Finite Dimensionality and Upper Semicontinuity of Kernel Sections for the Discrete Zakharov Equations

Abstract

In this paper, the authors first prove an upper bound of fractal dimension of kernel sections for the discrete Zakharov equations for plasmas with a quantum correction. Then they establish the upper semicontinuity of the kernel sections when the infinite lattice system is approximated by finite ordinary differential equations.

1 Introduction

In order to describe the interaction between high-frequency Langmuir waves and low frequency ion-acoustic waves, a set of coupled nonlinear wave equations was first derived by Zakharov [31]. In one dimension, the classical Zakharov equations can be written as

$$\begin{aligned} i\psi _{t}+\psi _{xx}&=u\psi , \end{aligned}$$

(1.1)

$$\begin{aligned} \frac{1}{\varrho ^2} u_{tt}-u_{xx}&=(|\psi |^{2})_{xx}, \end{aligned}$$

(1.2)

where the parameter $\varrho$ is proportional to the ion-acoustic speed. However, the classical Zakharov equations did not consider the quantum effects. The importance of quantum effects in ultrasmall electronic devices, in dense astrophysical plasmas systems and in laser plasmas has produced an increasing interest on the investigation of the quantum counterpart of some of the classical plasmas physics phenomena [26]. By making use of a quantum fluid approach, the modified Zakharov equations for plasmas with a quantum correction in the dissipative case can be written as

$$\begin{aligned}&\displaystyle i\psi _{t}+\psi _{xx}-h^2\psi _{xxxx}-\psi u+i\gamma \psi =g(x,t), \end{aligned}$$

(1.3)

$$\begin{aligned}&\displaystyle u_{tt}-u_{xx}+h^2u_{xxxx}-(|\psi |^2)_{xx}+\alpha u_{t}+\mu u=f(x,t), \end{aligned}$$

(1.4)

where the complex function $\psi (x,t)$ denotes the envelope of the high-frequency electric field and the real function u(x, t) represents the plasmas density measured from its equilibrium value. The dissipative mechanism of the system is introduced by the terms $i\gamma \psi , \alpha u_{t}$, and $\mu u$. The external forces f(x, t) and g(x, t) are complex and real-valued functions which are dependent on the time, respectively. The quantum parameter h expresses the ratio between the ion plasmon energy and electron thermal energy.

There are many works concerning the Cauchy problem and the initial boundary value problem of the continuous model of Zakharov equations or its related version, see [15–19] and the references therein. The soliton solution and numerical simulation were given to the continuous Zakharov equation with power law and dual-power law nonlinearities, see [10, 27]. However, there are few papers discussing the discrete modified Zakharov equations with a quantum correction.

Lattice dynamical systems (LDSs) are spatiotemporal systems with discretization in some variables which include coupled map lattices and coupled ordinary differential equations and cellular automata [6, 7]. Sometimes, LDSs arise as the spatial discretization of partial differential equations on unbounded domains. LDSs occur in a wide variety of applications, such as in chemical reaction theory [13, 22], in electrical engineering [11], in biology [23], image processing and pattern recognition [4, 5, 9], material science [21], laser systems [14], etc.

Recently, mathematicians and physicists have paid great attentions to the dynamics of infinite lattice systems, see [1–3, 6–8, 20, 28, 29, 32–37]. For example, the existence and upper semicontinuity of global attractors for autonomous lattice systems and other various properties of solutions for LDSs have been investigated [3, 6, 7, 28, 29, 32–35, 37] and the references therein. The exponential, uniform exponential, and pullback exponential attractors for nonautonomous lattice systems were investigated by [1, 2, 20, 36].

In this paper, we consider the following nonautonomous lattice system

$$\begin{aligned} i\dot{\psi }_{m}+(A\psi )_{m}-h^2(D\psi )_{m}-u_{m}\psi _{m}+i\gamma \psi _{m}&=g_{m}(t),\quad m\in \mathbf{\mathbb {Z}}, \quad t>\tau , \end{aligned}$$

(1.5)

$$\begin{aligned} \ddot{u}_{m}-(Au)_{m}+h^2(Du)_{m}-(A(|\psi |^2))_{m}+\alpha \dot{u}_{m}+\mu u_{m}&=f_{m}(t),\quad m\in \mathbf{\mathbb {Z}},\quad t>\tau , \end{aligned}$$

(1.6)

with initial conditions

$$\begin{aligned} \psi _{m}(\tau )=\psi _{m,\tau },\quad u_{m}(\tau )=u_{m,\tau },\quad \dot{u}_{m}(\tau ) =u_{1m,\tau }, \quad m\in \mathbf{\mathbb {Z}}, \quad \tau \in \mathbf{\mathbb {R}}, \end{aligned}$$

(1.7)

where $\psi _{m}(\cdot )\in \mathbb {C}, u_{m}(\cdot )\in \mathbb {R}$, and $\mathbb {C}$ and $\mathbb {R}$ are the sets of complex and real numbers, respectively, $\mathbb {Z}$ is the set of integer numbers, $i=\sqrt{-1}$ is the unit of the imaginary numbers, $h, \gamma ,\alpha$, and $\mu$ are positive constants, and A and D are both linear operators defined as

$$\begin{aligned} (Au)_{m}&=u_{m+1}-2u_{m}+u_{m-1},\quad \forall u=(u_{m})_{m\in \mathbf{\mathbb {Z}}}, \end{aligned}$$

(1.8)

$$\begin{aligned} (Du)_{m}&=u_{m+2}-4u_{m+1}+6u_{m}-4u_{m-1}+u_{m-2},\quad \forall u=(u_{m})_{m\in \mathbf{\mathbb {Z}}}. \end{aligned}$$

(1.9)

Equations (1.5)–(1.6) can be regarded as a discrete analogue of Eqs. (1.3)–(1.4). So we also call it as discrete modified Zakharov equations with a quantum correction.

There are some references concerning the discrete Zakharov equations. For example, article [38] proved the existence and upper semicontinuity of the compact uniform attractor, and article [30] established the existence of global attractor. Very recently, Liang et al. [25] proved the existence of compact kernel sections for the lattice system (1.5)–(1.7). Also, they gave an upper bound of the Kolmogorov $\varepsilon$-entropy for the obtained kernel sections.

In this paper, we further discuss the property of the kernel sections obtained by [25]. Firstly, we give an upper bound of the fractal dimension for the kernel sections. Secondly, we establish the upper semicontinuity of the kernel sections when the infinite lattice system is approximated by the finite ordinary differential equations (ODEs).

We want to point out that our idea concerning the existence of the fractal dimension originates from article [24, 41], which is a minor extension of the criteria of [12]. It is worthy mentioning that this minor extension is valid for a type of lattice system which consists of the type of term $(Bu)_m$ or $(Au)_m$, such as lattice long-wave-short-wave resonance equations [40], lattice KGS-type equations [39], and lattice Zakharov equations discussed in this paper.

Compared to the lattice Zakharov equations discussed in [30], the lattice modified Zakharov equations with a quantum correction contains the additional term $(D\psi )_m$. It is this term and the nonlinear term $(A|\psi |^2)_m$ that cause some difficulty in deriving the fractal dimension of the kernel sections. We need do some rigorous analysis and technical estimations to deal with these two terms.

The rest of the paper is organized as follows. In Sect. 2, we first introduce some spaces and operators, then we recall some results on the existence, uniqueness, and some estimations of the solutions. Section 3 is devoted to obtain an upper bound of fractal dimension of the kernel sections to the discrete Zakharov equations for plasmas with a quantum correction. In the last section, we verify the upper semicontinuity of the kernel sections.

2 Preliminaries

We first introduce some notations and operators. Set

$$\begin{aligned}&\ell ^{2} = \Big \{u=(u_{m})_{m\in \mathbf{\mathbb {Z}}}, \,\,u_{m}\in \mathbf{\mathbb {C}}:\sum \limits _{m\in \mathbf{\mathbb {Z}}}|u_{m}|^{2}<+\infty \Big \}, \end{aligned}$$

(2.1)

$$\begin{aligned}&l^{2} = \Big \{v=(v_{m})_{m\in \mathbf{\mathbb {Z}}}, \,\,v_{m}\in \mathbf{\mathbb {R}}:\sum \limits _{m\in \mathbf{\mathbb {Z}}}v_{m}^{2}<+\infty \Big \}. \end{aligned}$$

(2.2)

For brevity, we use X to denote $\ell ^{2}$ or $l^{2}$, and equip X with the inner product and norm as

$$\begin{aligned} (u,v) = \sum \limits _{m\in \mathbf{\mathbb {Z}}}u_{m}\bar{v}_{m},\quad \Vert u\Vert ^{2}=(u,u), \quad \forall u=(u_{m})_{m\in \mathbf{\mathbb {Z}}},\,v=(v_{m})_{m\in \mathbf{\mathbb {Z}}}\in X, \end{aligned}$$

where $\bar{v}_{m}$ denotes the conjugate of $v_{m}$. For any two elements $u,v \in X$, we define a bilinear form on X by

$$\begin{aligned} (u,v)_{\mu } = (Bu,Bv)+\mu (u,v), \end{aligned}$$

(2.3)

where $\mu$ is the constant from equation (1.6) and B is a linear operator defined as

$$\begin{aligned} (Bu)_{m} = u_{m+1}-u_{m}, \,\,\, m \in \mathbf{\mathbb {Z}},\quad \forall u=(u_{m})_{m\in \mathbf{\mathbb {Z}}}\in X. \end{aligned}$$

We also define a linear operator $B^{*}$ from X to X via

$$\begin{aligned} (B^{*}u)_{m} = u_{m-1}-u_{m}, \,\,\, m \in \mathbf{\mathbb {Z}}, \quad \forall u=(u_{m})_{m\in \mathbf{\mathbb {Z}}}\in X. \end{aligned}$$

In fact, $B^{*}$ is the adjoint operator of B and one can check that

$$\begin{aligned} (Bu,v)&=(u,B^*v), \quad (Au,v)=-(Bu,Bv), \quad (Du,v)=(Au,Av), \,\forall u, v\in X,\nonumber \\ \Vert B^*u\Vert ^2&=\Vert B u\Vert ^2 \leqslant 4\Vert u\Vert ^2, \quad \Vert A u\Vert ^2 \leqslant 16\Vert u\Vert ^2, \quad \Vert D u\Vert ^2 \leqslant 256\Vert u\Vert ^2, \,\forall u\in X. \end{aligned}$$

(2.4)

Clearly, the bilinear form $(\cdot ,\cdot )_{\mu }$ defined by (2.3) is also an inner product in X. Since

$$\begin{aligned} \mu \Vert u\Vert ^{2}\leqslant \mu \Vert u\Vert ^{2}+\Vert Bu\Vert ^{2} = \Vert u\Vert _{\mu }^{2}\leqslant (\mu +4)\Vert u\Vert ^{2}, \quad \forall u \in X, \end{aligned}$$

the norm $\Vert \cdot \Vert _{\mu }$ induced by $(\cdot ,\cdot )_{\mu }$ is equivalent to the norm $\Vert \cdot \Vert$. Write

$$\begin{aligned} \ell ^{2} = (\ell ^{2},(\cdot ,\cdot ),\Vert \cdot \Vert ),\quad l^{2}_{\mu } = (l^{2},(\cdot ,\cdot )_{\mu },\Vert \cdot \Vert _{\mu }),\quad l^{2}=(l^{2},(\cdot ,\cdot ),\Vert \cdot \Vert ), \end{aligned}$$

then $\ell ^{2}$, $\ell ^{2}_{\mu }$, and $l^{2}$ are all Hilbert spaces. Set

$$\begin{aligned} E_{\mu }=\ell ^{2}\times l^{2}_{\mu }\times l^{2}. \end{aligned}$$

For any two elements $z^{(j)}=(u^{(j)},v^{(j)},\varphi ^{(j)})^{T}\in E_{\mu }$, $j=1,2$, the inner product and norm of $E_{\mu }$ are defined as

$$\begin{aligned} (z^{(1)},z^{(2)})_{E_{\mu }}&= (u^{(1)},u^{(2)})+(v^{(1)},v^{(2)})_{\mu }+(\varphi ^{(1)},\varphi ^{(2)})\nonumber \\&= \sum \limits _{m\in \mathbf{\mathbb {Z}}}\left( u_{m}^{(1)}\overline{u}_{m}^{(2)}+ (Bv^{(1)})_{m}(Bv^{(2)})_{m} + \mu v_{m}^{(1)}v_{m}^{(2)} + \varphi _{m}^{(1)}{\varphi _{m}^{(2)}}\right) , \nonumber \\ \Vert z\Vert _{E_{\mu }}^{2}&= (z,z)_{E_{\mu }}, \quad \,\forall z\in E_{\mu }, \end{aligned}$$

(2.5)

where $\overline{u}_{m}^{(2)}$ stands for the conjugate of $u_{m}^{(2)}$.

For convenience, we shall express Eqs. (1.5)–(1.7) as an abstract Cauchy problem of first-order ODE with respect to time t in $E_{\mu }$. To this end, we put $\psi =(\psi _{m})_{m \in \mathbf{\mathbb {Z}}}$, $u=(u_{m})_{m \in \mathbf{\mathbb {Z}}}$, $\psi u=(\psi _{m}u_{m})_{m \in \mathbf{\mathbb {Z}}}$, $A|\psi |^2=\left( (A|\psi |^2)_m\right) _{m\in \mathbb {Z}}$, $f(t)=(f_{m}(t))_{m\in \mathbf{\mathbb {Z}}}$, $g(t)=(g_{m}(t))_{m \in \mathbf{\mathbb {Z}}}$, $\psi _{\tau }=(\psi _{m,\tau })_{m \in \mathbf{\mathbb {Z}}}$, $u_{\tau }=(u_{m,\tau })_{m \in \mathbf{\mathbb {Z}}}$, $u_{1\tau }=(u_{1m,\tau })_{m \in \mathbf{\mathbb {Z}}}$. Then Eqs. (1.5)–(1.7) can be written as

$$\begin{aligned}&i\dot{\psi }+A\psi -h^2D\psi -u\psi +i\gamma \psi =g(t), \quad t>\tau , \end{aligned}$$

(2.6)

$$\begin{aligned}&\ddot{u}-Au+h^2Du-A|\psi |^2+\alpha \dot{u}+\mu =f(t),\quad t>\tau , \end{aligned}$$

(2.7)

$$\begin{aligned}&\psi (\tau )=\psi _{\tau },\,\,u(\tau )=u_{\tau },\,\,\dot{u}(\tau )=u_{1\tau }, \,\,\tau \in \mathbf{\mathbb {R}}. \end{aligned}$$

(2.8)

We further set

$$\begin{aligned} \varphi =\dot{u}+\lambda u, \quad \mathrm{where}\,\,\lambda = \frac{\mu \alpha }{4\mu +\alpha ^{2}}\in \left( 0,\frac{\alpha }{4}\right) , \end{aligned}$$

(2.9)

$z=(\psi ,u,\varphi )^T$, $F(z,t) =(-i\psi u-ig(t),0,f(t)+A |\psi |^2)^T$ and

$$\begin{aligned} \begin{array}{ccc} \Theta = \left( \begin{array}{ccc} \gamma I- i A + i h^{2}D&{} 0 &{} 0\\ 0 &{}\lambda I &{} -I \\ 0&{}\lambda (\lambda -\alpha )I+\mu I-A +h^{2}D \quad &{}(\alpha -\lambda )I\end{array}\right) , \end{array} \end{aligned}$$

(2.10)

where I is the identity operator. Then Eqs. (2.6)–(2.8) can be written as

$$\begin{aligned} \dot{z}+\Theta z&=F(z,t), \quad t>\tau , \end{aligned}$$

(2.11)

$$\begin{aligned} z(\tau )&=z_{\tau }=(\psi _{\tau },u_{\tau },\varphi _{\tau })^{T}=(\psi _{\tau },u_{\tau },u_{1\tau }+\lambda u_{\tau })^{T}, \quad \tau \in \mathbf{\mathbb {R}}. \end{aligned}$$

(2.12)

We next introduce the space which the external forces functions belong to. Let ${\mathcal {C}}_{b}(\mathbf{\mathbb {R}},X)$ be the set of continuous and bounded functions from $\mathbf{\mathbb {R}}$ into X, then for each function $f(t)\in {\mathcal {C}}_{b}(\mathbf{\mathbb {R}},X)$, we have $\sup _{t\in \mathbf{\mathbb {R}}}\sum _{m\in \mathbf{\mathbb {Z}}}|f_{m}(t)|^{2}<+\infty$. Write

$$\begin{aligned}&{\mathcal {H}}=\Big \{f(t)=( f_{m}(t))_{m \in \mathbf{\mathbb {Z}}}\in {\mathcal {C}}_{b}(\mathbf{\mathbb {R}}, X){:}\, \mathrm{for\;each} \,\tau \in \mathbf{\mathbb {R}} \, \mathrm{and} \, \forall \,\varepsilon > 0, \,\exists M(\varepsilon ,\tau )\, \in \mathbf{\mathbb {N}},\nonumber \\&\qquad \quad \,\,\mathrm{such} \, \mathrm{that} \sum \limits _{|m|\geqslant M(\varepsilon ,\tau )}|f_{m}(s)|^{2}\leqslant \varepsilon \,\, \mathrm{for} \, \mathrm{any} \, s\leqslant \tau \Big \}. \end{aligned}$$

(2.13)

Note that we use $\mathbb {N}$ to denote the set of positive integers throughout this paper.

In this paper, we need the following assumptions on the parameters.

Assumption (H) Assume the parameters $h, \mu$, and $\alpha$ satisfying

$$\begin{aligned} 4h(4\mu +\alpha ^2)\leqslant \mu \alpha . \end{aligned}$$

We next recall some known results of solutions to Eqs. (2.11)–(2.12).

Lemma 2.1

([25]) Let assumption (H) hold and $g(t)=(g_{m}(t))_{m \in \mathbf{\mathbb {Z}}}\in {\mathcal {C}}_{b}(\mathbf{\mathbb {R}},\ell ^{2})$, $f(t)=(f_{m}(t))_{m \in \mathbf{\mathbb {Z}}}\in {\mathcal {C}}_{b}(\mathbf{\mathbb {R}},l^{2})$. Then for any initial value $z_{\tau }=(\psi _{\tau },u_{\tau },\varphi _{\tau })^{T}\in E_{\mu }$, there exists a unique solution $z(t)=(\psi (t),u(t),\varphi (t))^{T}\in E_{\mu }$ of Eqs. (2.11)–(2.12) such that $z(t)\in {\mathcal {C}}([\tau , +\infty ),E_{\mu })\cap {\mathcal {C}}^{1}((\tau , +\infty ),E_{\mu })$. Moreover, the mapping

$$\begin{aligned} U(t,\tau ):z_{\tau } = (\psi _{\tau },u_{\tau },\varphi _{\tau })^{T}\in E_{\mu }\mapsto z(t) = (\psi (t),u(t),\varphi (t))^{T}\in E_{\mu },\quad \forall t\geqslant \tau \end{aligned}$$

(2.14)

generates a continuous process $\{U(t,\tau )\}_{t\geqslant \tau }$ on $E_{\mu }$, where $\varphi _\tau =u_{1\tau }+\lambda u_\tau$.

Lemma 2.2

([25]) Let the conditions of Lemma 2.1 hold. Then the solution $z(t)=(\psi (t),u(t),\varphi (t))^{T}=U(t,t-s)z_{t-s}\in E_{\mu }$ with initial value $z_{t-s}=(\psi _{t-s},u_{t-s},\varphi _{t-s})^{T}\in E_{\mu }$ of Eqs. (2.11)–(2.12) satisfies

$$\begin{aligned} \Vert z(t)\Vert ^{2}_{E_{\mu }} \leqslant C_{0}{\mathrm e}^{-2\vartheta s}+\frac{r^{2}_{0}}{2\vartheta },\quad \forall s>0, \end{aligned}$$

where $C_{0}, \vartheta$ and $r_{0}$ are positive constants independent of t and s.

Lemma 2.3

([25]) Let the conditions of Lemma 2.1 hold. Then the process $\{U(t,\tau )\}_{t\geqslant \tau }$ corresponding to Eqs. (2.11)–(2.12) is uniformly pullback bounded dissipative in the sense that for any bounded set ${\mathcal {B}}$ of $E_{\mu }$, there exists a time $s({\mathcal {B}})$ yielding

$$\begin{aligned} U(t,t-s){\mathcal {B}}\subset {\mathcal {B}}_0, \quad \forall s\geqslant s({\mathcal {B}}), \end{aligned}$$

where ${\mathcal {B}}_0={\mathcal {B}}(0,R_{0})\subset E_{\mu }$ is a closed ball centered at 0 with radius $\displaystyle R_{0}=\frac{r_{0}}{\sqrt{\vartheta }}$.

Lemma 2.3 shows that there exists a time $s_{0}= s_{0}({\mathcal {B}}_0)$ such that

$$\begin{aligned} U(t,t-s){\mathcal {B}}_0\subset {\mathcal {B}}_0, \quad \forall s\geqslant s_{0}. \end{aligned}$$

(2.15)

Lemma 2.4

([25]) Let assumption (H) hold and $g(t)=(g_{m}(t))_{m \in \mathbf{\mathbb {Z}}}\in {\mathcal {H}}$ with $X=\ell ^{2}$ and $f(t)=(f_{m}(t))_{m \in \mathbf{\mathbb {Z}}}\in {\mathcal {H}}$ with $X=l^{2}$, respectively. Then for any $\varepsilon >0$, there exist $T(\varepsilon ,t,{\mathcal {B}}_0)>0$ and $M(\varepsilon ,t,{\mathcal {B}}_0)\in {\mathbb {N}}$, such that when $s\geqslant T(\varepsilon ,t,{\mathcal {B}}_0)$ the solution $z(t)=U(t,t-s)z_{t-s}\in E_{\mu }$ with initial value $z_{t-s}\in {\mathcal {B}}_0$ of Eqs. (2.11)–(2.12) satisfies

$$\begin{aligned} \sum \limits _{|m|>M(\varepsilon ,t,{\mathcal {B}}_0)}|(U(t,t-s)z_{t-s})_{m}|^{2}_{E_{\mu }} = \sum \limits _{|m|>M(\varepsilon ,t,{\mathcal {B}}_0)}|z_{m}(t)|^{2}_{E_{\mu }}\leqslant \varepsilon ^{2}, \end{aligned}$$

(2.16)

hereinafter $|z_{m}|^{2}_{E_{\mu }}=|\psi _m|^2+(Bu)^2_m+\mu u_m^2+\varphi _m^2$.

Definition 2.1

A function $z(s)\in \mathcal {C}({\mathbb {R}},E_\mu )$ is said to be a complete trajectory of the process $\{U(t,\tau )\}_{t\geqslant \tau }$ provided $U(t,\tau )z(\tau )=z(t), \forall t\geqslant \tau , \tau \in \mathbb {R}$. The kernel ${\mathcal {K}}$ of the process $\{U(t,\tau )\}_{t\geqslant \tau }$ consists of all bounded complete trajectories of it, and the set ${\mathcal {K}}(s)=\{z(s)\in E_\mu \,\big |\,z(s)\in {\mathcal {K}}\}$ is called the kernel section of the kernel ${\mathcal {K}}$ at time $s\in \mathbb {R}$.

Lemma 2.5

([25]) Let the conditions of Lemma 2.4 hold. Then the process $\{U(t,\tau )\}_{t\geqslant \tau }$ corresponding to Eqs. (2.11)–(2.12) possesses a family of compact kernel sections $\{\mathcal {K}(\tau )\}_{\tau \in \mathbb {R}}\subset E_{\mu }$.

We end this section with the definition of fractal dimension of the kernel sections $\{\mathcal {K}(\tau )\}_{\tau \in \mathbb {R}}$.

Definition 2.2

For each $\tau \in \mathbb {R}$, the fractal dimension $\mathrm{dim}_F \mathcal {K}(\tau )$ of $\mathcal {K}(\tau )$ is defined by

$$\begin{aligned} \mathrm{dim}_F \mathcal {K}(\tau )=\limsup \limits _{\varepsilon \rightarrow 0}\frac{\ln {\mathcal {N}} (\mathcal {K}(\tau ), \varepsilon )}{\ln (1/ \varepsilon )}, \end{aligned}$$

where ${\mathcal {N}} (\mathcal {K}(\tau ), \varepsilon )$ is the minimal number of closed sets of the diameter $2\varepsilon$ which cover the set $\mathcal {K}(\tau )$.

3 Finite Fractal Dimension of the Kernel Sections

In this section, we will use the criteria presented in [24] to estimate the upper bound of fractal dimension of the kernel sections $\{\mathcal {K}(\tau )\}_{\tau \in \mathbb {R}}$ obtained by [25].

Write

$$\begin{aligned} E_{\mu }^{(N)} = \{z=(z_{m})_{m \in \mathbf{\mathbb {Z}}}\,\big ||m|\leqslant N\}, \end{aligned}$$

(3.1)

then $E_{\mu }^{(N)}$ is a $4(2N+1)$-dimensional space. Define a bounded projection $P_{N}: E_{\mu }\mapsto E_{\mu }^{(N)}$ by

$$\begin{aligned} \quad (P_{N}z)_{m}=\left\{ \begin{array}{ll} z_{m}, \,\,|m|\leqslant N; \\ 0, \, \quad |m|> N, \end{array} \right. \quad z=(z_m)_{m\in \mathbb {Z}}\in E_\mu . \end{aligned}$$

Theorem 3.1

Let the conditions of Lemma 2.4 hold and $\{\mathcal {K}(\tau )\}_{\tau \in \mathbb {R}}$ be the kernel sections of the process $\{U(t,\tau )\}_{t\geqslant \tau }$. Then for each $\tau \in \mathbb {R}$, the fractal dimension of $\mathcal {K}(\tau )$ satisfies

$$\begin{aligned} \dim _F\mathcal {K}(\tau ) \leqslant 4(4N^*+1) \cdot \ln \big (1+\frac{8(1+\mathrm {e}^{(C_1+C_2)T^*})}{\frac{1}{2}-\eta }\big ) \cdot \left( \ln \frac{2}{\frac{3}{2}+\eta }\right) ^{-1}, \end{aligned}$$

(3.2)

where $N^*,\ T^*,\ \eta ,\ C_1$, and $C_2$ are constants depending on $\alpha ,\gamma ,\mu$, and h, respectively.

Proof

According to [24, Theorem 4.1] or [41, Theorem 2.1], we prove Theorem 3.1 by two steps.

Step 1 According to [25, Theorem 3.1], there exists a uniform finite covering of closed subsets with diameter 2 of ${\mathcal {K}}(\tau )$ for each $\tau \in {\mathbb {R}}$. We next prove that $U(t,\tau )$ is Lipschitz on ${\mathcal {K}}(\tau )$. In fact, for each $\tau \in {\mathbb {R}}$, let

$$\begin{aligned} z^{(1)}(t)=U(t,\tau )z^{(1)}_{\tau },\quad z^{(2)}(t)=U(t,\tau )z^{(2)}_{\tau }, \quad \forall t\geqslant \tau \end{aligned}$$

be two solutions of Eqs. (2.11)–(2.12) with initial data $z^{(1)}_{\tau },z^{(2)}_{\tau }\in {\mathcal {K}}(\tau )\subset {\mathcal {B}}_{0}$, respectively. Then $z^{(1)}(t),z^{(2)}(t) \in {\mathcal {K}}(t)\subset {\mathcal {B}}_{0}$ for $t-\tau \geqslant s_{0}$. Set

$$\begin{aligned} \psi _{d}(t)= & {} \psi ^{(1)}(t)-\psi ^{(2)}(t),\quad u_{d}(t)=u^{(1)}(t)-u^{(2)}(t),\nonumber \\ \varphi _{d}(t)= & {} \varphi ^{(1)}(t)-\varphi ^{(2)}(t), \quad z_{d}(t)=z^{(1)}(t)-z^{(2)}(t). \end{aligned}$$

From (2.11)–(2.12), we get

$$\begin{aligned} \dot{z}_{d}+\Theta z_{d}&=F(z^{(1)},t)-F(z^{(2)},t), \quad \forall t\geqslant \tau ,\nonumber \\ z_{d}(\tau )&= z^{(1)}_{\tau }-z^{(2)}_{\tau }. \end{aligned}$$

(3.3)

Taking the real part of the inner product of (3.3) with $z_{d}$ in $E_{\mu }$ yields

$$\begin{aligned} \frac{1}{2}\frac{\mathrm {d}}{{\mathrm {d}}t}\Vert z_{d}\Vert ^{2}_{E_{\mu }} + {\mathbf{Re}}\big (\Theta z_{d},z_{d}\big )_{E_{\mu }} ={\mathbf{Re}}\big (F(z^{(1)},t)-F(z^{(2)},t),z_{d}\big )_{E_{\mu }},\quad \forall t\geqslant \tau . \end{aligned}$$

(3.4)

Since $\Theta : E_{\mu }\mapsto E_{\mu }$ is a bounded linear operator, $F: E_{\mu }\times \mathbf{\mathbb {R}}\mapsto E_{\mu }$ is a locally Lipschitz continuous operator (see [25, Lemma 2.2]) and ${\mathcal {B}}_0$ is a bounded set in $E_{\mu }$, we see that there exist two positive constants $C_1$ and $C_{2}=C_{2}({\mathcal {B}}_0)$ such that

$$\begin{aligned} {\mathbf{Re}}\big (-&\Theta z_{d}+F(z^{(1)},t)-F(z^{(2)},t),z_{d}\big )_{E_{\mu }} \leqslant C_1\Vert z_{d}\Vert ^2_{E_{\mu }}+C_2 \Vert z_{d}\Vert ^2_{E_{\mu }} \nonumber \\&=(C_1+C_{2})\Vert z_{d}\Vert _{E_{\mu }}^{2},\quad \forall t\geqslant \tau .\quad \end{aligned}$$

(3.5)

Then (3.4) and (3.5) give

$$\begin{aligned} \frac{\mathrm {d}}{{\mathrm {d}}t}\Vert z_{d}\Vert ^{2}_{E_{\mu }} \leqslant 2(C_{1}+C_{2})\Vert z_{d}\Vert ^{2}_{E_{\mu }}, \quad \forall t\geqslant \tau . \end{aligned}$$

(3.6)

Applying Gronwall inequality to (3.6) gives

$$\begin{aligned} \Vert z^{(1)}(t)-z^{(2)}(t)\Vert _{E_{\mu }} = \Vert z_{d}(t)\Vert _{E_{\mu }} \leqslant {\mathrm e}^{(C_{1}+C_{2})(t-\tau )}\Vert z^{(1)}_{\tau }-z^{(2)}_{\tau }\Vert _{E_{\mu }} ,\quad \forall t\geqslant \tau , \end{aligned}$$

(3.7)

which implies that $U(t,\tau )$ is Lipschitz on ${\mathcal {K}}(\tau )$.

Step 2 Define a function $\chi (x)\in {\mathcal {C}}^{1}(\mathbf{\mathbb {R}}_+,[0,1])$ such that

$$\begin{aligned} \quad \chi (x)=\left\{ \begin{array}{ll} 0, \,\,0\leqslant x\leqslant 1; \\ 1, \,\,x\geqslant 2, \end{array} \right. \, \mathrm{and }\,\,|\chi '(x)|\leqslant \chi _0 \, (\mathrm{positive \,\,constant}), \,\,\forall x\in \mathbf{\mathbb {R}}_+. \end{aligned}$$

Set

$$\begin{aligned} \xi _d&=(\xi _{dm})_{m\in \mathbb {Z}}\,\,\mathrm{with}\,\,\xi _{dm}=\chi \left( \frac{|m|}{M}\right) \psi _{dm},\\ v_d&=(v_{dm})_{m\in \mathbb {Z}}\,\,\mathrm{with}\,\,v_{dm}=\chi \left( \frac{|m|}{M}\right) u_{dm}, \\ w_d&=(w_{dm})_{m\in \mathbb {Z}}\,\,\mathrm{with}\,\,w_{dm}=\chi \left( \frac{|m|}{M}\right) \varphi _{dm}, y_{d}\\&=(y_{dm})_{m\in \mathbb {Z}}\,\,\mathrm{with}\,\,y_{dm}=(\xi _{dm},v_{dm},w_{dm}), \end{aligned}$$

where M is a positive integer that will be specified later. By (2.6), we get

$$\begin{aligned} i\dot{\psi }_{d}+A\psi _{d}+i\gamma \psi _{d}-h^2D\psi _{d}=\psi ^{(1)}u^{(1)}-\psi ^{(2)}u^{(2)}. \end{aligned}$$

(3.8)

Taking the imaginary part of the inner product of (3.8) with $\xi _{d}$ in $\ell ^{2}$ yields

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm {d}}{{\mathrm {d}}t}\sum \limits _{m\in \mathbf{\mathbb {Z}}} \chi \left( \frac{|m|}{M}\right) |\psi _{dm}|^{2} + {\mathbf{Im}}(A\psi _d,\xi _d) + \gamma \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |\psi _{dm}|^{2}\nonumber \\&- h^2{\mathbf{Im}}(D\psi _{d},\xi _{d}) = {\mathbf{Im}}(\psi ^{(1)}u^{(1)}-\psi ^{(2)}u^{(2)},\xi _{d}). \end{aligned}$$

(3.9)

According to Lemma 2.4, there exist $t_{1}>0$ and $M_{1}(\varepsilon ,\tau ,{\mathcal {B}}_0)\in \mathbb {N}$, such that when $t-\tau \geqslant t_{1}$ and $M>M_{1}(\varepsilon ,\tau ,{\mathcal {B}}_0)$, we obtain

$$\begin{aligned}&{\mathbf{Im}}\sum _{m\in \mathbb {Z}}\big (\psi ^{(1)}_{m}u^{(1)}_{m}-\psi ^{(2)}_{m}u^{(2)}_{m}\big ) \chi \left( \frac{|m|}{M}\right) \left( \bar{\psi }^{(1)}_{m}-\bar{\psi }^{(2)}_{m}\right) \nonumber \\&\quad \leqslant \sum _{m\in \mathbb {Z}}\chi \left( \frac{|m|}{M}\right) |\psi ^{(1)}_{m}u^{(1)}_{m} -\psi ^{(2)}_{m}u^{(2)}_{m}||\psi ^{(1)}_{m}-\psi ^{(2)}_{m}|\nonumber \\&\quad \leqslant \frac{\gamma }{4}\sum _{m\in \mathbb {Z}}\chi \left( \frac{|m|}{M}\right) |\psi ^{(1)}_{m}\!-\!\psi ^{(2)}_{m}|^{2} \!+\!\frac{\sqrt{\gamma \mu \lambda }}{2}\sum _{m\in \mathbb {Z}} \chi \!\left( \frac{|m|}{M}\right) |u^{(1)}_{m} \!-\!u^{(2)}_{m}||\psi ^{(1)}_{m}\!-\! \psi ^{(2)}_{m}|\nonumber \\&\quad \leqslant \frac{\gamma }{2}\sum _{m\in \mathbb {Z}}\chi \left( \frac{|m|}{M}\right) |\psi _{dm}|^{2} +\frac{\mu \lambda }{4}\sum _{m\in \mathbb {Z}}\chi \left( \frac{|m|}{M}\right) |u_{dm}|^{2}. \end{aligned}$$

(3.10)

Similar to [25, (52) and (61)], we can get

$$\begin{aligned} {\mathbf{Im}}(A\psi _{d},\xi _{d})&\leqslant \frac{\chi _{0}}{M}\Vert z_{d}\Vert ^2_{E_{\mu }}, \end{aligned}$$

(3.11)

$$\begin{aligned} h^2{\mathbf{Im}}(D\psi _{d},\xi _{d})&\leqslant \frac{6\chi _{0}h^2}{M}\Vert z_{d}\Vert ^2_{E_{\mu }}. \end{aligned}$$

(3.12)

Then taking (3.9)–(3.12) into account, we have for $t-\tau \geqslant t_{1}$ and $M>M_{1}(\varepsilon ,\tau ,{\mathcal {B}}_0)$ that

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm {d}}{{\mathrm {d}}t} \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |\psi _{dm}|^{2} +\frac{\gamma }{2}\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |\psi _{dm}|^{2}\nonumber \\&\quad \leqslant \frac{\mu \lambda }{4}\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) u_{dm}^{2} + \frac{\chi _{0}+6\chi _{0}h^2}{M}\Vert z_{d}\Vert ^{2}_{E_{\mu }}. \end{aligned}$$

(3.13)

Now, by (2.7) and (2.9),

$$\begin{aligned} \dot{\varphi }_{d}+(\alpha -\lambda )\varphi _{d} + (\lambda (\lambda -\alpha )+\mu )u_{d}-Au_{d}+h^2Du_{d} -A(|\psi ^{(1)}|^{2}-|\psi ^{(2)}|^{2})=0. \end{aligned}$$

(3.14)

Taking the inner product of (3.14) with $w_{d}$ in $\ell ^{2}$ gives

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm {d}}{{\mathrm {d}}t}\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \varphi _{dm}^{2} + (\alpha -\lambda )\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \varphi _{dm}^{2} + \big ((\lambda (\lambda -\alpha )+\mu )u_{d},w_{d}\big )\nonumber \\&\quad -(Au_{d},w_{d})+h^2(Du_{d},w_{d})-\big (A(|\psi ^{(1)}|^{2} -|\psi ^{(2)}|^{2}),w_{d}\big )=0. \end{aligned}$$

(3.15)

It is clear that $w_{d}=\dot{v}_{d}+\lambda v_{d}$. Thus (3.15) can be written as

$$\begin{aligned}&\frac{\mathrm {d}}{{\mathrm {d}}t}\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) (\mu u_{dm}^{2}+\varphi _{dm}^{2}) + 2(\alpha -\lambda )\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \varphi _{dm}^{2}\nonumber \\&\quad + 2\lambda (\lambda -\alpha )(u_{d},w_{d}) + 2\mu \lambda \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) u_{dm}^{2}\nonumber \\&\quad - 2(Au_{d},w_{d}) + 2h^2(Du_{d},w_{d}) - 2\big (A(|\psi ^{(1)}|^{2}-|\psi ^{(2)}|^{2}),w_{d}\big )=0. \end{aligned}$$

(3.16)

By some computations, we have

$$\begin{aligned} -(Au_{d},w_{d})&= -\big ((Au_{d},\dot{v}_{d})+\lambda (Au_{d},v_{d})\big ) = (Bu_{d},B\dot{v}_{d})+\lambda (Bu_{d},Bv_{d}), \end{aligned}$$

(3.17)

$$\begin{aligned} (Bu_{d},B\dot{v}_{d})&= \sum \limits _{m\in \mathbf{\mathbb {Z}}}(Bu_{d})_{m}(B\dot{v}_{d})_{m} \nonumber \\&= \sum \limits _{m\in \mathbf{\mathbb {Z}}}(Bu_{d})_{m} \Big [\chi \Big (\frac{|m+1|}{M}\Big )\dot{u}_{dm+1}-\chi \Big (\frac{|m|}{M}\Big ) \dot{u}_{dm}\Big ]\nonumber \\&\geqslant \frac{1}{2}\frac{\mathrm {d}}{{\mathrm {d}}t}\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |(Bu_{d})_{m}|^{2} -\frac{2\chi _{0}(\mu +\lambda +1)}{M\mu }\Vert z_{d}\Vert ^{2}_{E_{\mu }}, \quad \forall t\geqslant \tau , \end{aligned}$$

(3.18)

$$\begin{aligned} \lambda (Bu_{d},Bv_{d})&\geqslant \lambda \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |(Bu_{d})_{m}|^{2} -\frac{2\chi _{0}\lambda }{M\mu }\Vert z_{d}\Vert ^{2}_{E_{\mu }}, \quad \forall t\geqslant \tau . \end{aligned}$$

(3.19)

Inserting (3.18) and (3.19) into (3.17) yields

$$\begin{aligned} -(Au_{d},w_{d})&\geqslant \frac{1}{2}\frac{\mathrm {d}}{{\mathrm {d}}t}\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |(Bu_{d})_{m}|^{2} + \lambda \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |(Bu_{d})_{m}|^{2}\nonumber \\&\quad - \frac{2\chi _{0}(\mu +2\lambda +1)}{M\mu }\Vert z_{d}\Vert ^{2}_{E_{\mu }}, \quad \forall t\geqslant \tau . \end{aligned}$$

(3.20)

Similarly, we have

$$\begin{aligned} (Du_{d},w_{d})&= (Du_{d},\dot{v_{d}}+\lambda v_{d}) = (Au_{d},A\dot{v_{d}})+\lambda (Au_{d},Av_{d}), \end{aligned}$$

(3.21)

$$\begin{aligned} (Au_{d},A\dot{v}_{d})&= \sum \limits _{m\in \mathbf{\mathbb {Z}}}(Au_{d})_{m} \Big [(A\dot{u}_{d})_{m}\chi \left( \frac{|m|}{M}\right) \!+\!\left( \chi \left( \frac{|m+1|}{M}\right) \!-\!\chi \left( \frac{|m|}{M}\right) \right) \dot{u}_{dm+1}\nonumber \\&\quad \,\,+ \left( \chi \left( \frac{|m-1|}{M}\right) -\chi \left( \frac{|m|}{M}\right) \right) \dot{u}_{dm-1}\Big ]\nonumber \\&=\sum \limits _{m\in \mathbf{\mathbb {Z}}}(u_{dm+1}-2u_{dm}\,+u_{dm-1}) \left( \chi '\left( \frac{\tilde{m_{1}}}{M}\right) \frac{1}{M}\dot{u}_{dm+1}\right. \nonumber \\&\quad \left. \,+\,\chi '\left( \frac{{\tilde{m_{2}}}}{M}\right) \frac{1}{M}\dot{u}_{dm-1}\right) \,\,+\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) (Au_{d})_{m}(A\dot{u}_{d})_{m}\nonumber \\&\geqslant \frac{1}{2}\frac{\mathrm {d}}{{\mathrm {d}}t}\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |(Au_{d})_{m}|^{2} -\frac{8\chi _{0}(\mu +\lambda +1)}{M\mu }\Vert z_{d}\Vert ^{2}_{E_{\mu }}, \quad \forall t\geqslant \tau , \end{aligned}$$

(3.22)

and

$$\begin{aligned} \lambda (Au_{d},Av_{d})&=\lambda \sum \limits _{m\in \mathbf{\mathbb {Z}}}(Au_{d})_{m} \Big [(Au_{d})_{m}\chi \left( \frac{|m|}{M}\right) +\left( \chi \left( \frac{|m+\!1\!|}{M}\right) -\chi \left( \frac{|m|}{M}\right) \right) u_{dm+1}\nonumber \\&\quad \,\,+\left( \chi \left( \frac{|m-1|}{M}\right) -\chi \left( \frac{|m|}{M}\right) \right) u_{dm-1}\Big ]\nonumber \\&= \lambda \sum \limits _{m\in \mathbf{\mathbb {Z}}}(u_{dm+1}-2u_{dm}+u_{dm-1})\left( \chi '\left( \frac{\tilde{m_{3}}}{M}\right) \frac{1}{M}u_{dm+1}\right. \nonumber \\&\qquad \left. +\chi '\left( \frac{{\tilde{m_{4}}}}{M}\right) \frac{1}{M}u_{dm-1}\right) \,\,+\lambda \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |(Au_{d})_{m}|^2\nonumber \\&\geqslant \lambda \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |(Au_{d})_{m}|^2 -\frac{8\chi _{0}\lambda }{M\mu }\Vert z_{d}\Vert ^{2}_{E_{\mu }}, \quad \forall t\geqslant \tau , \end{aligned}$$

(3.23)

here $\tilde{m_{1}}$ and $\tilde{m_{3}}$ locate between $|m+1|$ and |m|, $\tilde{m_{2}}$ and $\tilde{m_{4}}$ locate between $|m-1|$ and |m|. Inserting (3.22) and (3.23) into (3.21) gives

$$\begin{aligned} (Du_{d},w_{d})&\geqslant \frac{1}{2}\frac{\mathrm {d}}{{\mathrm {d}}t}\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |(Au_{d})_{m}|^{2} + \lambda \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |(Au_{d})_{m}|^{2} \nonumber \\&\quad - \frac{8\chi _{0}(1+2\lambda +\mu )}{M\mu }\Vert z_{d}\Vert ^{2}_{E_{\mu }}, \quad \forall t\geqslant \tau . \end{aligned}$$

(3.24)

According to Lemma 2.3, we see that for every $t-\tau \geqslant s_{0}$,

$$\begin{aligned}&\big (A(|\psi ^{(1)}|^{2}-|\psi ^{(2)}|^{2}),w_{d}\big )= \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \big (A(|\psi ^{(1)}|^{2}-|\psi ^{(2)}|^{2}) \big )_{m}\varphi _{dm}\nonumber \\&\quad = \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \big (|\psi ^{(1)}_{m+1}|^2-|\psi ^{(2)}_{m+1}|^2 -2|\psi ^{(1)}_{ m}|^2+2|\psi ^{(2)}_{m}|^2\nonumber \\&\qquad +|\psi ^{(1)}_{m-1}|^2-|\psi ^{(2)} _{m-1}|^2\big )\varphi _{dm}\nonumber \\&\quad \leqslant \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \Big [(|\psi ^{(1)}_{m+1}| +|\psi ^{(2)}_{m+1}|)|\psi _{dm+1}| +2(|\psi ^{(1)}_{ m}|+|\psi ^{(2)}_{m}|)|\psi _{dm}|\nonumber \\&\qquad +(|\psi ^{(1)}_{m-1}|+|\psi ^{(2)} _{m-1}|)|\psi _{dm-1}|\Big ]|\varphi _{dm}|\nonumber \\&\quad \leqslant \Big (\frac{\alpha }{12}\sum \limits _{|m|\geqslant M}\chi \Big (\frac{|m|}{M}\Big ) \varphi ^2_{dm} +\frac{12 R^2_0}{\alpha }\sum \limits _{|m|\geqslant M}\chi \Big (\frac{|m|}{M} \Big )|\psi _{dm+1}|^2\Big ) \nonumber \\&\qquad +\Big (\frac{\alpha }{12}\sum \limits _{|m|\geqslant M}\chi \Big (\frac{|m|}{M}\Big ) \varphi ^2_{dm}+\frac{48 R^2_0}{\alpha }\sum \limits _{|m|\geqslant M}\chi \Big (\frac{|m|}{M}\Big )|\psi _{dm}|^2\Big ) \nonumber \\&\qquad +\Big (\frac{\alpha }{12}\sum \limits _{|m|\ge M}\chi \Big (\frac{|m|}{M}\Big ) \varphi ^2_{dm} +\frac{12 R^2_0}{\alpha }\sum \limits _{|m|\geqslant M}\chi \Big (\frac{|m|}{M}\Big )|\psi _{dm-1}|^2\Big ) \nonumber \\&\quad = \frac{\alpha }{4}\sum \limits _{|m|\geqslant M}\chi \left( \frac{|m|}{M}\right) \varphi ^2_{dm} + \frac{12 R^2_0}{\alpha }I_1 +\frac{48 R^2_0}{\alpha }I_2 +\frac{12 R^2_0}{\alpha }I_3, \end{aligned}$$

(3.25)

where

$$\begin{aligned} I_1&:=\sum \limits _{|m|\geqslant M}\chi \left( \frac{|m|}{M}\right) |\psi _{dm+1}|^2,\,\,I_2:=\sum \limits _{|m|\geqslant M} \chi \left( \frac{|m|}{M}\right) |\psi _{dm}|^2,\,\,\\ I_3&:=\sum \limits _{|m|\geqslant M}\chi \left( \frac{|m|}{M}\right) |\psi _{dm-1}|^2. \end{aligned}$$

By [25, (64)], we have

$$\begin{aligned} I_1&\leqslant 2\sum \limits _{|m|\geqslant M}\chi \left( \frac{|m|}{M}\right) \left( |\psi ^{(1)}_{m+1}|^2+ |\psi ^{(2)}_{m+1}|^2\right) \nonumber \\&\leqslant \displaystyle \frac{4}{\gamma }\int \limits _{\tau }^{t} \left( \sum \limits _{|m|\geqslant M}|g_{m+1}(y)|^{2}\right) {\mathrm e}^{-\gamma (t-y)}{\mathrm d}y +4R^2_{0}{\mathrm e}^{-\gamma (t-\tau )} + \frac{8\chi _0 R^2_0(1+6h^2)}{\gamma M}. \end{aligned}$$

(3.26)

Similar to (3.26),

$$\begin{aligned}&I_2 \leqslant 2\sum \limits _{|m|\geqslant M}\chi \left( \frac{|m|}{M} \right) \left( |\psi ^{(1)}_{m }|^2+ |\psi ^{(2)}_{m }|^2\right) \nonumber \\&\quad \leqslant \displaystyle \frac{4}{\gamma }\int \limits _{\tau }^{t} \left( \sum \limits _{|m|\geqslant M}|g_{m}(y)|^{2}\right) {\mathrm e}^{-\gamma (t-y)}{\mathrm d}y +4R^2_{0}{\mathrm e}^{-\gamma (t-\tau )} +\frac{8\chi _0 R^2_0(1+6h^2)}{\gamma M}, \end{aligned}$$

(3.27)

$$\begin{aligned}&I_3 \leqslant 2\sum \limits _{|m|\geqslant M}\chi \left( \frac{|m|}{M}\right) (|\psi ^{(1)}_{m-1}|^2+ |\psi ^{(2)}_{m-1}|^2)\nonumber \\&\quad \leqslant \displaystyle \frac{4}{\gamma }\int \limits _{\tau }^{t} \left( \sum \limits _{|m|\geqslant M}|g_{m-1}(y)|^{2}\right) {\mathrm e}^{-\gamma (t-y)}{\mathrm d}y +4R^2_{0}{\mathrm e}^{-\gamma (t-\tau )} + \frac{8\chi _0 R^2_0(1+6h^2)}{\gamma M}. \end{aligned}$$

(3.28)

Combining (3.25)–(3.28), we have for any $t-\tau \geqslant s_{0}$ that

$$\begin{aligned}&\big (A(|\psi ^{(1)}|^2-|\psi ^{(2)}|^2),w_{d}\big )\nonumber \\&\quad \leqslant \frac{\alpha }{4}\sum \limits _{|m|\geqslant M}\chi \left( \frac{|m|}{M} \right) \varphi ^2_{dm}+ \frac{288 R^4_0 {\mathrm e}^{-\gamma (t-\tau )}}{\alpha } +\frac{576\chi _{0} R^4_0(1+6h^2)}{\alpha \gamma M}\nonumber \\&\qquad +\frac{192R^2_{0}}{\alpha \gamma } \left[ \int \limits _{\tau }^{t} \left( \sum \limits _{|m|\geqslant M}(|g_{m+1}(y)|^{2} +|g_{m}(y)|^{2}+|g_{m-1}(y)|^{2})\right) {\mathrm e}^{-\gamma (t-y)}{\mathrm d}y \right] . \end{aligned}$$

(3.29)

Taking (3.16), (3.20), (3.24), and (3.29) into account, we get

$$\begin{aligned}&\frac{\mathrm {d}}{{\mathrm {d}}t}\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \big [|(Bu_{d})_{m}|^{2} +\mu u_{dm}^{2}+\varphi _{dm}^{2}+h^2 (Au_{d})^2_{m}\big ]\nonumber \\&\qquad + 2\lambda \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |(Bu_{d})_{m}|^{2} + 2\mu \lambda \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) u_{dm}^{2}\nonumber \\&\qquad +2(\alpha - \lambda )\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \varphi _{dm}^{2}\nonumber \\&\qquad +2\lambda (\lambda -\alpha )(u_{d},w_{d}) + 2\lambda h^2\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) (Au_d)^2_{m}\nonumber \\&\quad \leqslant \frac{\alpha }{2}\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |\varphi _{dm}|^{2} + \frac{4\chi _{0}(\mu +2\lambda +1+4h^2+8\lambda h^2+4\mu h^2)}{M\mu }\Vert z_{d}\Vert ^{2}_{E_{\mu }}\nonumber \\&\qquad + \frac{576 R^4_0 {\mathrm e}^{-\gamma (t-\tau )}}{\alpha } +\frac{1152\chi _{0} R^4_0(1+6h^2)}{\alpha \gamma M} \nonumber \\&\qquad + \frac{384 R^2_{0}}{\alpha \gamma } \left[ \int \limits _{\tau }^{t} \left( \sum \limits _{|m|\geqslant M}(|g_{m+1}(y)|^{2} + |g_{m}(y)|^{2} + |g_{m-1}(y)|^{2})\right) {\mathrm e}^{-\gamma (t-y)}{\mathrm d}y \right] . \end{aligned}$$

(3.30)

Set $\displaystyle \delta =\frac{\mu \alpha }{\sqrt{\alpha ^2+4\mu }\left( \sqrt{\alpha ^2+4\mu }+\alpha \right) }$, then $\displaystyle 4(\lambda -\delta )\left( \frac{\alpha }{2}-\lambda -\delta \right) =\frac{\lambda ^2\alpha ^2}{\mu }$ and thus

$$\begin{aligned}&\lambda \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |(Bu_{d})_{m}|^{2}\ + \mu \lambda \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) u_{dm}^{2}\nonumber \\&\qquad + (\alpha - \lambda )\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \varphi _{dm}^{2} +\lambda (\lambda -\alpha )(u_{d},w_{d})\nonumber \\&\quad =\sum \limits _{m\in {\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \Big [\delta \big ((B u_{d})^2_{m} +\mu u^2_{dm}+\varphi ^2_{dm}\big ) +\frac{\alpha }{2}\varphi ^2_{dm}\Big ]\nonumber \\&\qquad + \sum \limits _{m\in {\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \Big [(\lambda -\delta ) \big ((B u_{d})^2_{m}+\mu u^2_{dm}\big )\nonumber \\&\qquad +\big (\frac{\alpha }{2}-\lambda -\delta \big )\varphi ^2_{dm}+\lambda (\lambda -\alpha )u_{dm}\varphi _{dm}\Big ]\nonumber \\&\quad \geqslant \sum \limits _{m\in {\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \Big [\delta \big ((B u_{d})^2_{m} +\mu u^2_{dm}+\varphi ^2_{dm}\big ) +\frac{\alpha }{2}\varphi ^2_{dm}\Big ]. \end{aligned}$$

(3.31)

Then, (3.30) and (3.31) give

$$\begin{aligned}&\frac{\mathrm {d}}{{\mathrm {d}}t}\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \Big [|(Bu_{d})_{m}|^{2} +\mu u_{dm}^{2}+\varphi _{dm}^{2}+h^2 (Au_{d})^2_{m}\Big ]\nonumber \\&\qquad + 2\sum \limits _{m\in {\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \Big [\delta \big ((B u_{d})^2_{m} +\mu u^2_{dm}+\varphi ^2_{dm}\big ) +\frac{\alpha }{2}\varphi ^2_{dm}\Big ] \nonumber \\&\qquad + 2\lambda h^2\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) (Au_d)^2_{m}\nonumber \\&\quad \leqslant \frac{\alpha }{2}\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |\varphi _{dm}|^{2} + \frac{4\chi _{0}(\mu +2\lambda +1+4h^2+8\lambda h^2+4\mu h^2)}{M\mu }\Vert z_{d}\Vert ^{2}_{E_{\mu }}\nonumber \\&\qquad + \frac{576 R^4_0 {\mathrm e}^{-\gamma (t-\tau )}}{\alpha } +\frac{1152\chi _{0} R^4_0(1+6h^2)}{\alpha \gamma M}\nonumber \\&\qquad +\frac{192R^2_{0}}{\alpha \gamma } \left[ \int \limits _{\tau }^{t} \left( \sum \limits _{|m|\geqslant M}(|g_{m+1}(y)|^{2} + |g_{m}(y)|^{2} + |g_{m-1}(y)|^{2})\right) {\mathrm e}^{-\gamma (t-y)}{\mathrm d}y \right] . \end{aligned}$$

(3.32)

Combining (3.13) and (3.32), we get that

$$\begin{aligned}&\frac{\mathrm {d}}{{\mathrm {d}}t}\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \Big [|\psi _{dm}|^2+|(Bu_{d})_{m}|^{2} +\mu u_{dm}^{2}+\varphi _{dm}^{2}+h^2 (Au_{d})^2_{m}\Big ]\nonumber \\&\quad + 2\delta \sum \limits _{m\in {\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) (B u_{d})^2_{m} +\left( 2\mu \delta -\frac{\lambda \mu }{2}\right) \sum \limits _{m\in {\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) u^2_{dm}\nonumber \\&\quad + \left( 2\delta +\frac{\alpha }{2}\right) \sum \limits _{m\in {\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \varphi ^2_{dm}+ \gamma \sum \limits _{m\in {\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |\psi _{dm}|^2\nonumber \\&\quad + 2\lambda h^2\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) (Au_d)^2_{m}\nonumber \\&\,\leqslant \frac{4\chi _{0}(\mu +2\lambda +1+4h^2+8\lambda h^2+4\mu h^2)}{M\mu }\Vert z_{d}\Vert ^{2}_{E_{\mu }}\nonumber \\&\quad + \frac{576 R^4_0 {\mathrm e}^{-\gamma (t-\tau )}}{\alpha } +\frac{1152\chi _{0} R^4_0(1+6h^2)}{\alpha \gamma M} + \frac{2\chi _{0}+12\chi _{0}h^2}{M}\Vert z_{d}\Vert ^{2}_{E_{\mu }}\nonumber \\&\quad + \frac{384 R^2_{0}}{\alpha \gamma }\left[ \int \limits _{\tau }^{t} \Big (\sum \limits _{|m|\geqslant M}(|g_{m+1}(y)|^{2} + |g_{m}(y)|^{2} + |g_{m-1}(y)|^{2})\Big ) {\mathrm e}^{-\gamma (t-y)}{\mathrm d}y \right] . \end{aligned}$$

(3.33)

Letting $\beta =\min \left\{ 2\delta -\frac{\lambda }{2},\gamma \right\}$, we obtain by (3.33) that

$$\begin{aligned}&\frac{\mathrm {d}}{{\mathrm {d}}t} \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \Big [|z_{dm}|^{2}_{E_{\mu }}+h^2(Au_{d})^2_{m}\Big ] + \beta \sum \limits _{m\in \mathbf{\mathbb {Z}}} \chi \left( \frac{|m|}{M}\right) \Big [|z_{dm}|^{2}_{E_{\mu }}+h^2(Au_{d})^2_{m}\Big ]\nonumber \\&\quad \leqslant \frac{\chi _{0}(16h^2+32\lambda h^2+28\mu h^2+6\mu +8\lambda +4)}{M\mu } \Vert z_{d}\Vert ^{2}_{E_{\mu }}\nonumber \\&\qquad + \frac{384 R^2_{0}}{\alpha \gamma }\left[ \int \limits _{\tau }^{t} \left( \sum \limits _{|m|\geqslant M}(|g_{m+1}(y)|^{2} + |g_{m}(y)|^{2} + |g_{m-1}(y)|^{2})\right) {\mathrm e}^{-\gamma (t-y)}{\mathrm d}y \right] \nonumber \\&\qquad + \frac{576 R^4_0 {\mathrm e}^{-\gamma (t-\tau )}}{\alpha } +\frac{1152\chi _{0} R^4_0(1+6h^2)}{\alpha \gamma M}. \end{aligned}$$

(3.34)

Since $g\in {\mathcal {H}}$, for $\varepsilon >0$(given in Definition 2.2), there exist $t_2(\varepsilon )>0$, $M_2(\varepsilon ,\tau ,{\mathcal {B}}_{0})\in \mathbb {N}$ and $M_3(\varepsilon ,\tau )\in \mathbb {N}$ with $M_2(\varepsilon ,\tau ,{\mathcal {B}}_{0})\geqslant M_1(\varepsilon ,\mathcal {B}_0)$ and $M_3(\varepsilon ,\tau )\geqslant M_1(\varepsilon ,\mathcal {B}_0)$, such that

$$\begin{aligned} \frac{576 R^4_0 {\mathrm e}^{-\gamma (t-\tau )}}{\alpha }&\leqslant \frac{\beta }{8}\cdot 4\varepsilon ^2,\quad \forall t-\tau \geqslant t_2(\varepsilon ), \end{aligned}$$

(3.35)

$$\begin{aligned} \frac{1152\chi _{0} R^4_0(1+6h^2)}{\alpha \gamma M}&\leqslant \frac{\beta }{16}\cdot 4\varepsilon ^2,\quad \forall M\geqslant M_{2}(\varepsilon ,\tau ,{\mathcal {B}}_{0}), \end{aligned}$$

(3.36)

and for any $M\geqslant M_3(\varepsilon ,\tau )$,

$$\begin{aligned} \frac{384 R^2_{0}}{\alpha \gamma }\left[ \int \limits _{\tau }^{t} \left( \sum \limits _{|m|\geqslant M}(|g_{m+1}(y)|^{2} + |g_{m}(y)|^{2} + |g_{m-1}(y)|^{2})\right) {\mathrm e}^{-\gamma (t-y)}{\mathrm d}y \right] \leqslant \frac{\beta }{16}\cdot 4\varepsilon ^2. \end{aligned}$$

(3.37)

Choosing $N=\max \{M_{2}(\varepsilon ,\tau ,{\mathcal {B}}_{0}),M_{3}(\varepsilon ,\tau )\}$, $t_{3}(\varepsilon ,{\mathcal {B}}_{0})=\max \{t_{1},t_{2}(\varepsilon ),s_{0}\}$, then we conclude from (3.34)–(3.37) that when $t-\tau \geqslant t_{3}(\varepsilon ,{\mathcal {B}}_{0})$ and $M\geqslant N$,

$$\begin{aligned}&\frac{\mathrm {d}}{{\mathrm {d}}t} \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \left[ |z_{dm}|^{2}_{E_{\mu }} +h^2(Au_{d})^2_{m}\right] + \beta \sum \limits _{m\in \mathbf{\mathbb {Z}}} \chi \left( \frac{|m|}{M}\right) \left[ |z_{dm}|^{2}_{E_{\mu }}+h^2(Au_{d})^2_{m}\right] \nonumber \\&\quad \leqslant \frac{C_3}{M}\Vert z_{d}\Vert ^{2}_{E_{\mu }} +\frac{\beta }{4}\cdot 4\varepsilon ^2, \end{aligned}$$

(3.38)

where

$$\begin{aligned} C_3=\frac{\chi _{0}(6\mu +8\lambda +4+16h^2+32\lambda h^2+28\mu h^2)}{\mu }. \end{aligned}$$

Applying Gronwall inequality to (3.38) from $\tau$ to t with $t-\tau >t_{3}(\varepsilon ,{\mathcal {B}}_{0})$, we get that when $M>N(\varepsilon ,\tau ,{\mathcal {B}}_0)$,

$$\begin{aligned}&\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \left[ |z_{dm}|^{2}_{E_{\mu }} +h^2(Au_{d})^2_{m}\right] \nonumber \\&\quad \leqslant \! {\mathrm e}^{-\beta (t-\tau )}\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \Big [|z_{dm}(\tau )|^{2}_{E_{\mu }} +h^2(Au_{d}(\tau ))^2_{m}\Big ] +\varepsilon ^2\nonumber \\&\qquad + \frac{C_3}{M }\int \limits _{ \tau }^{ t}\Vert z_{d}(y)\Vert ^{2}_{E_{\mu }} {\mathrm e}^{-\beta (t-y)}{\mathrm {d}}y\nonumber \\&\quad \leqslant \! \frac{\mu +16h^2}{\mu }{\mathrm e}^{-\beta (t-\tau )}\sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \!\left( \frac{|m|}{M}\right) |z_{dm}(\tau )|^{2}_{E_{\mu }} +\varepsilon ^2 + \frac{C_3}{M }\int \limits _{\tau }^{ t}\Vert z_{d}(y)\Vert ^{2}_{E_{\mu }} {\mathrm e}^{-\beta (t-y)}{\mathrm {d}}y. \end{aligned}$$

(3.39)

By (3.7), we have

$$\begin{aligned} \frac{C_3}{M}\int \limits _{\tau }^{ t}\Vert z_{d}(y)\Vert ^{2}_{E_{\mu }} {\mathrm e}^{-\beta (t-y)}{\mathrm {d}}y&\leqslant \frac{C_3}{M }\Vert z_{d}(\tau )\Vert ^{2}_{E_{\mu }}\int \limits _{\tau }^{ t} {\mathrm e}^{-\beta (t-y)\,+\,2(C_{1}\,+\,C_{2})(y-\tau )}{\mathrm {d}}y\nonumber \\&= \frac{C_3}{M}\Vert z_{d}(\tau )\Vert ^{2}_{E_{\mu }}{\mathrm e}^{-\beta t -2(C_{1}\,+\,C_{2})\tau } \int \limits _{\tau }^{ t}{\mathrm e}^{(\beta \,+\,2C_{1}\,+\,2C_{2})y}{\mathrm {d}}y\nonumber \\&= \frac{C_3}{M (\beta +2C_{1}\,+\,2C_{2})}\Vert z_{d}(\tau )\Vert ^{2}_{E_{\mu }}{\mathrm e}^{2(C_{1}\,+\,C_{2})(t-\tau )}, \end{aligned}$$

(3.40)

hereinafter $C_1$ and $C_2$ come from (3.5). Thus, it follows from (3.39)–(3.40) that for any $t-\tau \geqslant t_{3}(\varepsilon ,{\mathcal {B}}_{0})$ and $M>N(\varepsilon ,\tau ,{\mathcal {B}}_0)$,

$$\begin{aligned} \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) |z_{dm}|^{2}_{E_{\mu }}&\leqslant \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \left[ |z_{dm}|^{2}_{E_{\mu }} +h^2(Au_{d})^2_{m}\right] \nonumber \\&\leqslant \frac{\mu +16h^2}{\mu } {\mathrm e}^{-\beta (t-\tau )}\Vert z_{d}(\tau )\Vert ^{2}_{E_{\mu }} +\varepsilon ^2\nonumber \\&\quad \quad +\frac{C_{3}}{M(\beta +2C_{1}\,+\,2C_{2})}\Vert z_{d}(\tau ) \Vert ^{2}_{E_{\mu }}{\mathrm e}^{2(C_{1}\,+\,C_{2})(t-\tau )}. \end{aligned}$$

(3.41)

That is,

$$\begin{aligned}&\sum \limits _{m\in \mathbf{\mathbb {Z}}} \chi \left( \frac{|m|}{M}\right) |z_{dm}(t_{3}(\varepsilon ,{\mathcal {B}}_{0}) +\tau ,\tau )|^{2}_{E_{\mu }}\nonumber \\&\,\leqslant \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{M}\right) \left[ |z_{dm}(t_{3}(\varepsilon ,{\mathcal {B}}_{0})+\tau ,\tau )|^{2}_{E_{\mu }}+h^2(Au_{d}(t_{3}(\varepsilon ,{\mathcal {B}}_{0})+\tau ,\tau ))^2_{m}\right] \nonumber \\&\,\leqslant \frac{\mu +16h^2}{\mu } {\mathrm e}^{-\beta t_{3}(\varepsilon ,{\mathcal {B}}_{0})}\Vert z_{d}(\tau )\Vert ^{2}_{E_{\mu }}+\varepsilon ^2\nonumber \\&\qquad +\frac{C_{3}}{M(\beta +2C_{1}+2C_{2})}\Vert z_{d}(\tau )\Vert ^{2}_{E_{\mu }}{\mathrm e}^{2(C_{1}\,+\,C_{2})t_{3}(\varepsilon ,{\mathcal {B}}_{0})}. \end{aligned}$$

(3.42)

Set

$$\begin{aligned} N^*= \max \left\{ N(\varepsilon ,\tau ,{\mathcal {B}}_0),\frac{C_{3}\mathrm e^{2(2C_{1}\,+\,2C_{2})t_{3}(\varepsilon ,{\mathcal {B}}_0)}}{\Big (\frac{1}{4}-\frac{\mu +16h^2}{\mu }\mathrm e^{-\beta t_{3}(\varepsilon ,{\mathcal {B}}_{0})}\Big )(\beta +2C_{1}+2C_{2})}\right\} , \end{aligned}$$

(3.43)

then by (3.42) it follows that for $T^{*}=t_{3}(\varepsilon ,{\mathcal {B}}_0)$,

$$\begin{aligned} \sum \limits _{|m|>2N^*}|z_{dm}(T^{*})|^{2}_{E_{\mu }} \leqslant \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{N^*}\right) |z_{dm}(T^{*})|^{2}_{E_{\mu }} \leqslant \eta ^{2}\Vert z_{d}(\tau )\Vert ^{2}_{E_{\mu }}+\theta ^2\cdot 4\varepsilon ^2,\nonumber \\ \end{aligned}$$

(3.44)

where

$$\begin{aligned}&\eta ^{2}=\frac{\mu +16h^2}{\mu }\mathrm {e}^{-\beta T^{*}} + \frac{C_{3}}{N^*(\beta +2C_{1}+2C_{2})}\mathrm {e}^{2(C_{1}+C_{2})T^*} < \frac{1}{4}, \end{aligned}$$

(independent of $\tau$), and set $\displaystyle \theta ^2=\frac{1}{4}\in \Big (0,\frac{1}{2}\Big )$. Then (3.44) gives

$$\begin{aligned} \sum \limits _{|m|>2N^*}|z_{dm}(T^{*})|^{2}_{E_{\mu }} \leqslant \sum \limits _{m\in \mathbf{\mathbb {Z}}}\chi \left( \frac{|m|}{N^*}\right) |z_{dm}(T^{*} )|^{2}_{E_{\mu }} \leqslant \eta ^{2}\Vert z_{d}(\tau )\Vert ^{2}_{E_{\mu }}+\theta ^2\cdot 4\varepsilon ^2, \end{aligned}$$

(3.45)

and thus

$$\begin{aligned} \big \Vert (I-P_{2N^*})\big [U(T^{*}+\tau ,\tau )z^{(1)}_{\tau } -U(T^{*}+\tau ,\tau )z^{(2)}_{\tau }\big ]\big \Vert _{E_{\mu }}\leqslant \eta \Vert z^{(1)}_{\tau }-z^{(2)}_{\tau }\Vert _{E_{\mu }}+\theta \cdot 2\varepsilon . \end{aligned}$$

By [24, Theorem 4.1], the fractal dimension of ${\mathcal {K}}(\tau )$ satisfies

$$\begin{aligned} \dim _F\mathcal {K}(\tau ) \leqslant 4(4N^*+1) \cdot \ln \left( 1+\frac{8(1+\mathrm {e}^{(C_1+C_2)T^*})}{\frac{1}{2}-\eta }\right) \cdot \left( \ln \frac{2}{\frac{3}{2}+\eta }\right) ^{-1}. \end{aligned}$$

The proof is complete. $\square$

4 Upper Semicontinuity of the Kernel Sections

In this section, we consider the approximation of the kernel sections $\{{\mathcal {K}}(\tau )\}_{\tau \in {\mathbb {R}}}$, by using the kernel sections of the following finite-dimensional truncated ODEs:

$$\begin{aligned}&i\dot{\psi }_m+(A\psi )_m+i\gamma \psi _m-\psi _m u_m-h^2(D\psi )_{m}=g_m(t), \quad |m|\leqslant n, \,\,t>\tau , \end{aligned}$$

(4.1)

$$\begin{aligned}&\ddot{u}_m-(A u)_m+h^2(D u)_m-(A|\psi |^2)_m+\alpha \dot{u}_m+\mu u_m =f_m(t), \quad |m|\leqslant n, \, t>\tau , \end{aligned}$$

(4.2)

with the initial conditions

$$\begin{aligned} \psi _m(\tau )=\psi _{m,\tau }, \quad u_m(\tau )=u_{m,\tau }, \quad \dot{u}_m(\tau )=u_{1m,\tau }, \quad |m|\leqslant n, \end{aligned}$$

(4.3)

where the functions $g_m(t)$ and $f_m(t)$ with $|m|\leqslant n$ are exactly chosen as the same as in (1.5) and (1.6), respectively. We set $\varphi _m=\dot{u}_m+\lambda u_m\,\,(|m|\leqslant n)$, where $\lambda$ comes from (2.9), then Eqs. (4.1)–(4.2) can be written as

$$\begin{aligned} \dot{z}^{(n)}+\Theta _n z^{(n)}&= F_n(z^{(n)},t),\quad t>\tau , \end{aligned}$$

(4.4)

$$\begin{aligned} z^{(n)}(\tau )&=\big (\psi ^{(n)}_\tau ,u^{(n)}_\tau ,\varphi ^{(n)}_\tau \big )^T= \big (\psi ^{(n)}_\tau ,u^{(n)}_\tau , u^{(n)}_{1\tau }+\lambda u^{(n)}_\tau \big )^T, \end{aligned}$$

(4.5)

where

$$\begin{aligned} z^{(n)}&=(\psi ^{(n)},u^{(n)},\varphi ^{(n)})^T,\quad \psi ^{(n)}=(\psi _m)_{|m|\leqslant n},\quad u^{(n)}=(u_m)_{|m|\leqslant n},\\ \varphi ^{(n)}&=\dot{u}^{(n)}+\lambda u^{(n)}, \psi ^{(n)}u^{(n)}=(\psi _m u_m)_{|m|\leqslant n}, A_n|\psi ^{(n)}|^2=\left( (A|\psi |^2)_m\right) _{|m|\leqslant n},\\ F_n(z^{(n)},t)&=\big (-i\psi ^{(n)} u^{(n)}-ig^{(n)}(t),0,f^{(n)}(t)+ A_{n}|\psi ^{(n)}|^2\big )^T, \end{aligned}$$

and

$$\begin{aligned}&\begin{array}{ccc} \Theta _{n}={\left( \begin{array}{ccc} \gamma I_n-i A_n+i h^2 D_{n} &{} \mathbf{0} &{} \mathbf{0} \\ \mathbf{0} &{} \lambda I_n &{} -I_n \\ \mathbf{0} &{} \lambda (\lambda -\alpha )I_n+\mu I_n-A_n+h^2 D_n &{} (\alpha -\lambda )I_n\end{array}\right) }_{3(2n+1)\times 3(2n+1)}, \end{array}\\&\begin{array}{ccccccccc} A_n={\left( \begin{array}{ccccccccc} -2 &{} \quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 1 &{}\quad -2 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad -2 &{}\quad 1 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad 0 &{}\quad 0\\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad -2 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 1 &{}\quad -2 &{}\quad 1 \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad 1 &{}\quad -2 \end{array}\right) }_{(2n+1)\times (2n+1)}, \end{array}\\&\begin{array}{cccccccccc} D_n={\left( \begin{array}{cccccccccc} 6 &{}\quad -4 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 4\\ -4 &{}\quad 6 &{}\quad -4 &{}\quad 1 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 1 &{}\quad -4 &{}\quad 6 &{}\quad -4 &{}\quad 1 &{}\quad \cdots &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad -4 &{}\quad 6 &{}\quad -4 &{}\quad 1\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 1 &{}\quad -4 &{}\quad 6 &{}\quad -4\\ -4 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad 1 &{}\quad -4 &{}\quad 6 \end{array}\right) }_{(2n+1)\times (2n+1)}, \end{array} \end{aligned}$$

$I_n$ is the $(2n+1)$-order identity matrix and $\mathbf{0}$ is the $(2n+1)$-order zero matrix. Write

$$\begin{aligned} \begin{array}{ccccccc} B_n={\left( \begin{array}{ccccccc} -1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad 0\\ 0 &{}\quad -1 &{}\quad 1 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad -1 &{}\quad 1 &{}\quad \cdots &{}\quad 0 &{}\quad 0\\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad -1 &{}\quad 1\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 &{}\quad -1 \end{array}\right) }_{(2n+1)\times (2n+1)}, \end{array} \end{aligned}$$

then we have

$$\begin{aligned} D_n=A_nA^T_n=A^T_nA_n, \qquad A_n=-B_nB^T_n=-B^T_nB_n, \end{aligned}$$

where $B^T_n$ is the transpose matrix of $B_n$. For any two elements $\psi ^{(n)}=(\psi _m)_{|m|\leqslant n}$, $u^{(n)}=(u_m)_{|m|\leqslant n}\in {\mathbb {R}}^{2n+1}$ or ${\mathbb {C}}^{2n+1}$, define

$$\begin{aligned} \big (\psi ^{(n)},u^{(n)}\big )&=\sum \limits _{|m|\leqslant n}\psi _m\overline{u}_m, \qquad \Vert \psi ^{(n)}\Vert ^2=\big (\psi ^{(n)},\psi ^{(n)}\big ),\\ \big (\psi ^{(n)},u^{(n)}\big )_\mu&=\sum \limits _{|m|\leqslant n}\Big (\big (B_n\psi ^{(n)}\big )_m\big (B_n\overline{u}^{(n)}\big )_m +\mu \psi _m\overline{u}_m\Big ), \\ \Vert \psi ^{(n)}\Vert ^2_\mu&=\big (\psi ^{(n)},\psi ^{(n)}\big )_\mu . \end{aligned}$$

Then

$$\begin{aligned} {\mathbb {R}}^{2n+1}=\left( {\mathbb {R}}^{2n+1},(\cdot ,\cdot ),\Vert \cdot \Vert \right) , \quad {\mathbb {C}}^{2n+1}=\left( {\mathbb {C}}^{2n+1},(\cdot ,\cdot ),\Vert \cdot \Vert \right) , \end{aligned}$$

and

$$\begin{aligned} {\mathbb {R}}^{2n+1}_\mu =\left( {\mathbb {R}}^{2n+1},(\cdot ,\cdot )_\mu , \Vert \cdot \Vert _\mu \right) , \end{aligned}$$

are all Hilbert spaces. Consider the truncated space $E^{(n)}_\mu ={\mathbb {C}}^{2n+1}\times {\mathbb {R}}^{2n+1}_\mu \times {\mathbb {R}}^{2n+1}$ of $E_\mu$. For any two elements $z^{(n)}=\big (\psi ^{(n)},u^{(n)},\varphi ^{(n)}\big )^T$, $p^{(n)}=\big (\xi ^{(n)},\zeta ^{(n)},\eta ^{(n)}\big )^T\in E^{(n)}_\mu$, define

$$\begin{aligned} \big (z^{(n)},p^{(n)}\big )_{E^{(n)}_\mu }&= \big (\psi ^{(n)},\xi ^{(n)}\big ) +\big (u^{(n)},\zeta ^{(n)}\big )_\mu +\big (\varphi ^{(n)},\eta ^{(n)}\big ),\\ \Vert z^{(n)}\Vert ^2_{E^{(n)}_\mu }&= \big (z^{(n)},z^{(n)}\big )_{E^{(n)}_\mu }, \end{aligned}$$

then $E^{(n)}_\mu$ is a finite-dimensional Hilbert space.

Clearly, if both $g(t)=(g_m(t))_{m\in {\mathbb {Z}}}$ and $f(t)=(f_m(t))_{m\in {\mathbb {Z}}}$ belong to ${\mathcal {C}}_{b}({\mathbb {R}},\ell ^2 )$, then also $g^{(n)}(t)=(g_m(t))_{|m|\leqslant n}\in {\mathcal {C}}_{b}({\mathbb {R}},{\mathbb {C}}^{2n+1})$ and $f^{(n)}(t)=(f_m(t))_{|m|\leqslant n}\in {\mathcal {C}}_{b}({\mathbb {R}},{\mathbb {R}}^{2n+1})$. Thus, under the assumptions of [25, Theorem 3.1], Eqs. (4.4)–(4.5) are well-posed in $E^{(n)}_\mu$. Also, from Lemma 4.1 below, we see that the solution $z^{(n)}(t)$ of Eqs. (4.4)–(4.5) is bounded in finite time, so $z^{(n)}(t)$ exists globally on $[\tau ,+\infty )$. Hence, for any initial value $z^{(n)}(\tau )\in E^{(n)}_\mu$, there exists a unique solution $z^{(n)}(t)\in {\mathcal {C}}([\tau ,+\infty ),E^{(n)}_\mu )\cap {\mathcal {C}}^{1}((\tau ,+\infty ),E^{(n)}_\mu )$, and one can check that the maps of solution operators

$$\begin{aligned} U^{(n)}(t,\tau ): z^{(n)}(\tau )\in E^{(n)}_\mu \mapsto z^{(n)}(t)=U^{(n)}(t,\tau )z^{(n)}(\tau )\in E^{(n)}_\mu ,\quad t\geqslant \tau , \end{aligned}$$

generate a continuous process $\left\{ U^{(n)}(t,\tau )\right\} _{t\geqslant \tau }$ on $E^{(n)}_\mu$.

Similar to Lemmas 2.3 and 2.4, we have the following results.

Lemma 4.1

Let the conditions of Lemma 2.3 hold. Then the process $\left\{ U^{(n)}(t,\tau )\right\} _{t\geqslant \tau }$ corresponding to Eqs. (4.4)–(4.5) is uniformly pullback bounded dissipative in the sense that for any bounded set ${\mathcal {B}}^{(n)}$ of $E^{(n)}_\mu$, there exists a time $s({\mathcal {B}}^{(n)})> 0$ yielding

$$\begin{aligned} U^{(n)}(t,t-s){\mathcal {B}}^{(n)}\subset {\mathcal {B}}^{(n)}_0, \quad s\geqslant s({\mathcal {B}}^{(n)}), \end{aligned}$$

where ${\mathcal {B}}^{(n)}_0={\mathcal {B}}^{(n)}(0,R_0)\subset E^{(n)}_\mu$ is a bounded ball centered at 0 with radius $R_0=\frac{r_0}{\sqrt{\vartheta }}$ being independent of n. Moreover, the process $\left\{ U^{(n)}(t,\tau )\right\} _{t\geqslant \tau }$ possesses a nonempty bounded kernel ${\mathcal {K}}^{(n)}$, and ${\mathcal {K}}^{(n)}(\tau )\subseteq {\mathcal {B}}^{(n)}_{0}\subset E^{(n)}_\mu$ for any $\tau \in {\mathbb {R}}$.

Lemma 4.2

Let the conditions of Lemma 2.4 hold. Then for every $\varepsilon >0$, there exist $T(\varepsilon ,t,{\mathcal {B}}^{(n)}_0)>0$ and $M(\varepsilon ,t,{\mathcal {B}}^{(n)}_0)\in {\mathbb {N}}$ such that the solution $U^{(n)}(t,t-s)z^{(n)}_{t-s}$ corresponding to the initial value $z^{(n)}_{t-s}$ of Eqs. (4.4)–(4.5) satisfies

$$\begin{aligned} \sum \limits _{|m|>M(\varepsilon ,t, {\mathcal {B}}^{(n)}_0)}\big | \big (U^{(n)}(t,t-s)z^{(n)}_{t-s}\big )_m(t)\big |^{2}_{E^{(n)}_\mu } \leqslant \varepsilon ^{2}, \quad \forall s\geqslant T(\varepsilon ,t,{\mathcal {B}}^{(n)}_0). \end{aligned}$$

We next prove that the kernel sections $\{{\mathcal {K}}^{(n)}(\tau )\}_{\tau \in \mathbb {R}}$ of the process $\left\{ U^{(n)}(t,\tau )\right\} _{t\geqslant \tau }$ converge to the kernel sections $\{\mathcal {K}(\tau )\}_{\tau \in \mathbb {R}}$ of the process $\{U(t,\tau )\}_{t\geqslant \tau }$. We use $z^{(n)}=\big (z^{(n)}_m\big )_{m\in {\mathbb {Z}}}\in E_\mu$ to denote the extension of $z^{(n)}=\big (z^{(n)}_m\big )_{|m|\leqslant n}\in E^{(n)}_\mu$ such that $z^{(n)}_m=0$ for $|m|>n$. In this sense, we have

$$\begin{aligned} {\mathcal {K}}^{(n)}(\tau )\subseteq {\mathcal {B}}^{(n)}_0\subset {\mathcal {B}}_0, \,\,E^{(n)}_\mu \subset E_\mu ,\,\,\Vert z^{(n)}\Vert _{E^{(n)}_\mu }=\Vert z^{(n)}\Vert _{E_\mu }, \,\,\forall z^{(n)}\in E^{(n)}_\mu , \,\,\forall n\in {\mathbb {N}}.\nonumber \\ \end{aligned}$$

(4.6)

Clearly, we can deduce from Lemma 2.4 and (4.6) that for any $\varepsilon >0$, there holds

$$\begin{aligned} \sup \limits _{z^{(n)}\in {\mathcal {K}}^{(n)}(\tau )}\sum \limits _{|m|\geqslant M(\varepsilon ,t,{\mathcal {B}}_0)}|z^{(n)}_m|^{2}_{E_\mu } \leqslant \sup \limits _{z\in {\mathcal {B}}_{0}}\sum \limits _{|m|\geqslant M(\varepsilon ,t,{\mathcal {B}}_0)}\left| z_m\right| ^{2}_{E_\mu } \leqslant \varepsilon ^{2},\quad \forall \tau \in {\mathbb {R}}. \end{aligned}$$

(4.7)

Lemma 4.3

Let the conditions of Lemma 2.4 hold and $z^{(n)}(\tau )\in {\mathcal {K}}^{(n)}(\tau )$ for $\tau \in {\mathbb {R}}$, then there is a subsequence $\left\{ z^{(n_k)}(\tau )\right\}$ of $\left\{ z^{(n)}(\tau )\right\}$ and $z(\tau )\in {\mathcal {K}}(\tau )$ such that

$$\begin{aligned} z^{(n_k)}(\tau )\longrightarrow z(\tau ) \,\,\mathrm{strongly\,\,in}\,\,E_\mu \,\,\mathrm{as}\,\,n_k\rightarrow +\infty . \end{aligned}$$

(4.8)

Proof

Given $\tau \in {\mathbb {R}}$, denote ${\mathbb {R}}_{\tau }=[\tau ,+\infty )$ and let

$$\begin{aligned} z^{(n)}(t,\tau )=U^{(n)}(t,\tau )z^{(n)}(\tau )=\big (\psi ^{(n)}(t,\tau ),u^{(n)}(t,\tau ), \varphi ^{(n)}(t,\tau )\big )^T\in E^{(n)}_\mu , \end{aligned}$$

be the solution of Eqs. (4.4)–(4.5) corresponding to the initial value $z^{(n)}(\tau )$. Since $z^{(n)}(\tau )\in {\mathcal {K}}^{(n)}(\tau )$, we see that $z^{(n)}(t,\tau )\in {\mathcal {K}}^{(n)}$ for all $t\geqslant \tau$. By Lemma 4.2 and (4.6), $z^{(n)}(t,\tau )\in {\mathcal {K}}^{(n)}\subseteq {\mathcal {B}}^{(n)}_{0}\subset E^{(n)}_\mu$ for all $t\geqslant \tau$. Thus,

$$\begin{aligned} \Vert z^{(n)}(t,\tau )\Vert ^{2}_{E^{(n)}_\mu }=\Vert z^{(n)}\Vert ^{2}_{E_\mu }\leqslant R^2_0, \,\,\text { for any } \, t\in {\mathbb {R}}_{\tau }, \quad n=1, 2, \ldots . \end{aligned}$$

(4.9)

By equation (4.4),

$$\begin{aligned} \big \Vert \dot{z}^{(n)}(t,\tau )\big \Vert ^2_{E^{(n)}_\mu }\leqslant 2\big \Vert \Theta _n z^{(n)}(t,\tau )\big \Vert ^2_{E^{(n)}_\mu }+2\big \Vert F_n\big (z^{(n)}(t,\tau ),t\big ) \big \Vert ^2_{E^{(n)}_\mu }. \end{aligned}$$

(4.10)

It follows from (2.4) and (4.9) that

$$\begin{aligned}&\big \Vert \Theta _n z^{(n)}(t,\tau )\big \Vert ^2_{E^{(n)}_\mu }\nonumber \\&\quad =\Vert \gamma \psi ^{(n)}-iA_n\psi ^{(n)}+ih^2D_{n}\psi ^{(n)}\Vert ^2 +\Vert \lambda u^{(n)}-\varphi ^{(n)}\Vert ^2_\mu \nonumber \\&\qquad \,+\,\Vert \lambda (\lambda -\alpha )u^{(n)}+\mu u^{(n)}-A_n u^{(n)}+h^2 D_n u^{(n)}+(\alpha -\lambda )\varphi ^{(n)}\Vert ^2\nonumber \\&\quad \leqslant 2\gamma ^2\Vert \psi ^{(n)}\Vert ^{2}+4h^2\Vert D_{n}\psi ^{n}\Vert ^2+ 4\Vert A_n\psi ^{(n)}\Vert ^2+2\lambda ^2\Vert u^{(n)}\Vert ^2_\mu +2(4\,+\,\mu )\Vert \varphi ^{(n)}\Vert ^2\nonumber \\&\qquad +\,8\Vert A_n u^{(n)}\Vert ^2\,+\,4\left( \lambda ^2\,-\lambda \alpha \,+\mu \right) ^2\Vert u^{(n)}\Vert ^2\,+\,8h^4\Vert D_n u^{(n)}\Vert ^2 \nonumber \\&\qquad +\,2(\alpha \,-\,\lambda )^2 \Vert \varphi ^{(n)}\Vert ^2\nonumber \\&\quad \leqslant C_4, \quad \forall \, t\in {\mathbb {R}}_{\tau }, \,\,n=1, 2, \ldots , \end{aligned}$$

(4.11)

where

$$\begin{aligned} C_4&= \Big \{\frac{1}{\mu }\big [4(\lambda ^2-\lambda \alpha +\mu )^2+2048h^4+ 128\big ]+2(\alpha -\lambda )^2+2\gamma ^2+2\lambda ^2\\&\quad \quad +2\mu +72+1024h^4\Big \}R^2_0. \end{aligned}$$

Similarly,

$$\begin{aligned}&\big \Vert F_n\big (z^{(n)}(t),t\big )\big \Vert ^2_{E^{(n)}_\mu }\nonumber \\&\quad =\Vert -i\psi ^{(n)}u^{(n)}-ig^{(n)}(t)\Vert ^2+\Vert f^{(n)}(t)+ A_n|\psi ^{(n)}|^{2}\Vert ^{2}\nonumber \\&\quad \leqslant \Vert \psi ^{(n)}\Vert ^4+\Vert u^{(n)}\Vert ^4+2\Vert g^{(n)}(t)\Vert ^2+2\Vert f^{(n)}(t)\Vert ^2+2 \Vert A_n|\psi ^{(n)}|^{2}\Vert ^{2}\nonumber \\&\quad \leqslant 33\Vert \psi ^{(n)}\Vert ^{4}+\frac{1}{\mu ^2}\Vert u^{(n)}\Vert ^{4}_{\mu }+ 2\Vert f^{(n)}(t)\Vert ^2+2\Vert g^{(n)}(t)\Vert ^2\nonumber \\&\quad \leqslant C_5, \quad \forall \, t\in {\mathbb {R}}_{\tau }, \,\,n=1, 2, \ldots , \end{aligned}$$

(4.12)

where

$$\begin{aligned} C_5=\left( \frac{1}{\mu ^2}+33\right) R^4_0+2\Vert |f\Vert |^2+2\Vert |g\Vert |^2. \end{aligned}$$

It then follows from (4.10), (4.11), and (4.12) that there exists a constant $C_6=C_6(C_4,C_5)$ such that

$$\begin{aligned} \Vert \dot{z}^{(n)}(t)\Vert ^2_{E^{(n)}_\mu }\leqslant C_6, \quad \forall \, t\in {\mathbb {R}}_{\tau }, \,\,n=1, 2, \ldots . \end{aligned}$$

(4.13)

Let $I_j=[-j,j]\bigcap {\mathbb {R}}_{\tau }$ for $j\in {\mathbb {N}}$ be a sequence of compact interval of ${\mathbb {R}}_{\tau }$. Considering $s, t\in I_j$, we have

$$\begin{aligned} \Vert z^{(n)}(t,\tau )-z^{(n)}(s,\tau )\Vert _{E^{(n)}_\mu } \leqslant C_6(C_4,C_5)|t-s|, \end{aligned}$$

(4.14)

which gives the equicontinuity of $\left\{ z^{(n)}(t,\tau )\right\} ^{\infty }_{n=1}$ in $\mathcal {C}(I_j,E^{(n)}_\mu )$. Equation (4.9) implies that, for fixed t, $\left\{ z^{(n)}(t)\right\} ^{\infty }_{n=1}$ is bounded in $E_\mu$. By the fact that $E_\mu$ is a Hilbert space, there exist a subsequence (still denoted by $\left\{ z^{(n)}(t,\tau )\right\} ^{\infty }_{n=1}$) of $\left\{ z^{(n)}(t,\tau )\right\} ^{\infty }_{n=1}$ and $z_t(\tau )=(z_{t,m}(\tau ))_{m\in {\mathbb {Z}}}\in E_\mu$ such that

$$\begin{aligned} z^{(n)}(t,\tau )\rightharpoonup z_t(\tau )\,\,\mathrm{weakly}\,\,\mathrm{in}\,\,E_\mu \,\,\mathrm{as}\,\,n\rightarrow +\infty . \end{aligned}$$

(4.15)

It follows from (4.9) and (4.15) that

$$\begin{aligned} \Vert z_t(\tau )\Vert _{E_\mu }\leqslant \liminf \limits _{n}\Vert z^{(n)}(t,\tau )\Vert _{E_\mu } \leqslant R_0, \quad \text {for any fixed } \,t\geqslant \tau . \end{aligned}$$

(4.16)

We next prove that the weak convergence in (4.15) is strong. In fact, for any $\varepsilon >0$ and any fixed $t\geqslant \tau$, by (4.7) and Lemma 2.4, there exists an $M(\varepsilon ,t,{\mathcal {B}}_0)\in {\mathbb {N}}$ such that

$$\begin{aligned} \sum \limits _{|m|\geqslant M(\varepsilon ,t,{\mathcal {B}}_{0})}\big |z^{(n)}_m(t,\tau )-z_{t,m}(\tau )\big |^2_{E_\mu }&\leqslant 2\sum \limits _{|m|\geqslant M(\varepsilon ,t,{\mathcal {B}}_{0})}\big (\big |z^{(n)}_m(t,\tau )\big |^2_{E_\mu }+ \big |z_{t,m}(\tau )\big |^2_{E_\mu }\big )\nonumber \\&\leqslant \frac{{\varepsilon }^2}{2}. \end{aligned}$$

(4.17)

At the same time, by (4.15),

$$\begin{aligned}&\big (z^{(n)}_m(t,\tau )\big )_{|m|\leqslant M(\varepsilon ,t,{\mathcal {B}}_{0})}\\&\quad \longrightarrow \big (z_{m,t}(\tau )\big )_{|m|\leqslant M(\varepsilon ,t,{\mathcal {B}}_{0})} \hbox { strongly in } E^{2M(\varepsilon ,t,{\mathcal {B}}_{0})+1}_{\mu } \hbox { as } n\rightarrow +\infty , \end{aligned}$$

from which we infer that there exists an $M(\varepsilon )>0$ such that

$$\begin{aligned} \sum \limits _{|m|\leqslant M(\varepsilon ,t,{\mathcal {B}}_{0})}\big |z^{(n)}_m(t,\tau ) -z_{t,m}(\tau )\big |^2_{E_\mu }&= \big \Vert z^{(n)}(t,\tau )-z_{t}(\tau )\big \Vert ^2_ {E^{2M(\varepsilon ,t,{\mathcal {B}}_{0})+1}_{\mu }}\nonumber \\&\leqslant \frac{{\varepsilon }^2}{2}, \quad n\geqslant M(\varepsilon ). \end{aligned}$$

(4.18)

(4.17) and (4.18) imply that for any fixed $t\geqslant \tau$,

$$\begin{aligned} z^{(n)}(t,\tau )\longrightarrow z_t(\tau ) \text { strongly in } E_\mu \text { as } n\rightarrow +\infty . \end{aligned}$$

(4.19)

Using Arzela-Ascoli’s theorem and the technique of diagonal subsequence, there exist a subsequence $\left\{ z^{(n_k)}(t,\tau )\right\}$ of $\left\{ z^{(n)}(t,\tau )\right\}$ and

$$\begin{aligned} z(t,\tau )=\left( z_m(t,\tau )\right) _{m\in {\mathbb {Z}}}=\left( \psi _m(t,\tau ),u_m(t,\tau ), \varphi _m(t,\tau )\right) ^{T}_{m\in {\mathbb {Z}}}\in {\mathcal {C}}\left( I_j,E_\mu \right) , \end{aligned}$$

such that

$$\begin{aligned} z^{(n_k)}(t,\tau )\longrightarrow z(t,\tau ) \text { strongly in } {\mathcal {C}}\left( I_j,E_\mu \right) \text { as } n_k \rightarrow +\infty . \end{aligned}$$

(4.20)

We next prove that $\psi (t,\tau )\in {\mathcal {K}}(t)$ for any $t\geqslant \tau$. To this end, we establish that $z(t,\tau )$ with $t\in {\mathbb {R}}_{\tau }$ is a bounded solution of Eqs. (2.11)–(2.12). For brevity, we denote $\left\{ z^{(n_k)}(t,\tau )\right\}$ by $\left\{ z^{(n)}(t,\tau )\right\}$. It then follows from (4.6), (4.13), and (4.15) that

$$\begin{aligned} \dot{z}^{(n)}(t,\tau )\rightharpoonup \dot{z}(t,\tau ) \text { weak star in } L^{\infty }\left( {\mathbb {R}}_{\tau },E_\mu \right) \text { as } n\rightarrow +\infty . \end{aligned}$$

(4.21)

Since

$$\begin{aligned} z^{(n)}(t,\tau )\,=\,\big (z^{(n)}_m(t,\tau )\big )_{|m|\leqslant n}\,=\,\big (\psi ^{(n)}_m(t,\tau ), u^{(n)}_m(t,\tau ),\varphi ^{(n)}_m(t,\tau )\big )^{T}_{|m|\leqslant n}{\in }{\mathcal {K}}^{(n)}(t)\,\subset \,E^{(n)}_\mu \end{aligned}$$

is a solution of Eqs. (4.4)–(4.5) with initial data $z^{(n)}(\tau )\in {\mathcal {K}}^{(n)}(\tau )$, we can obtain for any $|m|\leqslant n$ and any $t\in I_j$ that

$$\begin{aligned}&i\dot{\psi }^{(n)}_m+(A_n\psi ^{(n)})_m+i\gamma \psi ^{(n)}_m-\psi ^{(n)}_m u^{(n)}_m-h^2(D_{n}\psi ^{(n)})_{m}=g_m(t), \end{aligned}$$

(4.22)

$$\begin{aligned}&\ddot{u}^{(n)}_m-(A_{n}u^{(n)})_{m}+h^2(D_{n}u^{(n)})_{m} -\big (A_n(|\psi ^{(n)}|^2)\big )_m+\mu u^{(n)}_m +\alpha \dot{u}^{(n)}_m =f_m(t). \end{aligned}$$

(4.23)

Thus, for any $\theta (t)\in {\mathcal {C}}^{\infty }_{0}(I_j)$, we get for $|m|\leqslant n$ that

$$\begin{aligned}&\int \limits _{I_j}i\dot{\psi }^{(n)}_m\theta (t)\mathrm {d}t+\int \limits _{I_j} (A_n\psi ^{(n)})_m\theta (t)\mathrm {d}t+ \int \limits _{I_j}i\gamma \psi ^{(n)}_m\theta (t)\mathrm {d}t -\int \limits _{I_j}\psi ^{(n)}_m u^{(n)}_m\theta (t)\mathrm {d}t\nonumber \\&\quad -\int \limits _{I_j} h^2(D_n\psi ^{(n)})_m\theta (t)\mathrm {d}t= \int \limits _{I_j}g_m(t)\theta (t)\mathrm {d}t, \end{aligned}$$

(4.24)

$$\begin{aligned}&\int \limits _{I_j}\ddot{u}^{(n)}_m\theta (t)\mathrm {d}t+ \int \limits _{I_j}\alpha \dot{u}^{(n)}_m\theta (t)\mathrm {d}t- \int \limits _{I_j}(A_n u^{(n)})_m\theta (t)\mathrm {d}t+ \int \limits _{I_j}\mu u^{(n)}_m\theta (t)\mathrm {d}t\nonumber \\&\quad -\int \limits _{I_j}(A_n|\psi ^{(n)}|^2)_m\theta (t)\mathrm {d}t+ \int \limits _{I_j}h^2(D_n u^{(n)})_m\theta (t)\mathrm {d}t =\int \limits _{I_j}f_m(t)\theta (t)\mathrm {d}t. \end{aligned}$$

(4.25)

From (4.20), we see that

$$\begin{aligned} \sup \limits _{t\in I_j}\Vert \psi ^{(n)}-\psi \Vert \longrightarrow 0 \text { and }\sup \limits _{t\in I_j}\Vert u^{(n)}-u\Vert _\mu \longrightarrow 0 \text { as } n\rightarrow +\infty . \end{aligned}$$

(4.26)

Obviously, for any $n\in {\mathbb {N}}$ and any $x\in {\mathbb {R}}^{2n+1}_{x}$, there holds

$$\begin{aligned} \Vert A_n x\Vert _{{\mathbb {R}}^{2n+1}_{x}}\leqslant 4\Vert x\Vert _{{\mathbb {R}}^{2n+1}_{x}}, \qquad \Vert D_n x\Vert _{{\mathbb {R}}^{2n+1}_{x}}\leqslant 16\Vert x\Vert _{{\mathbb {R}}^{2n+1}_{x}}. \end{aligned}$$

(4.27)

Thus, we get from (4.26) that

$$\begin{aligned}&\Big |\int \limits _{I_j}\big (A_n(\psi ^{(n)}-\psi )\big )_{m} \theta (t)\mathrm {d}t\Big | \leqslant \sup \limits _{t\in I_j}4\Vert \psi ^{(n)}-\psi \Vert \int \limits _{I_j}|\theta (t)|\mathrm {d}t\longrightarrow 0 \text { as}\,n\rightarrow +\infty , \end{aligned}$$

(4.28)

$$\begin{aligned}&\Big |\int \limits _{I_j}\big (D_n(u^{(n)}-u)\big )_{m} \theta (t)\mathrm {d}t\Big | \leqslant \sup \limits _{t\in I_j}\frac{16}{\sqrt{\mu }}\Vert u^{(n)}-u\Vert _{\mu }\int \limits _{I_j} |\theta (t)|\mathrm {d}t\longrightarrow 0\, \text { as } n\rightarrow +\infty , \end{aligned}$$

(4.29)

and

$$\begin{aligned}&\Big |\int \limits _{I_j}\psi ^{(n)}_m u^{(n)}_m\theta (t)\mathrm {d}t- \int \limits _{I_j}\psi _m u_m\theta (t)\mathrm {d}t\Big |\nonumber \\&\quad \leqslant \int \limits _{I_j}|\psi ^{(n)}_m||u^{(n)}_m-u_m||\theta (t)|\mathrm {d}t +\int \limits _{I_j}|u^{(n)}_m||\psi ^{(n)}_m-\psi _m||\theta (t)|\mathrm {d}t\nonumber \\&\quad \leqslant \Big (\sup \limits _{t\in I_j}\Vert \psi ^{(n)}-\psi \Vert +\frac{1}{\sqrt{\mu }} \sup \limits _{t\in I_j}\Vert u^{(n)}-u\Vert _\mu \Big )R_0 \int \limits _{I_j}|\theta (t)|\mathrm {d}t\longrightarrow 0 \text { as } n\rightarrow +\infty , \end{aligned}$$

(4.30)

$$\begin{aligned}&\Big |\int \limits _{I_j}\big (A_n(|\psi ^{(n)}|^2-|\psi |^2) \big )_m\theta (t)\mathrm {d}t\Big | \leqslant \sup \limits _{t\in {I_j}}4\big ||\psi ^{(n)}|^2-|\psi |^2\big | \int \limits _{I_j}|\theta (t)|\mathrm {d}t\nonumber \\&\quad \leqslant 8R_0\sup \limits _{t\in {I_j}}\Vert \psi ^{(n)}-\psi \Vert \int \limits _{I_j}|\theta (t)|\mathrm {d}t \longrightarrow 0 \text { as } n\rightarrow +\infty . \end{aligned}$$

(4.31)

Letting $n\rightarrow +\infty$ and $j\rightarrow +\infty$ in (4.22) and (4.23), we obtain, by using (4.21), (4.24), (4.25), and (4.28)–(4.29), for all $t\in I_j$ and $m\in {\mathbb {Z}}$ that

$$\begin{aligned} \left\{ \begin{array}{ll} i\dot{\psi }_m+(A\psi )_m+i\gamma \psi _m-\psi _m u_m-h^2(D\psi )_{m}=g_m(t), \\ \ddot{u}_m+\alpha \dot{u}_m-(Au)_m+\mu u_m-(A|\psi |^2)_m +h^2(Du)_m=f_m(t). \end{array} \right. \end{aligned}$$

(4.32)

Since $I_j$ is arbitrary, (4.32) holds for all $t\in {\mathbb {R}}_{\tau }$, which means that $z(t,\tau )$ with $t\in {\mathbb {R}}_{\tau }$ is a solution of (2.11)–(2.12). From (4.16), it follows that $z(t,\tau )$ is bounded for $t\in {\mathbb {R}}_{\tau }$. Thus, $z(t,\tau )\in {\mathcal {K}}(t)$ for all $t\in {\mathbb {R}}_{\tau }$. Now (4.19) implies $z^{(n)}(\tau ,\tau )\longrightarrow z_\tau (\tau )=z(\tau ,\tau )\in {\mathcal {K}}(\tau )$ strongly in $E_{\mu }$ as $n\rightarrow +\infty$. The proof is complete. $\square$

Using the argument of contradiction and Lemma 4.3, we can obtain the following upper semicontinuity of the kernel sections $\{{\mathcal {K}}(\tau )\}_{\tau \in \mathbb {R}}$.

Theorem 4.1

Let the conditions of Lemma 2.4 hold. Then

$$\begin{aligned} \lim \limits _{n\rightarrow +\infty }\mathrm{dist}_{E_\mu }\big ({\mathcal {K}}^ {(n)}(\tau ),\mathcal {K}(\tau )\big )=0,\quad \forall \tau \in {\mathbb {R}}. \end{aligned}$$