Random attractor for second-order stochastic delay lattice sine-Gordon equation

Abstract

In this paper, we prove the existence of random \(\mathcal{D}\)-attractor for the second-order stochastic delay sine-Gordon equation on infinite lattice with certain dissipative conditions, and then establish the upper bound of Kolmogorov ε-entropy for the random \(\mathcal{D}\)-attractor.

Introduction

This paper deals with the following second-order stochastic delay lattice sine-Gordon equation:

$$ \textstyle\begin{cases} \ddot{u}_{i}+\alpha \dot{u}_{i}+(Au)_{i}+\lambda u_{i}+\beta \sin {u_{i}}=f_{i}(u_{it})+g_{i}+ \epsilon _{i}\dot{w}_{i},\quad t>0, \\ u_{i0}=u_{i}(\tau ),\qquad \dot{u}_{i0}=\dot{u_{i}}(\tau ),\quad \tau \in [-h,0], \end{cases} $$
(1.1)

where \(i\in \mathbb{Z}^{k}\) with \(\mathbb{Z}\) being the set of integers and \(k\in \mathbb{N}\) a fixed positive integer, \(u_{it}=u_{it}( \tau )=u_{i}(t+\tau )\) is the delay term with the interval of delay time \([-h,0]\), and \(u_{i0}=u_{i}(\tau )\), \(\dot{u}_{i0}=\dot{u_{i}}(\tau )\) is the initial data, A is a linear operator defined by (3.1), \(u=(u_{i})_{i\in \mathbb{Z}^{k}}\), \(g=(g_{i})_{i\in \mathbb{Z}^{k}}\in l^{2}\), α, β, h are positive constants, \(\epsilon =(\epsilon _{i})_{i\in \mathbb{Z}^{k}}\in l^{2}\), \(f_{i}\) is a smooth function satisfying some dissipative conditions (see the hypotheses \(H_{1}\)\(H_{3}\) in Sect. 3), \(\{w_{i}(t):i\in \mathbb{Z}^{k}\}\) is independent two-sided real valued standard Wiener processes.

Lattice dynamical systems, whose the spatial structure has a discrete character, arise from a variety of applications such as electrical engineering [1], biology [2, 3], chemical reaction [4], and pattern formation [5]. As a matter of fact, systems in the process of evolution are always influenced by the external environment, those influence may be random or the time delay. If the system add the random or the time delay terms, it makes up for the defects of some deterministic systems, and explains new evolutionary rules. Many researchers have discussed broadly the deterministic models in [68]. Stochastic lattice equations, driven by additive independent white noise, were discussed for the first time in [9], and then intensive researched in [1020]. Furthermore, a kind of stochastic delay lattice systems were considered in [2124], and these interesting results have attracted wide attention of scholars.

However, to the best of our knowledge, there is little literature about the existence of random attractors for a second-order stochastic delay lattice sine-Gordon equation on \(\mathbb{Z}^{k}\). To this end, this paper is devoted to study this problem. The main ideas and methods used in the proofs are motivated by [21, 22, 2427].

This paper is organized as follows. In Sect. 2, we introduce some basic concepts and propositions. In Sect. 3, we present some sufficient conditions for the existence of continuous stochastic dynamical systems. Section 4 is devoted to proving the existence of a random \(\mathcal{D}\)-attractor for a stochastic lattice sine-Gordon equation. In Sect. 5, we study the upper bound of the Kolmogorov ε-entropy for the random \(\mathcal{D}\)-attractor.

Preliminaries

In this section, we recall some basic concepts and propositions related to random attractors for stochastic dynamical systems (SDS), more details can be found in the literature [9, 28, 29].

Let \((X,\|\cdot \|_{X})\) be a separable Hilbert space, and \((\Omega ,\mathcal{F},P)\) a probability space.

Definition 2.1

Let \(\mathcal{D}\) be a collection of random subsets of X, stochastic process \(S\{(t,\omega )\}_{t\geq 0,\omega \in \Omega }\) a continuous random dynamical system and \(\{B_{0}(\omega )\}_{\omega \in \Omega }\in \mathcal{D}\). Then \(\{B_{0}(\omega )\}_{\omega \in \Omega }\) is said to be a random absorbing set for S in \(\mathcal{D}\) if for every \(B\in \mathcal{D}\) for P-a.e. \(\omega \in \Omega \), there exists some \(t_{B}(\omega )>0\) such that

$$ S(t,\theta _{-t}\omega )B(\theta _{-t}\omega )\subset B_{0}(\omega ),\quad \forall t\geq t_{B}(\omega ). $$

Definition 2.2

Let \(\mathcal{D}\) be a collection of random subsets of X. Then a random set \(\{\mathcal{A}(\omega )\}_{\omega \in \Omega } \) in X is said to be a \(\mathcal{D}\)-random attractor for S, if the following conditions are satisfied, for P-a.e. \(\omega \in \Omega \):

  1. (1)

    \(\{\mathcal{A}(\omega )\}\) is compact, and \(\omega \rightarrow d(x,\mathcal{A}(\omega ))\) is measurable for every \(x\in X\);

  2. (2)

    \(\{\mathcal{A}(\omega )\} \) is invariant;

  3. (3)

    \(\{\mathcal{A}(\omega )\}\) attracts every set in \(\mathcal{D}\), i.e., for all \(B=\{B(\omega )\}_{\omega \in \Omega }\in \mathcal{D}\),

    $$ \lim_{t\rightarrow \infty }d \bigl(S(t,\theta _{-t} \omega )B( \theta _{-t}\omega ),\mathcal{A}(\omega ) \bigr)=0, $$

where d denotes the Hausdorff semi-metric.

Definition 2.3

Let \(B(\omega )\subset X\) be a random set. For any \(\varepsilon >0\), \(\omega \in \Omega \), let \(\mathcal{N}_{\varepsilon ,\omega }(B(\omega ),X)=\mathcal{N}_{ \varepsilon ,\omega }(B(\omega ))\) be the minimal number of deterministic open balls in X with radii ε that is necessary to cover \(B(\omega )\). The number \(\mathcal{K}_{\varepsilon }(B(\omega ))=\mathcal{K}_{\varepsilon }(B( \omega ),X) =\ln \mathcal{N}_{\varepsilon ,\omega }(B(\omega ))\) is called the Kolmogorov ε-entropy of \(B(\omega )\) in X.

Proposition 2.1

(See [30])

Let \(n\in \mathbb{N}\) and \(\Upsilon =\{x=(x_{i})_{|i|\leq n}:x_{i}\in \mathbb{R},|x_{i}|\leq r \}\subset \mathbb{R}^{2n+1}\) be a regular polyhedron. Then ϒ can be covered by \(\mathcal{N}_{\varepsilon }(\Upsilon )= ([2r\cdot \frac{1}{\varepsilon }\sqrt{2n+1}]+1 )^{2n+1}\) balls in \(\mathbb{R}^{2n+1}\) with radii \(\frac{\varepsilon }{2}\), where \([\cdot ]\) denotes the integer-valued function.

Proposition 2.2

(See [29])

If \(r(\omega )>0\) is tempered and \(r(\theta _{t}\omega )\) is continuous in t for P-a.e. \(\omega \in \Omega \), then

  1. (1)

    for any \(t\in \mathbb{R}\), \(r(\theta _{t}\omega )\) is tempered. Moreover, for any \(h>0\), \(\max_{\tau \in [-h,0]}r(\theta _{\tau }\omega )\) is also tempered;

  2. (2)

    for any \(\beta >0\) and P-a.e. \(\omega \in \Omega \), \(R(\omega )=\int _{-\infty }^{0}e^{\beta s}r(\theta _{s}\omega )\,ds< \infty \) is tempered, and \(R(\theta _{t}\omega )\) is also continuous in t.

Proposition 2.3

(See [9])

Suppose that \(B_{0}(\omega )\in \mathcal{D}(X)\) is a closed random absorbing set for the continuous \(\operatorname{SDS}\{S(t,\omega )\}_{t\geq 0,\omega \in \Omega }\), and, for a.e. \(\omega \in \Omega \), each sequence \(x_{n} \in S(t_{n},\theta _{-t_{n}}\omega ) B_{0}(\theta _{-t_{n}} \omega )\) with \(t_{n}\rightarrow \infty \) has a convergent subsequence in X. Then the \(\operatorname{SDS}\{S(t,\omega )\}_{t\geq 0,\omega \in \Omega }\) has a unique \(\mathcal{D}\)-random attractor \(A(\omega )\), which is given by

$$ A(\omega )=\bigcap_{s\geq 0} \overline{\bigcup _{t\geq s}S(t,\theta _{-t}\omega ) B_{0}(\theta _{-t}\omega )}. $$

Second-order stochastic delay lattice sine-Gordon equation

Denote \(l^{p}\) (\(p\geq 1\)) defined by

$$ l^{p}= \biggl\{ u\Big|u=(u_{i})_{i\in \mathbb{Z}^{k}},i=(i_{1},i_{1}, \ldots ,i_{k}) \in {\mathbb{Z}}^{k},u_{i}\in \mathbb{R} , \sum_{i\in \mathbb{Z}^{k}} \vert u_{i} \vert ^{p}< + \infty \biggr\} , $$

with the norm

$$ \Vert u \Vert _{p}^{p}=\sum _{i\in \mathbb{Z}^{k}} \vert u_{i} \vert ^{p}. $$

In particular, \(l^{2}\) is a Hilbert space with the inner product \((\cdot ,\cdot )\) and norm \(\|\cdot \|\) given by

$$ (u,v)=\sum_{i\in \mathbb{Z}^{k}}u_{i}v_{i},\qquad \Vert u \Vert ^{2}=\sum_{i \in \mathbb{Z}^{k}} \vert u_{i} \vert ^{2}, $$

for any \(u=(u_{i})_{i\in \mathbb{Z}^{k}}\), \(v=(v_{i})_{i\in \mathbb{Z}^{k}} \in l^{2}\).

Define a linear operator in the following way:

$$\begin{aligned}& \begin{aligned} &(Au_{i})_{i\in \mathbb{Z}^{k}}=2ku_{(i_{1},\ldots ,i_{j},\ldots ,i_{k})}-u_{(i_{1}-1, \ldots ,i_{j},\ldots ,i_{k})} -\cdots -u_{(i_{1},\ldots ,i_{j}-1, \ldots ,i_{k})}-\cdots -u_{(i_{1},\ldots ,i_{j},\ldots ,i_{k}-1)} \\ &\hphantom{(Au_{i})_{i\in \mathbb{Z}^{k}}={}}{} -u_{(i_{1}+1,\ldots ,i_{j},\ldots ,i_{k})} - \cdots -u_{(i_{1},\ldots ,i_{j}+1,\ldots ,i_{k})}-\cdots -u_{(i_{1}, \ldots ,i_{j},\ldots ,i_{k}+1)}, \end{aligned} \\& (B_{j}u)_{i}=u_{(i_{1},\ldots ,i_{j}+1,\ldots ,i_{k})}-u_{(i_{1}, \ldots ,i_{j},\ldots ,i_{k})},\qquad \bigl(B^{\ast }_{j}u \bigr)_{i}=u_{(i_{1},\ldots ,i_{j}, \ldots ,i_{k})}-u_{(i_{1},\ldots ,i_{j}-1,\ldots ,i_{k})}, \\& (A_{j}u)_{i}=2u_{(i_{1},\ldots ,i_{j},\ldots ,i_{k})}-u_{(i_{1}, \ldots ,i_{j}+1,\ldots ,i_{k})}-u_{(i_{1},\ldots ,i_{j}-1,\ldots ,i_{k})}. \end{aligned}$$
(3.1)

Then \(B_{j}^{\ast }\) is the adjoint operator of \(B_{j}\), and

$$ A=A_{1}+A_{2}+\cdots +A_{k},\qquad A_{j}=B_{j}B_{j}^{\ast }=B_{j}^{ \ast }B_{j},\quad \mbox{for }j=1,2,\ldots ,k. $$

By using the above equalities, we have

$$ (Au,v)=\sum_{j=1}^{k}(B_{j}u,B_{j}v)= \sum_{j=1}^{k} \bigl(B_{j}^{\ast }u,B_{j}^{ \ast }v \bigr). $$

For any \(u=(u_{i})_{i\in \mathbb{Z}^{k}}\), \(v=(v_{i})_{i\in \mathbb{Z}^{k}} \in l^{2}\), we define a new inner product and norm on \(l^{2}\) by

$$ (u,v)_{\lambda }=\sum_{j=1}^{k}(B_{j}u,B_{j}v)+ \lambda (u,v), \qquad \Vert u \Vert ^{2}_{\lambda }=(u,u)_{\lambda }= \sum_{j=1}^{k} \Vert B_{j}u \Vert ^{2}+ \lambda \Vert u \Vert ^{2}. $$

It is obvious that

$$ \Vert B_{j}u \Vert ^{2}\leq 4 \Vert u \Vert ^{2},\qquad \lambda \Vert u \Vert ^{2}\leq \Vert u \Vert ^{2}_{ \lambda }\leq (4k+\lambda ) \Vert u \Vert ^{2}. $$

Denote

$$ l^{2}= \bigl(l^{2},(\cdot ,\cdot ), \Vert \cdot \Vert \bigr),\qquad l^{2}_{\lambda }= \bigl(l^{2},( \cdot ,\cdot )_{\lambda }, \Vert \cdot \Vert _{\lambda } \bigr). $$

Then the norms \(\|\cdot \|\) and \(\|\cdot \|_{\lambda }\) are equivalent.

Let \(H=l^{2}_{\lambda }\times l^{2}\) be endowed with the inner product and norm

$$ (\psi _{1},\psi _{2})_{H}= \bigl(u^{(1)},u^{(2)} \bigr)_{\lambda }+ \bigl(v^{(1)},v^{(2)} \bigr),\qquad \Vert \psi \Vert _{H}^{2}= \Vert u \Vert ^{2}_{\lambda }+ \Vert v \Vert ^{2}, $$

for \(\psi _{j}=(u^{(j)},v^{(j)})^{T}= ((u^{(j)}_{i},v^{(j)}_{i}) )^{T}_{i \in \mathbb{Z}^{k}}\in H\), \(j=1, 2\), \(\psi =(u,v)^{T}= ((u_{i},v_{i}) )^{T}_{i\in \mathbb{Z}^{k}}\in H\). In addition, the space \(H_{0}=C([-h,0],H)\) is endowed with \(\|\psi \|_{H_{0}}=\max_{\tau \in [-h,0]}\| \psi (\tau )\|_{H}\).

In the following, we consider the probability space \((\Omega ,\mathcal{F},P)\), where

$$ \Omega = \bigl\{ \omega \in C \bigl(\mathbb{R},l^{2} \bigr):\omega (0)=0 \bigr\} , $$

\(\mathcal{F}\) is the Borel σ-algebra induced by the compact-open topology of Ω, and P the corresponding Wiener measure on \((\Omega ,\mathcal{F})\). We will identify ω with

$$ \omega (t)=w(t), \quad t\in \mathbb{R}. $$

Define the time shift by

$$ \theta _{t}\omega (\cdot )=\omega (\cdot +t)-\omega (t), \quad \omega \in \Omega , t \in \mathbb{R}. $$

Then \((\Omega ,\mathcal{F},P,(\theta _{t})_{t\in \mathbb{R}})\) is a metric dynamical system with the filtration

$$ \mathcal{F}=\bigvee_{s\leq t} \mathcal{F}^{t}_{s},\quad t\in \mathbb{R}, $$

where \(\mathcal{F}^{t}_{s}=\sigma \{w(t_{2})-w(t_{1}):s\leq t_{1}\leq t_{2} \leq t\}\) is the smallest σ-algebra generated by \(w(t_{2})-w(t_{1})\) for all \(s\leq t_{1}\leq t_{2}\leq t\).

For convenience, we rewrite Eq. (1.1) as

$$\begin{aligned} \textstyle\begin{cases} \ddot{u}+\alpha \dot{u}+Au+\lambda u+\beta \sin {u}=f(u_{t})+g+\dot{w},\quad t>0, \\ u_{0}=u(\tau ),\qquad \dot{u}_{0}=\dot{u}(\tau ),\quad \tau \in [-h,0], \end{cases}\displaystyle \end{aligned}$$
(3.2)

where \(u=(u_{i})_{i\in \mathbb{ Z}^{k}}\), \(\dot{u}=(\dot{u}_{i})_{i\in \mathbb{ Z}^{k}}\), \(\ddot{u}=(\ddot{u}_{i})_{i\in \mathbb{ Z}^{k}}\), \(u_{0}=(u_{0i})_{i\in \mathbb{ Z}^{k}}\), \(\dot{u}_{0}=(\dot{u}_{0i})_{i\in \mathbb{ Z}^{k}}\), \(Au=(Au_{i})_{i\in \mathbb{ Z}^{k}}\), \(\lambda u=(\lambda u_{i})_{i\in \mathbb{ Z}^{k}}\), \(\beta \sin {u}=\beta (\sin {u_{i}})_{i\in \mathbb{ Z}^{k}}\), \(f(u_{t})=(f_{i}(u_{it}))_{i\in \mathbb{ Z}^{k}}\), \(g=(g_{i})_{i\in \mathbb{ Z}^{k}}\) and \(\dot{w}=(\epsilon _{i}\dot{w}_{i})_{i\in \mathbb{ Z}^{k}}\). Let \(\bar{v}=\dot{u}+\delta u\), where δ is a positive constant and satisfies

$$ 4\delta +\frac{2\delta \alpha ^{2}}{\lambda }+ \frac{4\beta ^{2}}{\lambda \delta }-2\alpha < 0, $$
(3.3)

then Eq. (3.2) can be rewritten as

$$\begin{aligned} \textstyle\begin{cases} \dot{\varphi }(t)+D\varphi (t)=F(\varphi (t))+G(t),\quad t>0, \\ \varphi _{0}=(u_{0},\bar{v}_{0})^{T}=(u(\tau ),\dot{u}(\tau )+\delta u( \tau ))^{T},\quad \tau \in [-h,0], \end{cases}\displaystyle \end{aligned}$$
(3.4)

where \(\varphi =(u,\bar{v})^{T}\), \(\varphi _{t}=(u_{t},\bar{v}_{t})^{T}\), \(F(\varphi (t))=(0,f(u_{t})+g-\beta \sin {u})^{T}\), \(G=(0,\dot{w})^{T}\) and

$$ D\varphi (t)= \begin{pmatrix} \delta u-\bar{v} \\ Au+\lambda u +(\delta -\alpha )(\delta u-\bar{v}) \end{pmatrix} . $$

Also, we make the following assumptions:

(\(H_{1}\)):

\(f_{i}:C([-h,0];\mathbb{R})\rightarrow \mathbb{R}\) is continuous and \(f_{i}(0)=0\);

(\(H_{2}\)):

\(|f_{i}(\xi )|\leq M_{0,i}+M_{1,i}\max_{\tau \in [-h,0]}|\xi (\tau )|\) for all \(\xi \in C([-h,0];\mathbb{R})\), where \(M_{r,i}\geq 0\), \((M_{r,i})_{i\in \mathbb{Z}^{k}}\in l^{2}\), \(M_{r}^{2}:=\sum_{i\in \mathbb{Z}^{k}}M_{r,i}^{2}\) (\(r=0,1\));

(\(H_{3}\)):

for any bounded set \(Y\subset l^{2}\), there exists a constant \(L_{f}>0\), such that

$$ \bigl\Vert f(u)-f(v) \bigr\Vert \leq L_{f} \Vert u-v \Vert , \quad \forall u,v\in Y. $$

Lemma 3.1

Suppose (\(H_{1}\))(\(H_{3}\)) hold. For any \(T>0\) and an initial data \(\varphi _{0}\in H_{0}\), there exists a unique solution \(\varphi _{t}\in L^{2}(\Omega ,C([0,T]);H)\) of Eq. (3.4) with \(\varphi _{t}(\cdot ,\varphi _{0})\in H_{0}\) for \(t\in [0,T]\) and \(\varphi _{0}(\cdot ,\varphi _{0})= \varphi _{0}\). Moreover, \(\varphi _{t}(\cdot ,\varphi _{0})\) depends continuously on the initial data \(\varphi _{0}\) for each \(\omega \in \Omega \).

Proof

Rewriting (3.4) as

$$ \varphi (t)=\varphi (0)+ \int _{0} ^{t} \bigl(-D\varphi (s)+F \bigl( \varphi (s) \bigr)+G(s) \bigr) \,ds, \quad t>0. $$

By (\(H_{1}\)) and (\(H_{3}\)), we know that

$$ \begin{aligned} \bigl\Vert F \bigl(\varphi (t) \bigr) \bigr\Vert _{H}^{2} &= \bigl\Vert f(u_{t})+g-\beta \sin u \bigr\Vert ^{2} \leq 3L_{f}^{2} \Vert u_{t} \Vert ^{2}+3 \Vert g \Vert ^{2}+3\beta ^{2} \Vert u \Vert ^{2} \\ & \leq \frac{3L_{f}^{2}+3\beta ^{2}}{\lambda } \Vert \varphi _{t} \Vert _{H_{0}}^{2}+3 \Vert g \Vert ^{2} \end{aligned} $$

and

$$ \begin{aligned} \bigl\Vert D\varphi (t) \bigr\Vert _{H}^{2} & = \Vert \delta u-\bar{v} \Vert _{\lambda }^{2}+ \bigl\Vert A u+ \lambda u+(\delta -\alpha ) (\delta u-\bar{v}) \bigr\Vert ^{2} \\ & \leq 2\delta ^{2} \Vert u \Vert _{\lambda }^{2}+2 \Vert \bar{v} \Vert _{\lambda }^{2}+3 \Vert A u \Vert ^{2}+3 \lambda ^{2} \Vert u \Vert ^{2}+3( \delta -\alpha )^{2}\|\delta u- \bar{v})\|^{2} \\ & \leq \biggl(2\delta ^{2}+\frac{54k^{2}}{\lambda }+3\lambda + \frac{6\delta ^{2}(\delta -\alpha )^{2}}{\lambda } \biggr) \Vert u \Vert _{\lambda }^{2}+ \bigl(6(\delta -\alpha )^{2}+2(4k+\lambda ) \bigr) \Vert \bar{v} \Vert ^{2} \\ &\leq d_{1} \bigl\Vert \varphi (t) \bigr\Vert _{H}^{2}\leq d_{1} \Vert \varphi _{t} \Vert _{H_{0}}^{2}, \end{aligned} $$

where \(d_{1}=\max \{2\delta ^{2}+\frac{54k^{2}}{\lambda }+3\lambda + \frac{6\delta ^{2}(\delta -\alpha )^{2}}{\lambda }, 6(\delta -\alpha )^{2}+2(4k+ \lambda )\}\). Thus, F and D map the bounded sets into bounded sets. In this way, by the standard theory of differential equations, we find that there exists a unique local solution. Then calculations in blow shows that this local solution is actually global. Indeed, suppose the solutions \(\varphi ^{(1)}(t)\), \(\varphi ^{(2)}(t)\) of Eq. (3.4) with the initial data \(\varphi _{0}^{(1)}\), \(\varphi _{0}^{(2)}\in H_{0}\), respectively, we have

$$ \begin{aligned} & \frac{d}{dt} \bigl\Vert \varphi ^{(1)}(t)- \varphi ^{(2)}(t) \bigr\Vert _{H}^{2} \\ &\quad = -2 \bigl(D \bigl(\varphi ^{(1)}(t)-\varphi ^{(2)}(t) \bigr),\varphi ^{(1)}(t)- \varphi ^{(2)}(t) \bigr)_{H} \\ &\qquad {}+2 \bigl(F \bigl(\varphi ^{(1)}(t) \bigr)-F \bigl( \varphi ^{(2)}(t) \bigr), \varphi ^{(1)}(t)- \varphi ^{(2)}(t) \bigr)_{H} \\ &\quad \leq \bigl\Vert D \bigl(\varphi ^{(1)}(t)- \varphi ^{(2)}(t) \bigr) \bigr\Vert _{H}^{2}+2 \bigl\Vert \varphi ^{(1)}(t)- \varphi ^{(2)}(t) \bigr\Vert _{H}^{2}+ \bigl\Vert F \bigl(\varphi ^{(1)}(t) \bigr)- F \bigl(\varphi ^{(2)}(t) \bigr) \bigr\Vert _{H}^{2} \\ &\quad \leq (d_{1}+2) \bigl\Vert \varphi ^{(1)}(t)- \varphi ^{(2)}(t) \bigr\Vert _{H}^{2}+ \bigl\Vert F \bigl( \varphi ^{(1)}(t) \bigr)- F \bigl(\varphi ^{(2)}(t) \bigr) \bigr\Vert _{H}^{2}, \end{aligned} $$

which implies that

$$ \begin{aligned} & \bigl\Vert \varphi ^{(1)}(t)- \varphi ^{(2)}(t) \bigr\Vert _{H}^{2} \\ &\quad \leq \bigl\Vert \varphi ^{(1)}(0)- \varphi ^{(2)}(0) \bigr\Vert _{H}^{2}+(d_{1}+2) \int _{0} ^{t} \bigl\Vert \varphi ^{(1)}(s)- \varphi ^{(2)}(s) \bigr\Vert _{H}^{2} \\ &\qquad {}+ \int _{0} ^{t} \bigl\Vert F \bigl(\varphi ^{(1)}(s) \bigr)- F \bigl(\varphi ^{(2)}(s) \bigr) \bigr\Vert _{H}^{2}\,ds \\ &\quad \leq \bigl\Vert \varphi ^{(1)}(0)- \varphi ^{(2)}(0) \bigr\Vert _{H}^{2}+ \biggl(d_{1}+2+ \frac{2\beta ^{2}}{\lambda } \biggr) \int _{0} ^{t} \bigl\Vert \varphi ^{(1)}(s)- \varphi ^{(2)}(s) \bigr\Vert _{H}^{2} \\ &\qquad {}+ \frac{2L_{f}^{2}}{\lambda } \int _{0} ^{t} \bigl\Vert \varphi ^{(1)}_{s}- \varphi ^{(2)}_{s} \bigr\Vert _{H}^{2}\,ds \\ &\quad = \biggl(1+\frac{2hL_{f}^{2}}{\lambda } \biggr) \bigl\Vert \varphi ^{(1)}(0)- \varphi ^{(2)}(0) \bigr\Vert _{H_{0}}^{2}+ \biggl(d_{1}+2+ \frac{2\beta ^{2}+2L_{f}^{2}}{\lambda } \biggr) \int _{0} ^{t} \bigl\Vert \varphi ^{(1)}(s)- \varphi ^{(2)}(s) \bigr\Vert _{H}^{2}. \end{aligned} $$

Applying the Gronwall inequality, we have

$$ \bigl\Vert \varphi ^{(1)}(t)- \varphi ^{(2)}(t) \bigr\Vert _{H}^{2}\leq \biggl(1+ \frac{2hL_{f}^{2}}{\lambda } \biggr)e^{(d_{1}+2+ \frac{2\beta ^{2}+2L_{f}^{2}}{\lambda })t} \bigl\Vert \varphi ^{(1)}(0)- \varphi ^{(2)}(0) \bigr\Vert _{H_{0}}^{2},\quad \forall t\in [0,T], $$

from which we get

$$ \sup_{t\in [0,T]} \bigl\Vert \varphi ^{(1)}(t)- \varphi ^{(2)}(t) \bigr\Vert _{H}^{2}\leq \biggl(1+\frac{2hL_{f}^{2}}{\lambda } \biggr)e^{(d_{1}+2+ \frac{2\beta ^{2}+2L_{f}^{2}}{\lambda })T} \bigl\Vert \varphi ^{(1)}(0)- \varphi ^{(2)}(0) \bigr\Vert _{H_{0}}^{2}. $$

The proof is complete. □

Lemma 3.2

Suppose (\(H_{1}\))(\(H_{3}\)) hold, Eq. (3.4) generates a continuous random dynamical system \(\varphi _{t}\) over \((\Omega ,\mathcal{F},P,(\theta _{t})_{t\in R})\).

Proof

The proof is similar to that of Theorem 3.2 in [9], so here it is omitted. □

Existence of random attractor

This section will be devoted to prove the existence of a \(\mathcal{D}\)-random attractor for \(\{S(t,\omega )\}_{t\geq 0,\omega \in \Omega }\) in \(H_{0}\). Firstly, we introduce an Ornstein–Uhlenbeck process in \(l^{2}\) on the metric dynamical system \((\Omega ,\mathcal{F},P,(\theta _{t})_{t\in R})\) given by the Wiener process:

$$ z(\theta _{t}\omega )=-\alpha \int _{-\infty }^{0}e^{\alpha s}\theta _{t} \omega (s)\,ds, $$

where \(\alpha >0\), the above integral solves the following Itô equation:

$$ dz+\alpha z\,dt=dw. $$

In fact, there exists a \(\theta _{t}\)-invariant set \(\Omega '\subset \Omega \) such that

  1. (i)

    the mapping \(t\rightarrow z(\theta _{t}\omega )\) is continuous for P-a.s. \(\omega \in \Omega '\);

  2. (ii)

    the random variable \(\|z(\theta _{t}\omega )\|\) is tempered.

Denote

$$ \psi (t)= \bigl(u(t),v(t) \bigr)^{T}=\varphi (t)- \bigl(0,z(\theta _{t}\omega ) \bigr)^{T}, $$

where \(\varphi (t)\) is the solution of Eq. (3.4). Then \(\psi (t)\) satisfies

$$ \textstyle\begin{cases} \dot{\psi }(t)+D\psi (t)=C(\psi (t),t,\omega ),\quad t>0, \\ \psi _{0}=\varphi _{0}-(0,z(\theta _{0_{-}}\omega ))^{T}, \end{cases} $$
(4.1)

where \(z(\theta _{t_{-}}\omega )=z(\theta _{t+\tau }\omega )\), \(\tau \in [-h,0]\), for any \(t\geq 0\), \(\psi _{t}=\varphi _{t}-(0,z(\theta _{t_{-}}\omega ))^{T}\), \(C(\psi (t),t, \omega )=(z(\theta _{t}\omega ),f(u_{t})+g-\beta \sin {u}+\delta z( \theta _{t}\omega ))^{T}\), and

$$ D\psi = \begin{pmatrix} \delta u-v \\ Au+\lambda u +(\delta -\alpha )(\delta u-v) \end{pmatrix} . $$

Lemma 4.1

Suppose (\(H_{1}\))(\(H_{3}\)) hold, and \(\delta >\frac{2M_{1}e^{\delta h}}{\sqrt{\lambda }} \), Then there exists a random absorbing set \(B_{0}(\omega )\in \mathcal{D}(H_{0})\) for \(\{S(t,\omega )\}_{t\geq 0,\omega \in \Omega }\).

Proof

Taking the inner product \((\cdot ,\cdot )_{H}\) on both sides of (4.1) with \(\psi (t)\), we get

$$ \frac{d}{dt} \bigl\Vert \psi (t) \bigr\Vert ^{2}_{H}+2 \bigl(D\psi (t),\psi (t) \bigr)_{H}=2 \bigl(C,\psi (t) \bigr)_{H}. $$
(4.2)

Now, we estimate the terms of (4.2) one by one. We have

$$ \begin{aligned} 2 \bigl(D\psi (t),\psi (t) \bigr)_{H} & = 2 \bigl(\delta \Vert u \Vert ^{2}_{\lambda }+( \alpha -\delta ) \Vert v \Vert ^{2}-\delta (\alpha -\delta ) (u,v) \bigr) \\ &\geq 2 \biggl(\delta \Vert u \Vert ^{2}_{\lambda }+( \alpha -\delta ) \Vert v \Vert ^{2}- \frac{\delta \alpha }{\sqrt{\lambda }} \Vert u \Vert _{\lambda } \Vert v \Vert \biggr) \\ & \geq 2\delta \Vert u \Vert ^{2}_{\lambda }+2(\alpha - \delta ) \Vert v \Vert ^{2}- \frac{\delta }{2} \Vert u \Vert ^{2}_{\lambda }- \frac{2\delta \alpha ^{2}}{\lambda } \Vert v \Vert ^{2} \\ & =\frac{3\delta }{2} \Vert u \Vert ^{2}_{\lambda }+2 \biggl(\alpha -\delta - \frac{\delta \alpha ^{2}}{\lambda } \biggr) \Vert v \Vert ^{2} \end{aligned} $$
(4.3)

and

$$ \begin{aligned} &2 \bigl(C,\psi (t) \bigr)_{H} \\ &\quad =2 \bigl(u,z(\theta _{t}\omega ) \bigr)_{\lambda }+2 \bigl(f(u_{t}),v \bigr)+2(g,v)-2(\beta \sin u,v)+2 \bigl(\delta z( \theta _{t}\omega ),v \bigr) \\ &\quad \leq \frac{\delta }{4} \Vert u \Vert ^{2}_{\lambda }+ \frac{4}{\delta }\bigl\| z(\theta _{t} \omega )\bigr\| ^{2}_{\lambda } +4\delta \bigl\| z(\theta _{t}\omega ) \bigr\Vert ^{2}+ \frac{\delta }{4} \Vert v \Vert ^{2}+2\sum _{i\in \mathbb{Z}^{k}} \vert g_{i} \vert \vert v_{i} \vert \\ &\qquad {} +2\sum_{i\in \mathbb{Z}^{k}} \Bigl(M_{0,i}+M_{1,i} \max_{\tau \in [-h,0]} \vert u_{it} \vert \Bigr) \vert v_{i} \vert +2\beta \sum_{i\in \mathbb{Z}^{k}} \vert \sin u_{i} \vert \vert v_{i} \vert \\ &\quad \leq \frac{\delta }{2} \Vert u \Vert ^{2}_{\lambda }+ \biggl( \frac{16k+4\lambda }{\delta }+4\delta \biggr) \bigl\Vert z(\theta _{t}\omega ) \bigr\Vert ^{2} + \biggl( \frac{3\delta }{4}+\frac{4\beta ^{2}}{\delta \lambda } \biggr) \Vert v \Vert ^{2} \\ &\qquad {}+ \frac{4}{\delta } \bigl(M_{0}^{2}+ \Vert g \Vert ^{2} \bigr)+\frac{2M_{1}}{\sqrt{\lambda }} \Vert \psi _{t} \Vert ^{2}_{H_{0}}. \end{aligned} $$
(4.4)

By (4.2)–(4.4) and (3.3), we have

$$ \begin{aligned} &\frac{d}{dt} \bigl\Vert \psi (t) \bigr\Vert ^{2}_{H}+\delta \bigl\Vert \psi (t) \bigr\Vert ^{2}_{H} \\ &\quad \leq \biggl(4\delta +\frac{2\delta \alpha ^{2}}{\lambda } + \frac{4\beta ^{2}}{\delta \lambda }-2\alpha \biggr) \Vert v \Vert ^{2} + \biggl( \frac{16k+4\lambda }{\delta }+4\delta \biggr) \bigl\Vert z(\theta _{t}\omega ) \bigr\Vert ^{2} \\ &\qquad {}+ \frac{2M_{1}}{\sqrt{\lambda }} \Vert \psi _{t} \Vert ^{2}_{H_{0}} + \frac{4}{\delta } \bigl(M_{0}^{2}+ \Vert g \Vert ^{2} \bigr) \\ &\quad \leq \biggl(\frac{16k+4\lambda }{\delta }+4\delta \biggr) \bigl\Vert z(\theta _{t}\omega ) \bigr\Vert ^{2}+ \frac{2M_{1}}{\sqrt{\lambda }} \Vert \psi _{t} \Vert ^{2}_{H_{0}} + \frac{4}{\delta } \bigl(M_{0}^{2}+ \Vert g \Vert ^{2} \bigr), \end{aligned} $$
(4.5)

which gives

$$ \begin{aligned} \frac{d}{dt} \bigl(e^{\delta t} \bigl\Vert \psi (t) \bigr\Vert ^{2}_{H} \bigr) \leq&{} \biggl( \frac{16k+4\lambda }{\delta }+4\delta \biggr)e^{\delta t} \bigl\Vert z( \theta _{t} \omega ) \bigr\Vert ^{2} \\ &{}+ \frac{2M_{1}e^{\delta t}}{\sqrt{\lambda }} \Vert \psi _{t} \Vert ^{2}_{H_{0}} + \frac{4 e^{\delta t}}{\delta } \bigl(M_{0}^{2}+ \Vert g \Vert ^{2} \bigr). \end{aligned} $$
(4.6)

One deduces by integrating the above inequality on [0,t] that

$$ \begin{aligned} e^{\delta t} \bigl\Vert \psi (t) \bigr\Vert ^{2}_{H} \leq{}& \bigl\Vert \psi (0) \bigr\Vert ^{2}_{H}+ \biggl( \frac{16k+4\lambda }{\delta }+4\delta \biggr) \int _{0}^{t}e^{\delta s} \bigl\Vert z( \theta _{s}\omega ) \bigr\Vert ^{2}\,ds \\ &{} +\frac{2M_{1}}{\sqrt{\lambda }} \int _{0}^{t}e^{\delta s} \Vert \psi _{s} \Vert ^{2}_{H_{0}}\,ds + \frac{4 e^{\delta t}}{\delta ^{2}} \bigl(M_{0}^{2}+ \Vert g \Vert ^{2} \bigr). \end{aligned} $$
(4.7)

For fixed \(\tau \in [-h,0]\), we have, for all \(t\geq 0\),

$$\begin{aligned} e^{\delta t} \bigl\Vert \psi (t+\tau ) \bigr\Vert ^{2}_{H} \leq{}& e^{-\delta \tau } \bigl\Vert \psi (0) \bigr\Vert ^{2}_{H}+e^{-\delta \tau } \biggl( \frac{16k+4\lambda }{\delta }+4\delta \biggr) \int _{0}^{t+\tau }e^{\delta s} \bigl\Vert z( \theta _{s}\omega ) \bigr\Vert ^{2}\,ds \\ &{} +e^{-\delta \tau }\frac{2M_{1}}{\sqrt{\lambda }} \int _{0}^{t+ \tau }e^{\delta s} \Vert \psi _{s} \Vert ^{2}_{H_{0}}\,ds +e^{-\delta \tau } \frac{4 e^{\delta (t+\tau )}}{\delta ^{2}} \bigl(M_{0}^{2}+ \Vert g \Vert ^{2} \bigr) \\ \leq{}& e^{\delta h} \bigl\Vert \psi (0) \bigr\Vert ^{2}_{H}+e^{\delta h} \biggl( \frac{16k+4\lambda }{\delta }+4\delta \biggr) \int _{0}^{t}e^{\delta s} \bigl\Vert z( \theta _{s}\omega ) \bigr\Vert ^{2}\,ds \\ &{} +e^{\delta h}\frac{2M_{1}}{\sqrt{\lambda }} \int _{0}^{t}e^{ \delta s} \Vert \psi _{s} \Vert ^{2}_{H_{0}}\,ds + \frac{4 e^{\delta t}}{\delta ^{2}} \bigl(M_{0}^{2}+ \Vert g \Vert ^{2} \bigr). \end{aligned}$$
(4.8)

This implies that

$$ \begin{aligned} e^{\delta t} \Vert \psi _{t} \Vert ^{2}_{H_{0}} \leq{}& e^{\delta h} \bigl\Vert \psi (0) \bigr\Vert ^{2}_{H_{0}}+e^{ \delta h} \biggl(\frac{16k+4\lambda }{\delta }+4\delta \biggr) \int _{0}^{t}e^{ \delta s} \bigl\Vert z( \theta _{s}\omega ) \bigr\Vert ^{2}\,ds \\ &{} +e^{\delta h}\frac{2M_{1}}{\sqrt{\lambda }} \int _{0}^{t}e^{ \delta s} \Vert \psi _{s} \Vert ^{2}_{H_{0}}\,ds + \frac{4 e^{\delta t}}{\delta ^{2}} \bigl(M_{0}^{2}+ \Vert g \Vert ^{2} \bigr). \end{aligned} $$
(4.9)

Let us introduce the following notations:

$$ c_{1}=e^{\delta h} \biggl(\frac{16k+4\lambda }{\delta }+4\delta \biggr),\qquad c_{2}= \frac{4}{\delta ^{2}} \bigl(M_{0}^{2}+ \Vert g \Vert ^{2} \bigr), \qquad c_{3}= \frac{2M_{1}e^{\delta h}}{\sqrt{\lambda }}. $$

Thus, the inequality (4.9) can be rewritten as

$$ e^{\delta t} \Vert \psi _{t} \Vert ^{2}_{H_{0}} \leq e^{\delta h} \bigl\Vert \psi (0) \bigr\Vert ^{2}_{H_{0}}+c_{1} \int _{0}^{t}e^{\delta s} \bigl\Vert z( \theta _{s} \omega ) \bigr\Vert ^{2}\,ds+c_{2}e^{\delta t} +c_{3} \int _{0}^{t}e^{\delta s} \Vert \psi _{s} \Vert ^{2}_{H_{0}}\,ds. $$
(4.10)

Applying the Gronwall inequality to the above inequality yields

$$ \begin{aligned} e^{\delta t} \Vert \psi _{t} \Vert ^{2}_{H_{0}} \leq{}& e^{\delta h} \bigl\Vert \psi (0) \bigr\Vert ^{2}_{H_{0}}+c_{1} \int _{0}^{t}e^{\delta s} \bigl\Vert z( \theta _{s} \omega ) \bigr\Vert ^{2}\,ds +c_{2}e^{\delta t} \\ &{} + c_{3} \int _{0}^{t} \biggl(e^{\delta h} \bigl\Vert \psi (0) \bigr\Vert ^{2}_{H_{0}} +c_{1} \int _{0}^{s}e^{\delta \upsilon } \bigl\Vert z( \theta _{\upsilon }\omega ) \bigr\Vert ^{2}\,d\upsilon +c_{2}e^{\delta s} \biggr)e^{c_{3}(t-s)}\,ds \\ \leq{}& e^{\delta h} \bigl\Vert \psi (0) \bigr\Vert ^{2}_{H_{0}}e^{c_{3}t}+ \frac{c_{2}\delta }{\delta -c_{3}}e^{\delta t} +c_{1}e^{c_{3} t} \int _{0}^{t}e^{( \delta -c_{3})s} \bigl\Vert z( \theta _{s}\omega ) \bigr\Vert ^{2}\,ds, \end{aligned} $$
(4.11)

and hence

$$ \begin{aligned} \bigl\Vert \psi _{t} \bigl(\cdot ,\omega ,\psi _{0}(\omega ) \bigr) \bigr\Vert ^{2}_{H_{0}} \leq{}& e^{ \delta h} \bigl\Vert \psi (0) \bigr\Vert ^{2}_{H_{0}}e^{-(\delta -c_{3})t} \\ &{}+ \frac{c_{2}\delta }{\delta -c_{3}} +c_{1} \int _{0}^{t}e^{(\delta -c_{3})(s-t)} \bigl\Vert z( \theta _{s}\omega ) \bigr\Vert ^{2}\,ds. \end{aligned} $$
(4.12)

Replacing ω by \(\theta _{-t}\omega \) in (4.12), we have

$$ \begin{aligned} & \bigl\Vert \psi _{t} \bigl(\cdot ,\theta _{-t}\omega ,\psi _{0}(\theta _{-t}\omega ) \bigr) \bigr\Vert ^{2}_{H_{0}} \\ &\quad \leq e^{\delta h} \bigl\Vert \psi _{0}(\theta _{-t}\omega ) \bigr\Vert ^{2}_{H_{0}}e^{-( \delta -c_{3})t}+ \frac{c_{2}\delta }{\delta -c_{3}} +c_{1} \int _{0}^{t}e^{( \delta -c_{3})(s-t)} \bigl\Vert z( \theta _{s-t}\omega ) \bigr\Vert ^{2}\,ds \\ &\quad \leq e^{\delta h} \bigl\Vert \psi _{0}(\theta _{-t}\omega ) \bigr\Vert ^{2}_{H_{0}}e^{-( \delta -c_{3})t}+ \frac{c_{2}\delta }{\delta -c_{3}} +c_{1} \int _{- \infty }^{0}e^{(\delta -c_{3})s} \bigl\Vert z( \theta _{s}\omega ) \bigr\Vert ^{2}\,ds. \end{aligned} $$
(4.13)

Since \(\varphi _{t}(\cdot ,\omega ,\varphi _{0}(\omega ))=\psi _{t} ( \cdot ,\omega ,\varphi _{0}(\omega )-(0,z(\theta _{0_{-}}\omega ))^{T} ) +(0,z(\theta _{t_{-}}\omega ))^{T}\), it follows from (4.13) that

$$ \begin{aligned} & \bigl\Vert \varphi _{t} \bigl(\cdot ,\theta _{-t}\omega ,\varphi _{0}(\theta _{-t} \omega ) \bigr) \bigr\Vert ^{2}_{H_{0}} \\ &\quad = \bigl\Vert \psi _{t} \bigl(\cdot ,\theta _{-t} \omega ,\varphi _{0}(\theta _{-t} \omega )- \bigl(0,z( \theta _{-t}\omega ) \bigr)^{T} \bigr) + \bigl(0,z(\theta _{0_{-}} \omega ) \bigr)^{T} \bigr\Vert ^{2}_{H_{0}} \\ &\quad \leq 2 \bigl\Vert \psi _{t} \bigl(\cdot ,\theta _{-t}\omega ,\varphi _{0}( \theta _{-t}\omega )- \bigl(0,z(\theta _{-t}\omega ) \bigr)^{T} \bigr) \bigr\Vert ^{2}_{H_{0}}+ 2 \bigl\Vert \bigl(0,z(\theta _{0_{-}}\omega ) \bigr)^{T} \bigr\Vert ^{2}_{H_{0}} \\ &\quad \leq 4e^{\delta h} \Bigl( \bigl\Vert \varphi _{0}(\theta _{-t}\omega ) \bigr\Vert ^{2}_{H_{0}} + \max _{\tau \in [-h,0]} \bigl\Vert z(\theta _{-t+\tau } \omega ) \bigr\Vert ^{2} \Bigr) e^{-(\delta -c_{3})t}+ \frac{2c_{2}\delta }{\delta -c_{3}} \\ &\qquad {} + 2c_{1} \int _{-\infty }^{0}e^{(\delta -c_{3})s} \bigl\Vert z( \theta _{s} \omega ) \bigr\Vert ^{2}\,ds +2\max _{\tau \in [-h,0]} \bigl\Vert z( \theta _{\tau }\omega ) \bigr\Vert ^{2}. \end{aligned} $$
(4.14)

By assumption, \(B(\omega )\in \mathcal{D}(H_{0})\) is tempered. On the other hand, by Proposition 2.2, we know that \(\max_{\tau \in [-h,0]}\|z(\theta _{\tau } \omega )\|^{2}\), \(\max_{\tau \in [-h,0]}\|z( \theta _{-t+\tau }\omega )\|^{2}\), and \(\int _{-\infty }^{0}e^{(\delta -c_{3})s}\|z(\theta _{s}\omega )\|^{2}\,ds\) are also tempered. Thus, if \(\varphi _{0}(\theta _{-t}\omega )\in B(\theta _{-t}\omega )\), then there exists some \(T_{B}(\omega )>0\) such that, for all \(t\geq T_{B}(\omega )\),

$$ \begin{aligned} & \bigl\Vert \varphi _{t} \bigl(\cdot ,\theta _{-t}\omega ,\varphi _{0}(\theta _{-t} \omega ) \bigr) \bigr\Vert ^{2}_{H_{0}} \\ &\quad \leq \frac{4c_{2}\delta }{\delta -c_{3}}+2c_{1} \int _{-\infty }^{0}e^{( \delta -c_{3})s} \bigl\Vert z( \theta _{s}\omega ) \bigr\Vert ^{2}\,ds +2\max _{\tau \in [-h,0]} \bigl\Vert z(\theta _{\tau }\omega ) \bigr\Vert ^{2}\equiv R_{0}^{2}( \omega ), \end{aligned} $$
(4.15)

that is, \(B_{0}(\omega )=\{\xi \in H_{0}:\|\xi \|_{H_{0}}\leq R_{0}(\omega )\}\) is a random absorbing set for \(\{S(t,\omega )\}_{t\geq 0,\omega \in \Omega }\). The proof is complete. □

For convenience, we denote \(\|\psi \|_{H}^{2}=\sum_{i\in \mathbb{Z}^{k}}|\psi _{i}|_{H}^{2}\), where \(|\psi _{i}|_{H}^{2}=\sum_{j=1}^{k}(B_{j}u)_{i}^{2}+\lambda u_{i}^{2}+v_{i}^{2}\), for any \(\psi =(\psi _{i})_{i\in \mathbb{Z}^{k}}=((u_{i},v_{i}))^{T}_{i\in \mathbb{Z}^{k}}\).

Lemma 4.2

Suppose the conditions of Lemma 4.1hold, and \(\varphi _{0}(\omega )\in B_{0}(\omega )\). Then, for any \(\varepsilon >0\), there exist \(M(\varepsilon ,\omega )\in N\) and \(T(\varepsilon ,\omega )>0\) such that the solution \(\varphi (t,\omega ,\varphi _{0}(\omega ))\) of Eq. (3.4) satisfies

$$ \max_{\tau \in [-h,0]}\sum_{ \Vert i \Vert >M( \varepsilon ,\omega )} \bigl\vert \varphi _{i}\bigl(t+\tau ,\theta _{-t}\omega , \varphi _{0}(\theta _{-t}\omega )\bigr) \bigr\vert _{H}^{2}< \varepsilon ,\quad \forall t \geq T(\varepsilon , \omega ), $$

where \(i\in \mathbb{Z}^{k}\) and \(\|i\|=\max_{1\leq j\leq k}|i_{j}|\).

Proof

Define a smooth increasing function \(\eta (x)\in C(R_{+},[0,1])\bigcap C^{1}(R_{+},R_{+}) \) such that

$$ \eta (x)= \textstyle\begin{cases} 0 , & 0\leq x\leq 1; \\ 1, & x \geq 2, \end{cases} $$
(4.16)

and \(|\eta (x)'|\leq \eta _{0}\) (constant) for all \(x\in R_{+}\). Set \(\xi =(\xi _{i})_{i\in \mathbb{Z}^{k}}=(p,q)^{T}\) with \(\xi _{i}=(p_{i},q_{i})^{T}=(\eta (\frac{\|i\|}{M})u_{i},\eta ( \frac{\|i\|}{M})v_{i})^{T} \), where M is a fixed positive integer. Taking the inner product \((\cdot ,\cdot )_{H}\) of Eq. (4.1) with ξ, we have

$$ (\dot{\psi },\xi )_{H}+(D\psi ,\xi )_{H}= \bigl(C(\psi ,t,\omega ),\xi \bigr)_{H}. $$
(4.17)

Next, we estimate the terms of (4.17) one by one. Firstly,

$$ \begin{aligned} (\dot{\psi },\xi )_{H} & = \sum_{i\in \mathbb{Z}^{k}}\sum_{j=1}^{k}(B_{j} \dot{u})_{i}(B_{j}p)_{i}+\sum _{i\in \mathbb{Z}^{k}}\lambda \dot{u}_{i}p_{i}+ \sum_{i\in \mathbb{Z}^{k}} \dot{v}_{i}q_{i} \\ & = \sum_{i\in \mathbb{Z}^{k}}\sum_{j=1}^{k}(B_{j} \dot{u})_{i} \biggl[ \eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) (B_{j}u)_{i}+(B_{j}p)_{i}- \eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) (B_{j}u)_{i} \biggr] \\ &\quad {}+\sum_{i\in \mathbb{Z}^{k}}\lambda \dot{u}_{i}p_{i}+ \sum_{i \in \mathbb{Z}^{k}} \dot{v}_{i}q_{i} \\ & = \frac{1}{2}\frac{d}{dt}\sum_{i\in \mathbb{Z}^{k}} \eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \Biggl[\sum _{j=1}^{k}(B_{j}u)_{i}^{2}+ \lambda u_{i}^{2}+v_{i}^{2} \Biggr] \\ &\quad {}+\sum_{i\in \mathbb{Z}^{k}}\sum_{j=1}^{k}(B_{j} \dot{u})_{i} \biggl[(B_{j}p)_{i}-\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) (B_{j}u)_{i} \biggr] \end{aligned} $$
(4.18)

and

$$ \begin{aligned} &\sum_{i\in \mathbb{Z}^{k}} \sum_{j=1}^{k}(B_{j} \dot{u})_{i} \biggl[(B_{j}p)_{i}- \eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) (B_{j}u)_{i} \biggr] \\ &\quad \leq \frac{2k\eta _{0}}{M} \bigl((1+\delta ) \Vert u \Vert ^{2}+ \Vert v \Vert ^{2}+ \bigl\Vert z(\theta _{t}\omega ) \bigr\Vert ^{2} \bigr), \end{aligned} $$
(4.19)

by (4.18) and (4.19) we have

$$ (\dot{\psi },\xi )_{H} \geq \frac{1}{2}\frac{d}{dt}\sum_{i\in \mathbb{Z}^{k}} \eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{i} \vert _{H}^{2} - \frac{2k\eta _{0}}{M} \bigl((1+\delta ) \Vert u \Vert ^{2}+ \Vert v \Vert ^{2}+ \bigl\Vert z(\theta _{t} \omega ) \bigr\Vert ^{2} \bigr). $$
(4.20)

Secondly,

$$ \begin{aligned} (D\psi ,\xi )_{H} & = \delta (u,p)_{\lambda }-(v,p)_{\lambda }+(Au,q)+ \lambda (u,q)+(\delta -\alpha )\delta (u,q)+(\alpha -\delta ) (v,q) \\ & = \delta \sum_{j=1}^{k}(B_{j}u,B_{j}p)+ \lambda \delta (u,p)-\sum_{j=1}^{k}(B_{j}v,B_{j}p)+ \sum_{j=1}^{k}(B_{j}u,B_{j}q) \\ &\quad {} +(\delta -\alpha )\delta (u,q)+(\alpha -\delta ) (v,q) \end{aligned} $$
(4.21)

and

$$\begin{aligned}& \delta \sum _{j=1}^{k}(B_{j}u,B_{j}p) = \delta \sum_{j=1}^{k}\sum _{i \in \mathbb{Z}^{k}}(B_{j}u)_{i}(B_{j}p)_{i} \geq \delta \sum_{j=1}^{k} \sum _{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) (B_{j}u)_{i}^{2}- \frac{2k\delta \eta _{0}}{M} \Vert u \Vert ^{2}, \end{aligned}$$
(4.22)
$$\begin{aligned}& \sum_{j=1}^{k}(B_{j}u,B_{j}q)- \sum_{j=1}^{k}(B_{j}v,B_{j}p) \geq - \frac{k\eta _{0}}{M} \biggl(\sqrt{\lambda } \Vert u \Vert ^{2}+\frac{1}{\sqrt{\lambda }} \Vert v \Vert ^{2} \biggr) \geq -\frac{k\eta _{0}}{\sqrt{\lambda }M} \Vert \psi \Vert _{H}^{2}, \end{aligned}$$
(4.23)
$$\begin{aligned}& \delta \lambda (u,p)=\delta \lambda \sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)u_{i}^{2}, \end{aligned}$$
(4.24)
$$\begin{aligned}& (\alpha -\delta ) (v,q)=(\alpha - \delta )\sum_{i\in \mathbb{Z}^{k}} \eta \biggl(\frac{ \Vert i \Vert }{M} \biggr)v_{i}^{2}, \end{aligned}$$
(4.25)
$$\begin{aligned}& \begin{aligned} (\delta -\alpha )\delta (u,q)&=(\delta -\alpha )\delta \sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)u_{i}v_{i} \\ &\geq - \frac{\lambda \delta }{4}\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)u_{i}^{2}- \frac{\alpha ^{2}\delta }{\lambda }\sum_{i \in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)v_{i}^{2}. \end{aligned} \end{aligned}$$
(4.26)

By (4.21)–(4.26) we obtain

$$ \begin{aligned} 2(D\psi ,\xi )_{H} \geq{}& 2\delta \sum_{i\in \mathbb{Z}^{k}}\sum _{j=1}^{k} \eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) (B_{j}u)_{i}^{2}- \frac{4k\delta \eta _{0}}{M} \Vert u \Vert ^{2}-\frac{2k\eta _{0}}{\sqrt{\lambda }M} \Vert \psi \Vert _{H}^{2} \\ &{} +2\delta \lambda \sum _{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr)u_{i}^{2}+2(\alpha -\delta )\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)v_{i}^{2} \\ &{}- \frac{\lambda \delta }{2}\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)u_{i}^{2}- \frac{2\alpha ^{2}\delta }{\lambda }\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)v_{i}^{2} \\ ={}& 2\delta \sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{i} \vert _{H}^{2}- \frac{4k\delta \eta _{0}}{M} \Vert u \Vert ^{2}- \frac{2k\eta _{0}}{\sqrt{\lambda }M} \Vert \psi \Vert _{H}^{2} \\ &{}- \frac{\lambda \delta }{2}\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)u_{i}^{2} + \biggl(2\alpha -4\delta -\frac{2\alpha ^{2}\delta }{\lambda } \biggr)\sum _{i \in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr)v_{i}^{2}. \end{aligned} $$
(4.27)

Thirdly,

$$ \bigl(C(\psi ,t,\omega ),\xi \bigr)_{H}= (z,p)_{\lambda }+ \bigl(f(u_{t}),q \bigr)+(g,q)+(- \beta \sin u,q)+(\delta z ,q) $$
(4.28)

and

$$\begin{aligned}& \begin{aligned} (z,p)_{\lambda } &=\sum _{j=1}^{k}(B_{j}z,B_{j}p)+ \lambda (z,p) \\ &\leq \frac{\delta }{2}\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)\sum_{j=1}^{k}(B_{j}u)_{i}^{2}+ \frac{\lambda \delta }{8}\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)u_{i}^{2}+ \frac{k\eta _{0}}{2M} \bigl( \Vert u \Vert ^{2}+4 \Vert z \Vert ^{2} \bigr) \\ &\quad {} +\frac{2k+2\lambda }{\delta }\sum_{ \Vert i \Vert \geq M-1}z_{i}^{2}, \end{aligned} \end{aligned}$$
(4.29)
$$\begin{aligned}& \delta (z,q) = \delta \sum _{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)z_{i}v_{i} \leq \frac{\delta }{6}\sum _{i\in \mathbb{Z}^{k}} \eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)v_{i}^{2}+ \frac{3\delta }{2}\sum _{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)z_{i}^{2}, \end{aligned}$$
(4.30)
$$\begin{aligned}& \begin{aligned} (-\beta \sin u,q) &=-\beta \sum _{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)\sin u_{i}v_{i} \\ &\leq \frac{\lambda \delta }{8}\sum _{i \in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr)u_{i}^{2}+ \frac{2\beta ^{2}}{\lambda \delta }\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)v_{i}^{2}, \end{aligned} \end{aligned}$$
(4.31)
$$\begin{aligned}& (g,q) =\sum _{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr)g_{i}v_{i} \leq \frac{\delta }{6}\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)v_{i}^{2}+ \frac{3}{2\delta }\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)g_{i}^{2}, \end{aligned}$$
(4.32)
$$\begin{aligned}& \begin{aligned} \bigl(f(u_{t}),q \bigr) &= \sum_{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr)f_{i}(u_{it})v_{i} \leq \sum _{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \Bigl(M_{0,i}+M_{1,i} \max_{\tau \in [-h,0]} \vert u_{it} \vert \Bigr) \vert v_{i} \vert \\ &=\sum_{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr)M_{0,i} \vert v_{i} \vert + \sum _{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr)M_{1,i} \max_{\tau \in [-h,0]} \vert u_{it} \vert \vert v_{i} \vert \\ &\leq \sum_{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \biggl( \frac{3}{2\delta }M_{0,i}^{2}+ \frac{\delta }{6}v_{i}^{2} \biggr)+ \frac{M_{1}}{\sqrt{\lambda }}\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{it} \vert _{H_{0}}^{2}. \end{aligned} \end{aligned}$$
(4.33)

Combining (4.28)–(4.33), we have

$$ \begin{aligned} &2 \bigl(C(\psi ,t\omega ),\xi \bigr)_{H} \\ &\quad \leq \delta \sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \Biggl(\sum_{j=1}^{k}(B_{j}u)_{i}^{2} \Biggr)+ \frac{\lambda \delta }{2}\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)u_{i}^{2}+\delta \sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)v_{i}^{2} \\ &\qquad {}+\frac{k\eta _{0}}{M} \bigl( \Vert u \Vert ^{2}+4 \Vert z \Vert ^{2} \bigr) +\frac{4k+4\lambda }{\delta }\sum_{ \Vert i \Vert \geq M-1}z_{i}^{2} +3 \delta \sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)z_{i}^{2} \\ &\qquad {}+ \frac{4\beta ^{2}}{\lambda \delta }\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)v_{i}^{2}+ \frac{3}{\delta }\sum_{i\in \mathbb{Z}^{k}} \eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)g_{i}^{2} \\ &\qquad {} +\frac{3}{\delta }\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)M_{0,i}^{2}+ \frac{2M_{1}}{\sqrt{\lambda }}\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{it} \vert _{H_{0}}^{2}. \end{aligned} $$
(4.34)

By (4.17), (4.20), (4.27) and (4.34), we get

$$ \begin{aligned} &\frac{d}{dt}\sum _{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{i} \vert _{H}^{2}+ \delta \sum _{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{i} \vert _{H}^{2} \\ &\quad \leq \frac{k\eta _{0}}{M} \bigl((5+8\delta ) \Vert u \Vert ^{2}+4 \Vert v \Vert ^{2}+8 \Vert z \Vert ^{2} \bigr)+\frac{3}{\delta }\sum_{i\in \mathbb{Z}^{k}} \eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)g_{i}^{2} \\ &\qquad {} + \biggl(\frac{4k+4\lambda }{\delta }+3\delta \biggr)\sum _{ \Vert i \Vert \geq M-1}z_{i}^{2}+ \biggl( \frac{4\beta ^{2}}{\lambda \delta }+4\delta + \frac{2\alpha ^{2}\delta }{\lambda }-2\alpha \biggr)\sum _{i\in \mathbb{Z}^{k}} \eta \biggl(\frac{ \Vert i \Vert }{M} \biggr)v_{i}^{2} \\ &\qquad {} +\frac{3}{\delta }\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr)M_{0i}^{2}+ \frac{2M_{1}}{\sqrt{\lambda }}\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{it} \vert _{H_{0}}^{2}+ \frac{2k\eta _{0}}{\sqrt{\lambda }M} \Vert \psi \Vert _{H}^{2}. \end{aligned} $$
(4.35)

It follows from (3.3) that

$$\begin{aligned} &\frac{d}{dt}\sum _{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{i} \vert _{H}^{2}+\delta \sum _{i\in \mathbb{Z}^{k}} \eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{i} \vert _{H}^{2} \\ &\quad \leq \frac{k\eta _{0}}{M} \bigl((5+8\delta ) \Vert u \Vert ^{2}+4 \Vert v \Vert ^{2}+8 \Vert z \Vert ^{2} \bigr)+\frac{3}{\delta }\sum_{i\in \mathbb{Z}^{k}} \eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \bigl(g_{i}^{2}+M_{0i}^{2} \bigr) \\ &\qquad {} + \biggl(\frac{4k+4\lambda }{\delta }+3\delta \biggr)\sum _{ \Vert i \Vert \geq M-1}z_{i}^{2}+ \frac{2M_{1}}{\sqrt{\lambda }}\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{it} \vert _{H_{0}}^{2}+ \frac{2k\eta _{0}}{\sqrt{\lambda }M} \Vert \psi \Vert _{H}^{2} \\ &\quad = h_{1} \Vert \psi \Vert _{H}^{2}+ \frac{8k\eta _{0}}{M} \Vert z \Vert ^{2}+ \frac{3}{\delta } \sum_{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \bigl(g_{i}^{2}+M_{0i}^{2} \bigr) \\ &\qquad {}+h_{2}\sum_{ \Vert i \Vert \geq M-1}z_{i}^{2}+ \frac{2M_{1}}{\sqrt{\lambda }}\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{it} \vert _{H_{0}}^{2}, \end{aligned}$$
(4.36)

where \(h_{1}=\frac{k\eta _{0}}{M}\max \{\frac{5+8\delta }{\lambda },4\}+ \frac{2k\eta _{0}}{\sqrt{\lambda }M}\), \(h_{2}= \frac{4k+4\lambda }{\delta }+3\delta \). One deduces by integrating the above inequality on [0,t] that

$$\begin{aligned} &e^{\delta t}\sum _{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{i} \vert _{H}^{2} \\ &\quad \leq \sum_{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \bigl\vert \psi _{i}(0) \bigr\vert _{H}^{2} +h_{1} \int _{0}^{t}e^{\delta s} \Vert \psi \Vert _{H}^{2}\,ds + \frac{8k\eta _{0}}{M} \int _{0}^{t}e^{\delta s} \Vert z \Vert ^{2}\,ds \\ &\qquad {} + \frac{3}{\delta ^{2}}\sum _{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \bigl(g_{i}^{2}+M_{0i}^{2} \bigr)e^{ \delta t}+h_{2} \int _{0}^{t}e^{\delta s}\sum _{ \Vert i \Vert \geq M-1}z_{i}^{2}\,ds \\ &\qquad {}+ \frac{2M_{1}}{\sqrt{\lambda }} \int _{0}^{t}\sum_{i\in \mathbb{Z}^{k}}e^{ \delta s} \eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{it} \vert _{H_{0}}^{2}\,ds. \end{aligned} $$
(4.37)

Note that \(\|\psi (\tau )\|_{H}\leq \|\psi _{0}\|_{H_{0}}\) for fixed \(\tau \in [-h,0]\), hence we get, for all \(t\geq 0\),

$$ \begin{aligned} &e^{\delta t}\sum _{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \bigl\vert \psi _{i}(t+ \tau ) \bigr\vert _{H}^{2} \\ &\quad \leq e^{-\delta \tau }\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{i0} \vert _{H}^{2} +h_{1}e^{-\delta \tau } \int _{0}^{t}e^{ \delta s} \Vert \psi \Vert _{H}^{2}\,ds +\frac{8k\eta _{0}}{M}e^{-\delta \tau } \int _{0}^{t}e^{\delta s} \Vert z \Vert ^{2}\,ds \\ &\qquad {}+h_{2}e^{-\delta \tau } \int _{0}^{t}e^{\delta s}\sum _{ \Vert i \Vert \geq M-1}z_{i}^{2}\,ds+ \frac{2M_{1}}{\sqrt{\lambda }}e^{-\delta \tau } \int _{0}^{t}\sum_{i \in \mathbb{Z}^{k}}e^{\delta s} \eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{it} \vert _{H_{0}}^{2}\,ds \\ &\qquad {}+\frac{3}{\delta ^{2}}e^{\delta t}\sum_{i\in \mathbb{Z}^{k}} \eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \bigl(M_{0i}^{2}+g_{i}^{2} \bigr), \end{aligned} $$
(4.38)

from which we get

$$ \begin{aligned} &e^{\delta t}\sum _{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{it} \vert _{H_{0}}^{2} \\ &\quad \leq e^{\delta h}\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{i0} \vert _{H_{0}}^{2}+k_{1}e^{\delta t} +k_{2} \int _{0}^{t}e^{ \delta s} \Vert \psi \Vert _{H}^{2}\,ds +k_{3} \int _{0}^{t}e^{\delta s} \Vert z \Vert ^{2}\,ds \\ &\qquad {}+k_{4} \int _{0}^{t}e^{\delta s}\sum _{ \Vert i \Vert \geq M-1}z_{i}^{2}\,ds+c_{3} \int _{0}^{t}\sum_{i\in \mathbb{Z}^{k}}e^{\delta s} \eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{it} \vert _{H_{0}}^{2}\,ds, \end{aligned} $$
(4.39)

where \(k_{1}=\frac{3}{\delta ^{2}}\sum_{i\in \mathbb{Z}^{k}}\eta ( \frac{\|i\|}{M})(M_{0i}^{2}+g_{i}^{2})\), \(k_{2}=h_{1}e^{\delta h}\), \(k_{3}= \frac{8k\eta _{0}}{M}e^{\delta h}\), \(k_{4}=h_{2}e^{\delta h}\). Using the Gronwall inequality yields

$$ \begin{aligned} &e^{\delta t}\sum _{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{it} \vert _{H_{0}}^{2} \\ &\quad \leq e^{\delta h}\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{i0} \vert _{H_{0}}^{2}e^{c_{3}t} +\frac{k_{1}\delta }{\delta -c_{3}} e^{ \delta t} +k_{2}e^{c_{3}t} \int _{0}^{t}e^{(\delta -c_{3})s} \Vert \psi \Vert _{H}^{2}\,ds \\ &\qquad {}+k_{3}e^{c_{3}t} \int _{0}^{t}e^{(\delta -c_{3})s} \Vert z \Vert ^{2}\,ds +k_{4}e^{c_{3}t} \int _{0}^{t}e^{(\delta -c_{3})s}\sum _{ \Vert i \Vert \geq M-1}z_{i}^{2}\,ds. \end{aligned} $$
(4.40)

This implies that

$$ \begin{aligned} &\sum_{i\in \mathbb{Z}^{k}} \eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{it} \vert _{H_{0}}^{2} \\ &\quad \leq e^{\delta h}\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \vert \psi _{i0} \vert _{H_{0}}^{2}e^{-(\delta -c_{3})t} + \frac{k_{1}\delta }{\delta -c_{3}} +k_{2}e^{-(\delta -c_{3})t} \int _{0}^{t}e^{( \delta -c_{3})s} \Vert \psi \Vert _{H}^{2}\,ds \\ &\qquad {} +k_{3}e^{-(\delta -c_{3})t} \int _{0}^{t}e^{(\delta -c_{3})s} \Vert z \Vert ^{2}\,ds +k_{4}e^{-(\delta -c_{3})t} \int _{0}^{t}e^{(\delta -c_{3})s} \sum _{ \Vert i \Vert \geq M-1}z_{i}^{2}\,ds. \end{aligned} $$
(4.41)

Replacing ω with \(\theta _{-t}\omega \), we have

$$ \begin{aligned} &\sum_{i\in \mathbb{Z}^{k}} \eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \bigl\vert \psi _{it} \bigl( \cdot , \theta _{-t}\omega ,\psi _{0}(\theta _{-t}\omega ) \bigr) \bigr\vert _{H_{0}}^{2} \\ &\quad \leq e^{\delta h}\sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \bigl\vert \psi _{i0}(\theta _{-t}\omega ) \bigr\vert _{H_{0}}^{2}e^{-(\delta -c_{3})t} + \frac{k_{1}\delta }{\delta -c_{3}} \\ &\qquad {} +k_{2}e^{-(\delta -c_{3})t} \int _{0}^{t}e^{(\delta -c_{3})s} \bigl\Vert \psi \bigl(s,\theta _{-t}\omega ,\psi _{0}(\theta _{-t}\omega ) \bigr) \bigr\Vert _{H}^{2}\,ds \\ &\qquad {} +k_{3} \int _{-\infty }^{0}e^{(\delta -c_{3})s} \bigl\Vert z( \theta _{s} \omega ) \bigr\Vert ^{2}\,ds +k_{4} \int _{-\infty }^{0}e^{(\delta -c_{3})s}\sum _{ \Vert i \Vert \geq M-1}z_{i}^{2}(\theta _{s}\omega )\,ds. \end{aligned} $$
(4.42)

Next, we estimate the terms of (4.42). Note that \(\|\varphi _{0}(\theta _{-t}\omega )\|^{2}\), \(\max_{\tau \in [-h,0]} \|z(\theta _{-t+\tau }\omega )\|^{2}\) are tempered, we find that, for every \(\varepsilon >0\), there exists some \(T_{1}(\varepsilon ,\omega )>0\) such that, for all \(t\geq T_{1}(\varepsilon ,\omega )\),

$$ \begin{aligned} &e^{\delta h}\sum_{i\in \mathbb{Z}^{k}} \eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \bigl\vert \psi _{i0}( \theta _{-t}\omega ) \bigr\vert _{H_{0}}^{2}e^{-( \delta -c_{3})t} \\ &\quad \leq e^{\delta h} \bigl\Vert \psi _{0}(\theta _{-t}\omega ) \bigr\Vert _{H_{0}}^{2}e^{-( \delta -c_{3})t} \\ &\quad \leq 2e^{\delta h} \Bigl( \bigl\Vert \varphi _{0}(\theta _{-t} \omega ) \bigr\Vert _{H_{0}}^{2}+\max _{\tau \in [-h,0]} \bigl\Vert z(\theta _{-t+\tau }\omega ) \bigr\Vert ^{2} \Bigr)e^{-(\delta -c_{3})t} < \frac{\varepsilon }{4}. \end{aligned} $$
(4.43)

Since \(M_{0}, g\in l^{2}\), there exists \(M_{1}(\varepsilon ,\omega )>0\) such that

$$ \frac{k_{1}\delta }{\delta -c_{3}}< \frac{\varepsilon }{4}, \quad \forall M\geq M_{1}(\varepsilon ,\omega ). $$
(4.44)

In the light of \(\int _{-\infty }^{0}e^{(\delta -c_{3})s}\|z(\theta _{s}\omega )\|^{2}\,ds< \infty \) and the Lebesgue theorem of dominated convergence, there exists some \(M_{2}(\varepsilon ,\omega )>0\) such that, for all \(M\geq M_{2}(\varepsilon ,\omega )\),

$$ k_{3} \int _{-\infty }^{0}e^{(\delta -c_{3})s} \bigl\Vert z( \theta _{s} \omega ) \bigr\Vert ^{2}\,ds +k_{4} \int _{-\infty }^{0}e^{(\delta -c_{3})s}\sum _{ \Vert i \Vert \geq M-1}z_{i}^{2}(\theta _{s}\omega )\,ds< \frac{\varepsilon }{4}. $$
(4.45)

Moreover, it follows from (4.13) that

$$ \begin{aligned} &k_{2}e^{-(\delta -c_{3})t} \int _{0}^{t}e^{(\delta -c_{3})s} \bigl\Vert \psi \bigl(s,\theta _{-t}\omega ,\psi _{0}(\theta _{-t}\omega ) \bigr) \bigr\Vert _{H}^{2}\,ds \\ &\quad \leq k_{2}e^{-(\delta -c_{3})t} \int _{0}^{t}e^{(\delta -c_{3})s} \biggl(e^{\delta h } \bigl\Vert \psi _{0}(\theta _{-t}\omega ) \bigr\Vert _{H_{0}}^{2}e^{-( \delta -c_{3})s}+ \frac{c_{2}\delta }{\delta -c_{3}} \biggr)\,ds \\ &\qquad {} + k_{2}e^{-(\delta -c_{3})t} \int _{0}^{t}e^{(\delta -c_{3})s}c_{1} \int _{-\infty }^{0}e^{(\delta -c_{3})\upsilon } \bigl\Vert z( \theta _{\upsilon } \omega ) \bigr\Vert ^{2}\,d\upsilon \,ds \\ &\quad \leq k_{2}e^{\delta h } \bigl\Vert \psi _{0}( \theta _{-t}\omega ) \bigr\Vert _{H_{0}}^{2}t e^{-(\delta -c_{3})t}+\frac{c_{2}k_{2}\delta }{(\delta -c_{3})^{2}}+ \frac{c_{1}k_{2}}{\delta -c_{3}} \int _{-\infty }^{0}e^{(\delta -c_{3}) \upsilon } \bigl\Vert z( \theta _{\upsilon }\omega ) \bigr\Vert ^{2}\,d\upsilon , \end{aligned} $$
(4.46)

which shows that there exist \(T_{2}(\varepsilon ,\omega )>0\) and \(M_{3}(\varepsilon ,\omega )>0\) such that

$$ \begin{aligned} &k_{2}e^{-(\delta -c_{3})t} \int _{0}^{t}e^{(\delta -c_{3})s} \bigl\Vert \psi \bigl(s,\theta _{-t}\omega ,\psi _{0}(\theta _{-t}\omega ) \bigr) \bigr\Vert _{H}^{2}\,ds< \frac{\varepsilon }{4}, \\ &\quad \forall t\geq T_{2}(\varepsilon ,\omega ), M \geq M_{3}(\varepsilon ,\omega ). \end{aligned} $$
(4.47)

We set

$$ \begin{aligned} &T(\varepsilon ,\omega )=\max \bigl\{ T_{1}( \varepsilon ,\omega ),T_{2}( \varepsilon ,\omega ) \bigr\} , \\ &M_{4}(\varepsilon ,\omega )=\max \bigl\{ M_{1}( \varepsilon ,\omega ),M_{2}(\varepsilon ,\omega ),M_{3}( \varepsilon , \omega ) \bigr\} , \end{aligned} $$
(4.48)

we have

$$ \sum_{i\in \mathbb{Z}^{k}}\eta \biggl( \frac{ \Vert i \Vert }{M} \biggr) \bigl\vert \psi _{it} \bigl( \cdot , \theta _{-t}\omega ,\psi _{0}(\theta _{-t} \omega ) \bigr) \bigr\vert _{H_{0}}^{2}< \varepsilon ,\quad \forall t\geq T(\varepsilon ,\omega ), M\geq M_{4}( \varepsilon , \omega ), $$
(4.49)

from which we get

$$ \begin{aligned} &\max_{\tau \in [-h,0]}\sum _{ \Vert i \Vert \geq 2K} \bigl\vert \varphi _{i} \bigl(t+\tau , \theta _{-t}\omega ,\varphi _{0} ( \theta _{-t}\omega ) \bigr) \bigr\vert _{H}^{2} \\ &\quad \leq 2\max_{\tau \in [-h,0]}\sum_{i\in \mathbb{Z}^{k}} \eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \bigl\vert \psi _{i} \bigl(t+\tau ,\theta _{-t} \omega ,\psi _{0}(\theta _{-t}\omega ) \bigr) \bigr\vert _{H}^{2} \\ &\qquad {}+2 \max_{\tau \in [-h,0]}\sum_{i\in \mathbb{Z}^{k}} \eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \bigl\vert z_{i}(\theta _{\tau }\omega ) \bigr\vert ^{2} \\ &\quad =2\sum_{i\in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \bigl\vert \psi _{it} \bigl( \cdot ,\theta _{-t} \omega ,\psi _{0}(\theta _{-t}\omega ) \bigr) \bigr\vert _{H_{0}}^{2} \\ &\qquad {}+2\max_{\tau \in [-h,0]}\sum _{i\in \mathbb{Z}^{k}} \eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \bigl\vert z_{i}(\theta _{\tau }\omega ) \bigr\vert ^{2}< 4 \varepsilon , \end{aligned} $$
(4.50)

where \(M(\varepsilon ,\omega )=\max \{M_{4}(\varepsilon ,\omega ),M_{5}( \varepsilon ,\omega )\}\), and \(M_{5}(\varepsilon ,\omega )\in N\) leads to

$$ \max_{\tau \in [-h,0]}\sum _{i \in \mathbb{Z}^{k}}\eta \biggl(\frac{ \Vert i \Vert }{M} \biggr) \bigl\vert z_{i}(\theta _{\tau }\omega ) \bigr\vert ^{2}< \varepsilon ,\quad M>M_{5}(\varepsilon ,\omega ). $$
(4.51)

The proof is complete. □

Theorem 4.1

Suppose that Lemma 4.1holds, then the \(\operatorname{SDS}\{S(t,\omega )\}_{t\geq 0,\omega \in \Omega }\) over \((\Omega ,F,P,(\theta _{t})_{t\in R})\) defined by Eq. (3.4) has a unique \(\mathcal{D}\)-random attractor \(\mathcal{A}(\omega )\).

Proof

In the light of Proposition 2.3 and Lemma 4.1, it suffices to prove that, for a.e. \(\omega \in \Omega \), each sequence \(\varphi _{t_{n}}(\cdot ,\theta _{-t_{n}}\omega ,\varphi _{0}(\theta _{-t_{n}} \omega ))=S(t_{n},\theta _{-t_{n}}\omega )\varphi _{0} (\theta _{-t_{n}} \omega )\) has a convergent subsequence in \(H_{0}\) as \(t_{n}\rightarrow \infty \) and \(\varphi _{0}(\theta _{-t_{n}}\omega )\in B_{0}(\theta _{-t_{n}} \omega )\). By (4.14) we have

$$ \bigl\Vert \varphi \bigl({t_{n}}+\tau ,\theta _{-t_{n}} \omega , \varphi _{0}(\theta _{-t_{n}} \omega ) \bigr) \bigr\Vert _{H} \leq \bigl\Vert \varphi _{t_{n}} \bigl(\cdot , \theta _{-t_{n}} \omega ,\varphi _{0}(\theta _{-t_{n}} \omega ) \bigr) \bigr\Vert _{H_{0}}\leq C, \quad \forall \tau \in [-h,0], $$

where \(C>0\) is a given constant. For a fixed \(\tau \in [-h,0]\), we can find a subsequence \(\{\varphi ({t_{n}}+\tau ,\theta _{-t_{n}}\omega ,\varphi _{0}( \theta _{-t_{n}}\omega ))\}\) and \(\mu (\tau )\in H\) such that

$$ \varphi \bigl({t_{n}}+\tau ,\theta _{-t_{n}}\omega ,\varphi _{0}(\theta _{-t_{n}} \omega ) \bigr)\rightarrow \mu (\tau ) \quad \text{weakly in } H, \text{as } n \rightarrow \infty . $$

Next, we prove the above convergence is also strong. We know, for any \(\varepsilon >0\), there exist \(N(\varepsilon ,\omega )\) and \(\hat{M}(\varepsilon ,\omega )\) such that

$$ \sum_{ \Vert i \Vert >\hat{M}(\varepsilon ,\omega )} \bigl\vert \varphi _{i} \bigl({t_{n}}+ \tau ,\theta _{-t_{n}}\omega ,\varphi _{0}(\theta _{-t_{n}}\omega ) \bigr) \bigr\vert ^{2}_{H} < \varepsilon ,\qquad \sum _{ \Vert i \Vert >\hat{M}(\varepsilon ,\omega )} \bigl\vert \mu _{i}( \tau ) \bigr\vert ^{2}_{H}< \varepsilon , $$
(4.52)

and for \(n\geq N(\varepsilon ,\omega )\) we have

$$ \sum_{ \Vert i \Vert \leq \hat{M}(\varepsilon ,\omega )} \bigl\vert \varphi _{i} \bigl({t_{n}}+ \tau ,\theta _{-t_{n}}\omega ,\varphi _{0} (\theta _{-t_{n}}\omega ) \bigr)- \mu _{i}(\tau ) \bigr\vert ^{2}_{H}< \varepsilon . $$
(4.53)

By (4.52) and (4.53), we have

$$ \begin{aligned} & \bigl\Vert \varphi \bigl({t_{n}}+ \tau ,\theta _{-t_{n}}\omega ,\varphi _{0} ( \theta _{-t_{n}}\omega ) \bigr)-\mu (\tau ) \bigr\Vert ^{2}_{H} \\ &\quad \leq \sum_{ \Vert i \Vert >\hat{M}(\varepsilon ,\omega )} \bigl\vert \varphi _{i} \bigl({t_{n}}+ \tau ,\theta _{-t_{n}} \omega ,\varphi _{0} (\theta _{-t_{n}}\omega ) \bigr)- \mu _{i}(\tau ) \bigr\vert ^{2}_{H} \\ &\qquad {} +\sum_{ \Vert i \Vert \leq \hat{M}(\varepsilon ,\omega )} \bigl\vert \varphi _{i} \bigl({t_{n}}+ \tau ,\theta _{-t_{n}} \omega ,\varphi _{0} (\theta _{-t_{n}}\omega ) \bigr)- \mu _{i}(\tau ) \bigr\vert ^{2}_{H}\leq 5 \varepsilon . \end{aligned} $$
(4.54)

This shows that, for any \(\tau \in [-h,0]\), \(\varphi ({t_{n}}+\tau ,\theta _{-t_{n}}\omega ,\varphi _{0} (\theta _{-t_{n}} \omega ))\rightarrow \mu (\tau )\) is strong in H as \(n\rightarrow \infty \). In addition, making use of the integral representation of solutions, we obtain for any \(\tau _{1},\tau _{2}\in [-h,0]\)

$$\begin{aligned} & \bigl\Vert \varphi \bigl({t_{n}+\tau _{1}},\theta _{-t_{n}}\omega , \varphi _{0} ( \theta _{-t_{n}}\omega ) \bigr)-\varphi \bigl({t_{n}+\tau _{2}},\theta _{-t_{n}} \omega , \varphi _{0} (\theta _{-t_{n}}\omega ) \bigr) \bigr\Vert _{H} \\ &\quad \leq \bigl\Vert \psi \bigl({t_{n}+\tau _{1}},\theta _{-t_{n}}\omega ,\varphi _{0} ( \theta _{-t_{n}} \omega ) \bigr)-\psi \bigl({t_{n}+\tau _{2}},\theta _{-t_{n}} \omega ,\varphi _{0} (\theta _{-t_{n}} \omega ) \bigr) \bigr\Vert _{H} \\ &\qquad {}+ \bigl\Vert z(\theta _{ \tau _{1}}\omega )-z(\theta _{\tau _{2}}\omega ) \bigr\Vert \\ &\quad \leq \int _{\tau _{2}}^{\tau _{1}} \bigl( \bigl\Vert D\psi \bigl({t_{n}+\tau },\theta _{-t_{n}} \omega ,\varphi _{0} (\theta _{-t_{n}}\omega ) \bigr) \bigr\Vert _{H}+ \bigl\Vert C\psi \bigl({t_{n}+ \tau },\theta _{-t_{n}}\omega ,\varphi _{0} (\theta _{-t_{n}}\omega ) \bigr) \bigr\Vert _{H} \bigr)\,ds \\ &\qquad {} + \bigl\Vert z(\theta _{\tau _{1}}\omega )-z(\theta _{\tau _{2}} \omega ) \bigr\Vert . \end{aligned}$$
(4.55)

It is obvious that D is a linear operator from H into itself and the operator C is bounded. Hence we have

$$\begin{aligned} &\lim_{ \vert \tau _{1}-\tau _{2} \vert \rightarrow 0} \int _{\tau _{2}}^{\tau _{1}} \bigl( \bigl\Vert D\psi \bigl({t_{n}+\tau },\theta _{-t_{n}} \omega ,\varphi _{0} (\theta _{-t_{n}}\omega ) \bigr) \bigr\Vert _{H}+ \bigl\Vert C\psi \bigl({t_{n}+ \tau },\theta _{-t_{n}}\omega ,\varphi _{0} (\theta _{-t_{n}}\omega ) \bigr) \bigr\Vert _{H} \bigr)\,ds \\ &\quad =0 \end{aligned}$$
(4.56)

and

$$ \lim_{ \vert \tau _{1}-\tau _{2} \vert \rightarrow 0} \bigl\Vert z( \theta _{\tau _{1}}\omega )-z(\theta _{\tau _{2}}\omega ) \bigr\Vert =0. $$
(4.57)

Hence

$$ \lim_{ \vert \tau _{1}-\tau _{2} \vert \rightarrow 0} \bigl\Vert \varphi \bigl({t_{n}+\tau _{1}},\theta _{-t_{n}} \omega ,\varphi _{0} ( \theta _{-t_{n}} \omega ) \bigr)-\varphi \bigl({t_{n}+\tau _{2}},\theta _{-t_{n}} \omega , \varphi _{0} (\theta _{-t_{n}} \omega ) \bigr) \bigr\Vert _{H}=0, $$
(4.58)

which is the required equicontinuity. In view of the Ascoli–Arzelá theorem, we conclude that there exists a subsequence \(\{\varphi _{t_{n_{k}}}(\cdot ,\theta _{-t_{n_{k}}}\omega ,\varphi _{0}( \theta _{-t_{n_{k}}}\omega ))\}\) of \(\{\varphi _{t_{n}}(\cdot ,\theta _{-t_{n}}\omega ,\varphi _{0}( \theta _{-t_{n}}\omega ))\}\) such that

$$ \varphi _{t_{n_{k}}} \bigl(\cdot ,\theta _{-t_{n_{k}}}\omega ,\varphi _{0}( \theta _{-t_{n_{k}}}\omega ) \bigr)\rightarrow \mu (\cdot )\quad \text{strongly in } H_{0}. $$

The proof is complete. □

An upper bound of the Kolmogorov ε-entropy

In this section, we study the upper bound of the Kolmogorov ε-entropy of the global random \(\mathcal{D}\)-attractor \(\mathcal{A}(\omega )\) given by Theorem 4.1.

Theorem 5.1

Under the same conditions of Theorem 4.1, for a.e. \(\omega \in \Omega \),

$$ \begin{aligned} K_{\varepsilon } \bigl(\mathcal{A}( \omega ) \bigr)\leq {}&\bigl(2\hat{M}( \varepsilon ,\omega )+1 \bigr)\ln \biggl( \biggl[ \frac{R_{0}(\omega )}{\sqrt{\lambda }}\cdot \frac{4k+\lambda }{\varepsilon } \cdot \sqrt{2\hat{M}( \varepsilon , \omega )+1} \biggr]+1 \biggr) \\ &{} + \bigl(2\hat{M}(\varepsilon ,\omega )+1 \bigr)\ln \biggl( \biggl[R_{0}( \omega )\cdot \frac{4k+\lambda }{\varepsilon } \cdot \sqrt{2 \hat{M}( \varepsilon ,\omega )+1} \biggr]+1 \biggr), \end{aligned} $$
(5.1)

where \(\hat{M}(\varepsilon ,\omega )\doteq \hat{M}( \frac{\sqrt{4k+\lambda }-\sqrt{2}}{\sqrt{4k+\lambda }}\varepsilon , \omega ,B_{0})\) is the minimal positive integer such that

$$ \sup_{(\varphi _{it})_{i\in \mathbb{Z}^{k}}= \varphi _{t}\in B_{0}} \biggl(\sum_{ \Vert i \Vert >\hat{M}(\varepsilon ,\omega )} \vert \varphi _{it} \vert _{H}^{2} \biggr)^{\frac{1}{2}}\leq \frac{\sqrt{4k+\lambda }-\sqrt{2}}{\sqrt{4k+\lambda }}\varepsilon . $$
(5.2)

Proof

By Lemma 4.1, we obtain \(\mathcal{A}(\omega )=\varphi _{t}(t,\theta _{-t}\omega )\mathcal{A}( \theta _{-t}\omega )\subset B_{0}(\omega )\) for \(t>T(\omega ,\mathcal{A})\). Thus, for any \(\varepsilon >0\) and \(\varphi _{t}=(\varphi _{it})_{i\in \mathbb{Z}^{k}}= ((u_{it},v_{it})_{i \in \mathbb{Z}^{k}} )^{T} =\varphi _{t}(t,\theta _{-t}\omega ) \varphi _{0}(\theta _{-t}\omega )\in \mathcal{A}\), where \(\varphi _{0}(\omega )\in \mathcal{A}(\omega )\subset B_{0}(\omega )\), and by Lemma 4.2 there exists some \(\hat{M}(\varepsilon ,\omega )\doteq \hat{M}( \frac{\sqrt{4k+\lambda }-\sqrt{2}}{\sqrt{4k+\lambda }}\varepsilon , \omega , B_{0})\in \mathbb{N}\) such that (5.2) holds. Next, we decompose φ into two parts as

$$ \varphi _{t}=\varphi _{t}^{(1)}+\varphi _{t}^{(2)}= \bigl(\varphi ^{(1)}_{it} \bigr)_{i \in \mathbb{Z}^{k}}+ \bigl( \varphi ^{(2)}_{it} \bigr)_{i\in \mathbb{Z}^{k}}, $$
(5.3)

where

$$ \varphi ^{(1)}_{it}=( \varsigma _{it},\mu _{it})^{T}= \textstyle\begin{cases} \varphi _{it}, & \Vert i \Vert \leq \hat{M}(\varepsilon ,\omega ); \\ 0, & \Vert i \Vert >\hat{M}(\varepsilon ,\omega ); \end{cases} $$
(5.4)

and

$$\begin{aligned} \varphi ^{(2)}_{it}=( \varsigma _{it},\mu _{it})^{T}= \textstyle\begin{cases} 0, & \Vert i \Vert \leq \hat{M}(\varepsilon ,\omega ); \\ \varphi _{it}, & \Vert i \Vert >\hat{M}(\varepsilon ,\omega ). \end{cases}\displaystyle \end{aligned}$$
(5.5)

By (5.2)–(5.5) we get

$$\begin{aligned}& \bigl\Vert \varphi _{t}^{(2)} \bigr\Vert _{H}= \biggl(\sum_{ \Vert i \Vert >\hat{M}( \varepsilon ,\omega )} \vert \varphi _{it} \vert _{H}^{2} \biggr)^{\frac{1}{2}} \leq \frac{\sqrt{4k+\lambda }-\sqrt{2}}{\sqrt{4k+\lambda }}\varepsilon , \\& \bigl\Vert \varphi _{t}^{(1)} \bigr\Vert _{H}^{2}=\sum_{ \Vert i \Vert \leq \hat{M}( \varepsilon ,\omega )} \bigl\vert \varphi _{it}^{(1)} \bigr\vert _{H}^{2}=\sum_{ \Vert i \Vert \leq \hat{M}(\varepsilon ,\omega )} \Biggl(\sum_{j=1}^{k}(B_{j} \varsigma _{t})_{i}^{2}+ \lambda \varsigma _{it}^{2}+\mu _{it}^{2} \Biggr) \leq \Vert \varphi _{t} \Vert _{H_{0}}^{2} \leq R_{0}^{2}(\omega ), \end{aligned}$$

this implies that

$$ \vert \varsigma _{it} \vert \leq \frac{R_{0}(\omega )}{\sqrt{\lambda }}, \qquad \vert \mu _{it} \vert \leq R_{0}( \omega ),\quad \forall \Vert i \Vert \leq \hat{M}(\varepsilon , \omega ). $$

Consider the regular polyhedron

$$ \Upsilon _{1}= \biggl\{ \varsigma _{t}=(\varsigma _{it})_{ \Vert i \Vert \leq \hat{M}(\varepsilon ,\omega )}:\varsigma _{it}\in \mathbb{R}, \vert \varsigma _{it} \vert \leq \frac{R_{0}(\omega )}{\sqrt{\lambda }} \biggr\} \subset \mathbb{R}^{2\hat{M}(\varepsilon ,\omega )+1}, $$

by Lemma 2.1 we see that \(\Upsilon _{1}\) can be covered by

$$ \mathcal{N}_{\varepsilon ,\omega }^{(1)}(\Upsilon _{1})= \biggl( \biggl[\frac{R_{0}(\omega )}{\sqrt{\lambda }}\cdot \frac{4k+\lambda }{\varepsilon } \cdot \sqrt{2\hat{M}(\varepsilon , \omega )+1} \biggr]+1 \biggr)^{2\hat{M}(\varepsilon ,\omega )+1} $$

balls in \(\mathbb{R}^{2\hat{M}(\varepsilon ,\omega )+1}\) with radii \(\frac{\varepsilon }{4k+\lambda } \). Next, we study the other regular polyhedron

$$ \begin{aligned} \Upsilon _{2}= \bigl\{ \mu _{t}=(\mu _{it})_{ \Vert i \Vert \leq \hat{M}( \varepsilon ,\omega )}: \mu _{it}\in \mathbb{R}, \vert \mu _{it} \vert \leq R_{0}( \omega ) \bigr\} \subset \mathbb{R}^{2\hat{M}(\varepsilon ,\omega )+1}, \end{aligned} $$

it can be covered by

$$ \mathcal{N}_{\varepsilon ,\omega }^{(2)}(\Upsilon _{2})= \biggl( \biggl[R_{0}(\omega )\cdot \frac{4k+\lambda }{\varepsilon } \cdot \sqrt{2\hat{M}(\varepsilon ,\omega )+1} \biggr]+1 \biggr)^{2\hat{M}( \varepsilon ,\omega )+1} $$

balls in \(\mathbb{R}^{2\hat{M}(\varepsilon ,\omega )+1}\) with radii \(\frac{\varepsilon }{4k+\lambda } \). Therefore, the polyhedron

$$ \begin{aligned} \Upsilon &=\Upsilon _{1}\times \Upsilon _{2}= \biggl\{ \varphi _{t}^{(1)}=(\varsigma _{it},\mu _{jt})_{ \Vert i \Vert , \Vert j \Vert \leq \hat{M}(\varepsilon ,\omega )}: \vert \varsigma _{it} \vert \leq \frac{R_{0}(\omega )}{\sqrt{\lambda }}, \vert \mu _{it} \vert \leq R_{0}(\omega ) \biggr\} \\ & \subset \mathbb{R}^{2\hat{M}(\varepsilon ,\omega )+1}\times \mathbb{R}^{2\hat{M}(\varepsilon ,\omega )+1} \end{aligned} $$

can be covered by

$$ \begin{aligned} \mathcal{N}_{\varepsilon ,\omega }(\Upsilon )&= \mathcal{N}_{\varepsilon ,\omega }^{(1)}(\Upsilon _{1}) \times \mathcal{N}_{\varepsilon ,\omega }^{(2)}(\Upsilon _{2}) \\ &= \biggl( \biggl[ \frac{R_{0}(\omega )}{\sqrt{\lambda }}\cdot \frac{4k+\lambda }{\varepsilon } \cdot \sqrt{2\hat{M}( \varepsilon , \omega )+1} \biggr]+1 \biggr)^{2\hat{M}(\varepsilon ,\omega )+1} \\ &\quad {} \times \biggl( \biggl[R_{0}(\omega )\cdot \frac{4k+\lambda }{\varepsilon } \cdot \sqrt{2\hat{M}(\varepsilon , \omega )+1} \biggr]+1 \biggr)^{2\hat{M}(\varepsilon ,\omega )+1} \end{aligned} $$

balls in \(\mathbb{R}^{2\hat{M}(\varepsilon ,\omega )+1}\times \mathbb{R}^{2 \hat{M}(\varepsilon ,\omega )+1}\) with radii \(\frac{\sqrt{2}\varepsilon }{4k+\lambda }\). Let the centers of those balls be

$$ \varphi ^{\ast }_{\iota t }= \bigl( \varsigma _{\iota i t}^{\ast }, \mu _{\iota jt}^{\ast } \bigr)_{\|i\|,\|j\| \leq \hat{M}(\varepsilon , \omega )}\subset \mathbb{R}^{2\hat{M}(\varepsilon ,\omega )+1}\times \mathbb{R}^{2\hat{M}(\varepsilon ,\omega )+1}, $$

where \(\iota =1,2,\ldots ,\mathcal{N}_{\varepsilon ,\omega }(\Upsilon )\). We choose

$$ \hat{\varphi }_{\iota t}=(\hat{\varphi }_{\iota ijt})_{i,j\in { \mathbb{Z}}^{k}}= \textstyle\begin{cases} \varphi _{\iota t}^{\ast },& \max \{ \Vert i \Vert , \Vert j \Vert \}\leq \hat{M}( \varepsilon ,\omega ); \\ 0, & \max \{ \Vert i \Vert , \Vert j \Vert \}>\hat{M}(\varepsilon ,\omega ). \end{cases} $$

Then there exists some ι (\(1\leq \iota \leq \mathcal{N}_{\varepsilon ,\omega }(\Upsilon )\)) such that

$$ \begin{aligned} \bigl\Vert \varphi _{t}^{(1)}- \hat{\varphi }_{\iota t} \bigr\Vert _{H}&\leq \sqrt{4k+ \lambda } \bigl\Vert (\varphi _{it})_{ \Vert i \Vert \leq \hat{M}(\varepsilon , \omega )}-\varphi _{\iota t}^{\ast } \bigr\Vert _{\mathbb{R}^{2\hat{M}( \varepsilon ,\omega )+1} \times \mathbb{R}^{2\hat{M}(\varepsilon , \omega )+1}} \\ & \leq \sqrt{4k+\lambda }\cdot \frac{\sqrt{2}}{4k+\lambda } \varepsilon = \frac{\sqrt{2}}{\sqrt{4k+\lambda }}\varepsilon . \end{aligned} $$

Thus, for any \(\varphi \in \mathcal{A}(\omega )\subset \mathcal{B}_{0}(\omega )\), there exists some ι (\(1\leq \iota \leq \mathcal{N}_{\varepsilon ,\omega }(\Upsilon )\)) such that

$$ \begin{aligned} \Vert \varphi _{t}-\hat{\varphi }_{\iota t} \Vert _{H}&= \bigl\Vert \varphi _{t}^{(1)}+ \varphi _{t}^{(2)}- \hat{\varphi }_{\iota t} \bigr\Vert _{H}\leq \bigl\Vert \varphi _{t}^{(2)} \bigr\Vert _{H}+ \bigl\Vert \varphi _{t}^{(1)}-\hat{\varphi }_{\iota t} \bigr\Vert _{H} \\ & \leq \frac{\sqrt{2}}{\sqrt{4k+\lambda }}\varepsilon + \frac{\sqrt{4k+\lambda }-\sqrt{2}}{\sqrt{4k+\lambda }}\varepsilon = \varepsilon , \end{aligned} $$

which means that the global random \(\mathcal{D}\)-attractor \(\mathcal{A}(\omega )\subset H\) can be covered by \(\mathcal{N}_{\varepsilon ,\omega }(\Upsilon )\) balls centered at \(\hat{\varphi }_{\iota t}\), \(\iota =1,2,\ldots , \mathcal{N}_{\varepsilon ,\omega }(\Upsilon )\), with radii ε. The proof is complete. □

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Acknowledgements

The authors would like to thank anonymous referees and editors for their valuable comments and constructive suggestions.

Funding

This work is supported by the Science and Technology Foundation of Guizhou Province ([2020]1Y007), the Natural Science Foundation of Education of Guizhou Province (KY[2019]139, KY[2019]143)) and School level Foundation of Liupanshui Normal University (LPSSYKJTD201907).

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Li, X., She, L. & Shan, Z. Random attractor for second-order stochastic delay lattice sine-Gordon equation. Bound Value Probl 2021, 14 (2021). https://doi.org/10.1186/s13661-021-01489-7

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MSC

  • 60H15
  • 34B35
  • 58F11
  • 58F36

Keywords

  • Stochastic lattice dynamical systems
  • Asymptotically compact
  • Random attractor
  • Kolmogorov ε-entropy