Random attractors for Ginzburg–Landau equations driven by difference noise of a Wiener-like process

Abstract

We consider a Wong–Zakai process, which is the difference of a Wiener-like process. We then prove that there are random attractors for non-autonomous Ginzburg–Landau equations driven by linear multiplicative noise in terms of Wong–Zakai process and Wiener-like process, respectively. Moreover, we establish the upper semi-continuity of random attractors as the size of difference noise tends to zero.

1 Introduction

Given a Wiener process, its δ-difference is called a Wong–Zakai process [40, 41]. Such difference noise was often used to study stochastic equations as an approximation of white noise [15, 17, 19, 28, 30, 31].

In this paper, we consider a so-called Wiener-like process. Let

$$ \varOmega = \mathcal{C}_{0}(\mathbb{R},\mathbb{R})={ \biggl\{ \omega \in C( \mathbb{R},\mathbb{R}):\omega (0)=0,\lim _{t\rightarrow \pm \infty }\frac{ \omega (t)}{t}=0\biggr\} } $$

(1.1)

and equip it with the Fréchet metric and the Borel σ-algebra $\mathcal{F}=\mathcal{B}(\varOmega )$. Then the shift $\theta _{t}:\varOmega \rightarrow \varOmega$, $\omega (\cdot )\mapsto \omega (\cdot +t)-\omega (t)$ is measurable for each $t\in \mathbb{R}$.

We then take an arbitrary probability measure P on the measurable space $(\varOmega ,\mathcal{F})$. On the probability space $(\varOmega , \mathcal{F},P)$, we obtain a stochastic process given by $W(t, \omega )=\omega (t)$, which is called a Wiener-like process [21]. If P is a Wiener measure, then the corresponding process is just the standard Wiener process; see [1, 3,4,5, 7, 24, 44].

In other words, a Wiener-like process only satisfies the properties on the right-hand side of (1.1). We do not require other properties (such as increment independence and Gauss distribution) of a Wiener process.

For each $\delta >0$ (the case of $\delta <0$ is similar), the δ-difference of the Wiener-like process determines the Wong–Zakai process given by

$$ \mathcal{G}_{\delta }(t,\omega ):=\mathcal{G}_{\delta }( \theta _{t} \omega )=\frac{1}{\delta }\bigl(\omega (t+\delta )-\omega (t)\bigr),\quad \forall t \in \mathbb{R}, \omega \in \varOmega . $$

(1.2)

The difference process is not a Wiener-like process since $\mathcal{G}_{\delta }(0,\omega )=\omega (\delta )/\delta \neq 0$. However, by (1.1), we have $\mathcal{G}_{\delta }(\cdot ,\omega )\in C(\mathbb{R},\mathbb{R})$ and $\mathcal{G}_{\delta }(\theta _{t} \omega )/t\to 0$ as $t\to \pm \infty$.

Recently, Lu and Wang [27] (see also [13, 14, 38]) have studied both the existence and approximation of random attractors for the reaction–diffusion equation driven by difference noise of a Wiener process.

In this paper, we consider the complex Ginzburg–Landau equation perturbed by difference noise of a Wiener-like process:

$$\begin{aligned}& \frac{\partial u_{\delta }}{\partial t}-\bigl(\lambda +i\mu (t)\bigr)\Delta u _{\delta }=\gamma u_{\delta }-\bigl(\kappa +i\beta (t)\bigr) \vert u_{\delta } \vert ^{2}u _{\delta }+f(t,x)+u_{\delta }\mathcal{G}_{\delta }( \theta _{t}\omega ), \end{aligned}$$

(1.3)

$$\begin{aligned}& u_{\delta }(t,0)=u_{\delta }(t,1)=0, \qquad u_{\delta }(\tau ,x)=u_{\delta ,\tau }(x), \quad t\geq \tau , x\in \mathcal{I}, \end{aligned}$$

(1.4)

where $\mathcal{I}=(0,1)\subset \mathbb{R}$, $\lambda ,\gamma ,\kappa >0$, $\mu ,\beta \in {C}_{b}(\mathbb{R},\mathbb{R})$ and $f\in {L_{\mathrm{loc}} ^{2}}(\mathbb{R},\mathbb{L}^{2}(\mathcal{I}))$.

The first aim in this paper is to establish a random attractor $\mathcal{A}_{\delta }$ for the problem (1.3)–(1.4). In view of both the non-autonomous and the random nature, the attractor is actually a bi-parametric set formulated by $\mathcal{A}_{\delta }=\{ \mathcal{A}_{\delta }(\tau ,\omega )\}$ and called a pullback random attractor, which was first introduced by Crauel et al. [8] and by Wang [32] independently, with developments [2, 9, 10, 18, 20, 26, 36, 37, 42, 43, 45].

The second aim is to prove the upper semi-continuity of the attractors:

$$ \lim_{\delta \rightarrow 0}\operatorname{dist}_{\mathbb{L}^{2}(\mathcal{O})}\bigl( \mathcal{A}_{\delta }(\tau ,\omega ),\mathcal{A}_{0}(\tau ,\omega ) \bigr)=0, \quad \forall \tau \in \mathbb{R}, \omega \in \varOmega , $$

(1.5)

where $\mathcal{A}_{0}$ is the random attractor for the following limiting equation perturbed by the Wiener-like process:

$$ \frac{\partial u}{\partial t}-\bigl(\lambda +i\mu (t)\bigr)\Delta u=\gamma u- \bigl( \kappa +i\beta (t)\bigr) \vert u \vert ^{2}u+f(t,x)+u\circ \frac{dW}{dt} $$

(1.6)

with the same initial-boundary conditions as in (1.4).

By an abstract combined result on both existence and upper semi-continuity of random attractors, given by Li et al. [23] (also see [12]), we have to verify three aspects: (a) the convergence of the solution operators from Eqs. (1.3) to (1.6), (b) the equi-absorption of the systems for all small size δ of difference noise and (c) the equi-asymptotic compactness in small size.

It is worth pointing out that all uniform estimates depend on the convergence of $\mathcal{G}_{\delta }(\theta _{t}\omega )$ as $\delta \to 0$. However, since the Wiener-like process $\omega ( \cdot )$ may be nowhere differential, it is easy from (1.2) to see that $\mathcal{G}_{\delta }(\theta _{t}\omega )$ generally diverges as $\delta \to 0$. Instead of this convergence, we must prove a convergence in the sense of the integrals of $\mathcal{G}_{\delta }(\theta _{t} \omega )$, which can be deduced from the convergence of the Wiener-like process as given in (1.1).

2 Uniform absorption in size for approximate equations

2.1 The cocycle generated from the approximate equation

A standard method can show the well-posed property of the problem (1.3)–(1.4) and the existence of a family of cocycles given by $\varPhi _{\delta }: \mathbb{R}^{+}\times \mathbb{R}\times \varOmega \times \mathbb{L}^{2}(\mathcal{I}) \rightarrow \mathbb{L}^{2}( \mathcal{I})$,

$$ \varPhi _{\delta }(t,\tau ,\omega )u_{\delta ,\tau }= u_{\delta }(t+ \tau ,\tau ,\theta _{-\tau }\omega ,u_{\delta ,\tau }). $$

(2.1)

The same method as in [11] can show the measurability of $\varPhi _{\delta }$ in ω, and the cocycle property (see [32]) can be deduced from the uniqueness of solutions.

We consider a universe $\mathfrak{D}$ of all tempered bi-parametric sets in $\mathbb{L}^{2}(\mathcal{I})$, that is, for $\mathcal{D}={\{ \mathcal{D}(\tau ,\omega ) :\tau \in \mathbb{R},\omega \in \varOmega }\}$, we have $\mathcal{D}\in \mathfrak{D}$ if and only if

$$ \lim_{t\rightarrow \infty }e^{-\alpha t} \bigl\Vert \mathcal{D}(\tau -t,\theta _{-t}\omega ) \bigr\Vert ^{2}=0, \quad \forall \alpha >0, \tau \in \mathbb{R}, \omega \in \varOmega , $$

(2.2)

where the norm of a set means the maximum of $\mathbb{L}^{2}$-norms of all elements.

In order to obtain a $\mathfrak{D}$-pullback absorption set, we make some assumptions.

Assumption F

$f\in {L_{\mathrm{loc}}^{2}}(\mathbb{R},\mathbb{L}^{2}( \mathcal{I}))$ and there is a $\alpha _{0}>0$ such that

$$\begin{aligned}& \int ^{0}_{-\infty }e^{\alpha _{0} s}{ \bigl\Vert f(s,\cdot ) \bigr\Vert }^{2}\,ds< \infty , \end{aligned}$$

(2.3)

$$\begin{aligned}& \lim_{t\rightarrow {\infty }}e^{-\alpha t} \int ^{0}_{-\infty }e^{ \alpha _{0} s}\bigl({ \bigl\Vert f(s-t,\cdot ) \bigr\Vert }^{2}\bigr)\,ds=0, \quad \forall \alpha >0. \end{aligned}$$

(2.4)

We also need the following convergence from the Wong–Zakai process to a Wiener-like process.

Lemma 2.1

Let $\tau \in \mathbb{R}$, $\omega \in \varOmega$ and $T>0$. Then

$$ \lim_{\delta \rightarrow {0}}\sup_{t\in [\tau ,\tau +T]} \biggl\vert \int _{0}^{t}\mathcal{G}_{\delta }(\theta _{s}\omega )\,ds-\omega (t) \biggr\vert =0. $$

(2.5)

Moreover, for each $\varepsilon >0$, there exist $\delta _{0}(\varepsilon ,\omega )>0$ and $C_{0}(\varepsilon ,\omega )>0$ such that

$$ \biggl\vert \int ^{t}_{0}\mathcal{G}_{\delta }(\theta _{s}\omega )\,ds \biggr\vert \leq \varepsilon \vert t \vert +C_{0}(\varepsilon ,\omega ), \quad \forall t\in \mathbb{R}, \delta \in (0,\delta _{0}]. $$

(2.6)

Proof

By the mean value theorem, there is a $r_{t,\delta }\in [t,t+\delta ]$ such that

$$\begin{aligned} \biggl\vert \frac{1}{\delta } \int _{t}^{t+\delta } \omega (s)\,ds-\omega (t) \biggr\vert = \bigl\vert \omega (r_{t,\delta })-\omega (t) \bigr\vert . \end{aligned}$$

By (1.1), $\omega (\cdot )$ is continuous and thus uniformly continuous on $[\tau , \tau +T+1]$, which implies that

$$\begin{aligned} \lim_{\delta \to 0}\sup_{t\in [\tau ,\tau +T]} \biggl\vert \frac{1}{\delta } \int _{t}^{t+\delta } \omega (s)\,ds-\omega (t) \biggr\vert =0 \quad \text{and}\quad \lim_{\delta \to 0} \frac{1}{\delta } \biggl\vert \int _{0}^{\delta }\omega (s)\,ds \biggr\vert = 0 \end{aligned}$$

in view of $\omega (0)=0$. Therefore, by the definition (1.2),

$$\begin{aligned} &\sup_{t\in [\tau ,\tau +T]} \biggl\vert \int _{0}^{t} \mathcal{G}_{\delta }( \theta _{s}\omega )\,ds-\omega (t) \biggr\vert \\ &\quad \leq \sup_{t\in [\tau ,\tau +T]} \biggl\vert \frac{1}{\delta } \int _{t}^{t+ \delta } \omega (s)\,ds-\omega (t) \biggr\vert +\frac{1}{\delta } \biggl\vert \int _{0} ^{\delta }\omega (s)\,ds \biggr\vert \to 0 \end{aligned}$$

as $\delta \to 0$. Hence, (2.5) holds true.

Given now $\varepsilon >0$, there is a $\delta _{1}\in (0,1]$ such that

$$\begin{aligned} \sup_{0< \vert \delta \vert \leq \delta _{1}} \biggl\vert \frac{1}{\delta } \int _{0}^{ \delta }\omega (s)\,ds \biggr\vert \leq \epsilon . \end{aligned}$$

By (1.1), $|\omega (s)/s|\leq \varepsilon$ for all $|s|\geq s _{0}-1$ with a large $s_{0}(\epsilon )$. Then, for all $|t|\geq s_{0}$ and $\delta \in (0,\delta _{1}]$, there is a $r_{t,\delta }$ with $|r_{t,\delta }-t|\leq |\delta |$ such that

$$\begin{aligned} \biggl\vert \frac{1}{\delta } \int _{t}^{t+\delta } \omega (s)\,ds \biggr\vert = \bigl\vert \omega (r_{t,\delta }) \bigr\vert = \biggl\vert \frac{\omega (r_{t,\delta })}{r_{t,\delta }} \biggr\vert \vert r_{t,\delta } \vert \leq \varepsilon \bigl( \vert t \vert +1\bigr). \end{aligned}$$

By (1.2), we find that, for all $|t|\geq s_{0}$ and $0<|\delta |\leq \delta _{1}$,

$$\begin{aligned} \biggl\vert \int _{0}^{t}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr \biggr\vert \leq \biggl\vert \frac{1}{\delta } \int _{t}^{t+\delta } \omega (s)\,ds \biggr\vert + \biggl\vert \frac{1}{\delta } \int _{0}^{\delta }\omega (s)\,ds \biggr\vert \leq \varepsilon \bigl( \vert t \vert +2\bigr). \end{aligned}$$

By (2.5), there is $\delta _{0}\in (0, \delta _{1}]$ such that, for all $|t|\leq s_{0}$ and $\delta \in (0, \delta _{0}]$,

$$\begin{aligned} \biggl\vert \int _{0}^{t}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr \biggr\vert \leq &\sup_{ \vert s \vert \leq s_{0}} \biggl\vert \int _{0}^{s}\mathcal{G}_{\delta }( \theta _{r}\omega )\,dr-\omega (s) \biggr\vert + \sup_{ \vert s \vert \leq s_{0}} \bigl\vert \omega (s) \bigr\vert \leq \varepsilon +C(\omega ). \end{aligned}$$

Therefore, (2.6) holds true for all $t\in \mathbb{R}$.

In order to prove that the absorption is uniform in size, we consider a change of variables:

$$ v_{\delta }(t,\tau ,\omega )=g_{\delta }^{-1}(t, \omega )u_{\delta }(t, \tau ,\omega ), \quad \text{where } g_{\delta }(t, \omega )=e^{\int ^{t} _{0}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}. $$

(2.7)

Then from (1.3) we obtain a random equation:

$$ \frac{\partial v_{\delta }}{\partial t}-\bigl(\lambda +i\mu (t)\bigr)\Delta v _{\delta }=\gamma v_{\delta }-g_{\delta }^{2}(t,\omega ) \bigl(\kappa +i \beta (t)\bigr) \vert v_{\delta } \vert ^{2}v_{\delta }+g_{\delta }^{-1}(t,\omega )f(t, \cdot ) $$

(2.8)

with $v_{\delta }\equiv 0$ on $\partial \mathcal{I}$ and $v_{\delta }( \tau )=v_{\delta ,\tau }=g_{\delta }(\tau ,\omega )u_{\delta ,\tau }$.

By Lemma 2.1 and the inequality $|e^{a}-e^{b}|\leq e^{|a|+|b|}|b-a|$, we have

$$ \lim_{\delta \rightarrow {0}}\sup_{t\in [\tau ,\tau +T]}\bigl( \bigl\vert g_{ \delta }(t,\omega )-e^{\omega (t)} \bigr\vert + \bigl\vert g_{\delta }^{-1}(t, \omega )-e^{-\omega (t)} \bigr\vert \bigr)=0. $$

(2.9)

 □

2.2 Uniform absorption in size for approximate equations

Lemma 2.2

For each $\delta >0$, $\mathcal{D}_{\delta }\in \mathfrak{D}$, $\tau \in \mathbb{R}$ and $\omega \in \varOmega$, there are $T_{\delta }:=T(\mathcal{D}_{\delta }, \tau ,\omega )\geq 1$ such that, for all $t\geq T_{\delta }$ and $u_{\delta ,\tau -t}\in \mathcal{D}_{\delta }( \tau -t, \theta _{-t} \omega )$,

$$\begin{aligned} \bigl\Vert u_{\delta }(\tau ,\tau -t,\theta _{-\tau }\omega ,u_{\delta ,\tau -t}) \bigr\Vert ^{2}\leq R_{\delta }(\tau ,\omega )+1, \end{aligned}$$

(2.10)

where $u_{\delta }$ is a solution of the problem (1.3), and for a positive constant $c_{1}$,

$$\begin{aligned} R_{\delta }(\tau ,\omega ):=c_{1} \int ^{0}_{-\infty }e^{2\alpha _{0} s-2 \int _{0}^{s}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}\bigl( \bigl\Vert f(s+\tau ) \bigr\Vert ^{2}+1\bigr)\,ds. \end{aligned}$$

(2.11)

Proof

We multiply Eq. (2.8) with the conjugate function $\overline{v_{\delta }}$ and then take the real part to obtain

$$\begin{aligned} \frac{1}{2}\frac{d }{d s}{ \Vert v_{\delta } \Vert }^{2}+\lambda \Vert \nabla v_{ \delta } \Vert ^{2}=\gamma \Vert v_{\delta } \Vert ^{2}-\kappa g_{\delta }^{2}(s, \omega ) \Vert v_{\delta } \Vert ^{4}_{4} +g_{\delta }^{-1}(s,\omega ) \operatorname{Re}(f, v_{\delta }), \end{aligned}$$

(2.12)

where $\|\cdot \|_{4}$ denotes the norm in $\mathbb{L}^{4}( \mathcal{I})$. The Young inequality gives

$$ g_{\delta }^{-1}(s,\omega ) \bigl\vert \operatorname{Re}\bigl(f(s), v_{\delta }\bigr) \bigr\vert \leq \alpha _{0} \Vert v_{\delta } \Vert ^{2}+cg_{\delta }^{-2}(s,\omega ) \bigl\Vert f(s) \bigr\Vert ^{2}, $$

where $\alpha _{0}$ is the number in Assumption F. By the Young inequality again,

$$ (\gamma +2\alpha _{0}) \Vert v_{\delta } \Vert ^{2}-\frac{\kappa }{2} g_{\delta }^{2}(s,\omega ) \Vert v_{\delta } \Vert ^{4}_{4}\leq c\bigl( \vert \mathcal{O} \vert \bigr)g_{\delta }^{-2}(s,\omega ). $$

So, we can rewrite (2.12) for the solution $v_{\delta }(s)=v _{\delta }(s, \tau -t,\theta _{-\tau }\omega , v_{\delta ,\tau -t})$:

$$\begin{aligned}& \frac{d }{d s}{ \Vert v_{\delta } \Vert }^{2}+2\alpha _{0} \Vert v_{\delta } \Vert ^{2}+ \lambda \Vert \nabla v_{\delta } \Vert ^{2}+ \kappa g_{\delta }^{2}(s, \theta _{-\tau }\omega ) \Vert v_{\delta } \Vert ^{4}_{4} \\& \quad \leq c_{1} g_{\delta }^{-2}(s,\theta _{-\tau } \omega ) \bigl( \bigl\Vert f(s) \bigr\Vert ^{2}+1\bigr). \end{aligned}$$

(2.13)

Multiplying (2.13) by $e^{2\alpha _{0} s}$ and then integrating over $(\tau -t,\tau )$, we obtain

$$\begin{aligned}& \bigl\Vert v_{\delta }(\tau ,\tau -t,\theta _{-\tau }\omega ,v_{\delta , \tau -t}) \bigr\Vert ^{2}+\lambda \int ^{\tau }_{\tau -t}e^{2\alpha _{0} (s- \tau )} \bigl\Vert \nabla v_{\delta }(s,\tau -t,\theta _{-\tau }\omega ,v_{ \delta ,\tau -t}) \bigr\Vert ^{2}\,ds \\& \qquad {} +\kappa \int ^{\tau }_{\tau -t}e^{2\alpha _{0} (s-\tau )}g_{\delta } ^{2}(s,\theta _{-\tau }\omega ) \bigl\Vert v_{\delta }(s, \tau -t,\theta _{-\tau } \omega ,v_{\delta ,\tau -t}) \bigr\Vert ^{4}_{4}\,ds \\& \quad \leq e^{-2\alpha _{0} t} \Vert v_{\delta ,\tau -t} \Vert ^{2}+c_{1} \int ^{ \tau }_{\tau -t} e^{2\alpha _{0} (s-\tau )} g_{\delta }^{-2}(s, \theta _{-\tau }\omega ) \bigl( \bigl\Vert f(s) \bigr\Vert ^{2}+1\bigr)\,ds. \end{aligned}$$

(2.14)

By the change of variables (2.7), we have $v_{\delta ,\tau -t}=g _{\delta }^{-1}(\tau ,\theta _{-\tau }\omega )u_{\delta ,\tau -t}$ and

$$\begin{aligned} \bigl\Vert u_{\delta }(\tau ,\tau -t,\theta _{-\tau }\omega ,v_{\delta , \tau -t}) \bigr\Vert ^{2} =&g_{\delta }^{2}( \tau ,\theta _{-\tau }\omega ) \bigl\Vert v_{ \delta }(\tau ,\tau -t, \theta _{-\tau }\omega ,v_{\delta ,\tau -t}) \bigr\Vert ^{2} \\ \leq& e^{-2\alpha _{0} t} \Vert u_{\delta ,\tau -t} \Vert ^{2}+I, \end{aligned}$$

(2.15)

where

$$\begin{aligned} I : =&c_{1}g_{\delta }^{2}(\tau ,\theta _{-\tau } \omega ) \int ^{\tau } _{\tau -t} e^{2\alpha _{0} (s-\tau )} g_{\delta }^{-2}(s,\theta _{- \tau }\omega ) \bigl( \bigl\Vert f(s) \bigr\Vert ^{2}+1\bigr)\,ds \\ =&c_{1} \int ^{\tau }_{\tau -t}e^{2\alpha _{0} (s-\tau )+2\int _{0}^{ \tau }\mathcal{G}_{\delta }(\theta _{r-\tau }\omega )\,dr-2\int _{0}^{s} \mathcal{G}_{\delta }(\theta _{r-\tau }\omega )\,dr}\bigl( \bigl\Vert f(s) \bigr\Vert ^{2}+1\bigr)\,ds \\ =&c_{1} \int ^{\tau }_{\tau -t}e^{2\alpha _{0} (s-\tau )-2\int _{0}^{s- \tau }\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}\bigl( \bigl\Vert f(s) \bigr\Vert ^{2}+1\bigr)\,ds \\ \leq& c_{1} \int ^{0}_{-\infty }e^{2\alpha _{0} s-2\int _{0}^{s} \mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}\bigl( \bigl\Vert f(s+\tau ) \bigr\Vert ^{2}+1\bigr)\,ds=:R _{\delta }(\tau ,\omega ). \end{aligned}$$

On the other hand, since $u_{\delta ,\tau -t}\in \mathcal{D}_{\delta }(\tau -t,\theta _{-\tau }\omega )$, there is $T_{\delta }=T( \mathcal{D}_{\delta },\tau ,\omega )$ such that, for all $t\geq T_{ \delta }$,

$$ e^{-2\alpha _{0} t} \Vert u_{\delta ,\tau -t} \Vert ^{2}\leq e^{-2\alpha _{0} t} \bigl\Vert \mathcal{D}_{\delta }(\tau -t,\theta _{-\tau }\omega ) \bigr\Vert ^{2}\leq 1. $$

Substituting the above estimates into (2.15), we obtain (2.10) as desired. □

In addition, by (2.14) and (2.15), we have, for all $t\geq T_{\delta }$ and $u_{\delta ,\tau -t}\in \mathcal{D}_{\delta }( \tau -t,\theta _{-\tau }\omega )$,

$$\begin{aligned}& \int ^{\tau }_{\tau -t}e^{2\alpha _{0} (s-\tau )} \bigl\Vert \nabla v_{\delta }(s,\tau -t,\theta _{-\tau }\omega ,v_{\delta ,\tau -t}) \bigr\Vert ^{2}\,ds \\& \quad \leq g_{\delta }^{-2}(\tau ,\theta _{-\tau }\omega ) \bigl(e^{-2\alpha _{0} t} \Vert u_{\delta ,\tau -t} \Vert ^{2}+I\bigr) \leq \frac{c(R_{\delta }(\tau ,\omega )+1)}{g_{\delta }^{2}(\tau ,\theta _{-\tau }\omega )}. \end{aligned}$$

(2.16)

Similarly, we have the following useful estimate:

$$\begin{aligned}& \int ^{\tau }_{\tau -t}e^{2\alpha _{0} (s-\tau )}g_{\delta }^{2}(s, \theta _{-\tau }\omega ) \bigl\Vert v_{\delta }(s,\tau -t,\theta _{-\tau }\omega ,v_{\delta ,\tau -t}) \bigr\Vert ^{4}_{4}\,ds \\& \quad \leq c\bigl(R_{\delta }(\tau ,\omega )+1\bigr)g_{\delta }^{-2}( \tau , \theta _{-\tau }\omega ). \end{aligned}$$

(2.17)

Proposition 2.3

Under the Assumption F, for each $\delta >0$, the cocycle $\varPhi _{\delta }$ has a closed, $\mathfrak{D}$-pullback random absorbing set $\mathcal{K}_{\delta }\in \mathfrak{D}$ in $\mathbb{L} ^{2}(\mathcal{I})$, given by

$$\begin{aligned} \mathcal{K}_{\delta }(\tau ,\omega ):=\bigl\{ w\in \mathbb{L}^{2}( \mathcal{I}): \Vert w \Vert ^{2} \leq R_{\delta }(\tau ,\omega )+1\bigr\} , \quad \forall (\tau ,\omega )\in \mathbb{R} \times \varOmega , \end{aligned}$$

(2.18)

where $R_{\delta }(\tau ,\omega )$ is given in (2.11) and satisfies

$$\begin{aligned} \lim_{\delta \to 0}R_{\delta }(\tau ,\omega )= c_{1} \int ^{0}_{-\infty }e^{2\alpha _{0} s-2\omega (s)}\bigl( \bigl\Vert f(s+\tau ) \bigr\Vert ^{2}+1\bigr)\,ds=:R_{0}(\tau , \omega ). \end{aligned}$$

(2.19)

Proof

We first prove that each $R_{\delta }(\tau ,\omega )$ is finite. Notice that the formula (2.6) in Lemma 2.1 holds true for every $\delta >0$. Hence, for each $\varepsilon >0$ and $\omega \in \varOmega$, there is a $C_{\delta }(\varepsilon ,\omega )>0$ such that

$$ \biggl\vert \int ^{s}_{0}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr \biggr\vert \leq \varepsilon \vert s \vert +C_{\delta }(\varepsilon ,\omega ),\quad \forall s \in \mathbb{R}. $$

(2.20)

By taking $\varepsilon =\frac{\alpha _{0}}{2}$, there is a $C_{\delta }(\omega )$ such that

$$\begin{aligned} R_{\delta }(\tau ,\omega ) :=&c_{1} \int ^{0}_{-\infty }e^{2\alpha _{0} s-2 \int _{0}^{s}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}\bigl( \bigl\Vert f(s+\tau ) \bigr\Vert ^{2}+1\bigr)\,ds \\ \leq& c_{1}e^{2C_{\delta }(\omega )} \int ^{0}_{-\infty }e^{\alpha _{0} s}\bigl( \bigl\Vert f(s+\tau ) \bigr\Vert ^{2}+1\bigr)\,ds. \end{aligned}$$

By (2.3) in Assumption F,

$$\begin{aligned} \int ^{0}_{-\infty }e^{\alpha _{0} s} \bigl\Vert f(s+\tau ) \bigr\Vert ^{2}\,ds&= \int ^{ \tau }_{-\infty }e^{\alpha _{0} (s-\tau )} \bigl\Vert f(s) \bigr\Vert ^{2}\,ds \\ &=e^{-\alpha _{0} \tau } \int ^{0}_{-\infty }e^{\alpha s} \bigl\Vert f(s) \bigr\Vert ^{2}\,ds +e^{-\alpha _{0} \tau } \int ^{\tau }_{0}e^{\alpha _{0} s} \bigl\Vert f(s) \bigr\Vert ^{2}\,ds< + \infty \end{aligned}$$

and thus $R_{\delta }(\tau ,\omega )$ is finite.

The mapping $\omega \to R_{\delta }(\tau ,\omega )$ is obviously measurable and thus $\mathcal{K}_{\delta }$ is a family of random sets. By Lemma 2.2, $\mathcal{K}_{\delta }$ is a $\mathfrak{D}$-pullback absorbing set for $\varPhi _{\delta }$.

We then prove $\mathcal{K}_{\delta }\in \mathfrak{D}$. Indeed, for any $\alpha >0$, we take $\varepsilon =\min \{\frac{\alpha }{5},\frac{ \alpha _{0}}{2}\}$ in (2.20), then, by (2.4) in Assumption F, as $t\to +\infty$, i.e. as $\tilde{t}=t-\tau \to +\infty$,

$$\begin{aligned} &e^{-\alpha t}R_{\delta }(\tau -t,\theta _{-t} \omega ) \\ &\quad =c_{1}e^{-\alpha t} \int _{-\infty }^{0}e^{2\alpha _{0} s-2\int _{0} ^{s}\mathcal{G}_{\delta }(\theta _{r-t}\omega )\,dr}\bigl( \bigl\Vert f(s+\tau -t) \bigr\Vert ^{2}+1\bigr)\,ds \\ &\quad \leq c_{1}e^{4C_{\delta }(\omega )} e^{-\alpha t} \int _{-\infty } ^{0}e^{2\alpha _{0} s+2\varepsilon (t-s)+2\varepsilon t}\bigl( \bigl\Vert f(s+\tau -t) \bigr\Vert ^{2}+1\bigr)\,ds \\ &\quad = c_{1}e^{4C_{\delta }(\omega )} e^{-(\alpha -4\varepsilon )( \tilde{t}+\tau )} \int _{-\infty }^{0}e^{(2\alpha _{0}-2\varepsilon ) s}\bigl( \bigl\Vert f(s-\tilde{t}) \bigr\Vert ^{2}+1\bigr)\,ds\to 0 \end{aligned}$$

in view of the facts that $\alpha -4\varepsilon >0$ and $2\alpha _{0}-2 \varepsilon \geq \alpha _{0}$.

Finally, we show the convergence (2.19). By (2.6) in Lemma 2.1, there are $\delta _{0}>0$ and $C_{0}(\omega )>0$ (independent of δ) such that

$$ \sup_{\delta \in (0,\delta _{0}]} \biggl\vert \int ^{s}_{0}\mathcal{G}_{ \delta }(\theta _{r}\omega )\,dr \biggr\vert \leq \frac{\alpha }{2} \vert s \vert +C_{0}( \omega ), \quad \forall s\in \mathbb{R}. $$

(2.21)

Hence, by taking the supremum of $R_{\delta }(\tau ,\omega )$ on $\delta \in (0,\delta _{0}]$, we have

$$\begin{aligned} &\sup_{\delta \in (0,\delta _{0}]} \int ^{0}_{-\infty }e^{2\alpha s-2\int _{0}^{s}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}\bigl( \bigl\Vert f(s+\tau ) \bigr\Vert ^{2}+1\bigr)\,ds \\ &\quad \leq e^{2C_{0}(\omega )} \int ^{0}_{-\infty }e^{\alpha s}\bigl( \bigl\Vert f(s+ \tau ) \bigr\Vert ^{2}+1\bigr)\,ds< +\infty . \end{aligned}$$

Hence, by (2.5) in Lemma 2.1, the Lebesgue controlled convergence theorem gives

$$\begin{aligned} \lim_{\delta \to 0}R_{\delta }(\tau ,\omega ) &= c_{1} \int ^{0}_{- \infty }\lim_{\delta \to 0}e^{2\alpha _{0} s-2\int _{0}^{s}\mathcal{G} _{\delta }(\theta _{r}\omega )\,dr} \bigl( \bigl\Vert f(s+\tau ) \bigr\Vert ^{2}+1\bigr)\,ds \\ &=c_{1} \int ^{0}_{-\infty }e^{2\alpha _{0} s-2\omega (s)}\bigl( \bigl\Vert f(s+\tau ) \bigr\Vert ^{2}+1\bigr)\,ds=R_{0}(\tau ,\omega ). \end{aligned}$$

 □

3 Uniform compactness in size for approximate equations

3.1 Uniformly asymptotic compactness

Lemma 3.1

For each $\mathcal{D}_{\delta }\in \mathfrak{D}$, $\tau \in \mathbb{R}$ and $\omega \in \varOmega$, let $T_{\delta }\geq 1$ be the entrance time in Lemma 2.2. Then there is a $\delta _{0}>0$ such that, for all $\delta \in (0,\delta _{0}]$, $t\geq T_{\delta }$ and $u_{\delta ,\tau -t}\in \mathcal{D}_{\delta }(\tau -t, \theta _{-t} \omega )$,

$$ \bigl\Vert \nabla u_{\delta }(\tau ,\tau -t,\theta _{-\tau }\omega ,v_{\delta , \tau -t}) \bigr\Vert ^{2}\leq e^{\varUpsilon (M_{0}(\tau , \omega ))(R_{0}(\tau , \omega )+2)}< +\infty , $$

(3.1)

where $\varUpsilon (y)=a_{4}y^{4}+a_{2}y^{2}+a_{0}$ ($y>0$) with positive coefficients, $R_{0}(\tau ,\omega )$ is given in (2.19) and

$$ M_{0}(\tau , \omega ):=\sup_{s\in [\tau -1,\tau ]} e^{\pm (\omega (s- \tau )-\omega (-\tau ))}. $$

(3.2)

Proof

We multiply Eq. (2.8) by $-\overline{\Delta v _{\delta }}=-\Delta \overline{v_{\delta }}$ and then take the real part to find

$$\begin{aligned}& {\frac{1}{2}}\frac{d }{ds}{ \Vert \nabla v_{\delta } \Vert }^{2}+\lambda \Vert \Delta v_{\delta } \Vert ^{2} \\& \quad = \gamma \Vert \nabla v_{\delta } \Vert ^{2} +g_{\delta }^{2}(s,\omega )\operatorname{Re}\bigl(\kappa +i\beta (t) \bigr) \bigl( \vert v_{\delta } \vert ^{2}v_{\delta }, \Delta v_{\delta }\bigr)-g_{\delta }^{-1}(s,\omega ) \operatorname{Re}\bigl(f(s),\Delta v_{\delta }\bigr). \end{aligned}$$

(3.3)

By the Young inequality we obtain

$$ g_{\delta }^{-1}(s,\omega ) \bigl\vert \operatorname{Re}\bigl(f(s), \Delta v_{\delta }\bigr) \bigr\vert \leq \frac{\lambda }{4} \Vert \Delta v_{\delta } \Vert ^{2}+c_{3}g_{\delta } ^{-2}(s,\omega ) \bigl\Vert f(s) \bigr\Vert ^{2}. $$

Since $\mathcal{I}$ is a 1D-domain, by the compactness of Sobolev embedding and the interpolation inequality, we have the following inequality (see Temam [29]):

$$ \Vert \nabla w \Vert ^{2}_{4}\leq c \Vert \nabla w \Vert \bigl( \Vert w \Vert ^{2}+ \Vert \Delta w \Vert ^{2}\bigr)^{ \frac{1}{2}},\quad \forall w\in \mathbb{H}^{1}_{0}( \mathcal{I})\cap \mathbb{H}^{2}(\mathcal{I}). $$

By the initial assumption, $\beta \in C_{b}(\mathbb{R},\mathbb{R})$ and thus $\beta _{0}:=\sup_{t\in \mathbb{R}}|\beta (t)|<+\infty$. Hence,

$$\begin{aligned}& \bigl\vert \operatorname{Re}\bigl(\kappa +i\beta (t)\bigr) \bigl( \vert v_{\delta } \vert ^{2}v_{\delta },\Delta \overline{v_{\delta }}\bigr) \bigr\vert \\& \quad = \biggl\vert \operatorname{Re}\bigl(\kappa +i\beta (t)\bigr) \int _{\mathcal{I}}\bigl(2 \vert v_{\delta } \vert ^{2} \vert \nabla v_{\delta } \vert ^{2}+v_{\delta }^{2} \nabla \overline{v_{\delta }} \cdot \nabla \overline{v_{\delta }} \bigr)\,dx \biggr\vert \\& \quad \leq 3(\kappa +\beta _{0}) \int _{\mathcal{I}} \vert v_{\delta } \vert ^{2} \vert \nabla v_{\delta } \vert ^{2}\,dx \\& \quad \leq c \Vert v_{\delta } \Vert ^{2}_{4} \Vert \nabla v_{\delta } \Vert ^{2}_{4} \\& \quad \leq c \Vert v_{\delta } \Vert ^{2}_{4} \Vert \nabla v_{\delta } \Vert \bigl( \Vert v_{\delta } \Vert ^{2}+ \Vert \Delta v_{\delta } \Vert ^{2} \bigr)^{\frac{1}{2}}, \end{aligned}$$

which together with the Poincaré inequality implies that

$$\begin{aligned}& g_{\delta }^{2}(s,\omega ) \bigl\vert \operatorname{Re}\bigl(\kappa +i\beta (t)\bigr) \bigl( \vert v_{\delta } \vert ^{2}v_{\delta }, \Delta \overline{v_{\delta }}\bigr) \bigr\vert \\& \quad \leq \frac{\lambda }{4}\bigl( \Vert v_{\delta } \Vert ^{2}+ \Vert \Delta v_{\delta } \Vert ^{2}\bigr)+ cg_{\delta }^{4}(s,\omega ) \Vert v_{\delta } \Vert ^{4}_{4} \Vert \nabla v _{\delta } \Vert ^{2} \\& \quad \leq \frac{\lambda }{4} \Vert \Delta v_{\delta } \Vert ^{2}+ c\bigl(1+g_{\delta } ^{4}(s,\omega ) \Vert v_{\delta } \Vert ^{4}_{4}\bigr) \Vert \nabla v_{\delta } \Vert ^{2}. \end{aligned}$$

Substituting it into (3.3) at the sample $\theta _{-\tau }\omega$, we obtain

$$ \frac{d }{d s}{ \Vert \nabla v_{\delta } \Vert }^{2}\leq c_{4}\bigl(1+g_{\delta } ^{4}(s,\theta _{-\tau } \omega ) \Vert v_{\delta } \Vert ^{4}_{4}\bigr) \Vert \nabla v_{ \delta } \Vert ^{2}+ c_{3}g_{\delta }^{-2}(s, \theta _{-\tau }\omega ) \bigl\Vert f(s) \bigr\Vert ^{2}. $$

By the uniform Gronwall lemma [29] (also see [25, 36] for the non-autonomous version), we obtain

$$ \bigl\Vert \nabla v_{\delta }(\tau ,\tau -t,\theta _{-\tau }\omega ,v_{\delta , \tau -t}) \bigr\Vert ^{2}\leq e^{I_{1}+I_{2}(\delta )}\bigl(I_{3}(\delta )+I_{4}( \delta )\bigr), $$

(3.4)

where $I_{1}:=c_{4}\int _{\tau -1}^{\tau }\,ds=c_{4}$ and

$$\begin{aligned}& I_{2}(\delta ):=c_{4} \int _{\tau -1}^{\tau } g_{\delta }^{4}(s, \theta _{-\tau }\omega ) \bigl\Vert v_{\delta }(s,\tau -t,\theta _{-\tau }\omega ,v_{\delta ,\tau -t}) \bigr\Vert ^{4}_{4}\,ds, \\& I_{3}(\delta ):= \int _{\tau -1}^{\tau } \bigl\Vert \nabla v_{\delta }(s,\tau -t, \theta _{-\tau }\omega ,v_{\delta ,\tau -t}) \bigr\Vert ^{2}\,ds, \\& I_{4}(\delta ):=c_{3} \int _{\tau -1}^{\tau }g_{\delta }^{-2}(s, \theta _{-\tau }\omega ) \bigl\Vert f(s) \bigr\Vert ^{2}\,ds. \end{aligned}$$

We will use (2.17) to estimate $I_{2}(\delta )$. Indeed, by (2.9),

$$\begin{aligned} g_{\delta }(s,\theta _{-\tau }\omega ) =&e^{\int ^{s}_{0}\mathcal{G} _{\delta }(\theta _{r-\tau }\omega )\,dr} \\ =&e^{\int ^{s-\tau }_{0}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr- \int ^{-\tau }_{0}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}\to e^{ \omega (s-\tau )-\omega (-\tau )} \end{aligned}$$

as $\delta \to 0$ uniformly in $s\in [\tau -1,\tau ]$. Hence, there is a $\delta _{1}>0$ such that

$$ \sup_{\delta \in (0,\delta _{1}]}\sup_{s\in [\tau -1,\tau ]}g_{\delta }(s, \theta _{-\tau }\omega )\leq \sup_{s\in [\tau -1,\tau ]} e^{\omega (s-\tau )-\omega (-\tau )}+1\leq M_{0}(\tau , \omega )+1, $$

(3.5)

where $M_{0}(\tau , \omega )$ is defined by (3.2). So, for all $\delta \in (0,\delta _{1}]$ and $t\geq 1$,

$$\begin{aligned} I_{2}(\delta ) \leq& c_{4}e^{2\alpha _{0}}\sup _{s\in [\tau -1,\tau ]}g _{\delta }^{2}(s,\theta _{-\tau } \omega ) \\ &{}\times \int _{\tau -1}^{\tau } e^{2\alpha _{0} (s-\tau )}g_{\delta }^{2}(s, \theta _{-\tau }\omega ) \bigl\Vert v_{\delta }(s,\tau -t,\theta _{-\tau }\omega ,v_{\delta ,\tau -t}) \bigr\Vert ^{4}_{4}\,ds \\ \leq& c\bigl(M_{0}^{2}(\tau , \omega )+1\bigr) \int _{\tau -t}^{\tau } e^{2 \alpha (s-\tau )}g_{\delta }^{2}(s, \theta _{-\tau }\omega ) \bigl\Vert v_{\delta }(s,\tau -t,\theta _{-\tau }\omega ,v_{\delta ,\tau -t}) \bigr\Vert ^{4}_{4}\,ds. \end{aligned}$$

By (2.17), for all $\delta \in (0,\delta _{1}]$, $t\geq T_{ \delta }$ and $u_{\delta ,\tau -t}\in \mathcal{D}_{\delta }(\tau -t, \theta _{-t}\omega )$,

$$\begin{aligned} I_{2}(\delta )\leq c\bigl(M_{0}^{2}(\tau , \omega )+1\bigr) \bigl(R_{\delta }(\tau , \omega )+1\bigr) (\tau ,\omega )g_{\delta }^{-2}(\tau ,\theta _{-\tau } \omega ). \end{aligned}$$

By the convergence (2.19), $R_{\delta }(\tau ,\omega ) \leq R_{0}(\tau ,\omega )+1$ for all $\delta \in (0,\delta _{2}]$ with $\delta _{2}\leq \delta _{1}$. By the same method as in (3.5), there is a $\delta _{3}\in (0,\delta _{2}]$ such that, for all $\delta \in (0, \delta _{3}]$,

$$ \sup_{s\in [\tau -1,\tau ]}g_{\delta }^{-1}(s, \theta _{-\tau }\omega ) \leq \sup_{s\in [\tau -1,\tau ]} e^{-(\omega (s-\tau )-\omega (- \tau ))}+1\leq M_{0}(\tau , \omega )+1. $$

(3.6)

Hence, for all $\delta \in (0,\delta _{3}]$, $t\geq T_{\delta }$ and $u_{\delta ,\tau -t}\in \mathcal{D}_{\delta }(\tau -t, \theta _{-t} \omega )$,

$$ I_{2}(\delta )\leq \varUpsilon \bigl(M_{0}( \tau , \omega )\bigr) \bigl(R_{0}(\tau , \omega )+2\bigr), $$

(3.7)

where $\varUpsilon (\cdot )$ denotes the fourth-order polynomial with positive coefficients.

Similarly, by (2.16), there is a $\delta _{4}\in (0,\delta _{3}]$ such that, for all $\delta \in (0,\delta _{4}]$, $t\geq T_{\delta }$ and $u_{\delta ,\tau -t}\in \mathcal{D}_{\delta }(\tau -t, \theta _{-t} \omega )$,

$$ I_{3}(\delta )\leq \varUpsilon \bigl(M_{0}( \tau , \omega )\bigr) \bigl(R_{0}(\tau , \omega )+1\bigr). $$

(3.8)

By (3.6) and the Assumption F, we have

$$ \sup_{\delta \in (0,\delta _{3}]}I_{4}(\delta )\leq \varUpsilon \bigl(M_{0}( \tau , \omega )\bigr) \int _{\tau -1}^{\tau } \bigl\Vert f(s) \bigr\Vert ^{2}\,ds\leq \varUpsilon \bigl(M _{0}(\tau , \omega )\bigr) \bigl(R_{0}(\tau ,\omega )+1\bigr). $$

(3.9)

We substitute (3.7)–(3.9) into (3.4) to find that, for all $\delta \in (0,\delta _{4}]$, $t\geq T_{\delta }$ and $u_{\delta ,\tau -t}\in \mathcal{D}_{\delta }(\tau -t, \theta _{-t} \omega )$,

$$ \sup_{\delta \in (0,\delta _{5}]} \bigl\Vert \nabla v_{\delta }(\tau ,\tau -t, \theta _{-\tau }\omega ,v_{\delta ,\tau -t}) \bigr\Vert ^{2}\leq e^{\varUpsilon (M _{0}(\tau , \omega ))(R_{0}(\tau ,\omega )+2)}. $$

(3.10)

By using the relationship

$$ u_{\delta }(\tau ,\tau -t,\theta _{-\tau }\omega , u_{\delta ,\tau -t})=g _{\delta }^{2}(\tau , \theta _{-\tau } \omega )v_{\delta }(\tau ,\tau -t, \theta _{-\tau }\omega ,v_{\delta ,\tau -t}), $$

we see from (3.5) and (3.10) that (3.3) holds true for all $\delta \in (0,\delta _{4}]$, $t\geq T_{\delta }$ and $u_{\delta , \tau -t}\in \mathcal{D}_{\delta }(\tau -t, \theta _{-t}\omega )$. □

3.2 Random attractors for the equation with difference noise

A bi-parametric set $\mathcal{A}_{\delta }=\{\mathcal{A}_{\delta }( \tau ,\omega )\}\in \mathfrak{D}$ is called a $\mathfrak{D}$-pullback random attractor for the cocycle $\varPhi _{\delta }$ if $\mathcal{A}_{\delta }$ is random, compact, invariant and $\mathfrak{D}$-pullback attracting. The details and existence criteria can be found in [26, 32, 33].

Theorem 3.2

Each Ginzburg–Landau equation with δ-difference noise possesses a unique $\mathfrak{D}$-pullback random attractor $\mathcal{A}_{ \delta }=\{\mathcal{A}_{\delta }(\tau ,\omega )\}$ in $\mathbb{L}^{2}( \mathcal{I})$.

Proof

By Proposition 2.3, the cocycle $\varPhi _{ \delta }$ has a $\mathfrak{D}$-pullback random absorbing set $\mathcal{K}_{\delta }=\{\mathcal{K}_{\delta }(\tau ,\omega )\}\in \mathfrak{D}$.

We prove that for each $\delta >0$ the cocycle $\varPhi _{\delta }$ is $\mathfrak{D}$-pullback asymptotically compact in $\mathbb{L}^{2}( \mathcal{I})$. Indeed, let $t_{n}\to +\infty$ and $u_{\delta ,\tau -t _{n}}\in \mathcal{D}_{\delta }(\tau -t_{n}, \theta _{-t_{n}}\omega )$ with $\mathcal{D}_{\delta }\in \mathfrak{D}$, $\tau \in \mathbb{R}$ and $\omega \in \varOmega$. Then, by the same method as in Lemma 3.1, there is a large $N\in \mathbb{N}$ such that, for all $n\geq N$,

$$ \bigl\Vert \nabla u_{\delta }(\tau ,\tau -t_{n},\theta _{-\tau }\omega ,u_{ \delta ,\tau -t_{n}}) \bigr\Vert ^{2}\leq e^{\varUpsilon (M_{\delta }(\tau , \omega ))(R_{\delta }(\tau ,\omega )+1)}< +\infty , $$

where, by the continuity of $\mathcal{G}_{\delta }$,

$$ M_{\delta }(\tau , \omega ):=\sup_{s\in [\tau -1,\tau ]}e^{\pm \int ^{s}_{0}\mathcal{G}_{\delta }(\theta _{r-\tau }\omega )\,dr}< +\infty . $$

Therefore, the sequence

$$ \bigl\{ \varPhi _{\delta }(t_{n},\tau -t_{n},\theta _{-t_{n}}\omega )u_{\delta , \tau -t_{n}}\bigr\} = \bigl\{ u_{\delta _{n}}(\tau ,\tau -t_{n},\theta _{-\tau } \omega ,u_{\delta ,\tau -t_{n}})\bigr\} $$

(3.11)

is bounded in $\mathbb{H}_{0}^{1}(\mathcal{I})$. By the compactness of the Sobolev embedding $\mathbb{H}_{0}^{1}(\mathcal{I})\hookrightarrow \mathbb{L}^{2}(\mathcal{I})$, the sequence has a convergent subsequence in $\mathbb{L}^{2}(\mathcal{I})$. By the abstract result in [26, 32], there is a $\mathfrak{D}$-pullback random attractor such that $\{\mathcal{A}_{\delta }(\tau ,\omega )\subset \{ \mathcal{K}_{\delta }(\tau ,\omega )$. □

3.3 Random attractors for the equation with Wiener-like noise

We now consider the Ginzburg–Landau equation (1.6) with Wiener-like noise. Let

$$ v(t,\tau ,\omega ,v_{\tau })=e^{-\omega (t)}u(t,\tau , \omega ,u_{ \tau }). $$

(3.12)

We obtain a random equation:

$$ \frac{\partial v}{\partial t}-\bigl(\lambda +i\mu (t)\bigr)\Delta v=\gamma v-e ^{2\omega (t)}\bigl(\kappa +i\beta (t)\bigr) \vert v \vert ^{2}v+e^{-\omega (t)}f(t,x), $$

(3.13)

with the initial-boundary conditions

$$ v(t,0)=v(t,1)=0, \qquad v(\tau ,x)=v_{\tau }(x),\quad x\in \mathcal{I},t\geq \tau , $$

(3.14)

where $v_{\tau }(x)=e^{-\omega (\tau )}u_{\tau }(x)$. As in [35], it is standard to show that problem (3.13)–(3.14) has a unique solution

$$ v(\cdot ,\tau ,\omega ,v_{\tau })\in C\bigl([\tau ,\infty ),L^{2}( \mathcal{I})\bigr)\cap {L_{\mathrm{loc}}^{2}}\bigl([ \tau ,\infty),H^{1}_{0}(\mathcal{I})\bigr). $$

Passing to the variable u, we obtain a cocycle $\varPhi _{0}:\mathbb{R} ^{+}\times \mathbb{R}\times \varOmega \times \mathbb{L}^{2}(\mathcal{I}) \rightarrow \mathbb{L}^{2}(\mathcal{I})$ for the stochastic equation (1.6), given by

$$ \varPhi _{0}(t,\tau ,\omega ,u_{\tau })=u(t+\tau ,\tau ,\theta _{-\tau } \omega ,u_{\tau })=e^{\omega (t)-\omega (-\tau )}v(t+\tau ,\tau , \theta _{-\tau }\omega ,v_{\tau }). $$

(3.15)

The same method as given in Proposition 2.3 shows that the cocycle $\varPhi _{0}$ has a $\mathfrak{D}$-pullback random absorbing set $\mathcal{K}_{0}\in \mathfrak{D}$ in the space $\mathbb{L}^{2}( \mathcal{I})$, given by

$$ \mathcal{K}_{0}(\tau ,\omega ):=\bigl\{ w\in \mathbb{L}^{2}(\mathcal{I}): \Vert w \Vert ^{2} \leq R_{0}(\tau ,\omega )+2\bigr\} , \quad \forall (\tau ,\omega ) \in \mathbb{R} \times \varOmega , $$

(3.16)

where $R_{0}(\tau ,\omega )$ is just the limit of $R_{\delta }(\tau , \omega )$ as given in (2.19).

By the same method as given in Lemma 3.1, one can show that the cocycle $\varPhi _{0}$ has another $\mathfrak{D}$-pullback absorbing set $\widetilde{\mathcal{K}_{0}}(\tau ,\omega )\subset \mathbb{H}_{0}^{1}( \mathcal{I})$, given by

$$ \widetilde{\mathcal{K}_{0}}(\tau ,\omega ):=\bigl\{ w\in \mathbb{H}_{0}^{1}( \mathcal{I}): \Vert \nabla w \Vert ^{2} \leq e^{\varUpsilon (M_{0}(\tau , \omega ))(R_{0}(\tau ,\omega )+2)}\bigr\} . $$

(3.17)

By the compactness of the Sobolev embedding, $\varPhi _{0}$ is $\mathfrak{D}$-pullback asymptotically compact. So, we obtain

Theorem 3.3

The Ginzburg–Landau equation with Wiener-like noise possesses a unique $\mathfrak{D}$-pullback random attractor $\mathcal{A}_{0}=\{ \mathcal{A}_{0}(\tau ,\omega )\}$ in $\mathbb{L}^{2}(\mathcal{I})$.

4 Upper semi-continuity of random attractors

We need to prove the convergence from $\varPhi _{\delta }$ to $\varPhi _{0}$ as $\delta \to 0$.

Lemma 4.1

Let $u_{\delta }$ and u be the solutions of (1.3) and (1.6) with initial data $u_{\delta ,\tau },u_{\tau }\in \mathbb{L}^{2}(\mathcal{I})$, respectively. If $\|u_{\delta ,\tau }-u _{\tau }\|\rightarrow 0$ as $\delta \rightarrow 0$, then

$$ \lim_{\delta \rightarrow 0}\sup_{t\in [\tau ,\tau +T]} \bigl\Vert u_{\delta }(t, \tau ,\omega ,u_{\delta ,\tau })-u(t,\tau ,\omega ,u_{\tau }) \bigr\Vert =0, \quad \forall T>0. $$

(4.1)

Proof

For each $\delta \in (0,\delta _{0}]$ with the positive number $\delta _{0}$ in Lemma 3.1, we define

$$\begin{aligned} \xi _{\delta }(t):=v_{\delta }(t,\tau ,\omega ,v_{\delta ,\tau })-v(t, \tau ,\omega ,v_{\tau }), \quad t\in [\tau ,\tau +T]. \end{aligned}$$

(4.2)

By the difference between Eqs. (2.8) and (3.13), we obtain

$$\begin{aligned}& \frac{\partial \xi _{\delta }}{\partial t}-\bigl(\lambda +i\mu (t)\bigr)\Delta \xi _{\delta } \\& \quad =\gamma \xi _{\delta }+\bigl(e^{-\int ^{t}_{0}\mathcal{G}_{ \delta }(\theta _{r}\omega )\,dr}-e^{-\omega (t)} \bigr)f(t,\cdot ) \\& \qquad {}-\bigl(\kappa +i\beta (t)\bigr) \bigl(e^{2\int ^{t}_{0}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr} \vert v_{\delta } \vert ^{2}v_{\delta }-e^{2\omega (t)} \vert v \vert ^{2}v\bigr). \end{aligned}$$

(4.3)

Multiplying (4.3) with $\overline{\xi _{\delta }}$ and taking the real part, we obtain

$$\begin{aligned}& {\frac{1}{2}}\frac{d }{d t}{ \Vert \xi _{\delta } \Vert }^{2}+\lambda \Vert \nabla \xi _{\delta } \Vert ^{2} \\& \quad =\gamma \Vert \xi _{\delta } \Vert ^{2}+\bigl(e^{-\int ^{t}_{0} \mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}-e^{-\omega (t)}\bigr) \bigl(f(t), \xi _{\delta }\bigr) \\& \qquad {}-\operatorname{Re}\bigl[\bigl(\kappa +i\beta (t)\bigr) \bigl(e^{2\int ^{t}_{0}\mathcal{G} _{\delta }(\theta _{r}\omega )\,dr} \vert v_{\delta } \vert ^{2}v_{\delta }-e^{2 \omega (t)} \vert v \vert ^{2}v,\xi _{\delta }\bigr) \bigr]. \end{aligned}$$

(4.4)

We split the last term of (4.4) to obtain

$$\begin{aligned}& \bigl(e^{2\int ^{t}_{0}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr} \vert v_{ \delta } \vert ^{2}v_{\delta }-e^{2\omega (t)} \vert v \vert ^{2}v,\xi _{\delta }\bigr) \\& \quad =\bigl(e^{2\int ^{t}_{0}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}-e^{2 \omega (t)}\bigr) \bigl( \vert v_{\delta } \vert ^{2}v_{\delta },\xi _{\delta }\bigr) +e^{2\omega (t)}\bigl( \vert v_{\delta } \vert ^{2}v_{\delta }- \vert v \vert ^{2}v,\xi _{\delta }\bigr). \end{aligned}$$

(4.5)

By the Gagliardo–Nirenberg inequality, $\|w\|_{4}^{4}\leq c\|w\|^{2} \|\nabla w\|^{2}$, we have

$$\begin{aligned} \bigl\vert \bigl( \vert v_{\delta } \vert ^{2}v_{\delta },\xi _{\delta }\bigr) \bigr\vert \leq c \Vert \xi _{\delta } \Vert _{4}^{4}+ \Vert v_{\delta } \Vert _{4}^{4}\leq c\bigl( \Vert \nabla v_{\delta } \Vert ^{2}+ \Vert \nabla v \Vert ^{2}\bigr) \Vert \xi _{\delta } \Vert ^{2}+ \Vert v_{\delta } \Vert _{4}^{4}. \end{aligned}$$

(4.6)

By Lemma 2.1 or (2.9), we have, as $\delta \to 0$,

$$\begin{aligned} &C_{\delta ,1}(T):=\sup_{t\in [\tau ,\tau +T]} \bigl\vert e^{2\int ^{t}_{0} \mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}-e^{2\omega (t)} \bigr\vert \to 0, \\ &C_{\delta ,2}(T):=\sup_{t\in [\tau ,\tau +T]} \bigl\vert e^{-\int ^{t}_{0} \mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}-e^{-\omega (t)} \bigr\vert \to 0, \end{aligned}$$

which further implies

$$\begin{aligned} \sup_{\delta \in (0, \delta _{0}]}\sup_{t\in [\tau ,\tau +T]}\bigl(e^{2\int ^{t}_{0}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}+ e^{-2\int ^{t} _{0}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}\bigr)\leq C(T)< +\infty . \end{aligned}$$

Hence, by (4.6) and $\beta \in C_{b}(\mathbb{R},\mathbb{R})$,

$$\begin{aligned}& \bigl\vert \operatorname{Re}\bigl(\kappa +i\beta (t)\bigr) \bigl(e^{2\int ^{t}_{0}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}-e^{2\omega (t)}\bigr) \bigl( \vert v_{\delta } \vert ^{2}v_{\delta }, \xi _{\delta }\bigr) \bigr\vert \\& \quad \leq C(T) \bigl( \Vert \nabla v_{\delta } \Vert ^{2}+ \Vert \nabla v \Vert ^{2}\bigr) \Vert \xi _{ \delta } \Vert ^{2}+C_{\delta ,1}(T) \Vert v_{\delta } \Vert _{4}^{4}. \end{aligned}$$

(4.7)

Furthermore, on the 1D-domain, we have the Agmon inequality, $\|w\|^{2}_{\infty }\leq c\|w\|\|\nabla w\|$ for $w\in \mathbb{H}^{1} _{0}(\mathcal{I})$, and thus

$$\begin{aligned} \bigl\vert \bigl( \vert v_{\delta } \vert ^{2}v_{\delta }- \vert v \vert ^{2}v,\xi _{\delta }\bigr) \bigr\vert =& \biggl\vert \int _{\mathcal{O}}\bigl( \vert v_{\delta } \vert ^{2}v_{\delta }- \vert v \vert ^{2}v\bigr)\overline{ \xi _{\delta }}\,dx \biggr\vert \\ =& \biggl\vert \int _{\mathcal{O}} \vert v_{\delta } \vert ^{2} \vert \xi _{\delta } \vert ^{2}+\overline{v _{\delta }}v \vert \xi _{\delta } \vert ^{2}+v^{2}(\overline{\xi _{\delta }})^{2}\,dx \biggr\vert \\ \leq& \int _{\mathcal{O}} \bigl( \vert v_{\delta } \vert ^{2}+ \vert v_{\delta } \vert \vert v \vert + \vert v \vert ^{2}\bigr) \vert \xi _{\delta } \vert ^{2}\,dx \\ \leq& \frac{3}{2} \int _{\mathcal{O}} \bigl( \vert v_{\delta } \vert ^{2}+ \vert v \vert ^{2}\bigr) \vert \xi _{\delta } \vert ^{2}\,dx \\ \leq& 3 \int _{\mathcal{O}} \vert \xi _{\delta } \vert ^{4}\,dx+\frac{9}{2} \int _{\mathcal{O}} \vert v \vert ^{2} \vert \xi _{\delta } \vert ^{2}\,dx \\ \leq& 3 \Vert \xi _{\delta } \Vert _{4}^{4}+ \frac{9}{2} \Vert v \Vert _{\infty }^{2} \Vert \xi _{\delta } \Vert ^{2} \\ \leq& c\bigl( \Vert \nabla v_{\delta } \Vert ^{2}+ \Vert \nabla v \Vert ^{2}\bigr) \Vert \xi _{\delta } \Vert ^{2}+c \Vert v \Vert \Vert \nabla v \Vert \Vert \xi _{\delta } \Vert ^{2} \\ \leq& c\bigl( \Vert \nabla v_{\delta } \Vert ^{2}+ \Vert \nabla v \Vert ^{2}\bigr) \Vert \xi _{\delta } \Vert ^{2}. \end{aligned}$$

(4.8)

Hence, for all $t\in [\tau ,\tau +T]$,

$$\begin{aligned}& \bigl\vert \operatorname{Re}\bigl(\kappa +i\beta (t) \bigr)e^{2\omega (t)}\bigl( \vert v_{\delta } \vert ^{2}v _{\delta }- \vert v \vert ^{2}v,\xi _{\delta }\bigr) \bigr\vert \\& \quad \leq C(T) \bigl( \Vert \nabla v_{\delta } \Vert ^{2}+ \Vert \nabla v \Vert ^{2}\bigr) \Vert \xi _{ \delta } \Vert ^{2}. \end{aligned}$$

(4.9)

By (4.5), (4.7) and (4.9), we have

$$\begin{aligned}& \bigl\vert \operatorname{Re}\bigl[\bigl(\kappa +i\beta (t)\bigr) \bigl(e^{2\int ^{t}_{0}\mathcal{G} _{\delta }(\theta _{r}\omega )\,dr} \vert v_{\delta } \vert ^{2}v_{\delta }-e^{2 \omega (t)} \vert v \vert ^{2}v,\xi _{\delta }\bigr) \bigr] \bigr\vert \\& \quad \leq C(T) \bigl( \Vert \nabla v_{\delta } \Vert ^{2}+ \Vert \nabla v \Vert ^{2}\bigr) \Vert \xi _{ \delta } \Vert ^{2}+C_{\delta ,1}(T) \Vert v_{\delta } \Vert _{4}^{4}. \end{aligned}$$

(4.10)

On the other hand, the Young inequality gives

$$ \bigl\vert \bigl(e^{-\int ^{t}_{0}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}-e^{- \omega (t)}\bigr) \bigl(f(t),\xi _{\delta }\bigr) \bigr\vert \leq \frac{1}{4} \Vert \xi _{\delta } \Vert ^{2}+C_{\delta ,2}^{2}(T) \bigl\Vert f(t) \bigr\Vert ^{2}. $$

(4.11)

We substitute (4.10) and (4.11) into (4.4) to obtain

$$ \frac{d }{d t}{ \Vert \xi _{\delta } \Vert }^{2}\leq C_{0}\bigl( \Vert \nabla v_{\delta } \Vert ^{2}+ \Vert \nabla v \Vert ^{2}+1\bigr) \Vert \xi _{\delta } \Vert ^{2}+C_{\delta }\bigl( \Vert v_{ \delta } \Vert _{4}^{4}+ \bigl\Vert f(t) \bigr\Vert ^{2}\bigr), $$

(4.12)

where $C_{\delta }=C_{\delta ,1}(T)+C_{\delta ,2}^{2}(T)\to 0$ as $\delta \to 0$.

By applying the Gronwall inequality on (4.12), we obtain, for all $t\in [\tau ,\tau +T]$,

$$\begin{aligned} \bigl\Vert \xi _{\delta }(t) \bigr\Vert ^{2} \leq& e^{C_{0}\int _{\tau }^{\tau +T}( \Vert \nabla v_{\delta }(r) \Vert ^{2}+ \Vert \nabla v(r) \Vert ^{2}+1)\,dr} \\ & {}\times\biggl( \Vert \xi _{\tau ,\delta } \Vert ^{2}+C_{\delta } \int _{\tau }^{\tau +T}\bigl( \bigl\Vert v_{\delta }(s) \bigr\Vert _{4}^{4}+ \bigl\Vert f(s) \bigr\Vert ^{2}\bigr)\,ds \biggr). \end{aligned}$$

(4.13)

By (2.16)–(2.17) in Lemma 2.2, there is a $\delta _{0}>0$ such that

$$ \sup_{\delta \in (0,\delta _{0}]} \int _{\tau }^{\tau +T}\bigl( \bigl\Vert \nabla v _{\delta }(r,\tau ,\omega ,v_{\tau }) \bigr\Vert ^{2}+ \bigl\Vert v_{\delta }(r) \bigr\Vert _{4} ^{4} \bigr)\,dr \leq C(T)< +\infty . $$

Since $v\in L^{2}(\tau , \tau +T, \mathbb{H}_{0}^{1}(\mathcal{I})$, we have

$$ \int _{\tau }^{\tau +T} \bigl\Vert \nabla v(r,\tau ,\omega ,v_{\tau }) \bigr\Vert ^{2} _{\mathbb{H}^{2}}\,ds \leq C(T,\tau , \omega )< +\infty . $$

(4.14)

Noticing that f is locally integrable, we have, for all $\delta \in (0,\delta _{0}]$,

$$\begin{aligned} \sup_{t\in [\tau , \tau +T]} \bigl\Vert \xi _{\delta }(t) \bigr\Vert ^{2}\leq C(T) \bigl( \Vert \xi _{\tau ,\delta } \Vert ^{2}+C_{\delta }\bigr). \end{aligned}$$

(4.15)

By Lemma 2.1 and $\|u_{\delta ,\tau }-u_{\tau }\|\rightarrow 0$ as $\delta \rightarrow 0$, we have

$$\begin{aligned} \Vert \xi _{\tau ,\delta } \Vert ^{2} =& \Vert v_{\tau ,\delta }-v_{\tau } \Vert ^{2} \leq e^{-2\int ^{\tau }_{0}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr} \Vert u_{\tau ,\delta }-u_{\tau } \Vert ^{2} \\ & {}+e^{-2\int ^{\tau }_{0}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}\bigl(e ^{2\int ^{\tau }_{0}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}-e^{2 \omega (\tau )} \bigr)e^{-2\omega (\tau )} \Vert u_{\tau } \Vert ^{2}\to 0 \end{aligned}$$

as $\delta \rightarrow 0$. On the other hand,

$$\begin{aligned} u_{\delta }(t,\tau ,\omega ,u_{\delta ,\tau })- u(t,\tau ,\omega ,u _{\tau }) =&e^{\int ^{t}_{0}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}v _{\delta }(t)-e^{\omega (t)}v(t) \\ =&e^{\int ^{t}_{0}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}\xi _{ \delta }(t)+\bigl(e^{\int ^{t}_{0}\mathcal{G}_{\delta }(\theta _{r}\omega )\,dr}-e ^{\omega (t)}\bigr)v(t,\tau ,\omega ,v_{\tau }). \end{aligned}$$

Notice $C_{\delta }\to 0$ in (4.15), we finish the proof. □

Remark

In a two-dimensional domain, the estimates in (4.8) may not be true and so we cannot prove the convergence of the system. This is the reason why we restrict the equation on the one-dimensional domain. In fact, the existence of a random attractor holds true in a two-dimensional domain.

Finally, we show the upper semi-continuity of attractors as the size of noise tends to zero, which is different from the case of varying density of noise [6, 16, 22, 39].

Theorem 4.2

Let $\mathcal{A}_{\delta }$ and $\mathcal{A}_{0}$ be random attractors for Ginzburg–Landau equations with difference noise and Wiener-like noise, as given in Theorems 3.2 and 3.3, respectively. Then

$$ {\lim_{\delta \rightarrow {0}}}\operatorname{dist}_{\mathbb{L}^{2}( \mathcal{I})}\bigl( \mathcal{A}_{\delta }(\tau ,\omega ),\mathcal{A}_{0}( \tau ,\omega )\bigr)=0, \quad \forall \tau \in \mathbb{R}, \omega \in \varOmega . $$

(4.16)

Proof

By all previous uniform estimates, the abstract results as given in [23, 34] seems to be applied. However, we give a direct proof for completeness.

Suppose (4.16) is not true, then there are $\varepsilon _{0}>0$, $\delta _{n}\to 0$ and $z_{n}\in \mathcal{A}_{\delta _{n}}(\tau ,\omega )$ with $\tau \in \mathbb{R}$, $\omega \in \varOmega$ such that

$$ \operatorname{dist}_{\mathbb{L}^{2}(\mathcal{I})}\bigl(z_{n}, \mathcal{A}_{0}( \tau ,\omega )\bigr)\geq \varepsilon _{0},\quad \forall n\in \mathbb{N}. $$

(4.17)

We assume without loss of generality that $\delta _{n}\leq \delta _{0}( \tau ,\omega )$ for all $n\in \mathbb{N}$, where $\delta _{0}$ is given in Lemma 3.1. For each fixed $n\in \mathbb{N}$, we have $\mathcal{A}_{\delta _{n}}\in \mathfrak{D}$, let $T_{\delta _{n}}=T( \mathcal{A}_{\delta _{n}}, \tau ,\omega )$ as given in Lemma 3.1. By the invariance of $\mathcal{A}_{\delta _{n}}$ and by Lemma 3.1,

$$ z_{n}\in \mathcal{A}_{\delta _{n}}(\tau ,\omega )=\varPhi _{\delta _{n}}(T _{\delta _{n}}, \tau -T_{\delta _{n}},\theta _{-T_{\delta _{n}}}\omega ) \mathcal{A}_{\delta _{n}}(\tau -T_{\delta _{n}}, \theta _{-T_{\delta _{n}}} \omega )\subset \widetilde{\mathcal{K}_{0}}(\tau ,\omega ), $$

where $\widetilde{\mathcal{K}_{0}}(\tau ,\omega )$ is the bounded ball in $\mathbb{H}_{0}^{1}(\mathcal{I})$, as given in (3.17). By the compactness of the Sobolev embedding, $\widetilde{\mathcal{K}_{0}}( \tau ,\omega )$ is pre-compact in $\mathbb{L}^{2}(\mathcal{I})$ and thus, passing to a subsequence, we can assume that $\|z_{n}- z_{0}\| \to 0$ for some $z_{0}\in \mathbb{L}^{2}(\mathcal{I})$.

Next, we intend to prove $z_{0}\in \mathcal{A}_{0}(\tau ,\omega )$, which will be a contradiction with (4.17). For $m=1$, the invariance shows that there are $y_{n}^{1}\in \mathcal{A}_{\delta _{n}}( \tau -1, \theta _{-1}\omega )$ such that

$$ \varPhi _{\delta _{n}}(1, \tau -1, \theta _{-1}\omega )y_{n}^{1}=z_{n},\quad \forall n\in \mathbb{N}. $$

By the same method as above, there is a $N\in \mathbb{N}$ such that $\delta _{n}\leq \delta _{0}(\tau -1, \theta _{-1}\omega )$ for all $n\geq N$ and thus Lemma 3.1 gives

$$ \bigl\{ y_{n}^{1}: n\geq N\bigr\} \subset \widetilde{ \mathcal{K}_{0}}(\tau -1, \theta _{-1}\omega ). $$

By the compactness of the Sobolev embedding, the sequence $\{y_{n} ^{1}\}$ has a convergent subsequence $\{y_{n1}^{1}\}$ such that

$$ \bigl\Vert y_{n1}^{1}- y^{1} \bigr\Vert \to 0,\quad \text{for some } y^{1}\in \mathbb{L}^{2}( \mathcal{I}). $$

Repeating this process, there are $y_{n,m-1}^{m}\in \mathcal{A}_{ \delta _{n, m-1}}(\tau -m, \theta _{-m}\omega )$ such that

$$ \varPhi _{\delta _{n,m-1}}(m, \tau -m, \theta _{-m}\omega )y_{n,m-1}^{m}=z _{n, m-1},\quad \forall n\in \mathbb{N}, $$

and, for an index subsequence $\{nm\}$ of $\{n,m-1\}$,

$$ \bigl\Vert y_{nm}^{m}- y^{m} \bigr\Vert \to 0,\quad \text{for some } y^{m}\in \mathbb{L}^{2}( \mathcal{I}). $$

We consider the diagonal subsequence $\{nn\}$ of $\{n\}$ to obtain

$$ \bigl\Vert y_{nn}^{m}- y^{m} \bigr\Vert \to 0, \quad \text{and}\quad \varPhi _{\delta _{nn}}(m, \tau -m, \theta _{-m} \omega )y_{nn}^{m}=z_{nn}\ (\to z_{0}). $$

By the convergence (4.1) in Lemma 4.1, we have

$$ z_{nn}\to \varPhi _{0}(m, \tau -m, \theta _{-m} \omega )y^{m}\quad \text{and so}\quad \varPhi _{0}(m, \tau -m, \theta _{-m}\omega )y^{m}=z_{0}. $$

On the other hand, by Proposition 2.3,

$$\begin{aligned} \bigl\Vert y^{m} \bigr\Vert ^{2} &\leq \limsup _{n\to \infty } \bigl\Vert \mathcal{A}_{\delta _{nn}}( \tau -m, \theta _{-m}\omega ) \bigr\Vert ^{2} \\ &\leq \limsup_{n\to \infty }R_{\delta _{nn}}(\tau -m, \theta _{-m} \omega ) +1 \leq R_{0}(\tau -m, \theta _{-m}\omega ) +2. \end{aligned}$$

Since $R_{0}(\tau , \omega ) +2$ is tempered (i.e. $\mathcal{K}_{0} \in \mathfrak{D}$), it follows from the attraction of $\mathcal{A} _{0}$ that

$$ \operatorname{dist}\bigl(z_{0}, \mathcal{A}_{0}(\tau ,\omega ) \bigr)=\operatorname{dist}\bigl(\varPhi _{0}(m, \tau -m, \theta _{-m} \omega )y^{m}, \mathcal{A}_{0}(\tau , \omega )\bigr)\to 0 $$

as $m\to \infty$. Hence, $z_{0}\in \mathcal{A}_{0}(\tau ,\omega )$ as desired. □