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Random exponential attractor for second order non-autonomous stochastic lattice dynamical systems with multiplicative white noise in weighted spaces

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Abstract

This paper is concerned with the random exponential attractor for second order non-autonomous stochastic lattice system with multiplicative white noise and unbounded nonlinearity. Firstly, we transfer the stochastic lattice system into a random lattice system without noise term whose solutions generate a continuous cocycle on a weighted space of infinite sequences. Then we present some sufficient conditions for the existence of a random exponential attractor for a continuous cocycle in the product weighted space of sequences, which improved the existing conditions. Finally, we prove the existence of a random exponential attractor for the considered system in weighted space of sequences.

Introduction

It is well known that the dynamics of a random dynamical system can be determined by the random attractor. However, the speed of a random attractor attracting orbits is sometimes relatively slow and the dimension of a random attractor is maybe infinite [1]. To this end, Shirikyan and Zelik [2] introduced the random exponential attractor (a positively invariant measurable set with a finite fractal dimension and attracting any trajectory exponentially) for an autonomous random dynamical system. The random exponential attractor (if it exists) contains the random attractor with finite fractal dimension, which implies that the dynamics of a random dynamical system can be described by finite independent parameters. Recently, Zhou in [3] established the existence of a random exponential attractor for a continuous cocycle (non-autonomous random dynamical system) on a separable Banach space and the first order stochastic lattice system driven by linear multiplicative white noise.

In this paper, we consider the following second order non-autonomous stochastic lattice system with multiplicative white noise and initial conditions: for every \(\tau\in\mathbb{R}\) and \(t>\tau\),

$$ \textstyle\begin{cases} \ddot{u}_{i}+\gamma(A\dot{u})_{i}+\alpha\dot{u}_{i}+(Au)_{i}+\lambda _{i}u_{i}+f_{i}(u_{i},t)=g_{i}(t)+cu_{i}\circ\dot{W}(t),\quad i\in \mathbb{Z}, \\ u_{i}(\tau)=u_{0,i\tau}, \qquad \dot{u}_{i}(\tau)=u_{1,i\tau},\quad i\in \mathbb{Z}, \end{cases} $$
(1)

where \(i\in\mathbb{Z}\), \(u=(u_{i})_{i\in\mathbb{Z}}\), \(\dot{u}=(\dot {u}_{i})_{i\in\mathbb{Z}}\), α, \(c>0\), \(\gamma\geq0\), \(\lambda _{i}>0\), \(u_{i}\), \(g_{i}(t)\), \(f_{i}(u_{i},t)\in\mathbb{R}\), and A is a non-negative linear coupled operator; \(W(t)\) is a two-sided real-valued Wiener process on a probability space \((\varOmega,\mathcal{F},\mathbb{P})\), where \(\varOmega=\{\omega\in C(\mathbb{R},\mathbb{R}):\omega(0)=0\}\), the Borel σ-algebra \(\mathcal{F}\) on Ω is generated by the compact open topology, and \(\mathbb{P}\) is the Wiener measure on \((\varOmega,\mathcal{F})\) [4]; “” in (1) denotes the Stratonovich sense in the stochastic term.

For the second order autonomous stochastic lattice system (1) (\(f_{i}(u_{i},t)=f_{i}(u_{i})\), \(g_{i}(t)=g_{i}\) independent of t) with multiplicative and additive white noise, the existence and upper semi-continuity of the random attractor have been studied by some authors; see [5,6,7,8,9,10,11,12]. But as is well known, there is no result concerning the dimension of random attractor and existence of random exponential attractors for second order autonomous and non-autonomous stochastic lattice system (1). Based on the ideas of [3, 5, 9], in this paper, we aim to prove that under certain conditions, the system (1) possesses a random exponential attractor in weighted spaces of sequences, which implies that the corresponding autonomous and non-autonomous stochastic systems (1) have the random attractors with finite fractal dimension.

Random exponential attractor for second order non-autonomous stochastic lattice system

Let ρ: \(\mathbb{Z}\rightarrow\mathbb{(}0,+\infty)\), \(\rho _{i}=\rho(i)\) for \(i\in\mathbb{Z}\) be a positive weight function. Write

$$ l_{\rho}^{2}= \biggl\{ u=(u_{i})_{i\in\mathbb{Z}}:u_{i} \in\mathbb{R},\sum_{i\in\mathbb{Z}} \rho_{i} \vert u_{i} \vert ^{2}< \infty \biggr\} $$

with inner product \((u,v)_{\rho}=\sum_{i\in\mathbb{Z}}\rho_{i}u_{i}v_{i}\) and norm \(\Vert v\Vert_{\rho}^{2}=(v,v)_{\rho}\) for \(u=(u_{i})_{i\in \mathbb{Z}}\), \(v=(v_{i})_{i\in\mathbb{Z}}\in l_{\rho}^{2}\).

Consider the system (1), which can be written in the following vector form:

$$ \textstyle\begin{cases} \ddot{u}+\gamma A\dot{u}+\alpha\dot{u}+Au+\lambda u+f(u,t)=g(t)+cu\circ \dot{W}(t),\quad t>\tau, \\ u(\tau)=u_{\tau}, \qquad \dot{u}(\tau)=u_{1\tau},\quad \tau\in\mathbb{R},\end{cases} $$
(2)

where \(A\dot{u}=((A\dot{u})_{i})_{i\in\mathbb{Z}}\), \(Au=(Au_{i})_{i\in \mathbb{Z}}\), \(\lambda u=(\lambda_{i}u_{i})_{i\in\mathbb{Z}}\), \(f(u,t)=(f_{i}(u_{i},t))_{i\in\mathbb{Z}}\), \(g(t)=(g_{i}(t))_{i\in \mathbb{Z}}\), \(u\circ\dot{W}(t)=(u_{i}\circ\dot{W}(t))_{i\in\mathbb{Z}}\). The operator A has a decomposition \(A=\overline{D}D=D\overline{D}\), where D is a bounded linear operator defined by

$$ (Du)_{i}=\sum_{l=-m_{0}}^{m_{0}}d_{l}u_{i+l}, \quad \vert d_{l} \vert \leq c_{0}\ (\mathrm{constant}),\forall u=(u_{i})_{i\in\mathbb{Z}}, m_{0} \in\mathbb{N}, $$

is the adjoint of D in \(l^{2}=l_{\rho(i)\equiv 1,\forall i\in\mathbb{Z}}^{2}\).

Let \(i\in\mathbb{Z}\), \(G_{i}(s,t)=\int_{0}^{s}f_{i}(r,t)\,dr\), \(\overline {G}(u,t)=\sum_{i\in\mathbb{Z}}\rho_{i}G_{i}(u_{i},t)\) and we make the following assumptions on \(\rho_{i}\), \(f_{i}\), \(g_{i}\), \(\lambda_{i}\):

  1. (A1)

    \(0<\bar{\rho}\leq\rho_{i}=\rho(i)\leq a_{0}<+\infty\), \(c_{1}\rho(i)\leq\rho(i\pm1)\leq c_{2}\rho(i)\), \(|\rho(i\pm 1)-\rho (i)|\leq c_{3}\rho(i)\), \(\forall i\in\mathbb{Z}\), for some positive constants ρ̄, \(a_{0}\), \(c_{1}\), \(c_{2}\), \(c_{3}\).

  2. (A2)

    \(\forall i\in\mathbb{Z}\), t, \(s\in\mathbb{R}\), \(f_{i}\), \(f_{i,s}^{\prime}\in C(\mathbb{R}\times\mathbb{R}, \mathbb{R})\), and there exist positive constants \(a_{1}\), \(a_{2}\), \(a_{3}\), \(a_{4}\), \(\underline {\lambda}\), \(\bar{\lambda}>0\) and functions \(\beta_{i}\in C^{1}(\mathbb {R}, \mathbb{R})\) such that

    $$ \textstyle\begin{cases} f_{i}(0,t)=0,\qquad \vert f_{i,s}^{\prime}(s,t) \vert \leq a_{1} \vert s \vert (1+ \vert s \vert ^{p-1}),\quad p>1, \\ a_{2}f_{i}(s,t)s\geq G_{i}(s,t)\geq a_{3} \vert s \vert ^{p+2}-\beta _{i}^{2}(t), \\ G_{i,t}^{\prime}(s,t)\leq a_{4}G_{i}(s,t),\quad 0\leq a_{4}\leq\frac {\varepsilon }{2a_{2}}, \\ \underline{\lambda}\leq\lambda_{i}\leq\bar{\lambda}< +\infty.\end{cases} $$
  3. (A3)

    \(\forall t\in\mathbb{R}\), \(\beta(t)= ( \beta _{i}(t) ) _{i\in\mathbb{Z}}, \beta^{\prime}(t)= ( \beta _{i}^{\prime}(t) ) _{i\in\mathbb{Z}}, g(t)= ( g_{i}(t) ) _{i\in\mathbb{Z}}\in{\mathbf{G}}\), where

    $$ {\mathbf{G}}= \biggl\{ g\in C_{b} \bigl( \mathbb{R},l_{\rho}^{2} \bigr) : \forall\eta>0,\exists I(\eta)\in\mathbb{N}\text{ such that } { \sup}_{t\in\mathbb{R}}\sum_{ \vert i \vert >I(\eta)}\rho _{i}g_{i}^{2}(t)< \eta \biggr\} , $$

    and \(C_{b}(\mathbb{R},l_{\rho}^{2})\) is the space of all continuous bounded functions from \(\mathbb{R}\) into \(l_{\rho}^{2}\).

(A4):

\(c_{3}\) in (A1) satisfies

$$ 0\leq c_{3}\leq\min \biggl\{ \frac{\varepsilon}{2c_{0}c_{4}},\frac {\alpha}{2c_{0}c_{4}c_{2}^{m_{0}}(2m_{0}+1)^{2}(\delta+\gamma)}, \frac {2}{c_{0}c_{4}} \biggr\} , $$

where

$$\begin{aligned} &c_{4} =c_{2}^{m_{0}-1}+c_{2}^{m_{0}-2}+ \cdots+c_{2}+1, \\ &0 < \varepsilon= \textstyle\begin{cases} \frac{\alpha\underline{\lambda}}{2\alpha^{2}+3\underline{\lambda}}, & \gamma=0, \\ \min \{ \frac{1}{\gamma},\frac{\alpha\underline{\lambda }}{2\alpha ^{2}+3\underline{\lambda}} \}, & \gamma\neq0.\end{cases}\displaystyle \end{aligned}$$

For any u, \(v\in l_{\rho}^{2}\), define new inner products on \(l_{\rho }^{2}\) by

$$\begin{aligned} &(u,v)_{\lambda,\rho}=\sum_{i\in\mathbb{Z}} \rho_{i}\lambda _{i}u_{i}v_{i}, \qquad \Vert u \Vert _{\lambda,\rho}^{2}=\sum _{i\in\mathbb{Z}}\rho_{i}\lambda_{i}u_{i}^{2}, \\ &(u,v)_{\delta,\lambda,\rho}=\delta(Du,Dv)_{\rho}+(u,v)_{\lambda ,\rho },\\ & \Vert u \Vert _{\delta,\lambda,\rho}^{2}=\delta \Vert Du \Vert _{\rho}^{2}+ \Vert u \Vert _{\lambda,\rho}^{2},\quad \delta =1-\varepsilon \gamma\in{}[0,1], \end{aligned}$$

then the norms \(\Vert \cdot \Vert _{\rho}\), \(\Vert \cdot \Vert _{\lambda,\rho}\) and \(\Vert \cdot \Vert _{\delta,\lambda,\rho}\) are equivalent to each other because

$$ \underline{\lambda} \Vert u \Vert _{\rho}^{2}\leq \Vert u \Vert _{\lambda ,\rho}^{2}\leq \Vert u \Vert _{\delta,\lambda,\rho }^{2}\leq\frac{\delta \bar{\lambda}c_{0}^{2}c_{2}^{m_{0}}(2m_{0}+1)^{2}}{\underline{\lambda}} \Vert u \Vert _{\rho}^{2}. $$

Let \(l_{\delta,\lambda,\rho}^{2}= ( l_{\rho}^{2},(\cdot,\cdot )_{\delta,\lambda,\rho}, \Vert \cdot \Vert _{\delta ,\lambda,\rho} ) \), \(H=l_{\delta,\lambda,\rho}^{2}\times l_{\rho}^{2}\), then H is a separable Hilbert space.

Let \((\varOmega,\mathcal{F},\mathbb{P})\) be defined in Sect. 1. Define a family of mappings \((\theta_{t})_{t\in\mathbb{R}}\) on Ω: \(\theta_{t}\omega (\cdot)=\omega(\cdot+t)-\omega(t)\) for \(\omega\in\varOmega\), then \((\varOmega,\mathcal{F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})\) is an ergodic metric dynamical system [4]. Let us consider the Ornstein–Uhlenbeck stationary process \(z(\theta_{t}\omega)=-\alpha\int_{-\infty}^{0}e^{\alpha s}(\theta_{t}\omega)(s)\,ds\) for \(t\in\mathbb{R}\) and \(\omega\in \varOmega\), which solves the Itô stochastic equation \(dz(\theta_{t}\omega )+\alpha z(\theta_{t}\omega)\,dt=dW(t,\omega)\), where \(W(t,\omega)=\omega (t)\). It is well known that the random variable \(|z(\omega)|\) is tempered and there exists \(\varOmega_{0}\subseteq\varOmega\) with \(\mathbb{P}(\varOmega_{0})=1\), such that for every \(\omega\in\varOmega_{0}\), \(t\mapsto z(\theta_{t}\omega)\) is continuous in t and

$$ \lim_{t\rightarrow\pm\infty}\frac{ \vert z(\theta _{t}\omega) \vert }{t}=\lim _{t \rightarrow\pm\infty}\frac{\int_{0}^{t}z(\theta _{s}\omega)\,ds}{t}=0. $$
(3)

For convenience, we still write \(\varOmega_{0}\) as Ω.

Introduce a variable transformation\(v=u_{t}+\varepsilon u-cuz(\theta _{t}\omega)\), then the problem (2) is equivalent to the following random system:

φ ˙ +Lφ=F(φ, θ t ω), φ τ (ω)= ( u τ v τ ( ω ) ) = ( u τ u 1 , τ + ε u τ c u τ z ( θ τ ω ) ) ,
(4)

where

φ(t,τ,ω)= ( u v ) ,L= ( ε 1 λ + A ε ( α ε ) γ ε A α ε + γ A ) I,
(5)
F(φ, θ t ω)= ( c u z ( θ t ω ) [ 2 ε c z ( θ t ω ) c 2 z 2 ( θ t ω ) ] u c z ( θ t ω ) v γ c z ( θ t ω ) A u f ( u , t ) + g ( t ) ) .
(6)

It follows from assumptions (A1)–(A4) and theory of ordinary differential equations that, for every \(\omega\in\varOmega\), \(\tau\in\mathbb{R}\) and φ τ (ω)= ( u τ v τ ( ω ) ) H, the problem (4) has a unique solution \(\varphi (\cdot,\tau,\omega,\varphi_{\tau})\in C([\tau,\tau+T),H)\) for any \(T>0\), where \(\varphi(\tau,\tau,\omega,\varphi_{\tau})=\varphi _{\tau } \) and \(\varphi(t,\tau,\omega,\varphi_{\tau})\) is continuous in \(\varphi_{\tau}\in H\), which defines a continuous cocycle Φ: \(\mathbb{R}^{+}\times\mathbb{R}\times\varOmega\times H\rightarrow H\) by

$$ (t,\tau,\omega,\varphi_{\tau})\rightarrow\varPhi(t,\tau,\omega ,\varphi _{\tau})=\varPhi(t,\tau,\omega)\varphi_{\tau}=\varphi \bigl(t+ \tau,\tau ,\theta_{-\tau}\omega,\varphi_{\tau}( \theta_{-\tau}\omega) \bigr) $$

over \(\mathbb{R}\) and \((\varOmega,\mathcal{F},\mathbb{P},(\theta _{t})_{t\in \mathbb{R}})\) with state space H.

Random exponential attractor

In this subsection, we present the definition and existence conditions of a random exponential attractor for the continuous cocycle \(\{\varPhi(t,\tau ,\omega)\}_{t\geq0,\omega\in\varOmega,\tau\in\mathbb{R}}\) in H. Let \(\mathcal{D}(H)\) be the collection of all the tempered families of nonempty subsets of H [3]. A family \(D=\{D(\tau,\omega)\subset H\} _{\tau \in\mathbb{R},\omega\in\varOmega}\) of nonempty subsets of H is said to be tempered with respective to \((\theta_{t})_{t\in\mathbb{R}}\) if, for every \(\epsilon>0, \tau\in\mathbb{R}\), and a.e. \(\omega\in\varOmega\), \(\lim_{t\rightarrow\infty}e^{-\epsilon|t|}\Vert D(\tau+t,\theta _{t}\omega)\Vert_{H}=0\), where \(\Vert D(\tau,\omega)\Vert _{H}=\sup_{\varphi\in D(\tau,\omega)}\Vert\varphi\Vert_{H}\).

Definition 2.1

([3])

A family \(\{\mathcal{A}(\tau,\omega)\}_{\tau\in\mathbb{R},\omega\in\varOmega}\) of subsets of H is called a random exponential attractor in \(\mathcal{D}(H)\) for a continuous cocycle \(\{\varPhi(t,\tau,\omega)\}_{t\geq0,\tau\in\mathbb {R},\omega\in\varOmega}\) on H over \(\mathbb{R}\) and \((\varOmega,\mathcal {F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})\) if, for any \(\tau\in \mathbb{R}\) and \(\omega\in\varOmega\), (i) \(\mathcal{A}(\tau,\omega)\) is compact subset in H and measurable in ω; (ii) \(\varPhi(t,\tau-t,\theta _{-t}\omega )\mathcal{A}(\tau-t,\theta_{-t}\omega)\subseteq\mathcal{A}(\tau ,\omega ) \) for all \(t\geq0\); (iii) there exists a random variable \(\zeta _{\omega } \) \((<\infty)\) such that \(\sup_{\tau\in\mathbb{R}}\dim_{f}\mathcal {A}(\tau,\omega)\leq\zeta_{\omega}<\infty\), where \(\dim_{f}\mathcal {A}(\tau,\omega)=\lim\sup_{\varepsilon\rightarrow0^{+}}\frac{\ln N_{\varepsilon}(\mathcal{A}(\tau,\omega))}{-\ln\varepsilon}\) is the fractal dimension of \(\mathcal{A}(\tau,\omega)\) and \(N_{\varepsilon}( \mathcal{A}(\tau,\omega))\) is the minimal number of balls with radius ε covering \(\mathcal{A}(\tau,\omega)\) in H; (iv) there exists a constant \(\tilde{a}>0\) such that, for any \(D\in\mathcal{D}(H)\), there exist random variables \(t_{D}(\tau,\omega)\geq0\), \(Q(\tau ,\omega ,D)>0\) satisfying \(\mathrm{d}_{h}(\varPhi(t,\tau-t,\theta_{-t}\omega )D(\tau -t,\theta_{-t}\omega),\mathcal{A}(\tau,\omega))\leq Q(\tau,\omega ,D)e^{-\widetilde{a}t}\) for all \(t\geq t_{D}(\tau,\omega)\), where \({d}_{h}\) denotes the Hausdorff semidistance between two subsets.

Based on Theorem 2.8 in [13] and Theorems 2.1–2.4 in [3], by making some slightly revision in the proof, we have the following theorem.

Theorem 2.1

Consider the continuous cocycle \(\{\varPhi(t,\tau,\omega)\}_{t\geq0,\tau\in\mathbb{R},\omega\in \varOmega}\). Assume that:

  1. (B1)

    There exist a family of uniformly (with respect to \(\tau\in\mathbb{R}\)) tempered closed measurable set \(D_{0}=\{D_{0}(\omega)\}_{\tau\in\mathbb{R},\omega\in\varOmega}\in \mathcal{D}(H)\) satisfying that, for any \(\tau\in\mathbb{R}\), \(\omega\in\varOmega\) and \(D(\tau,\omega)\in\mathcal{D}(H)\), there exists a \(t_{D}(\tau,\omega)\geq0\) such that \(\varPhi(t,\tau-t,\omega)D(\tau-t,\theta_{-t}\omega)\subseteq D_{0}(\omega)\) for all \(t\geq t_{D}(\tau,\omega)\). Particularly, there exists a \(t_{D_{0}}(\omega)\geq0\) (independent of τ) such that \(\varPhi(t,\tau-t,\omega )D_{0}(\theta_{-t}\omega)\subseteq D_{0}(\omega)\) for all \(t\geq t_{D_{0}}(\omega)\). For any \(\omega\in\varOmega\), \(\tau\in\mathbb{R}\), set

    $$ \mathcal{X}(\tau,\omega)=\bigcup_{t\geq t_{D_{0}}(\omega)}{ \varPhi(t, \tau-t,\theta_{-t}\omega)}D_{0}( \theta_{-t} \omega)\subseteq D_{0}(\omega). $$
  2. (B2)

    There exist some positive numbers \(\hat{\lambda}>0\), \(\hat{\delta}>0\), k \((\in\mathbb{N})\) (independent of \((\tau,\omega)\), but δ̂, k maybe depend on λ̂) and random variables \(C_{0}(\omega)\), \(C_{1}(\omega)\geq0\) such that for any \(\tau\in\mathbb{R}\), \(\omega\in\varOmega\) and any u, \(v\in\mathcal{X}(\tau,\omega)\subset H\), we have

    $$\begin{aligned} & \bigl\Vert \varPhi (t,\tau,\omega )u-\varPhi (t,\tau,\omega )v \bigr\Vert _{H}\leq e^{\int_{0}^{t}C_{0}(\theta_{s}\omega )\,ds} \Vert u-v \Vert _{H},\quad \forall t\in \biggl[0,\frac{8\ln2}{\hat {\lambda}} \biggr], \\ &\sum_{ \vert i \vert >k}\rho_{i} \biggl(\varPhi \biggl(\frac{8\ln2}{\hat {\lambda}},\tau,\omega \biggr)u-\varPhi \biggl(\frac{8\ln2}{\hat{\lambda}}, \tau,\omega \biggr)v \biggr)_{i}^{2} \\ &\quad\leq \biggl(e^{-8\ln 2+\int_{0}^{\frac{8\ln2}{\hat{\lambda}}}C_{1}(\theta_{s}\omega )\,ds}+\frac{\hat{\delta}}{2}e^{\int_{0}^{\frac{8\ln2}{\hat{\lambda}}}C_{0}(\theta _{s}\omega)\,ds} \biggr)^{2} \Vert u-v \Vert _{H}^{2} \end{aligned}$$

    and

    $$ \textstyle\begin{cases} 0\leq\mathbf{E[}C_{1}(\omega)]\leq\frac{\hat{\lambda}}{16},\quad 0\leq \mathbf{E}[C_{0}^{2}(\omega)]< \infty, \\ 0< \hat{\delta}\leq\min \{ \frac{1}{16},e^{-\frac{128\ln ^{2}2}{\hat{\lambda}^{2}\ln\frac{3}{2}} ( \mathbf{E[}C_{0}^{2}(\omega)]+\hat{\lambda}\mathbf{E[}C_{0}(\omega)] ) } \},\end{cases} $$

    where E denotes the expectation.

  3. (B3)
    $$\textstyle\begin{cases} \lim_{t\rightarrow0^{+}}\sup_{u\in\mathcal{X}(\tau,\omega)} \Vert \varPhi (t,\tau,\omega)u-u \Vert _{H}=0, \\ \lim_{t\rightarrow0^{+}}\sup_{u\in\mathcal{X}(\tau-t,\theta _{-t}\omega )} \Vert \varPhi(0,\tau-t,\theta_{-t}\omega)u-u \Vert _{H}=0.\end{cases} $$

Then \(\{\varPhi(t,\tau,\omega)\}_{t\geq0,\tau\in\mathbb{R},\omega\in\varOmega}\) has a random exponential attractor \(\{\mathcal{A}(\tau,\omega)\}_{\tau\in\mathbb{R},\omega\in\varOmega}\) such that, for every \(\tau\in\mathbb{R}\), \(\omega\in \varOmega\) and \(t\geq0\),

  1. (i)

    \(\mathcal{A}(\tau,\omega)\) \((\subseteq\{\overline{\mathcal{X}(\tau,\omega)}\})\) is a compact set of H and measurable in ω;

  2. (ii)

    \(\varPhi(t,\tau,\omega)\mathcal{A}(\tau,\omega)\subseteq \mathcal{A}(\tau+t,\theta_{t}\omega)\);

  3. (iii)

    \(\dim_{f}\mathcal{A}(\tau,\omega)\leq\frac{4(2k+1)\ln(\frac{2\sqrt{2(2k+1)}}{\hat{\delta} }+1)}{\ln\frac{4}{3}}<\infty\);

  4. (iv)

    for every set \(D\in\mathcal{D}(H)\), there exist a random variable \(T_{\omega,D}\geq0\) and a tempered random variable \(b_{\omega,D}>0\) such that

    $$ \mathrm{d}_{h} \bigl( \varPhi(t,\tau,\omega)D(\tau,\omega), \mathcal{A}(t+\tau,\theta_{t}\omega) \bigr) \leq b_{\omega,D}e^{-\frac{\hat {\lambda}\ln\frac{4}{3}}{32\ln2}t},\quad t\geq T_{\omega,D}; $$
  5. (v)

    for any \(\tau\in\mathbb{R}\), \(\omega\in\varOmega\), \(\lim_{t\rightarrow0}\mathrm{d}_{h}(\mathcal{A}(\tau+t,\theta _{t}\omega),\mathcal{A}(\tau,\omega))=0\).

Remark 2.1

The condition (B2) in Theorem 2.1 above is different from the corresponding condition in the well-known publication [3].

In the following, we will prove the existence of a random exponential attractor for Φ based on Theorem 2.1.

Estimations of bound and tail of solutions

Write \(\Vert g \Vert _{\rho}=\sup_{r\in\mathbb{R}} \Vert g(r) \Vert _{\rho}\), \(\Vert \beta \Vert _{\rho}=\sup_{r\in\mathbb{R}} \Vert \beta(r) \Vert _{\rho}\) and

$$ M_{0}^{2}(\omega)=c_{5}K_{0}(\omega) , \quad K_{0}(\omega )= \int_{-\infty}^{0}e^{\mu s+a_{7}\int_{s}^{0}( \vert c \vert \vert z(\theta_{l}\omega ) \vert + \vert c \vert ^{2} \vert z(\theta_{l}\omega) \vert ^{2})\,dl}\,ds, $$
(7)

where

$$ \textstyle\begin{cases} a_{7}=\max \{ \frac{2a_{6}}{\underline{\lambda}}+\frac {2\varepsilon}{\sqrt{\underline{\lambda}}}+2,\frac{1}{\sqrt{\underline{\lambda }}}+\frac{8a_{5}^{2}}{\alpha\underline{\lambda}},\frac{a_{1}+1}{a_{3}} \} , \\ a_{5}= \vert \gamma \vert c_{0}^{2}c_{2}^{m_{0}}(2m_{0}+1)^{2},\qquad a_{6}=\frac{p-1}{p^{\frac{p}{p-1}}}, \\ c_{5}=\frac{8}{\alpha} \Vert g \Vert _{\rho}^{2}+2\Vert \beta \Vert _{\rho}^{2},\qquad \mu =\min\{\varepsilon,\frac{\varepsilon}{2a_{2}},1\}.\end{cases} $$

In the following part of this section, we always assume that conditions (A1)–(A4) and

$$ a_{7} \biggl( \frac{ \vert c \vert }{\sqrt{\pi\alpha}}+\frac{ \vert c \vert ^{2}}{2\alpha} \biggr) \leq \frac{\mu}{2} $$
(8)

hold.

Lemma 2.1

For every \(\tau\in\mathbb{R}\), \(\omega\in\varOmega\) and \(D\in\mathcal{D}(H)\), there exist \(T_{D}(\tau,\omega)\geq0\) and a tempered random variable \(M_{0}(\omega)\) (independent of τ), such that the solution \(\varphi(r,\tau-t,\theta_{-\tau}\omega ,\varphi_{\tau-t}(\theta_{-\tau}\omega))\in H\) (\(r\geq\tau -t \)) of (4) with \(\varphi_{\tau-t}(\theta_{-\tau }\omega)\in D(\tau-t,\theta_{-t}\omega)\) satisfies

$$ \varphi \bigl(\tau,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t}( \theta _{-\tau}\omega) \bigr)\leq M_{0}(\omega), \quad\forall t \geq T(\tau,\omega,D). $$

Proof

From (3) and (8), it is easy to see that \(M_{0}(\omega)\) is a tempered random variable. Taking the inner product of (4) with \(\varphi(r)=\varphi(r,\tau-t,\theta _{-\tau }\omega,\varphi_{\tau-t}(\theta_{-\tau}\omega))\) in H, we have

$$ \frac{1}{2}\frac{d}{dt} \Vert \varphi \Vert _{H}^{2}+ \bigl(L(\varphi),\varphi \bigr)_{H}= \bigl(f(\varphi,\theta_{r-\tau}\omega),\varphi \bigr)_{H}. $$
(9)

The second term of (9) is

$$\begin{aligned} (L\varphi,\varphi)_{H} ={}&\varepsilon \Vert u \Vert _{\delta,\lambda,\rho }^{2}-\delta(Du,Dv)_{\rho}-(v,u)_{\lambda,\rho}+ \lambda(u,v)_{\rho }+(1-\gamma\varepsilon) (Au,v)_{\rho} \\ &{}-\varepsilon(\alpha-\varepsilon) (u,v)_{\rho}+(\alpha-\varepsilon ) (v,v)_{\rho}+\gamma(Av,v)_{\rho}. \end{aligned}$$
(10)

From [14], we have

$$\begin{aligned} &(Au,v)_{\rho}\geq(Du,Dv)_{\rho}-\frac{1}{2}c_{0}c_{3}c_{4} \Vert Du \Vert _{\rho}^{2}-\frac{1}{2}c_{0}c_{2}^{m_{0}}c_{3}c_{4}(2m_{0}+1)^{2} \Vert v \Vert _{\rho}^{2}, \end{aligned}$$
(11)
$$\begin{aligned} &(Av,v)_{\rho}\geq \biggl(1-\frac{1}{2}c_{0}c_{3}c_{4} \biggr) \Vert Dv \Vert _{\rho}^{2}-\frac{1}{2}c_{0}c_{2}^{m_{0}}c_{3}c_{4}(2m_{0}+1)^{2} \Vert v \Vert _{\rho }^{2}. \end{aligned}$$
(12)

By (A4), (10)–(12), we have

$$\begin{aligned} (L\varphi,\varphi)_{H} \geq{}&\frac{3\varepsilon}{4} \Vert u \Vert _{\delta ,\lambda,\rho}^{2}-\varepsilon(\alpha-\varepsilon) (u,v)_{\rho }+ \biggl(\frac{3\alpha}{4}-\varepsilon \biggr) \Vert v \Vert _{\rho}^{2} \\ &{}+ \biggl( \frac{\delta\varepsilon}{4}-\frac{\delta }{2}c_{0}c_{3}c_{4} \biggr) \Vert Du \Vert _{\rho}^{2} +\gamma \biggl( 1- \frac{1}{2}c_{0}c_{3}c_{4} \biggr) \Vert Dv \Vert _{\rho }^{2} \\ &{}+ \biggl( \frac{\alpha}{4}-\frac{\delta+\gamma}{2}c_{0}c_{2}^{m_{0}}c_{3}c_{4}(2m_{0}+1)^{2} \biggr) \Vert v \Vert _{\rho}^{2} \\ \geq{}&\frac{\varepsilon}{2} \Vert u \Vert _{\delta,\lambda ,\rho}^{2}+\frac{\varepsilon}{2} \Vert v \Vert _{\rho}^{2}+ \frac{\alpha }{4} \Vert v \Vert _{\rho}^{2}+ \frac{\varepsilon}{4} \Vert u \Vert _{\delta,\lambda ,\rho}^{2}+ \biggl( \frac{3\alpha}{4}-\varepsilon \biggr) \Vert v \Vert _{\rho}^{2} \\ &{}+ \biggl(\frac{\alpha}{2}-\frac{3\varepsilon}{2} \biggr) \Vert v \Vert _{\rho}^{2}-\frac{\varepsilon\alpha}{\sqrt{\underline{\lambda}}} \Vert u \Vert _{\delta ,\lambda,\rho} \Vert v \Vert _{\rho}. \end{aligned}$$

From (A4), \(4\frac{\varepsilon}{4} ( \frac{\alpha}{2}-\frac{3\varepsilon}{2} ) \geq\frac{\varepsilon^{2}\alpha ^{2}}{\underline{\lambda}}\), we have

$$ (L\varphi,\varphi)_{H}\geq\frac{\varepsilon}{2} \bigl\Vert \varphi (r) \bigr\Vert _{H}^{2}+\frac{\alpha}{4} \Vert v \Vert _{\rho}^{2}. $$

The right-hand side of (9) is

$$ \bigl(f(\varphi,\theta_{r-\tau}\omega),\varphi \bigr)_{H}=(czu,u)_{\delta ,\lambda ,\rho}+ \bigl(-czv+2\varepsilon czu-c^{2}z^{2}u+g(r)-\gamma cAzu-f(u,r),v \bigr)_{\rho }. $$
(13)

For each term of (13), we have

$$\begin{aligned} &(czu,u)_{\delta,\lambda,\rho}-(czv,v)_{\rho}\leq \vert c \vert \vert z \vert \bigl\vert \varphi (r) \bigr\Vert _{H}^{2},\qquad (g,v)_{\rho}\leq\frac{4}{\alpha} \Vert g \Vert _{\rho}^{2}+ \frac{\alpha}{8} \Vert v \Vert _{\rho}^{2}, \end{aligned}$$
(14)
$$\begin{aligned} &(2\varepsilon czu,v)_{\rho}- \bigl(c^{2}z^{2}u,v \bigr)_{\rho}\leq \biggl( \frac{ \vert c \vert \vert \varepsilon \vert \vert z \vert }{\sqrt{\underline {\lambda}}}+\frac{ \vert c \vert ^{2} \vert z \vert ^{2}}{2\sqrt{\underline{\lambda}}} \biggr) \bigl\Vert \varphi(r) \bigr\Vert _{H}^{2}, \end{aligned}$$
(15)
$$\begin{aligned} &{-}(\gamma cAzu,v)_{\rho} \leq\frac{4 \vert \gamma cz \vert ^{2} \Vert Au \Vert _{\rho }^{2}}{\alpha}+\frac{\alpha}{8} \Vert v \Vert _{\rho}^{2} \\ &\phantom{-(\gamma cAzu,v)_{\rho}}\leq\frac{4 \vert \gamma cz \vert ^{2}c_{0}^{4}c_{2}^{2m_{0}}(2m_{0}+1)^{4}}{\alpha \underline{\lambda}} \Vert u \Vert _{\lambda,\rho}^{2}+ \frac {\alpha}{8} \Vert v \Vert _{\rho}^{2} \\ &\phantom{-(\gamma cAzu,v)_{\rho}}\leq\frac{4 \vert cz \vert ^{2}a_{5}^{2}}{\alpha\underline{\lambda }} \Vert \varphi \Vert _{H}^{2}+ \frac{\alpha}{8} \Vert v \Vert _{\rho}^{2}, \end{aligned}$$
(16)
$$\begin{aligned} &{-} \bigl(f(u,r),v \bigr)_{\rho}=- \bigl(f(u,r),\dot{u} \bigr)_{\rho}- \varepsilon \bigl(f(u,r),u \bigr)_{\rho }+cz \bigl(f(u,r),u \bigr)_{\rho}. \end{aligned}$$
(17)

By (A2) we have

$$\begin{aligned} & \bigl(f(u,r),\dot{u} \bigr)_{\rho} =\sum _{i\in\mathbb{Z}}\rho _{i}f_{i}(u_{i},r)\dot{u}_{i}=\frac{d}{dt} \biggl(\sum _{i\in\mathbb{Z}}\rho _{i}G_{i}(u_{i},r) \biggr)- \sum_{i\in\mathbb{Z}}\rho_{i}G'_{i,r}(u_{i},r) \\ &\phantom{ (f(u,r),\dot{u} )_{\rho}}\geq \frac{d}{dt}\overline{G}(u,r)-a_{4} \overline{G}(u,r)\geq \frac {d}{dt}\overline{G}(u,r)- \frac{\varepsilon}{2a_{2}} \overline{G}(u,r), \end{aligned}$$
(18)
$$\begin{aligned} &\varepsilon \bigl(f(u,r),u \bigr)_{\rho}=\varepsilon\sum _{i\in\mathbb{Z}} \rho _{i}f_{i}(u_{i},r)u_{i} \geq \frac{\varepsilon}{a_{2}}\sum_{i\in \mathbb{Z}} \rho_{i}G_{i}(u_{i},r)= \frac{\varepsilon}{a_{2}} \overline{G}(u,r). \end{aligned}$$
(19)

By the Young inequality, we obtain

$$ a_{1} \vert s \vert \leq \vert s \vert ^{p}+a_{1}^{\frac{p}{p-1}} \times \frac{p-1}{p^{\frac{p}{p-1}}}= \vert s \vert ^{p}+a_{6}, \quad p>1. $$
(20)

By (A4) and (20),

$$ \bigl\vert f_{i}^{\prime}(s,r) \bigr\vert \leq a_{1} \vert s \vert +a_{1} \vert s \vert ^{p}\leq a_{6}+(1+a_{1}) \vert s \vert ^{p}, $$
(21)

from (21), we get

$$ \bigl\vert f_{i}(s,r) \bigr\vert \leq a_{6} \vert s \vert +(1+a_{1}) \vert s \vert ^{p+1}. $$
(22)

From (22), we have

$$\begin{aligned} cz \bigl(f(u,r),u \bigr)_{\rho} ={}&cz\sum _{i\in\mathbb{Z}}\rho _{i}f_{i}(u_{i},r)u_{i} \leq \vert c \vert \vert z \vert \biggl\vert \sum _{i\in\mathbb{Z}}\rho _{i}f_{i}(u_{i},r)u_{i} \biggr\vert \\ \leq{}&a_{6} \vert c \vert \vert z \vert \Vert u \Vert _{\rho }^{2}+\frac{(a_{1}+1)}{a_{3}} \vert c \vert \vert z \vert \sum_{i\in\mathbb{Z}}\rho _{i} \bigl(G_{i}(u_{i},r)+\beta_{i}^{2}(r) \bigr) \\ \leq{}&a_{6} \vert c \vert \vert z \vert \Vert u \Vert _{\rho }^{2}+\frac{(a_{1}+1)}{a_{3}} \vert c \vert \vert z \vert \overline{G}(u,r) \\ &{}+\frac{(a_{1}+1)}{a_{3}} \bigl\vert c \Vert z \Vert \vert \beta \vert \bigr\vert _{\rho}^{2}. \end{aligned}$$
(23)

Thus, by all the above inequalities, we have

$$\begin{aligned} &\frac{d}{dt} \bigl[ \bigl\Vert \varphi(r) \bigr\Vert _{H}^{2}+2\overline{G}(u,r)+2 \Vert \beta \Vert _{\rho}^{2} \bigr] +\mu \bigl[ \bigl\Vert \varphi (r) \bigr\Vert _{H}^{2}+2\overline{G}(u,r)+2 \Vert \beta \Vert _{\rho}^{2} \bigr] \\ &\quad\leq \biggl( 2 \vert c \vert \vert z \vert +\frac{2 \vert c \vert \vert \varepsilon \vert \vert z \vert }{\sqrt{\underline{\lambda}}}+ \frac{ \vert c \vert ^{2} \vert z \vert ^{2}}{\sqrt{\underline{\lambda }}}+\frac{8 \vert cz \vert ^{2}|a_{5}^{2}}{\alpha\underline{\lambda}}+\frac {2a_{6} \vert c \vert \vert z \vert }{{\underline{\lambda}}} \biggr) \bigl\Vert \varphi(r) \bigr\Vert _{H}^{2} \\ &\qquad{}+\frac{2(a_{1}+1)}{a_{3}} \vert c \vert \vert z \vert \overline {G}(u,r)+ \frac{2(a_{1}+1)}{a_{3}} \vert c \vert \vert z \vert \beta \Vert _{\rho }^{2}+ \frac{8}{\alpha} \Vert g \Vert _{\rho}^{2}+2 \Vert \beta \Vert _{\rho}^{2}. \end{aligned}$$

Then, for \(r\geq\tau-t\), we have

$$ \frac{d}{dt}y(r)\leq(-\mu+a_{7} \bigl( \vert c \vert \bigl\vert z(\theta_{r-\tau }\omega ) \bigr\vert + \vert c \vert ^{2} \bigl\vert z(\theta_{r-\tau}\omega) \bigr\vert ^{2} \bigr)y(r)+c_{5}, $$
(24)

where

$$ y(r)= \bigl\Vert \varphi(r) \bigr\Vert _{H}^{2}+2 \overline {G}(u,r)+2 \vert \beta \vert _{\rho}^{2}\geq \bigl\Vert \varphi(r) \bigr\Vert _{H}^{2}. $$

By the Gronwall inequality applied to (24) on \([\tau-t,r]\), we obtain

$$\begin{aligned} \bigl\Vert \varphi(r) \bigr\Vert _{H}^{2} \leq{}&e^{\int_{\tau -t}^{r}(-\mu +a_{7}( \vert c \vert \vert z(\theta_{l-\tau}\omega) \vert + \vert c \vert ^{2} \vert z(\theta_{l-\tau}\omega ) \vert ^{2})\,dl}y(\tau-t) \\ &{}+c_{5} \int_{\tau-t}^{r}e^{\int_{s}^{r}(-\mu+a_{7}( \vert c \vert \vert z(\theta_{l-\tau }\omega) \vert + \vert c \vert ^{2} \vert z(\theta_{l-\tau}\omega ) \vert ^{2})\,dl}\,ds, \end{aligned}$$
(25)

and let \(r=\tau\), it follows that

$$ \bigl\Vert \varphi(\tau) \bigr\Vert _{H}^{2}\leq e^{\int_{-t}^{0} ( -\mu +a_{7}( \vert c \vert \vert z(\theta_{l }\omega) \vert + \vert c \vert ^{2} \vert z(\theta_{l }\omega ) \vert ^{2} ) \,dl}y(\tau-t)+\frac{1}{2}M_{0}^{2}( \omega). $$
(26)

By \(\varphi_{\tau-t}(\theta_{-\tau}\omega)\in D(\tau-t,\theta _{-t}\omega)\in\mathcal{D}(H)\) and (8), we have

$$\lim_{t\rightarrow +\infty}e^{\int_{-t}^{0} ( -\mu+a_{7}( \vert c \vert \vert z(\theta_{l }\omega ) \vert + \vert c \vert ^{2} \vert z(\theta_{l }\omega) \vert ^{2} ) \,dl}y(\tau-t)=0. $$

Thus, the proof is completed. □

Let

$$ D_{0}= \bigl\{ D_{0}(\omega)= \bigl\{ \varphi\in H: \Vert \varphi \Vert _{H}\leq M_{0}(\omega) \bigr\} \bigr\} \in D(H), $$

then \(D_{0}\) is a tempered family. By Lemma 2.1, there exists \(T_{D_{0}}(\omega)>0\) (independent of τ) such that, for any \(\omega \in\varOmega\), \(\tau\in R\),

$$ \varphi \bigl( r,\tau-t,\theta_{-\tau}\omega,D_{0}(\theta _{-t}\omega ) \bigr) \in{D_{0}(\omega}),\quad \forall r\geq \tau-t, t\geq T_{D_{0}}(\omega)\geq0. $$

Choosing a smooth increasing function \(\eta\in C^{1}(\mathbb {R}^{+},\mathbb{R})\) satisfies

$$ \textstyle\begin{cases} \eta(s)=0, & s\in{}[0,1); \\ 0\leq\eta(s)\leq1, & s\in{}[1,2); \\ \eta(s)=1, & s\in{}[2,+\infty); \\ \vert \eta^{\prime}(s) \vert \leq C_{0}, & \forall s\in\mathbb {R}^{+},C_{0}>0.\end{cases} $$

Lemma 2.2

For every \(\tau\in\mathbb{R}\), \(t\geq0\), \(\omega\in\varOmega\), \(I\in\mathbb{N}\) and \(\nu>0\), there exists \(T_{\nu}(\omega)>0\) (independent of τ) such that the solution \(\varphi (r)=\varphi(r,\tau-t,\theta_{-\tau}\omega,\varphi_{\tau-t}(\theta _{-\tau}\omega)\) (\(r\geq\tau-t\)) of (4) with \(\varphi_{\tau-t}(\theta_{-\tau}\omega)\in D_{0}(\theta _{-t}\omega)\) satisfies

$$ \sum_{ \vert i \vert \geq2I} \bigl\vert \varphi_{i}( \tau,\tau-t,\theta _{-\tau}\omega ,\varphi_{\tau-t}) \bigr\vert _{H}^{2}\leq\nu+ \biggl( \frac {c_{6}}{I}+\gamma _{I} \biggr) K_{2}(\omega),\quad t>T_{\nu}( \omega), I \in\mathbb{N}, $$
(27)

where

$$ \textstyle\begin{cases} \vert \varphi_{i} \vert _{H}^{2}=\rho_{i}[\delta (Du)_{i}^{2}+\lambda _{i}u_{i}^{2}+v_{i}^{2}],\qquad c_{6}=\frac{c_{5}}{2}; \\ \gamma_{I}=\sup_{r\in\mathbb{R}}\sum_{i\geq I}\rho_{i}(\frac {8}{\alpha}g_{i}^{2}(r)+4\beta_{i}^{2}(r)+4\beta_{i}^{\prime2}(r)); \\ K_{2}(\omega)=K_{0}(\omega)(1+K_{1}(\omega)); \\ K_{1}(\omega)=\int_{-\infty}^{0}\xi_{1}(\theta_{s}\omega)e^{\mu s+a_{7}\int_{s}^{0}( \vert c \vert \vert z(\theta_{l}\omega) \vert + \vert c \vert ^{2} \vert z(\theta_{l}\omega ) \vert ^{2})\,dl}\,ds; \\ \xi_{1}(\omega)=2c_{7} ( \underline{\lambda}+\varepsilon+2 \vert c z (\omega) \vert ) +2c_{8}(\delta+\gamma_{0}); \\ c_{7}=\frac{\delta m_{0}C_{0}c_{0}^{2}(2m_{0}+1)^{2}c_{2}^{m_{0}}}{\underline{\lambda}},\qquad c_{8}=m_{0}C_{0}c_{0}c_{2}^{\frac{m_{0}}{2}}(1+(2m_{0}+1)^{2}). \end{cases} $$
(28)

Proof

Let \(I\in\mathbb{N}\) be a suitable large integer, and set \(x_{i}=\eta(\frac{|i|}{I})u_{i}\), \(y_{i}=\eta(\frac {|i|}{I})v_{i}\), \(w= ( x,y ) =(x_{i},y_{i})_{i\in\mathbb{Z}}^{T}\). Taking the inner product \((\cdot,\cdot)_{H}\) of (4) with w, we have

$$ (\dot{\varphi},w)_{H}+(L\varphi,w)_{H}= \bigl(F( \varphi),w \bigr)_{H}. $$
(29)

By some computations, it follows that

$$\begin{aligned} &(\dot{\varphi},w)_{H}\geq\frac{1}{2}\frac{d}{dt}\sum _{i\in\mathbb {Z}}\eta \biggl( \frac{ \vert i \vert }{I} \biggr) \vert \varphi_{i} \vert _{H}^{2}-\frac{c_{7}}{I} \bigl(\underline{\lambda}+\varepsilon+ \bigl\vert cz( \theta_{r-\tau}\omega ) \bigr\vert \bigr) \Vert \varphi \Vert _{H}^{2}, \end{aligned}$$
(30)
$$\begin{aligned} &( L\varphi,w ) _{H}\geq\sum_{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{I} \biggr) \biggl( \frac{\varepsilon}{2} \vert \varphi _{i} \vert _{H}^{2}+\frac{\alpha}{4}\rho_{i} v_{i}^{2} \biggr) - \frac{c_{8}(\delta +\gamma)}{I} \Vert \varphi \Vert _{H}^{2}, \end{aligned}$$
(31)
$$\begin{aligned} & \bigl( cz(\theta_{r-\tau}\omega)u,x \bigr) _{\delta,\lambda,\rho }- \bigl( cz( \theta_{r-\tau}\omega)v,y \bigr) _{\rho} \\ &\quad\leq \vert c \vert \bigl\vert z(\theta _{r-\tau}\omega) \bigr\vert \biggl( \sum_{i\in\mathbb{Z}}\eta \biggl( \frac { \vert i \vert }{I} \biggr) \vert \varphi_{i} \vert _{H}^{2}+\frac{c_{7}}{I} \Vert \varphi \Vert _{H}^{2} \biggr), \end{aligned}$$
(32)
$$\begin{aligned} & \bigl( 2\varepsilon cz(\theta_{r-\tau}\omega)u,y \bigr) _{\rho} \leq \frac{\varepsilon \vert c \vert \vert z(\theta_{r-\tau}\omega) \vert }{\sqrt{\underline{\lambda } }}\sum_{i\in\mathbb{Z}} \eta \biggl( \frac{ \vert i \vert }{I} \biggr) \vert \varphi _{i} \vert _{H}^{2}, \end{aligned}$$
(33)
$$\begin{aligned} & \bigl( -c^{2}z^{2}(\theta_{r-\tau}\omega)u,y \bigr) _{\rho}\leq \frac{c^{2}z^{2}(\theta_{r-\tau}\omega)}{2\sqrt{\underline{\lambda} }}\sum _{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{I} \biggr) \vert \varphi _{i} \vert _{H}^{2}, \end{aligned}$$
(34)
$$\begin{aligned} & \bigl( -\gamma cAz(\theta_{r-\tau}\omega)u,y \bigr) _{\rho} \leq \frac{4 \vert cz \vert ^{2}a^{2}_{5}}{\underline{\lambda}\alpha}\sum_{i\in \mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{I} \biggr) \vert \varphi_{i} \vert _{H}^{2} \\ &\phantom{ ( -\gamma cAz(\theta_{r-\tau}\omega)u,y ) _{\rho}\leq}{}+\frac{\alpha}{8}\sum_{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{I} \biggr) \rho _{i} v_{i}^{2}, \end{aligned}$$
(35)
$$\begin{aligned} &(g,y)_{\rho}\leq\frac{4}{\alpha}\sum_{i\in\mathbb{Z}} \eta \biggl( \frac{ \vert i \vert }{I} \biggr) \rho_{i} g_{i}^{2}+ \frac{\alpha}{8}\sum_{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{I} \biggr) \rho_{i} v_{i}^{2}, \end{aligned}$$
(36)
$$\begin{aligned} & \bigl(f(u,r),y \bigr)_{\rho} \\ &\quad\geq\frac{d}{dt}\sum_{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{I} \biggr) \rho_{i}G_{i}(u_{i},r)+ \frac{\varepsilon}{2a_{2}}\sum_{i\in\mathbb {Z}}\eta \biggl( \frac{ \vert i \vert }{I} \biggr) \rho _{i}G_{i}(u_{i},r) \\ &\qquad{}-a_{6} \vert c \vert \vert z \vert \sum _{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{I} \biggr) \rho_{i}u_{i}^{2} -\frac{a_{1}+1}{a_{3}} \vert c \vert \vert z \vert \sum_{i\in\mathbb {Z}} \eta \biggl( \frac{ \vert i \vert }{I} \biggr) \rho_{i}G_{i}(u_{i},r) \\ &\qquad{}-\frac{a_{1}+1}{a_{3}} \vert c \vert \vert z \vert \sum _{i\in \mathbb{Z}} \eta \biggl( \frac{ \vert i \vert }{I} \biggr) \rho_{i}\beta _{i}^{2}(r). \end{aligned}$$
(37)

Thus, by putting (30)–(37) into (29), we obtain, for \(r\geq\tau-t\),

$$\begin{aligned} &\frac{d}{dt}\sum_{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{I} \biggr) \bigl[ \bigl\vert \varphi_{i}(r) \bigr\vert ^{2}_{H}+2\rho_{i}G_{i}(u_{i},r)+2 \rho _{i}\beta _{i}^{2}(r) \bigr] \\ &\quad\leq ( -\mu+a_{7} \bigl( \vert c \vert \bigl\vert z( \theta_{r-\tau}\omega ) \bigr\vert + \vert c \vert ^{2} \bigl\vert z(\theta _{r-\tau}\omega) \bigr\vert ^{2} \bigr) \\ &\qquad{}\times\sum_{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{I} \biggr) \bigl[ \bigl\vert \varphi_{i}(r) \bigr\vert ^{2}_{H}+2\rho _{i}G_{i}(u_{i},r)+2 \rho_{i} \beta_{i}^{2}(r) \bigr] \\ &\qquad{}+\frac{\xi_{1}(\theta_{r-\tau}\omega)}{I} \bigl\Vert \varphi (r) \bigr\Vert _{H}^{2}+ \sum_{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{I} \biggr) \rho_{i} \biggl[\frac{8}{\alpha}g_{i}^{2}(r)+4 \beta_{i}^{2}(r)+4 \beta_{i}^{\prime2}(r) \biggr]. \end{aligned}$$
(38)

By applying the Gronwall inequality to (38) on \([\tau-t,\tau ]\), we have

$$\begin{aligned} &\sum_{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{I} \biggr) \bigl[ \bigl\vert \varphi_{i}(\tau) \bigr\vert _{H}^{2}+2\rho_{i}G_{i}(u_{i,\tau}, \tau )+2\rho_{i} \beta_{i}^{2}(\tau) \bigr] \\ &\quad\leq \bigl[ \Vert \varphi_{\tau-t} \Vert _{H}^{2}+2G(u_{\tau-t}, \tau -t)+2 \bigl\Vert \beta(\tau-t) \bigr\Vert _{\rho}^{2} \bigr] \\ &\qquad{}\times e^{\int_{-t}^{0} [ -\mu+a_{7}( \vert c \vert \vert z(\theta_{l}\omega) \vert + \vert c \vert ^{2} \vert z(\theta_{l}\omega) \vert ^{2}) ] \,dl} \\ &\qquad{}+ \sup_{r\in\mathbb{R}}\sum_{i\geq I} \rho_{i} \biggl(\frac{8}{\alpha}g_{i}^{2}(r)+4 \beta_{i}^{2}(r) + 4\beta_{i}^{\prime 2}(r) \biggr) \\ &\qquad{} \times \int_{-t}^{0}e^{\int_{s}^{0} [ -\mu+a_{7}( \vert c \vert \vert z(\theta_{l}\omega ) \vert + \vert c \vert ^{2} \vert z(\theta_{l}\omega) \vert ^{2}) ] \,dl}\,ds \\ &\qquad{}+\frac{1}{I} \int_{\tau-t}^{\tau}\xi_{1}( \theta_{s-\tau}\omega ) \bigl\Vert \varphi(s) \bigr\Vert _{H}^{2} \\ &\qquad{}\times e^{\int_{s}^{\tau} [ -\mu+a_{7}( \vert c \vert \vert z(\theta _{l-\tau}\omega) \vert + \vert c \vert ^{2} \vert z(\theta_{l-\tau }\omega) \vert ^{2}) ] \,dl}\,ds. \end{aligned}$$
(39)

Since \(\xi_{1}(\omega)\) is tempered and \(\xi_{1}(\theta_{t}\omega )\) is continuous in t, by Proposition 4.3.3 of [15], for the fixed number \(\mu>0\), there exists a tempered random variable \(\sigma(\omega)\) such that \(\xi_{1}(\theta_{t}\omega)\leq\sigma(\omega)e^{\frac{\mu }{3}|t|}\) for \(t\in\mathbb{R}\). Thus, by (25),

$$\begin{aligned} & \int_{\tau-t}^{\tau}\xi_{1}( \theta_{s-\tau}\omega) \bigl\Vert \varphi (s) \bigr\Vert _{H}^{2}e^{\int_{s}^{\tau} [ -\mu+a_{7}( \vert c \vert \vert z(\theta_{l-\tau }\omega) \vert + \vert c \vert ^{2} \vert z(\theta_{l-\tau}\omega ) \vert ^{2}) ] \,dl}\,ds \\ &\quad \leq y(\tau-t)e^{\int_{\tau-t}^{\tau}(-\mu+a_{7}( \vert c \vert \vert z(\theta_{l-\tau }\omega) \vert + \vert c \vert ^{2} \vert z(\theta_{l-\tau}\omega ) \vert ^{2})\,dl} \int_{\tau-t}^{\tau }\xi_{1}( \theta_{s-\tau}\omega)\,ds \\ &\qquad{}+\frac{1}{2}M_{0}^{2}(\omega) \int_{\tau-t}^{\tau}\xi_{1}(\theta _{s-\tau}\omega)e^{\int_{s}^{\tau} [ -\mu+a_{7}( \vert c \vert \vert z(\theta _{l-\tau}\omega) \vert + \vert c \vert ^{2} \vert z(\theta_{l-\tau }\omega) \vert ^{2}) ] \,dl}\,ds. \end{aligned}$$
(40)

From (A4) and (22)

$$\begin{aligned} &y(\tau-t,\tau-t,\theta_{-t}\omega,\varphi_{\tau-t}) \\ &\quad= \bigl\Vert \varphi_{\tau-t}(\theta_{-\tau} \omega) \bigr\Vert _{H}^{2}+2 \sum_{i\in \mathbb{Z}} \rho_{i}f_{i}(u_{i,\tau-t}, \tau-t)u_{i,\tau-t}+2 \vert \beta \vert _{\rho}^{2} \\ &\quad\leq \bigl\Vert \varphi_{\tau-t}(\theta_{-t} \omega) \bigr\Vert _{H}^{2}+2a_{2} \bigl( a_{6} \Vert u_{\tau-t} \Vert _{\rho}^{2}+(1+a_{1}) \Vert u_{\tau-t} \Vert _{\rho}^{p+2} \bigr) +2 \vert \beta \vert _{\rho}^{2}, \end{aligned}$$

thus, (40) can be written as

$$\begin{aligned} & \int_{\tau-t}^{\tau}\xi_{1}( \theta_{s-\tau}\omega )y(s)e^{\int_{s}^{\tau} [ -\mu+a_{7}( \vert c \vert \vert z(\theta_{l-\tau}\omega ) \vert + \vert c \vert ^{2} \vert z(\theta_{l-\tau}\omega) \vert ^{2}) ] \,dl}\,ds \\ &\quad\leq\frac{\mu c_{9}}{3}\sigma(\omega) \bigl(M_{0}^{2}( \theta_{-t}\omega )+M_{0}^{p+2}( \theta_{-t}\omega)+1 \bigr)e^{-\frac{2\mu}{3}t+a_{7}\int_{-t}^{0}( \vert c \vert \vert z(\theta_{l}\omega) \vert + \vert c \vert ^{2} \vert z(\theta_{l}\omega ) \vert ^{2})\,dl} \\ &\qquad{}+\frac{1}{2}M_{0}^{2}(\omega) \int_{-\infty}^{0}\xi_{1}(\theta _{s}\omega)e^{\int_{s}^{0} [ -\mu+a_{7}( \vert c \vert \vert z(\theta_{l}\omega ) \vert + \vert c \vert ^{2} \vert z(\theta_{l}\omega) \vert ^{2}) ] \,dl}\,ds, \end{aligned}$$

where \(c_{9}=\max \{ 1+\frac{2a_{2}a_{6}}{\underline{\lambda }},\frac{2a_{2}(a_{1}+1)}{\underline{\lambda}^{\frac{p}{2}+1}},2 \Vert \beta \Vert _{\rho }^{2} \} \), and

$$\begin{aligned} & \int_{-\infty}^{0}\xi_{1}(\theta_{s} \omega)e^{\int_{s}^{0} [ -\mu +a_{7}( \vert c \vert \vert z(\theta_{l}\omega) \vert + \vert c \vert ^{2} \vert z(\theta_{l}\omega ) \vert ^{2}) ] \,dl}\,ds \\ &\quad\leq\sigma(\omega) \int_{-\infty}^{0}e^{\int_{s}^{0} [ -\frac {2\mu }{3}+a_{7}( \vert c \vert \vert z(\theta_{l}\omega) \vert + \vert c \vert ^{2} \vert z(\theta_{l}\omega ) \vert ^{2}) ] \,dl}\,ds< \infty. \end{aligned}$$

Thus, by (39) and \(I\geq1\), we have

$$\begin{aligned} &\sum_{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{I} \biggr) \bigl[ \bigl\vert \varphi_{i}(\tau) \bigr\vert _{H}^{2}+2\rho_{i}G_{i}(u_{i,\tau}, \tau )+2\rho_{i} \beta_{i}^{2}(\tau) \bigr] \\ &\quad\leq c_{9} \bigl(M_{0}^{2}( \theta_{-t}\omega)+M_{0}^{p+2}(\theta _{-t}\omega )+1 \bigr)e^{\int_{-t}^{0}(-\mu+a_{7}( \vert c \vert \vert z(\theta _{l}\omega) \vert + \vert c \vert ^{2} \vert z(\theta _{l}\omega) \vert ^{2})\,dl} \\ &\qquad{}+\frac{1}{I}\frac{3}{\mu}c_{9} \bigl(M_{0}^{2}( \theta_{-t}\omega )+M_{0}^{p+2}( \theta_{-t}\omega)+1 \bigr)e^{-\frac{2\mu}{3}t+a_{7}\int_{-t}^{0}( \vert c \vert \vert z(\theta_{l}\omega) \vert + \vert c \vert ^{2} \vert z(\theta_{l}\omega ) \vert ^{2})\,dl} \\ &\qquad{}+\frac{1}{2I}M_{0}^{2}(\omega) \int_{-\infty}^{0}\xi_{1}(\theta _{s}\omega)e^{\int_{s}^{0}(-\mu+a_{7}( \vert c \vert \vert z(\theta _{l}\omega ) \vert + \vert c \vert ^{2} \vert z(\theta_{l}\omega) \vert ^{2})\,dl}\,ds \\ &\qquad{}+\gamma_{I} \int_{-\infty }^{0}e^{\int_{s}^{0} [ -\mu+a_{7}( \vert c \vert \vert z(\theta _{l}\omega ) \vert + \vert c \vert ^{2} \vert z(\theta_{l}\omega) \vert ^{2}) ] \,dl}\,ds \\ &\quad\leq c_{9} \biggl( 1+\frac{3}{\mu} \biggr) \bigl(M_{0}^{2}(\theta_{-t}\omega )+M_{0}^{p+2}(\theta_{-t}\omega)+1 \bigr)e^{-\frac{2\mu}{3}t+a_{7}\int_{-t}^{0}( \vert c \vert \vert z(\theta_{l}\omega) \vert + \vert c \vert ^{2} \vert z(\theta_{l}\omega ) \vert ^{2})\,dl} \\ &\qquad{}+ \biggl( \frac{c_{6}}{I}+\gamma_{I} \biggr) K_{2}( \omega). \end{aligned}$$
(41)

By (3), we have

$$ \lim_{t\rightarrow+\infty} \biggl( \frac{2\mu}{3}-\frac{a_{7}}{t}\int_{-t}^{0} \bigl( \vert c \vert \bigl\vert z(\theta_{l}\omega) \bigr\vert + \vert c \vert ^{2} \bigl\vert z(\theta_{l}\omega ) \bigr\vert ^{2} \bigr) \,ds \biggr) \geq\frac{\mu}{6}>0. $$

Since \(M_{0}^{2}(\omega)+M_{0}^{p+2}(\omega)+1\) is tempered, we have

$$ \bigl(M_{0}^{2}(\theta_{-t}\omega)+M_{0}^{p+2}( \theta_{-t}\omega )+1 \bigr)e^{-\frac{2\mu}{3}t+a_{7}\int_{-t}^{0}( \vert c \vert \vert z(\theta_{l}\omega ) \vert + \vert c \vert ^{2} \vert z(\theta _{l}\omega) \vert ^{2}))\,dl}\rightarrow0 $$
(42)

as \(t\rightarrow+\infty\). Hence, by (41)–(42), for any \(\nu>0\), \(I\in\mathbb{N}\), there exists \(T_{\nu}(\omega)>0\), such that

$$ \sum_{ \vert i \vert >2I} \bigl\vert \varphi_{i}( \tau,\tau-t,\theta _{-\tau}\omega ,\varphi_{\tau-t}) \bigr\vert _{H}^{2}\leq\nu+ \biggl( \frac {c_{6}}{I}+\gamma _{I} \biggr) K_{2}(\omega),\quad t\geq T_{\nu}( \omega), I\in\mathbb{N}. $$

This completes the proof. □

It follows directly from Lemmas 2.1 and 2.2 and Theorem 3.6 in [1], Theorem 3.3 in [10] that

Theorem 2.2

The cocycle Φ possesses a random attractor \(R=\{R(\tau,\omega)\}_{\tau\in\mathbb{R},\omega \in \varOmega}\) in \(D(H)\) with the properties: for every \(\tau \in R\), \(\omega\in\varOmega\), (i) \(R(\tau,\omega )\subseteq D_{0}(\omega)\); (ii) \(R(\tau,\omega)\) is compact in H and measurable in ω; (iii) \(\varPhi (t,\tau,\omega,R(\tau,\omega))=R(t+\tau,\theta_{t}\omega)\), \(\forall t\geq0\); (iv) for every \(B=\{B(\tau,\omega)\}_{\tau \in\mathbb{R},\omega\in\varOmega}\in D(H)\), \(\lim_{t\rightarrow +\infty}\)d\(_{h}(\varPhi(t,\tau-t,\theta_{-t}\omega)B(\tau -t,\theta_{-t}\omega),R(\tau,\omega))=0\).

Existence of random exponential attractor

In this subsection, we prove the existence of a random exponential attractor for Φ based on Theorem 2.1. Obviously, the family of tempered closed random subsets \(D_{0}=\{D_{0}(\omega)\}\}_{\tau\in\mathbb{R},\omega \in \varOmega}\) satisfies condition (B1) in Theorem 2.1.

Taking a small enough positive constant \(\nu_{0}>0\) such that

$$ \frac{32a_{1}^{2}\nu_{0}}{\alpha\underline{\lambda}^{2}\overline {\rho}}+\frac{a_{1}^{2}2^{p+4}\nu_{0}^{p}}{\alpha\underline{\lambda}^{p+1}\overline{\rho}^{p}}\leq\frac{\varepsilon}{2}. $$
(43)

For any \(\omega\in\varOmega\), set

$$\begin{aligned} T_{\nu_{0}}(\omega)&=\min \biggl\{ t:c_{9} \biggl( 1+ \frac{3}{\mu} \biggr) M(\theta_{-t}\omega)e^{-\frac{2\mu}{3}t+a_{7}\int _{-t}^{0}( \vert c \vert \vert z(\theta _{l}\omega) \vert + \vert c \vert ^{2} \vert z(\theta_{l}\omega) \vert ^{2}))\,dl}\leq \nu_{0} \biggr\} \\ &< +\infty, \end{aligned}$$

where \(M(\omega)=M_{0}^{2}(\omega)+M_{0}^{p+2}(\omega)+1\).

For any \(\tau\in\mathbb{R}\), \(\omega\in\varOmega\), define a tempered bounded random subset \(\mathcal{X}_{1}(\tau,\omega)\) of H as

$$ \mathcal{X}_{1}(\tau,\omega)=\bigcup_{s\geq\max \{T_{D_{0}}(\omega ),T_{\nu_{0}}(\omega) \}} \varphi \bigl( \tau,\tau-s,\theta _{-\tau }\omega,D_{0}( \theta_{-s} \omega) \bigr)\subseteq D_{0}(\omega)\subset H. $$
(44)

Now let us show that \(\{\mathcal{X}_{1}(\tau,\omega)\}_{\tau\in \mathbb{R},\omega\in\varOmega}\) satisfies (B2) of Theorem 2.1.

For any \(\tau\in\mathbb{R}\), \(\omega\in\varOmega\), \(t\geq0\), and \(\varphi _{j,\tau-t}(\theta_{-\tau}\omega)\in\mathcal{X}_{1}(\tau-t,\theta _{-t}\omega)\subseteq D_{0}(\theta_{-t}\omega)\), \(j=1\), 2. Let \(\varphi _{j}(r)=\varphi(r,\tau-t,\theta_{-\tau}\omega,\varphi_{j,\tau -t}(\theta_{-\tau}\omega))=(u_{j}(r),v_{j}(r)), j=1,2\), \(\psi (r)=\varphi _{1}(r)-\varphi_{2}(r)=(\xi(r),\zeta(r))\), \(r\geq\tau-t\), then

$$ \textstyle\begin{cases} \dot{\psi}+L\psi=F(\varphi_{1},\theta_{r-\tau}\omega)-F(\varphi _{2},\theta_{r-\tau}\omega), \\ \psi_{\tau}(\omega)=(\xi_{\tau},\zeta_{\tau})=(u_{1\tau }-u_{2\tau },v_{1\tau}-v_{2\tau}),\quad r\geq\tau-t,\end{cases} $$
(45)

where

F ( φ 1 , θ r τ ω ) F ( φ 2 , θ r τ ω ) = ( c z ( θ r τ ω ) ξ ( 2 ε c z ) c z ξ γ c A z ξ c z ζ f ( u 1 , t ) + f ( u 2 , t ) ) .

By Lemma 2.2 and (44), we have

$$ \bigl\Vert \varphi_{1}(r) \bigr\Vert _{H}\leq M_{0}(\theta_{r-\tau }\omega), \qquad\bigl\Vert \varphi_{2}(r) \bigr\Vert _{H}\leq M_{0}( \theta_{r-\tau}\omega ),\quad \forall r\geq \tau-t $$
(46)

and

$$ \sum_{ \vert i \vert \geq2I} \bigl\vert \varphi_{i} \bigl(r,\tau-t,\theta_{-\tau }\omega,\varphi _{\tau-t}( \theta_{-\tau}\omega) \bigr) \bigr\vert _{H}^{2} \leq\nu_{0}+ \biggl( \frac{c_{6}}{I}+\gamma_{I} \biggr) K_{2}(\theta_{r-\tau}\omega),\quad I\in \mathbb{N}, r\geq\tau-t. $$
(47)

Lemma 2.3

For every \(\tau\in\mathbb{R}\), \(t\geq0\), \(\omega\in\varOmega\) and \(\varphi _{j,\tau-t}(\theta_{-\tau}\omega)\in\mathcal{X}_{1}(\tau-t,\theta _{-t}\omega)\), \(j=1\), 2, there exist random variables \(C_{2}(\omega)>0\), \(C_{3}(\omega)>0\), such that

$$\begin{aligned} & \bigl\Vert \varphi(\tau,\tau-t,\theta_{-\tau}\omega)\varphi _{1,\tau -t}(\theta_{-\tau}\omega)-\varphi(\tau,\tau-t, \theta_{-\tau }\omega )\varphi_{2,\tau-t}(\theta_{-\tau} \omega) \bigr\Vert _{H} \\ &\quad\leq e^{\int_{-t}^{0}C_{2}(\theta_{s}\omega)\,ds} \bigl\Vert \varphi _{1,\tau -t}( \theta_{-\tau}\omega)-\varphi_{2,\tau-t}(\theta_{-\tau}\omega ) \bigr\Vert _{H} \end{aligned}$$
(48)

and

$$\begin{aligned} &\sum_{ \vert i \vert >4I} \bigl\vert \bigl(\varphi \bigl(r, \tau-t,\theta_{-\tau }\omega,\varphi_{1,\tau -t}(\theta_{-\tau} \omega) \bigr)-\varphi \bigl(r,\tau-t,\theta_{-\tau}\omega , \varphi_{2,\tau-t}(\theta_{-\tau}\omega) \bigr) \bigr)_{i} \bigr\vert _{H}^{2} \\ &\quad\leq \biggl( e^{-\frac{\varepsilon}{4}t+\int_{-t}^{0}C_{3}(\theta _{s}\omega)\,ds}+\frac{\hat{\delta}_{I}}{2}e^{\int_{-t}^{0}C_{2}(\theta _{s}\omega)\,ds} \biggr) ^{2} \bigl\Vert \psi(\tau-t) \bigr\Vert _{H}^{2},\quad \forall I\in\mathbb{N}, \end{aligned}$$
(49)

where

$$ \hat{\delta}_{I}=\frac{2}{\sqrt[4]{\varepsilon}}\sqrt{ \frac {1}{I}+\gamma _{I}+\frac{1}{I^{p}}+ \gamma_{I}^{p}}. $$

Proof

Taking the inner product \((\cdot,\cdot)_{H}\) of (45) with \(\psi(r)\), we find that

$$\begin{aligned} &\frac{d \Vert \psi(r) \Vert _{H}^{2}}{dt} \\ &\quad\leq \biggl( -\varepsilon+2 \biggl(1+\frac{\varepsilon}{\sqrt{\underline{\lambda}}} \biggr) \vert c \vert \bigl\vert z(\theta_{r-\tau }\omega) \bigr\vert + \biggl(\frac{8a_{5}^{2}}{\underline{\lambda}\alpha}+\frac {1}{\sqrt{\underline{\lambda}}} \biggr) \vert c \vert ^{2} \bigl\vert z(\theta_{r-\tau }\omega) \bigr\vert ^{2} \biggr) \bigl\Vert \psi(r) \bigr\Vert _{H}^{2} \\ &\qquad{}+ \frac{2}{\alpha} \bigl\Vert f(u_{2},r)-f(u_{1},r) \bigr\Vert _{\rho}^{2}. \end{aligned}$$
(50)

By (A2) and (46), we have

$$ \bigl\Vert f(u_{2},r)-f(u_{1},r) \bigr\Vert _{\rho}^{2}=\sum_{i\in \mathbb{Z}}\rho _{i} \bigl\vert f_{i}(u_{2,i},r)-f_{i}(u_{1,i},r) \bigr\vert ^{2}\leq\frac {C_{0}(\theta_{r-\tau }\omega)}{\underline{\lambda}} \bigl\Vert \psi(r) \bigr\Vert _{H}^{2}, $$

where

$$ c_{10}=a_{1}^{2} \biggl( \frac{16}{\underline{\lambda}}+ \frac {2^{2(p-1)}}{\underline{\lambda}^{p-1}} \biggr),\qquad C_{0}(\omega )=c_{10}M_{0}^{2}( \omega) \bigl(1+M_{0}^{2(p-1)}(\omega) \bigr). $$
(51)

Thus,

$$ \frac{d \Vert \psi(r) \Vert _{H}^{2}}{dt}\leq \bigl( -\varepsilon +2C_{1}( \theta_{r-\tau}\omega) \bigr) \bigl\Vert \psi(r) \bigr\Vert _{H}^{2}, $$
(52)

where

$$ C_{1}(\omega)= \biggl( 1+\frac{\varepsilon}{\sqrt{\underline{\lambda}}} \biggr) \vert c \vert \bigl\vert z( \omega) \bigr\vert + \biggl( \frac {4a_{5}^{2}}{\underline{\lambda}\alpha}+\frac{1}{2\sqrt{\underline{\lambda}}} \biggr) \vert c \vert ^{2} \bigl\vert z( \omega) \bigr\vert ^{2}+\frac{C_{0}(\omega )}{\alpha \underline{\lambda}}. $$
(53)

By applying the Gronwall inequality to (52) on \([\tau-t,\tau ]\), we have

$$ \bigl\Vert \psi(\tau) \bigr\Vert _{H}^{2}\leq e^{\int_{-t}^{0} ( -\varepsilon +2C_{1}(\theta_{s}\omega) ) \,ds} \bigl\Vert \psi(\tau-t) \bigr\Vert _{H}^{2} \leq e^{2\int_{-t}^{0}C_{1}(\theta_{s}\omega)\,ds} \bigl\Vert \psi(\tau -t) \bigr\Vert _{H}^{2}. $$
(54)

Let \(I\in\mathbb{N}\) and M be a suitable large integer, and set

$$\omega =(\omega_{i})_{i\in\mathbb{Z}}= \biggl( \eta \biggl( \frac{ \vert i \vert }{M} \biggr) \xi _{i}{, }\eta \biggl( \frac{ \vert i \vert }{M} \biggr) \zeta _{i} \biggr) =(p_{i},q_{i}). $$

Taking the inner product \((\cdot,\cdot)_{H}\) of (45) with ω, we find that, for \(r\geq\tau-t\),

$$ (\dot{\psi},\omega)_{H}+ ( L\psi,\omega ) _{H}= \bigl( F \bigl(\varphi _{1}(r),\theta_{r-\tau}\omega \bigr)-F \bigl( \varphi_{2}(r),\theta_{r-\tau }\omega \bigr),\omega \bigr) _{H}. $$
(55)

Similar to (30)–(35), and by (A2), we obtain, for \(r\geq\tau-t\),

$$\begin{aligned} &2 \bigl(f(u_{2},r)-f(u_{1},r),q \bigr)_{\rho} \\ &\quad \leq\frac{\alpha}{4}\sum_{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{M} \biggr) \rho_{i} \vert \zeta_{i} \vert ^{2}+\frac{16a_{1}^{2}}{\alpha }\sum _{i\in \mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{M} \biggr) \rho _{i} \bigl( \vert u_{1,i} \vert ^{2}+ \vert u_{2,i} \vert ^{2} \bigr) \vert \xi_{i} \vert ^{2} \\ &\qquad{}+\frac{16a_{1}^{2}}{\alpha}\sum_{i\in\mathbb{Z}}\eta \biggl( \frac { \vert i \vert }{M} \biggr) \rho_{i} \bigl( \vert u_{1,i} \vert ^{2p}+ \vert u_{2,i} \vert ^{2p} \bigr) \vert \xi_{i} \vert ^{2}. \end{aligned}$$

By (47),

$$\begin{aligned} &\sum_{ \vert i \vert \geq 2I} \bigl[ \vert u_{1,i} \vert ^{2}+ \vert u_{2,i} \vert ^{2} \bigr] \leq \frac{2}{\underline{\lambda }\overline{\rho}}\nu_{0}+\frac{2}{\underline{\lambda}\overline{\rho }} \biggl(\frac{c_{6}}{I}+\gamma_{I} \biggr)K_{2}( \theta_{r-\tau}\omega), \\ &\sum_{ \vert i \vert \geq2I} \bigl[ \vert u_{1,i} \vert ^{2p}+ \vert u_{2,i} \vert ^{2p} \bigr] \leq \frac{2^{p}}{\underline{\lambda}^{p}\overline{\rho}^{p}}\nu_{0}^{p}+\frac{2^{p}}{\underline{\lambda}^{p}\overline{\rho}^{p}} \biggl( \frac{c_{6}}{I}+\gamma _{I} \biggr)^{p}K_{2}^{p}( \theta_{r-\tau}\omega). \end{aligned}$$

Then we have, for \(M\geq2I\),

$$\begin{aligned} &2 \bigl(f(u_{2},r)-f(u_{1},r),q \bigr)_{\rho} \\ &\quad\leq\frac{\alpha}{4}\sum_{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{M} \biggr) \rho_{i} \vert \zeta_{i} \vert ^{2}+ \biggl( \frac {32a_{1}^{2}\nu_{0}}{\alpha \underline{\lambda}^{2}\overline{\rho}}+ \frac{a_{1}^{2}2^{p+4}\nu _{0}^{p}}{\alpha\underline{\lambda}^{p+1}\overline{\rho}^{p}} \biggr) \sum_{i\in \mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{M} \biggr) \vert \psi _{i} \vert _{H}^{2} \\ &\qquad{}+\frac{32a_{1}^{2}}{\alpha\underline{\lambda}\overline{\rho}} \biggl(\frac{c_{6}}{I}+\gamma_{I} \biggr)K_{2}(\theta_{r-\tau}\omega)\sum _{i\in \mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{M} \biggr) \rho_{i} \vert \xi _{i} \vert ^{2} \\ &\qquad{}+\frac{a_{1}^{2}2^{p+4}}{\alpha\underline{\lambda}^{p}\overline {\rho}^{p}} \biggl(\frac{c_{6}}{I}+\gamma_{I} \biggr)^{p}K_{2}^{p}(\theta _{r-\tau}\omega )\sum_{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{M} \biggr) \rho _{i} \vert \xi _{i} \vert ^{2}, \end{aligned}$$
(56)

where \(| \psi_{i}| _{H}^{2}=\rho_{i}[\delta(D\xi)_{i}^{2}+\lambda _{i}\xi_{i}^{2}+\zeta_{i}^{2}]\). Thus, by (43) and the above inequality, we have, for \(M\geq2I\),

$$\begin{aligned} &\frac{d}{dt}\sum_{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{M} \biggr) \bigl\vert \psi _{i}(r) \bigr\vert _{H}^{2}+\frac{\varepsilon}{2}\sum _{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{M} \biggr) \bigl\vert \psi_{i}(r) \bigr\vert _{H}^{2} \\ &\quad\leq \biggl( \biggl(2+\frac{2\varepsilon}{\sqrt{\underline{\lambda}}} \biggr) \vert c \vert \bigl\vert z(\theta_{r-\tau }\omega) \bigr\vert + \biggl(\frac{8a_{5}^{2}}{\underline{\lambda}\alpha}+ \frac{1}{\sqrt{\underline{\lambda}}} \biggr) \vert c \vert ^{2} \bigl\vert z(\theta_{r-\tau}\omega) \bigr\vert ^{2} \biggr) \sum _{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{M} \biggr) \bigl\vert \psi_{i}(r) \bigr\vert _{H}^{2} \\ &\qquad{}+c_{11} \biggl( \frac{1}{I}+\gamma_{I}+ \frac{1}{I^{p}}+\gamma _{I}^{p} \biggr) \cdot{} \bigl[ \xi_{1}(\theta_{r-\tau}\omega )+K_{2}^{p}( \theta_{r-\tau}\omega) \bigr] \\ &\qquad{}\times e^{2\int_{r}^{\tau -t}C_{2}(\theta_{s-\tau}\omega)\,ds} \bigl\Vert \psi(\tau-t) \bigr\Vert _{H}^{2}, \end{aligned}$$
(57)

where

$$ c_{11}=\max \biggl\{ \frac{1}{2},\frac{32a_{1}^{2}}{\alpha\underline {\lambda }^{2}\overline{\rho}}, \frac{32c_{6}a_{1}^{2}}{\alpha\underline {\lambda}^{2}\overline{\rho}},\frac{2^{2p+3}c_{6}a_{1}^{2}}{\alpha\underline{\lambda}^{p+1}\overline{\rho}^{p}},\frac{2^{2p+3}a_{1}^{2}}{\alpha \underline{\lambda}^{p+1}\overline{\rho}^{p}} \biggr\} . $$

By applying the Gronwall inequality to (57) on \([\tau-t,\tau ]\) \((t\geq 0)\), we obtain

$$\begin{aligned} &\sum_{i\in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{M} \biggr) \bigl\vert \psi_{i}(\tau ) \bigr\vert _{H}^{2} \\ &\quad\leq \bigl\Vert \psi(\tau-t) \bigr\Vert _{E}^{2}e^{\int _{-t}^{0} [ -\frac{\varepsilon}{2}+2 (1+\frac{\varepsilon}{\sqrt{\underline{\lambda }}} ) \vert c \vert \vert z(\theta_{s}\omega) \vert + ( \frac {8a_{5}^{2}}{\underline{\lambda}\alpha}+\frac{1}{\sqrt{\underline{\lambda}}} ) \vert c \vert ^{2} \vert z(\theta _{s}\omega) \vert ^{2} ] \,ds} \\ &\qquad{}+ \biggl( \frac{1}{I}+\gamma_{I}+ \frac{1}{I^{p}}+ \gamma_{I}^{p} \biggr) \bigl\Vert \psi(\tau-t) \bigr\Vert _{H}^{2} \\ &\qquad{} \times e^{\int_{-t}^{0}2 [ C_{1}(\theta_{s}\omega )+ (1+\frac{\varepsilon}{\sqrt{\underline{\lambda}}} ) \vert c \vert \vert z(\theta _{s}\omega) \vert + ( \frac{4a_{5}^{2}}{\underline{\lambda}\alpha }+\frac{1}{2\sqrt{\underline{\lambda}}} ) \vert c \vert ^{2} \vert z(\theta _{s}\omega) \vert ^{2} ] \,ds} \\ &\qquad{} \times \int_{-t}^{0}c_{11} \bigl( \xi_{1}(\theta_{r}\omega )+K_{2}^{p}( \theta _{r}\omega) \bigr)e^{\frac{\varepsilon}{2}r}\,dr. \end{aligned}$$
(58)

Since \(\sqrt{x}\leq e^{x}\) for all \(x\geq0\), it follows that

$$\begin{aligned} & \int_{-t}^{0}c_{11} \bigl( \xi_{1}(\theta_{r}\omega)+K_{2}^{p}( \theta _{r}\omega) \bigr)e^{\frac{\varepsilon}{2}r}\,dr \\ &\quad\leq \biggl[ \int_{-t}^{0}c_{11}^{2} \bigl( \xi_{1}(\theta_{r}\omega )+K_{2}^{p}( \theta_{r}\omega) \bigr)^{2}\,dr \biggr] ^{\frac{1}{2}} \biggl[ \int_{-t}^{0}e^{\varepsilon r}\,dr \biggr] ^{\frac{1}{2}} \\ &\quad\leq\frac{1}{\sqrt{\varepsilon}}e^{\int_{-t}^{0}c_{11}^{2}(\xi _{1}(\theta_{r}\omega)+K_{2}^{p}(\theta_{r}\omega))^{2}\,dr}. \end{aligned}$$

By \(M\geq2I\) and (58),

$$\begin{aligned} &\sum_{ \vert i \vert >4I} \bigl\vert \psi_{i}( \tau) \bigr\vert _{H}^{2} \\ &\quad\leq\sum_{ \vert i \vert \in\mathbb{Z}}\eta \biggl( \frac{ \vert i \vert }{M} \biggr) \bigl\vert \psi_{i}(\tau) \bigr\vert _{H}^{2} \\ &\quad\leq \bigl( e^{\int_{-t}^{0}2(-\frac{\varepsilon}{4}+C_{3}(\theta_{s}\omega ))\,ds}+\hat{\delta}_{I}^{2}e^{2\int_{-t}^{0}C_{2}(\theta_{s}\omega)\,ds} \bigr) \bigl\Vert \psi(\tau-t) \bigr\Vert _{H}^{2}, \end{aligned}$$

where

$$\begin{aligned} &C_{2}(\omega)=C_{1}(\omega)+C_{3}( \omega)+c_{11}^{2} \bigl(\xi _{1}^{2}( \omega)+K_{2}^{2p}(\omega) \bigr), \end{aligned}$$
(59)
$$\begin{aligned} &C_{3}(\omega)= \biggl( 1+\frac{\varepsilon}{\sqrt{\underline{\lambda }}} \biggr) \vert c \vert \bigl\vert z(\omega) \bigr\vert + \biggl( \frac {4a_{5}^{2}}{\underline{\lambda}\alpha}+\frac{1}{2\sqrt{\underline{\lambda}}} \biggr) \vert c \vert ^{2} \bigl\vert z(\omega) \bigr\vert ^{2}, \end{aligned}$$
(60)

thus, (49) holds. By (54) and \(C_{2}(\omega)\geq C_{1}(\omega)\), it follows that (48) holds. The proof is completed. □

Lemma 2.4

Let the coefficient c of the random term in (2) satisfy

$$ \vert c \vert ^{2}< \min \biggl\{ \frac{\mu^{2}\pi\alpha }{64^{2}a_{7}^{2}}, \frac{\alpha\mu}{32a_{7}},\frac{\alpha^{2}}{64pa_{7}},\frac{\alpha^{3}}{32^{2}p^{2}a_{7}^{2}} \biggr\} ,\quad \mu= \min \biggl\{ \varepsilon,\frac{\varepsilon}{2a_{2}},1 \biggr\} , $$
(61)

then

$$ 0\leq\mathbf{E} \bigl[C_{3}(\omega) \bigr]\leq\frac{\varepsilon}{64}, \qquad 0 \leq \mathbf{E} \bigl[C_{2}(\omega) \bigr],\mathbf{E} \bigl[C_{2}^{2}( \omega) \bigr]< \infty. $$
(62)

Proof

From [13, 16], the Ornstein–Uhlenbeck stationary process \(z(\theta_{t}\omega)\) satisfies

$$\begin{aligned} &\mathbf{E} \bigl[ \bigl\vert z(\theta_{s}\omega) \bigr\vert ^{r} \bigr] =\frac {\varGamma(\frac{1+r}{2})}{\sqrt{\pi\alpha^{r}}} \quad(\forall r\geq0), \end{aligned}$$
(63)
$$\begin{aligned} &\textstyle\begin{cases} \mathbf{E}[e^{\zeta\int_{\tau}^{\tau+t} \vert z(\theta_{s}\omega ) \vert ^{2}\,ds}]\leq e^{\frac{\zeta}{\alpha}t},\quad 0\leq2\zeta\leq\alpha ^{2},\tau\in\mathbb{R},t\geq0, \\ \mathbf{E}[e^{\zeta\int_{\tau}^{\tau+t} \vert z(\theta_{s}\omega ) \vert \,ds}]\leq e^{\frac{\zeta}{\sqrt{\alpha}}t},\quad 0\leq\zeta^{2}\leq\alpha^{3},\tau \in \mathbb{R},t\geq0,\end{cases}\displaystyle \end{aligned}$$
(64)

where \(\varGamma(\cdot)\) is the Gamma function. It is easy to see that (61) implies (8). By (3), (59) and (60), we obtain

$$ \mathbf{E} \bigl[C_{3}(\omega) \bigr]\leq\frac{a_{7} \vert c \vert }{2\sqrt{\pi \alpha}}+ \frac{a_{7} \vert c \vert ^{2}}{4\alpha}\leq\frac{1}{16}\frac{\varepsilon}{4}. $$
(65)

By (7), (61) and (63)–(64),

$$\begin{aligned} \mathbf{E} \bigl[ M_{0}^{4p}(\omega) \bigr] &\leq c_{5}^{2p} \biggl( \int_{-\infty}^{0}e^{\frac{p\mu}{2p-1}s}\,ds \biggr) ^{2p-1} \int_{-\infty }^{0}e^{p\mu s}\mathbf{E}e^{2pa_{7}\int_{s}^{0}( \vert c \vert \vert z(\theta_{l}\omega ) \vert + \vert c \vert ^{2} \vert z \vert ^{2}(\theta_{l}\omega))\,dl} \,ds \\ &\leq c_{5}^{2p} \biggl( \frac{2p-1}{p\mu} \biggr) ^{2p-1} \int_{-\infty }^{0}e^{p\mu s} \bigl( \mathbf{E}e^{4pa_{7} \vert c \vert \int _{s}^{0} \vert z(\theta _{l}\omega) \vert \,dl}+\mathbf{E}e^{4pa_{7} \vert c \vert ^{2}\int_{s}^{0} \vert z \vert ^{2}(\theta _{l}\omega)\,dl} \bigr) \,ds \\ &\leq\frac{c_{5}^{2p}}{p} \biggl( \frac{2p-1}{p\mu} \biggr) ^{2p-1} \biggl( \frac{1}{\mu-\frac{4 \vert c \vert a_{7}}{\sqrt{\alpha}}}+\frac{1}{\mu -\frac{4 \vert c \vert ^{2}a_{7}}{\alpha}} \biggr) =k_{1}. \end{aligned}$$
(66)

Then, by (51), (53) and (66),

$$\begin{aligned} \mathbf{E} \bigl[C_{1}^{2}(\omega) \bigr] \leq{}&3\mathbf{E} \biggl( \biggl(1+\frac {\varepsilon }{\sqrt{\underline{\lambda}}} \biggr)^{2} \vert c \vert ^{2} \bigl\vert z(\omega) \bigr\vert ^{2}+ \biggl( \frac{4a_{5}^{2}}{\underline{\lambda}\alpha}+\frac{1}{2\sqrt{\underline{\lambda}}} \biggr)^{4} \vert c \vert ^{4} \bigl\vert z(\omega) \bigr\vert ^{4} \biggr) \\ &{}+3\mathbf{E} \biggl(\frac{2c_{10}^{2}}{\alpha^{2}\underline{\lambda}^{2}} \bigl(M_{0}^{4}( \omega)+M_{0}^{4p}(\omega) \bigr) \biggr) \\ \leq{}&3 \biggl( \biggl(1+\frac{\varepsilon}{\sqrt{\underline{\lambda }}} \biggr)^{2} \frac{ \vert c \vert ^{2}}{2\alpha}+ \biggl(\frac{4a_{6}^{2}}{\underline {\lambda}\alpha}+\frac{1}{2\sqrt{\underline{\lambda}}} \biggr)^{4}\frac{\varGamma(\frac{5}{2}) \vert c \vert ^{4}}{\alpha ^{2}\sqrt{\pi}} \biggr) \\ &{}+3 \biggl(\frac{2c_{10}^{2}}{\alpha^{2}\underline{\lambda}^{2}} \bigl(M_{0}^{4}( \omega)+M_{0}^{4p}(\omega) \bigr) \biggr) =k_{2}. \end{aligned}$$
(67)

Similar to (66), we have

$$\begin{aligned} \mathbf{E} \bigl[K_{0}^{4p}(\omega) \bigr]&=\mathbf{E} \biggl( \int_{-\infty }^{0}e^{\mu s+a_{7}\int_{s}^{0}[ \vert c \vert \vert z(\theta_{l}\omega) \vert + \vert c \vert ^{2} \vert z \vert ^{2}(\theta_{l}\omega )]\,dl}\,ds \biggr) ^{4p} \\ &\leq \biggl( \frac{4p-1}{2p\mu} \biggr) ^{4p-1} \int_{-\infty }^{0}e^{2p\mu s} \bigl( \mathbf{E}e^{8a_{7}p \vert c \vert \int_{s}^{0} \vert z(\theta _{l}\omega) \vert \,dl}+\mathbf{E}e^{8a_{7}pc^{2}\int_{s}^{0} \vert z \vert ^{2}(\theta _{l}\omega)\,dl} \bigr) \,ds \\ &\leq\frac{1}{2p} \biggl( \frac{4p-1}{2p\mu} \biggr) ^{4p-1} \biggl( \frac {1}{\mu-\frac{4 \vert c \vert a_{7}}{\sqrt{\alpha}}}+\frac{1}{\mu-\frac {4 \vert c \vert ^{2}a_{7}}{\alpha}} \biggr) =k_{3}. \end{aligned}$$
(68)

From (28),

$$\begin{aligned} &\mathbf{E} \bigl[\xi_{1}^{4}(\omega) \bigr]\leq\mathbf{E} \bigl[ c_{12} \bigl(1+ \bigl\vert z(\omega ) \bigr\vert ^{4} \bigr) \bigr] \leq c_{12} \biggl( 1+ \frac{\varGamma ( \frac {5}{2} ) }{\alpha^{2}\sqrt{\pi}} \biggr) =k_{4}, \end{aligned}$$
(69)
$$\begin{aligned} &\mathbf{E} \bigl[\xi_{1}^{16p}(\omega) \bigr]\leq\mathbf{E} \bigl[ c_{13} \bigl(1+ \bigl\vert z(\omega ) \bigr\vert ^{16p} \bigr) \bigr] \leq c_{13} \biggl( 1+ \frac{\varGamma ( \frac {16p+1}{2} ) }{\alpha^{8p}\sqrt{\pi}} \biggr) =k_{5}. \end{aligned}$$
(70)

Thus, by (28) and (70),

$$\begin{aligned} \mathbf{E} \bigl[K_{1}^{8p}(\omega) \bigr]={}&\mathbf{E} \biggl( \int_{-\infty }^{0}\xi _{1}( \theta_{s}\omega)e^{\mu s+a_{7}\int_{s}^{0}( \vert c \vert \vert z(\theta_{l}\omega ) \vert + \vert c \vert ^{2} \vert z(\theta_{l}\omega) \vert ^{2})\,dl}\,ds \biggr) ^{8p} \\ \leq{}& \biggl( \int_{-\infty}^{0}e^{\frac{4p}{8p-1}\mu s}\,ds \biggr) ^{8p-1}\mathbf{E} \int_{-\infty}^{0}\xi_{1}^{8p}( \theta_{s}\omega)e^{4p\mu s+8pa_{7}\int_{s}^{0}( \vert c \vert \vert z(\theta_{l}\omega) \vert + \vert c \vert ^{2} \vert z(\theta_{l}\omega ) \vert ^{2})\,dl}\,ds \\ \leq{}& \biggl( \frac{8p-1}{4p\mu} \biggr) ^{8p-1}\times \int_{-\infty }^{0}e^{4p\mu s} \bigl( \mathbf{E}e^{32pa_{7} \vert c \vert \int _{s}^{0} \vert z(\theta _{l}\omega) \vert \,dl}+\mathbf{E}e^{32pa_{7} \vert c \vert ^{2}\int _{s}^{0} \vert z(\theta _{l}\omega) \vert ^{2}\,dl} \bigr) \,ds \\ &{}+ \biggl( \frac{8p-1}{4p\mu} \biggr) ^{8p-1} \int_{-\infty}^{0}e^{4p\mu s}\mathbf{E} \xi_{1}^{16p}(\theta_{s}\omega)\,ds \\ \leq{}&\frac{1}{4p} \biggl( \frac{8p-1}{4p\mu} \biggr) ^{8p-1} \biggl( \frac {1}{\mu-\frac{8a_{7} \vert c \vert }{\sqrt{\alpha}}}+\frac{1}{\mu-\frac {8a_{7} \vert c \vert ^{2}}{\alpha}}+\frac{k_{5}}{\mu} \biggr) =k_{6}, \end{aligned}$$
(71)
$$\begin{aligned} \mathbf{E} \bigl[K_{2}^{4p}(\omega) \bigr] \leq{}&2^{4p-1} \bigl( \mathbf{E} \bigl[K_{0}^{4p}( \omega) \bigr]+ \mathbf{E} \bigl[K_{0}^{8p}(\omega) \bigr]+ \mathbf{E} \bigl[K_{1}^{8p}(\omega) \bigr] \bigr) =k_{7}. \end{aligned}$$
(72)

By (59), (67), (69) and (72), we have

$$ 0\leq\mathbf{E} \bigl[C_{2}^{2}(\omega) \bigr]\leq c_{14} ( 1+k_{2}+k_{4}+k_{7} ) < \infty. $$

Since \(0\leq\mathbf{E}[C_{2}(\omega)]\leq\frac{1}{2} [ 1+\mathbf {E}(C_{2}^{2}(\omega)) ] \), it follows that \(0\leq\mathbf{E} [ C_{2}(\omega) ] <\infty\). The proof is completed. □

Lemma 2.5

For any \(\tau\in\mathbb{R}\), \(\omega\in\varOmega\),

$$ \textstyle\begin{cases} \lim_{t\rightarrow0^{+}}\sup_{\varphi\in\mathcal{X}_{1}(\tau,\omega )} \Vert \varPhi(t,\tau,\omega)\varphi-\varphi \Vert _{H}=0, \\ \lim_{t\rightarrow0^{+}}\sup_{\varphi\in\mathcal{X}_{1}(\tau -t,\theta _{-t}\omega)} \Vert \varPhi(0,\tau-t,\theta_{-t}\omega)\varphi -\varphi \Vert _{H}=0.\end{cases} $$

Proof

By (5), (6) and (44), it follows that, for \(\varphi\in\mathcal{X}_{1}(\tau,\omega)\) and \(t\geq 0 \),

$$\begin{aligned} & \bigl\Vert F \bigl(\varphi(r),\theta_{r-\tau}\omega \bigr) \bigr\Vert _{H}^{2} \\ &\quad\leq \vert c \vert ^{2} \bigl\vert z(\theta_{r-\tau} \omega) \bigr\vert ^{2}M_{0}^{2}( \theta_{r-\tau }\omega)+\frac{5(2c\varepsilon z(\theta_{r-\tau}\omega)- \vert c \vert ^{2} \vert z(\theta _{r-\tau}\omega) \vert ^{2})^{2}}{\sqrt{\underline{\lambda }}}M_{0}^{2}( \theta _{r-\tau}\omega) \\ &\qquad{}+5 \vert c \vert ^{2} \bigl\vert z( \theta_{r-\tau} \omega) \bigr\vert ^{2}M_{0}^{2}( \theta_{r-\tau }\omega)+5 \vert c \vert ^{2}a_{5}^{2} \bigl\vert z(\theta_{r-\tau}\omega ) \bigr\vert ^{2}M_{0}^{2}( \theta _{r-\tau}\omega) \\ &\qquad{}+5a_{1}^{2} \bigl(1+M_{0}^{p-1}( \theta_{r-\tau}\omega ) \bigr)^{2}M_{0}^{4}( \theta _{r-\tau}\omega)+5 \Vert g \Vert _{\rho}^{2} \end{aligned}$$
(73)

and

$$\begin{aligned} \bigl\Vert L\varphi(r) \bigr\Vert _{H}^{2} \leq{}& \Vert \varepsilon u-v \Vert _{\delta ,\lambda,\rho}^{2}+5 \lambda^{2} \Vert u \Vert _{\rho }^{2}+5 \vert \varepsilon \vert (\alpha-\varepsilon)| \Vert u \Vert _{\rho}^{2} \\ &{}+5\delta \Vert Au \Vert _{\rho}^{2}+5(\alpha- \varepsilon)^{2} \Vert v \Vert _{\rho }^{2}+5 \gamma \Vert Av \Vert _{\rho}^{2} \\ \leq{}&c_{15}M_{0}^{2}(\theta_{r-\tau} \omega), \end{aligned}$$
(74)

therefore,

$$\begin{aligned} & \bigl\Vert \varPhi(t,\tau,\omega)\varphi-\varphi \bigr\Vert _{H}^{2} \\ &\quad\leq t \int_{\tau}^{\tau+t} \bigl\Vert F \bigl(\varphi(r), \theta_{r-\tau}\omega \bigr)-L\varphi(r) \bigr\Vert _{H}^{2} \,dr \\ &\quad\leq 2t \int_{0}^{t} \biggl[ \frac{5(2c\varepsilon z(\theta_{r}\omega )- \vert c \vert ^{2} \vert z(\theta_{r}\omega) \vert ^{2})^{2}}{\sqrt {\underline{\lambda}}}+c_{16} \vert c \vert ^{2} \bigl\vert z(\theta_{r}\omega) \bigr\vert ^{2} \biggr]M_{0}^{2}( \theta_{r}\omega)\,dr \\ &\qquad{}+2t \int_{0}^{t} \bigl[5a_{1}^{2} \bigl(1+M_{0}^{p-1}(\theta_{r}\omega ) \bigr)^{2}M_{0}^{4}(\theta_{r}\omega)+5 \Vert g \Vert _{\rho }^{2}+c_{15}M_{0}^{2}( \theta_{r}\omega) \bigr]\,dr \\ &\quad\rightarrow 0\quad\text{as }t\rightarrow0, \end{aligned}$$
(75)

where \(c_{16}=6+5a_{5}^{2}\), thus, \(\lim_{t\rightarrow0^{+}}\sup_{\varphi \in\mathcal{X}_{1}(\tau,\omega)} \Vert \varPhi(t,\tau,\omega)\varphi -\varphi \Vert _{H}=0\).

Similarly, \(\lim_{t\rightarrow0^{+}}\sup_{\varphi\in\mathcal {X}_{1}(\tau -t,\theta_{-t}\omega)} \Vert \varPhi(0,\tau-t,\theta_{-t}\omega )\varphi -\varphi \Vert _{H}=0\). The proof is completed. □

From Theorem 2.1 and Lemmas 2.32.5, we have the following result.

Theorem 2.3

Assume that (A1)–(A4) and (43), (61) hold. Then the continuous cocycle \(\{\varPhi(t,\tau,\omega)\}_{t\geq0,\tau\in\mathbb{R},\omega\in \varOmega}\) possesses a random exponential attractor \(\{\mathcal{K}(\tau ,\omega)\}_{\tau\in\mathbb{R},\omega\in\varOmega}\) with properties: for any \(\tau\in\mathbb{R}\), \(\omega\in\varOmega\),

  1. (i)

    \(\mathcal{R}(\tau,\omega)\subseteq\mathcal{K}(\tau,\omega )\subseteq\overline{\mathcal{X}_{1}(\tau,\omega)}\) and \(\mathcal{K}(\tau,\omega)\) is a compact set of H and measurable in ω.

  2. (ii)

    \(\varPhi(t,\tau,\omega)\mathcal{K}(\tau,\omega)\subseteq \mathcal{K}(t+\tau,\theta_{t}\omega)\) for all \(t\geq0\).

  3. (iii)

    There exists \(I_{0}=\min\{I:0<\hat{\delta}_{I}\leq \widetilde{\gamma}\}+1\in N\), where

    $$ \widetilde{\gamma}=\min \biggl\{ \frac{1}{16},e^{-\frac{2}{\ln\frac {3}{2}} ( \frac{32\ln2}{\boldsymbol{\varepsilon}} ) ^{2} ( \mathbf {E}[C_{2}^{2}(\omega)]+\frac{\varepsilon}{4}\mathbf{E}[C_{2}(\omega )] ) } \biggr\} , $$

    such that

    $$ \dim_{f}\mathcal{R}(\tau,\omega)\leq\dim_{f}\mathcal{K}( \tau ,\omega )\leq\frac{4(8I_{0}+1)\ln (\frac{2\sqrt{16I_{0}+2}}{\hat{\delta }_{I_{0}}}+1 )}{\ln\frac{4}{3}}< \infty. $$
  4. (iv)

    For every set \(D\in\mathcal{D}(H)\), there exist a random variable \(T_{\omega,D}\geq0\) and a tempered random variable \(b_{\omega,D}>0\) such that

    $$ \mathrm{d}_{h} \bigl( \varPhi(t,\tau,\omega)D(\tau,\omega), \mathcal{K}(t+\tau,\theta_{t}\omega) \bigr) \leq b_{\omega,D}e^{-\frac {\varepsilon \ln\frac{4}{3}}{128\ln2}t},\quad t\geq T_{\omega,D}. $$
  5. (v)

    For any \(\tau\in\mathbb{R}\), \(\omega\in\varOmega\), \(\lim_{t\rightarrow0}\mathrm{d}_{h}(\mathcal{K}(\tau+t,\theta _{t}\omega),\mathcal{K}(\tau,\omega))=0\).

Proof

Taking \(t=\frac{32\ln2}{\boldsymbol{\varepsilon}}\) in (48) and (49). From (62),

$$ 0< e^{-\frac{2}{\ln\frac{3}{2}} ( \frac{32\ln2}{\boldsymbol{\varepsilon}} ) ^{2} ( \mathbf{E}[C_{2}^{2}(\omega)]+\frac{\varepsilon}{4} \mathbf{E}[C_{2}(\omega)] ) }< \infty. $$
(76)

By \(\lim_{I\rightarrow+\infty}\frac{1}{I}=\lim_{I\rightarrow+\infty} \frac{1}{I^{p+1}}=0\) and \(g(t)\), \(\beta(t)\), \(\beta^{\prime}(t)\in \mathbf{G}\), it follows that \(\lim_{I\rightarrow+\infty}\gamma _{I}=0\) and

$$ 0< \hat{\delta}_{I}=\frac{2}{\sqrt[4]{\varepsilon}}\sqrt{ \frac {1}{I}+\gamma _{I}+\frac{1}{I^{p+1}}+ \gamma_{I}^{p+1}}\rightarrow0\quad\text{as }I \rightarrow+\infty. $$
(77)

Hence there exists a finite integer \(I_{0}\in\mathbb{N}\), such that

$$ 0< \hat{\delta}_{I_{0}}\leq\min \biggl\{ \frac{1}{16},e^{-\frac{2}{\ln \frac{3}{2}} ( \frac{32\ln2}{\boldsymbol{\varepsilon}} ) ^{2} ( \mathbf{E}[C_{2}^{2}(\omega)]+\frac{\varepsilon}{4}\mathbf{E}[C_{2}(\omega )] ) } \biggr\} . $$
(78)

Finally, by Theorem 2.1, the statements in Theorem 2.3 hold. The proof is completed. □

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Acknowledgements

The authors would like to express their sincere thanks to the anonymous referees for their time and comments.

Funding

This work is supported by the National Natural Science Foundation of China under Grant Nos. 11471290, 11871437.

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All authors contributed equally and significantly in writing this paper and typed, read, and approved the final manuscript.

Correspondence to Shengfan Zhou.

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MSC

  • 37L55
  • 60H15
  • 35B40

Keywords

  • Random exponential attractor
  • Stochastic lattice system
  • Multiplicative white noise
  • Cocycle
  • Weighted space of sequences