projection $P_1:{\rm X}\to {{\rm X}}_1,P_2:{\rm X}\to {{\rm X}}_2$, there is orthogonal decomposition ${\rm X}={{\rm X}}_1\oplus {{\rm X}}_2$, then the operator A satisfies the spectral interval condition.

Lemma 1 $g_i:V_{2m+k}\times V_{2m+k}\to V_{2m+k}\times V_{2m+k}\left(i=1,2\right)$ is a uniform bounded and integral Lipschitz continuous function.

Proof: $\forall \left(\tilde{u},\tilde{v}\right),\left(u,v\right)\in V_{2m+k}\times V_{2m+k}\left(k=1,2,\cdots ,2m\right)$,

$\begin{array}{l}{\Vert g_1\left(\tilde{u},\tilde{v}\right)-g_1\left(u,v\right)\Vert }_{V_{2m+k}\times V_{2m+k}}\\ =\Vert g_{1u}\left(u+\theta \left(\tilde{u}-u\right),v+\theta \left(\tilde{v}-v\right)\right)\left(\tilde{u}-u\right)\\ +{g_{1v}\left(u+\theta \left(\tilde{u}-u\right),v+\theta \left(\tilde{v}-v\right)\right)\left(\tilde{v}-v\right)\Vert }_{V_{2m+k}\times V_{2m+k}}\\ \le {\Vert g_{1u}\left(u+\theta \left(\tilde{u}-u\right),v+\theta \left(\tilde{v}-v\right)\right)\left(\tilde{u}-u\right)\Vert }_{V_{2m+k}\times V_{2m+k}}\\ +{\Vert g_{1v}\left(u+\theta \left(\tilde{u}-u\right),v+\theta \left(\tilde{v}-v\right)\right)\left(\tilde{v}-v\right)\Vert }_{V_{2m+k}\times V_{2m+k}}\\ \le l\left({\Vert \tilde{u}-u\Vert }_{V_{2m+k}}+{\Vert \tilde{v}-v\Vert }_{V_{2m+k}}\right).\end{array}$

Similarly, there are

${\Vert g_2\left(\tilde{u},\tilde{v}\right)-g_2\left(u,v\right)\Vert }_{V_{2m+k}\times V_{2m+k}}\le l\left({\Vert \tilde{u}-u\Vert }_{V_{2m+k}}+{\Vert \tilde{v}-v\Vert }_{V_{2m+k}}\right)$

where l is the Lipschitz constant of $g_i$, $\theta \in \left(0,1\right)$.

Lemma 2 [12] lets the sequence of eigenvalues ${\left\{{\mu }_j^{-}\right\}}_{j\ge 1}$ is a non-subtractive sequence, then $\exists N_0\in N_{+}$, for $\forall N\ge N_0$, ${\mu }_N^{-}$ and ${\mu }_{N+1}^{-}$ are consecutive adjacent values.

In order to verify that the operator satisfies the spectral interval condition, so as to draw the conclusion that there is a family of inertial manifolds in questions (1)-(5), the following definitions and assumptions can be made first.

Based on the above relevant conditions, consider the first-order development equation equivalent to Equations (1)-(5), as follows:

$U_t+A^{\prime }U=F\left(U\right)$ (7)

Of which $U=\left(u,z,v,q\right)$,

$A^{\prime }=\left(\begin{array}{cccc}0& -I& 0& 0\\ M\left(s\right){\left(-\Delta \right)}^{2m}& \beta {\left(-\Delta \right)}^{2m}& 0& 0\\ 0& 0& 0& -I\\ 0& 0& M\left(s\right){\left(-\Delta \right)}^{2m}& \beta {\left(-\Delta \right)}^{2m}\end{array}\right),$

$F\left(U\right)=\left(\begin{array}{c}0\\ f_1\left(x\right)-g_1\left(u,v\right)\\ 0\\ f_2\left(x\right)-g_2\left(u,v\right)\end{array}\right),$

$\begin{array}{l}D\left(A^{\prime }\right)=\left\{\left(u,v\right)\in V_{2m+k}\times V_{2m+k}|\left(u,v\right)\in V_0\times V_0,\left(D^{2m+k}u,D^{2m+k}v\right)\in V_0\times V_0\right\}\\ \times V_k\times V_k,\end{array}$

$s={\Vert D^{m}u\Vert }_p^p+{\Vert D^{m}v\Vert }_p^p.$

In order to determine the eigenvalue of matrix operator $A^{\prime }$, first consider graph module $\begin{array}{c}{\left(U,V\right)}_{E_k}=\left(M\left(s\right)D^{2m+k}u,D^{2m+k}\overline{u^{\prime }}\right)+\left(D^k\overline{z^{\prime }},D^{k}z)\right)\\ +\left(M\left(s\right)D^{2m+k}v,D^{2m+k}\overline{v^{\prime }}\right)+\left(D^k\overline{q^{\prime }},D^{k}q\right)\end{array}$ generated by inner product in $E_k$.

Where $U=\left(u,z,v,q\right),V=\left(u^{\prime },z^{\prime },v^{\prime },q^{\prime }\right)$, and $\overline{u^{\prime }},\overline{z^{\prime }},\overline{v^{\prime }},\overline{q^{\prime }}$ represent the conjugation of $u^{\prime },z^{\prime },v^{\prime },q^{\prime }$ respectively. In addition, operator $A^{\prime }$ is monotonic, and for $U\in D\left(A^{\prime }\right)$, there is

$\begin{array}{c}{\left(A^{\prime }U,U\right)}_{E_k}=-\left(M\left(s\right)D^{2m+k}z,D^{2m+k}\bar{u}\right)+\left(D^k\bar{z},M\left(s\right){\left(-\Delta \right)}^{2m}u+\beta {\left(-\Delta \right)}^{2m}z\right)\\ -\left(M\left(s\right)D^{2m+k}q,D^{2m+k}\bar{v}\right)+\left(\bar{q},M\left(s\right){\left(-\Delta \right)}^{2m}v+\beta {\left(-\Delta \right)}^{2m}q\right)\\ =-\left(M\left(s\right)D^{2m+k}z,D^{2m+k}\bar{u}\right)+\left(D^{2m+k}\bar{z},M\left(s\right)D^{2m+k}u\right)\\ +\left(D^{2m+k}\bar{z},\beta D^{2m+k}z\right)-\left(M\left(s\right)D^{2m+k}q,D^{2m+k}\bar{v}\right)\\ +\left(D^{2m+k}\bar{q},M\left(s\right)D^{2m+k}v\right)+\left(D^{2m+k}\bar{q},\beta D^{2m+k}q\right)\\ =\beta \left({\Vert D^{2m+k}z\Vert }^2+{\Vert D^{2m+k}q\Vert }^2\right)\ge 0,\end{array}$

Therefore, ${\left(A^{\prime }U,U\right)}_{E_k}$ is a nonnegative real number.

In order to further determine the eigenvalue of the matrix operator $A^{\prime }$, the following characteristic equation can be considered,

$A^{\prime }U=\lambda U,U=\left(u,z,v,q\right)\in E_k,$ (8)

That is

$\{\begin{array}{l}-z=\lambda u,\\ M\left(s\right){\left(-\Delta \right)}^{2m}u+\beta {\left(-\Delta \right)}^{2m}z=\lambda z,\\ -q=\lambda v,\\ M\left(s\right){\left(-\Delta \right)}^{2m}v+\beta {\left(-\Delta \right)}^{2m}q=\lambda q.\end{array}$

Thus $u,v$ meet the eigenvalue problem

$\{\begin{array}{l}{\lambda }^{2}u-\lambda \beta {\left(-\Delta \right)}^{2m}u+M\left(s\right){\left(-\Delta \right)}^{2m}u=0,\\ {\lambda }^{2}v-\lambda \beta {\left(-\Delta \right)}^{2m}v+M\left(s\right){\left(-\Delta \right)}^{2m}v=0,\\ {\frac{{\partial }^{i}u}{\partial n^i}|}_{\partial \Omega }={\frac{{\partial }^{i}v}{\partial n^i}|}_{\partial \Omega }=0,i=0,1,2,\cdots ,2m-1,\end{array}$

Take the inner product of ${\left(-\Delta \right)}^{k}u,{\left(-\Delta \right)}^{k}v$ and Equations (1) and (2) above respectively, with

$\{\begin{array}{l}{\lambda }^2{\Vert D^{k}u\Vert }^2-\lambda \beta {\Vert D^{2m+k}u\Vert }^2+M\left(s\right){\Vert D^{2m+k}u\Vert }^2=0,\\ {\lambda }^2{\Vert D^{k}v\Vert }^2-\lambda \beta {\Vert D^{2m+k}v\Vert }^2+M\left(s\right){\Vert D^{2m+k}v\Vert }^2=0,\end{array}$

That is

$\begin{array}{l}{\lambda }^2\left({\Vert D^{k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\right)-\lambda \beta \left({\Vert D^{2m+k}u\Vert }^2+{\Vert D^{2m+k}v\Vert }^2\right)\\ +M\left(s\right)\left({\Vert D^{2m+k}u\Vert }^2+{\Vert D^{2m+k}v\Vert }^2\right)=0.\end{array}$ (9)

Equation (9) is a univariate quadratic equation about $\lambda$. Replace $u,v$ with $u_j,v_j$. For each positive integer j, Equation (8) has paired eigenvalues

${\lambda }_j^{\pm }=\frac{\beta {\mu }_j\pm \sqrt{{\left(\beta {\mu }_j\right)}^2-4M\left(s\right){\mu }_j}}{2},$

where ${\mu }_j$ is the characteristic root of ${\left(-\Delta \right)}^{2m}$ in $V_{2m}\times V_{2m}$, then ${\mu }_j={\lambda }_1{j}^{\frac{2m}{n}}$.

If ${\left(\beta {\mu }_j\right)}^2\ge 4M\left(s\right){\mu }_j$, then ${\mu }_j\ge \frac{4M\left(s\right)}{{\beta }^2}$, the eigenvalues of operator $A^{\prime }$ are all real numbers, and the corresponding eigenfunction form is

$U_j^{\pm }=\left(u_j,-{\lambda }_j^{\pm }{u}_j,v_j,-{\lambda }_j^{\pm }{v}_j\right)$

For convenience, mark for any $j\ge 1$, there are

$\begin{array}{l}{\Vert D^{2m+k}{u}_j\Vert }^2+{\Vert D^{2m+k}{v}_j\Vert }^2={\mu }_j,{\Vert D^k{u}_j\Vert }^2+{\Vert D^k{v}_j\Vert }^2=1,\\ {\Vert D^{-2m-k}{u}_j\Vert }^2+{\Vert D^{-2m-k}{v}_j\Vert }^2=\frac{1}{{\mu }_j}.\end{array}$

Theorem 1: Assumes that l is the Lipschitz constant of $g_i\left(u,v\right)\left(i=1,2\right)$. When $N_0\in N_{+}$ is sufficiently large, for $\forall N\ge N_0$, the following inequality holds

$\left({\mu }_{N+1}-{\mu }_N\right)\left(\beta -\sqrt{{\beta }^{\text{2}}{\mu }_1-4M\left(s\right)}\right)\ge \frac{32l}{\sqrt{{\beta }^{\text{2}}{\mu }_1-4M\left(s\right)}}+1.$ (10)

Then all operators $A^{\prime }$ satisfy the spectral interval condition (6).

Proof. Because ${\mu }_j\ge \frac{4M\left(s\right)}{{\beta }^2}$ and the eigenvalues of $A^{\prime }$ are positive real numbers, ${\left\{{\lambda }_j^{-}\right\}}_{j\ge 1}$ and ${\left\{{\lambda }_j^{+}\right\}}_{j\ge 1}$ are single increment sequences.

The following four steps are taken to prove theorem 1:

Step 1: Because ${\left\{{\lambda }_j^{-}\right\}}_{j\ge 1}$ and ${\left\{{\lambda }_j^{+}\right\}}_{j\ge 1}$ are non subtractive columns, according to lemma 2, there are $\exists N_0\in N_{+}$, for $\forall N\ge N_0$, ${\lambda }_N^{-}$ and ${\lambda }_{N+1}^{-}$ are continuous adjacent values.

Therefore, there is N, so that ${\lambda }_N^{-}$ and ${\lambda }_{N+1}^{-}$ are continuous adjacent values, and the eigenvalue of $A^{\prime }$ can be decomposed into

${\sigma }_1=\left\{{\lambda }_r^{-}|1\le r\le N\right\},{\sigma }_2=\left\{{\lambda }_r^{+},{\lambda }_j^{\pm }|1\le r\le N\le j\right\}$

Step 2: Consider the corresponding decomposition of $E_k$ into

$\begin{array}{l}{E}_{k_1}=span\left\{U_r^{-}|{\lambda }_r^{-}\in {\sigma }_1\right\},\\ E_{k_2}=span\left\{U_r^{+},U_j^{\pm }|{\lambda }_r^{+},{\lambda }_j^{\pm }\in {\sigma }_2\right\}\end{array}$

The equivalent inner product ${\left(\left(U,V\right)\right)}_{E_k}$ given below makes $E_{k_1},E_{k_2}$ orthogonal.

Further decompose $E_{k_2}=E_H\oplus E_R$, of which

$E_H=span\left\{U_r^{+}|1\le r\le N\right\},E_R=span\left\{U_j^{\pm }|j\ge N\right\}$

Because $E_{k_1}$ and $E_H$ are finite dimensional subspaces, $U_N^{-}\in E_{k_1}$, $U_{N+1}^{-}\in E_R$, and $E_{k_1}$ and $E_R$ are orthogonal, while $E_{k_1}$ and $E_H$ are not orthogonal, $E_{k_1}$ and $E_{k_2}$ are not orthogonal. So we need to redefine the equivalent norm on $E_k$, so that $E_{k_1}$ and $E_H$ are orthogonal. Order $E_N=E_{k_1}\oplus E_H$.

Construct two functions $\Phi :E_N\to R,\Psi :E_R\to R$ of which,

$\begin{array}{c}\Phi \left(U,V\right)=2\beta \left(\beta -1\right)\left(D^{2m+k}u,D^{-\left(2m+k\right)}\overline{u^{\prime }}\right)+2\beta \left(D^{-\left(2m+k\right)}\overline{z^{\prime }},D^{2m+k}u\right)\\ +2\beta \left(D^{-\left(2m+k\right)}\bar{z},D^{2m+k}{u}^{\prime }\right)+4\left(D^{-\left(2m+k\right)}\overline{z^{\prime }},D^{-\left(2m+k\right)}z\right)\\ -4M\left(s\right)\left(D^{k}u,D^k\overline{u^{\prime }}\right)+2\beta \left(D^{2m+k}\bar{u},D^{2m+k}{u}^{\prime }\right)\\ +2\beta \left(\beta -1\right)\left(D^{2m+k}v,D^{2m+k}\overline{v^{\prime }}\right)+2\beta \left(D^{-\left(2m+k\right)}\overline{q^{\prime }},D^{2m+k}v\right)\\ +2\beta \left(D^{-\left(2m+k\right)}\bar{q},D^{2m+k}{v}^{\prime }\right)+4\left(D^{-\left(2m+k\right)}\overline{q^{\prime }},D^{-\left(2m+k\right)}q\right)\\ -4M\left(s\right)\left(D^{k}v,D^k\overline{v^{\prime }}\right)+2\beta \left(D^{2m+k}\bar{v},D^{2m+k}{v}^{\prime }\right),\end{array}$

$\begin{array}{c}\Psi \left(U,V\right)=2\beta \left(D^{2m+k}\bar{u},D^{2m+k}{u}^{\prime }\right)+\beta \left(D^{-\left(2m+k\right)}\overline{z^{\prime }},D^{2m+k}u\right)\\ +\beta \left(D^{-\left(2m+k\right)}\bar{z},D^{2m+k}{u}^{\prime }\right)+4\left(D^{-\left(2m+k\right)}\overline{z^{\prime }},D^{-\left(2m+k\right)}z\right)\\ -2M\left(s\right)\left(D^{k}u,D^k\overline{u^{\prime }}\right)+2\beta \left(\beta -1\right)\left(D^{2m+k}u,D^{-\left(2m+k\right)}\overline{u^{\prime }}\right)\\ +2\beta \left(D^{2m+k}\bar{v},D^{2m+k}{v}^{\prime }\right)+\beta \left(D^{-\left(2m+k\right)}\overline{q^{\prime }},D^{2m+k}v\right)\\ +\beta \left(D^{-\left(2m+k\right)}\bar{q},D^{2m+k}{v}^{\prime }\right)+4\left(D^{-\left(2m+k\right)}\overline{q^{\prime }},D^{-\left(2m+k\right)}q\right)\\ -2M\left(s\right)\left(D^{k}v,D^k\overline{v^{\prime }}\right)+2\beta \left(\beta -1\right)\left(D^{2m+k}v,D^{2m+k}\overline{v^{\prime }}\right).\end{array}$

Among them $U=\left(u,z,v,q\right),V=\left(u^{\prime },z^{\prime },v^{\prime },q^{\prime }\right)\in E_N$ or $E_R$.

For $U=\left(u,z,v,q\right)\in E_N$, then

$\begin{array}{c}\Phi \left(U,U\right)=2\beta \left(\beta -1\right)\left(D^{2m+k}u,D^{-\left(2m+k\right)}\bar{u}\right)+2\beta \left(D^{-\left(2m+k\right)}\bar{z},D^{2m+k}u\right)\\ +2\beta \left(D^{-\left(2m+k\right)}\bar{z},D^{2m+k}u\right)+4\left(D^{-\left(2m+k\right)}\bar{z},D^{-\left(2m+k\right)}z\right)\\ -4M\left(s\right)\left(D^{k}u,D^k\bar{u}\right)+2\beta \left(D^{2m+k}\bar{u},D^{2m+k}u\right)\\ +2\beta \left(\beta -1\right)\left(D^{2m+k}v,D^{2m+k}\bar{v}\right)+2\beta \left(D^{-\left(2m+k\right)}\bar{q},D^{2m+k}v\right)\\ +2\beta \left(D^{-\left(2m+k\right)}\bar{q},D^{2m+k}v\right)+4\left(D^{-\left(2m+k\right)}\bar{q},D^{-\left(2m+k\right)}q\right)\end{array}$

$\begin{array}{c}-4M\left(s\right)\left(D^{k}v,D^k\bar{v}\right)+2\beta \left(D^{2m+k}\bar{v},D^{2m+k}v\right)\\ \ge 2\beta \left(\beta -1\right)\left({\Vert D^{2m+k}u\Vert }^2+{\Vert D^{2m+k}v\Vert }^2\right)-4\beta (\Vert D^{-\left(2m+k\right)}z\Vert \Vert D^{2m+k}u\Vert \\ +\Vert D^{-\left(2m+k\right)}q\Vert \Vert D^{-\left(2m+k\right)}v\Vert )+4\left({\Vert D^{-\left(2m+k\right)}z\Vert }^2+{\Vert D^{-\left(2m+k\right)}q\Vert }^2\right)\\ -4M\left(s\right)\left({\Vert D^{k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\right)+2\beta \left({\Vert D^{2m+k}u\Vert }^2+{\Vert D^{2m+k}v\Vert }^2\right)\end{array}$

$\begin{array}{c}\ge 2\beta \left(\beta -1\right)\left({\Vert D^{2m+k}u\Vert }^2+{\Vert D^{2m+k}v\Vert }^2\right)-4\left({\Vert D^{-\left(2m+k\right)}z\Vert }^2+{\Vert D^{-\left(2m+k\right)}q\Vert }^2\right)\\ -{\beta }^2\left({\Vert D^{2m+k}u\Vert }^2+{\Vert D^{2m+k}v\Vert }^2\right)+4\left({\Vert D^{-\left(2m+k\right)}z\Vert }^2+{\Vert D^{-\left(2m+k\right)}q\Vert }^2\right)\\ -4M\left(s\right)\left({\Vert D^{k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\right)+2\beta \left({\Vert D^{2m+k}u\Vert }^2+{\Vert D^{2m+k}v\Vert }^2\right)\\ ={\beta }^2\left({\Vert D^{2m+k}u\Vert }^2+{\Vert D^{2m+k}v\Vert }^2\right)-4M\left(s\right)\left({\Vert D^{k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\right)\\ \ge \left({\beta }^2{\mu }_1-4M\left(s\right)\right)\left({\Vert D^{k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\right).\end{array}$

For any k, there is ${\beta }^2{\mu }_k\ge 4M\left({\mu }_k\right)$. According to hypothesis $1\le {\mu }_0\le M\left(s\right)\le {\mu }_1\le \frac{{\beta }^2{\mu }_k}{4}$, then $\Phi \left(U,U\right)\ge 0$, that is, $\Phi$ is positive definite.

Similarly, for any $U=\left(u,z,v,q\right)\in E_R$, there is

$\begin{array}{c}\Psi \left(U,U\right)=2\beta \left(D^{2m+k}\bar{u},D^{2m+k}u\right)+\beta \left(D^{-\left(2m+k\right)}\bar{z},D^{2m+k}u\right)\\ +\beta \left(D^{-\left(2m+k\right)}\bar{z},D^{2m+k}u\right)+4\left(D^{-\left(2m+k\right)}\bar{z},D^{-\left(2m+k\right)}z\right)\\ -2M\left(s\right)\left(D^{k}u,D^k\bar{u}\right)+2\beta \left(\beta -1\right)\left(D^{2m+k}u,D^{-\left(2m+k\right)}\bar{u}\right)\\ +2\beta \left(D^{2m+k}\bar{v},D^{2m+k}v\right)+\beta \left(D^{-\left(2m+k\right)}\bar{q},D^{2m+k}v\right)\\ +\beta \left(D^{-\left(2m+k\right)}\bar{q},D^{2m+k}v\right)+4\left(D^{-\left(2m+k\right)}\bar{q},D^{-\left(2m+k\right)}q\right)\end{array}$

$\begin{array}{c}-2M\left(s\right)\left(D^{k}v,D^k\bar{v}\right)+2\beta \left(\beta -1\right)\left(D^{2m+k}v,D^{2m+k}\bar{v}\right).\\ \ge 2\beta \left({\Vert D^{2m+k}u\Vert }^2+{\Vert D^{2m+k}v\Vert }^2\right)-2\beta (\Vert D^{-\left(2m+k\right)}z\Vert \Vert D^{2m+k}u\Vert \\ +\Vert D^{-\left(2m+k\right)}q\Vert \Vert D^{2m+k}v\Vert )+4\left({\Vert D^{-\left(2m+k\right)}z\Vert }^2+{\Vert D^{-\left(2m+k\right)}q\Vert }^2\right)\\ -2M\left(s\right)\left({\Vert D^{k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\right)+2\beta \left(\beta -1\right)\left({\Vert D^{2m+k}u\Vert }^2+{\Vert D^{2m+k}v\Vert }^2\right)\end{array}$

$\begin{array}{c}\ge 2\beta \left({\Vert D^{2m+k}u\Vert }^2+{\Vert D^{2m+k}v\Vert }^2\right)-4\left({\Vert D^{-\left(2m+k\right)}z\Vert }^2+{\Vert D^{-\left(2m+k\right)}q\Vert }^2\right)\\ -{\beta }^2\left({\Vert D^{2m+k}u\Vert }^2+{\Vert D^{2m+k}v\Vert }^2\right)+4\left({\Vert D^{-\left(2m+k\right)}z\Vert }^2+{\Vert D^{-\left(2m+k\right)}q\Vert }^2\right)\\ -2M\left(s\right)\left({\Vert D^{k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\right)+2\beta \left(\beta -1\right)\left({\Vert D^{2m+k}u\Vert }^2+{\Vert D^{2m+k}v\Vert }^2\right)\\ ={\beta }^2\left({\Vert D^{2m+k}u\Vert }^2+{\Vert D^{2m+k}v\Vert }^2\right)-2M\left(s\right)\left({\Vert D^{k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\right)\\ \ge \left({\beta }^2{\mu }_1-2M\left(s\right)\right)\left({\Vert D^{k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\right).\end{array}$

So there are $\forall U=\left(u,z,v,q\right)\in E_R$, $\Psi \left(U,U\right)\ge 0$, then $\Psi$ is also positive definite.

Redefine the inner product of $E_k$ :

${\left(\left(U,V\right)\right)}_{E_k}=\Phi \left(P_{N}U,P_{N}V\right)+\Psi \left(P_{R}U,P_{R}V\right)$ (11)

where $P_N$ and $P_R$ are projections of $E_k\to E_N$ and $E_k\to E_R$, respectively Here, Equation (11) is written as

${\left(\left(U,V\right)\right)}_{E_k}=\Phi \left(U,V\right)+\Psi \left(U,V\right)$

Under the redefined inner product of $E_k$, to prove that $E_{k_1}$ and $E_{k_2}$ are orthogonal, we only need to prove that $E_{k_1}$ and $E_H$ are orthogonal, that is,

${\left(\left(U_j^{-},U_j^{+}\right)\right)}_{E_k}=\Phi \left(U_j^{-},U_j^{+}\right)=0.$

Because there are $U_j^{-}\in E_{k_1},U_j^{+}\in E_H$, that is

$\begin{array}{c}\Phi \left(U_j^{-},U_j^{+}\right)=2\beta \left(\beta -1\right)\left(D^{2m+k}{u}_j,D^{2m+k}{\bar{u}}_j\right)-2\beta {\lambda }_j^{+}\left(D^{-\left(2m+k\right)}{\bar{u}}_j,D^{2m+k}{u}_j\right)\\ -2\beta {\lambda }_j^{-}\left(D^{-\left(2m+k\right)}{\bar{u}}_j,D^{2m+k}{u}_j\right)+4{\lambda }_j^{-}{\lambda }_j^{+}\left(D^{-\left(2m+k\right)}{\bar{u}}_j,D^{-\left(2m+k\right)}{u}_j\right)\\ -4M\left(s\right)\left(D^k{u}_j,D^k{\bar{u}}_j\right)+2\beta \left(D^{2m+k}{\bar{u}}_j,D^{2m+k}{u}_j\right)\\ +2\beta \left(\beta -1\right)\left(D^{2m+k}{v}_j,D^{2m+k}{\bar{v}}_j\right)-2\beta {\lambda }_j^{+}\left(D^{-\left(2m+k\right)}{\bar{v}}_j,D^{2m+k}{v}_j\right)\\ -2\beta {\lambda }_j^{-}\left(D^{-\left(2m+k\right)}{\bar{v}}_j,D^{2m+k}{v}_j\right)+4{\lambda }_j^{-}{\lambda }_j^{+}\left(D^{-\left(2m+k\right)}{\bar{v}}_j,D^{2m+k}{v}_j\right)\end{array}$

$\begin{array}{c}-4M\left(s\right)\left(D^k{v}_j,D^k{\bar{v}}_j\right)+2\beta \left(D^{2m+k}{\bar{v}}_j,D^{2m+k}{v}_j\right)\\ =2\beta \left(\beta -1\right)\left({\Vert D^{2m+k}{u}_j\Vert }^2+{\Vert D^{2m+k}{v}_j\Vert }^2\right)-2\beta \left({\lambda }_j^{-}+{\lambda }_j^{+}\right)({\Vert u_j\Vert }^2\\ +{\Vert v_j\Vert }^2)+4{\lambda }_j^{-}{\lambda }_j^{+}\left({\Vert D^{-\left(2m+k\right)}{u}_j\Vert }^2+{\Vert D^{-\left(2m+k\right)}{v}_j\Vert }^2\right)\\ -4M\left(s\right)\left({\Vert D^k{u}_j\Vert }^2+{\Vert D^k{v}_j\Vert }^2\right)+2\beta \left({\Vert D^{2m+k}{u}_j\Vert }^2+{\Vert D^{2m+k}{v}_j\Vert }^2\right)\\ =-4M\left({\mu }_j\right)+2{\beta }^2{\mu }_j-2\beta \left({\lambda }_j^{-}+{\lambda }_j^{+}\right)+4{\lambda }_j^{-}{\lambda }_j^{+}\cdot \frac{1}{{\mu }_j}.\end{array}$ (12)

Because of Equation (9), there are

${\lambda }_j^{+}+{\lambda }_j^{-}=\beta {\mu }_j,{\lambda }_j^{+}\cdot {\lambda }_j^{-}=M\left({\mu }_j\right){\mu }_j.$

So $\Phi \left(U_j^{-},U_j^{+}\right)=-4M\left({\mu }_j\right)+2{\beta }^2{\mu }_j-2\beta \left({\lambda }_j^{+}+{\lambda }_j^{-}\right)+4{\lambda }_j^{+}{\lambda }_j^{-}\cdot \frac{1}{{\mu }_j}=0$.

Step 3: according to the orthogonal decomposition established above, let’s prove that $A^{\prime }$ satisfies the spectral interval condition. First estimate the Lipschitz constant $l_F$ of F, where

$F\left(U\right)={\left(0,f_1\left(x\right)-g_1\left(u,v\right),0,f_2\left(x\right)-g_2\left(u,v\right)\right)}^{\text{T}}$

According to lemma 1, $g_i\left(u,v\right):V_{2m+k}\times V_{2m+k}\to V_{2m+k}\times V_{2m+k}$ are uniformly bounded and Lipschitz continuous, if $U=\left(u,z,v,q\right)\in E_k$, $U_i=\left(u_i,z_i,v_i,q_i\right)\in P_{i}U\left(i=1,2\right)$,

Then

$P_{1}u=u_1,P_{1}v=v_1,P_{2}u=u_2,P_{2}v=v_2.$

$\begin{array}{c}{\Vert U\Vert }_{E_k}^2=\Phi \left(P_{1}U,P_{2}U\right)+\Psi \left(P_{1}U,P_{2}U\right)\\ \ge \left({\beta }^2{\mu }_1-4M\left(s\right)\right)\left({\Vert D^k{P}_{1}u\Vert }^2+{\Vert D^k{P}_{1}v\Vert }^2\right)\\ +\left({\beta }^2{\mu }_1-2M\left(s\right)\right)\left({\Vert D^k{P}_{2}u\Vert }^2+{\Vert D^k{P}_{2}v\Vert }^2\right)\\ \ge \left({\beta }^2{\mu }_1-4M\left(s\right)\right)\left({\Vert D^{k}u\Vert }^2+{\Vert D^{k}v\Vert }^2\right).\end{array}$

Given $U=\left(u,z,v,q\right),V=\left(\tilde{u},\tilde{z},\tilde{v},\tilde{q}\right)\in E_k$, we can get

$\begin{array}{l}{\Vert F\left(U\right)-F\left(V\right)\Vert }_{E_k}\\ ={\Vert g_1\left(u,v\right)-g_1\left(\tilde{u},\tilde{v}\right)\Vert }_{V_{2m+k}\times V_{2m+k}}+{\Vert g_2\left(u,v\right)-g_2\left(\tilde{u},\tilde{v}\right)\Vert }_{V_{2m+k}\times V_{2m+k}}\\ \le 2l\left({\Vert u-\tilde{u}\Vert }_{V_{2m+k}}+{\Vert v-\tilde{v}\Vert }_{V_{2m+k}}\right)\\ \le \frac{\text{4}l}{\sqrt{{\beta }^2{\mu }_1-4M\left(s\right)}}{\Vert U-V\Vert }_{E_k}.\end{array}$

So

$l_F\le \frac{\text{4}l}{\sqrt{{\beta }^2{\mu }_1-4M\left(s\right)}}$ (13)

From (13), if

${\Lambda }_2-{\Lambda }_1={\lambda }_{N+1}^{-}-{\lambda }_N^{-}>\frac{\text{16}l}{\sqrt{{\beta }^2{\mu }_1-4M\left(s\right)}}$ (14)

Then the spectral interval condition (6) holds.

Step 4: according to the above paired eigenvalues, there are

${\Lambda }_2-{\Lambda }_1={\lambda }_{N+1}^{-}-{\lambda }_N^{-}=\frac{\beta }{2}\left({\mu }_{N+1}-{\mu }_N\right)+\frac{\sqrt{R\left(N\right)}-\sqrt{R\left(N+1\right)}}{2}.$ (15)

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

There are $N_0\in N_{+}$, for $\forall N\ge N_0$, let $R_0\left(N\right)=\sqrt{\frac{R\left(N\right)}{{\beta }^{\text{2}}{\mu }_1-4M\left(s\right)}}$, there are

$\begin{array}{l}\sqrt{R\left(N\right)}-\sqrt{R\left(N+1\right)}+\sqrt{{\beta }^{\text{2}}{\mu }_1-4M\left(s\right)}\left({\mu }_{N+1}-{\mu }_N\right)\\ =\sqrt{{\beta }^{\text{2}}{\mu }_1-4M\left(s\right)}\left(\left({\mu }_{N+1}-\frac{\sqrt{R\left(N+1\right)}}{\sqrt{{\beta }^{\text{2}}{\mu }_1-4M\left(s\right)}}\right)-\left({\mu }_N-\frac{\sqrt{R\left(N\right)}}{\sqrt{{\beta }^{\text{2}}{\mu }_1-4M\left(s\right)}}\right)\right)\\ =\sqrt{{\beta }^{\text{2}}{\mu }_1-4M\left(s\right)}\left(\left({\mu }_{N+1}-R_0\left(N+1\right)\right)-\left({\mu }_N-R_0\left(N\right)\right)\right)\end{array}$ (16)

And because of $\underset{N\to +\infty }{\lim }\left({\mu }_N-R_0\left(N\right)\right)=\underset{N\to +\infty }{\lim }\left({\mu }_N-\sqrt{\frac{R\left(N\right)}{{\beta }^{\text{2}}{\mu }_1-4M\left(s\right)}}\right)=0$, there are

$\underset{N\to +\infty }{\lim }\sqrt{R\left(N\right)}-\sqrt{R\left(N+1\right)}+\sqrt{{\beta }^{\text{2}}{\mu }_1-4M\left(s\right)}\left({\mu }_{N+1}-{\mu }_N\right)=0.$ (17)

According to the hypothesis (10) of Theorem 1 and Equations (13)-(17), there are

$\begin{array}{c}{\Lambda }_2-{\Lambda }_1\ge \frac{1}{2}\left(\left({\mu }_{N+1}-{\mu }_N\right)\left(\beta -\sqrt{{\beta }^{\text{2}}{\mu }_1-4M\left(s\right)}\right)-1\right)\\ \ge \frac{16l}{\sqrt{{\beta }^{\text{2}}{\mu }_1-4M\left(s\right)}}\ge 4l_F.\end{array}$ (18)

Theorem 1 is proved.

Theorem 2 [12] Through theorem1, operator $A^{\prime }$ satisfies the spectral interval condition, and problems (1)-(5) have a family of inertial manifolds ${\mu }_k$, and ${\mu }_k\in E_k$. The form is as follows,

${\mu }_k=graph\left(\Gamma \right)\in E_k:=\left\{\varsigma +\Gamma \left(\varsigma \right):\varsigma \in E_{k_1}\right\}$

where $\Gamma :E_{k_1}\to E_{k_2}$ is Lipschitz continuous and has Lipschitz constant $l_F$, and $graph\left(\Gamma \right)$ represents the graph of $\Gamma$.

Conflicts of Interest

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

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