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.
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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
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
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:
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:
where \(\mathcal{A}_{0}\) is the random attractor for the following limiting equation perturbed by the Wiener-like process:
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 \(\varPhi _{\delta }: \mathbb{R}^{+}\times \mathbb{R}\times \varOmega \times \mathbb{L}^{2}(\mathcal{I}) \rightarrow \mathbb{L}^{2}( \mathcal{I})\),
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
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
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
Moreover, for each \(\varepsilon >0\), there exist \(\delta _{0}(\varepsilon ,\omega )>0\) and \(C_{0}(\varepsilon ,\omega )>0\) such that
Proof
By the mean value theorem, there is a \(r_{t,\delta }\in [t,t+\delta ]\) such that
By (1.1), \(\omega (\cdot )\) is continuous and thus uniformly continuous on \([\tau , \tau +T+1]\), which implies that
in view of \(\omega (0)=0\). Therefore, by the definition (1.2),
as \(\delta \to 0\). Hence, (2.5) holds true.
Given now \(\varepsilon >0\), there is a \(\delta _{1}\in (0,1]\) such that
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
By (1.2), we find that, for all \(|t|\geq s_{0}\) and \(0<|\delta |\leq \delta _{1}\),
By (2.5), there is \(\delta _{0}\in (0, \delta _{1}]\) such that, for all \(|t|\leq s_{0}\) and \(\delta \in (0, \delta _{0}]\),
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:
Then from (1.3) we obtain a random equation:
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
□
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 \(u_{\delta ,\tau -t}\in \mathcal{D}_{\delta }( \tau -t, \theta _{-t} \omega )\),
where \(u_{\delta }\) is a solution of the problem (1.3), and for a positive constant \(c_{1}\),
Proof
We multiply Eq. (2.8) with the conjugate function \(\overline{v_{\delta }}\) and then take the real part to obtain
where \(\|\cdot \|_{4}\) denotes the norm in \(\mathbb{L}^{4}( \mathcal{I})\). The Young inequality gives
where \(\alpha _{0}\) is the number in Assumption F. By the Young inequality again,
So, we can rewrite (2.12) for the solution \(v_{\delta }(s)=v _{\delta }(s, \tau -t,\theta _{-\tau }\omega , v_{\delta ,\tau -t})\):
Multiplying (2.13) by \(e^{2\alpha _{0} s}\) and then integrating over \((\tau -t,\tau )\), we obtain
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
where
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 \(t\geq T_{ \delta }\),
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 \(u_{\delta ,\tau -t}\in \mathcal{D}_{\delta }( \tau -t,\theta _{-\tau }\omega )\),
Similarly, we have the following useful estimate:
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
where \(R_{\delta }(\tau ,\omega )\) is given in (2.11) and satisfies
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
By taking \(\varepsilon =\frac{\alpha _{0}}{2}\), there is a \(C_{\delta }(\omega )\) such that
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 \(\tilde{t}=t-\tau \to +\infty \),
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
Hence, by taking the supremum of \(R_{\delta }(\tau ,\omega )\) on \(\delta \in (0,\delta _{0}]\), we have
Hence, by (2.5) in Lemma 2.1, the Lebesgue controlled convergence theorem gives
□
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 \(u_{\delta ,\tau -t}\in \mathcal{D}_{\delta }(\tau -t, \theta _{-t} \omega )\),
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
Proof
We multiply Eq. (2.8) by \(-\overline{\Delta v _{\delta }}=-\Delta \overline{v_{\delta }}\) and then take the real part to find
By the Young inequality we obtain
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]):
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,
which together with the Poincaré inequality implies that
Substituting it into (3.3) at the sample \(\theta _{-\tau }\omega \), we obtain
By the uniform Gronwall lemma [29] (also see [25, 36] for the non-autonomous version), we obtain
where \(I_{1}:=c_{4}\int _{\tau -1}^{\tau }\,ds=c_{4}\) and
We will use (2.17) to estimate \(I_{2}(\delta )\). Indeed, by (2.9),
as \(\delta \to 0\) uniformly in \(s\in [\tau -1,\tau ]\). Hence, there is a \(\delta _{1}>0\) such that
where \(M_{0}(\tau , \omega )\) is defined by (3.2). So, for all \(\delta \in (0,\delta _{1}]\) and \(t\geq 1\),
By (2.17), for all \(\delta \in (0,\delta _{1}]\), \(t\geq T_{ \delta }\) and \(u_{\delta ,\tau -t}\in \mathcal{D}_{\delta }(\tau -t, \theta _{-t}\omega )\),
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 \(\delta \in (0, \delta _{3}]\),
Hence, for all \(\delta \in (0,\delta _{3}]\), \(t\geq T_{\delta }\) and \(u_{\delta ,\tau -t}\in \mathcal{D}_{\delta }(\tau -t, \theta _{-t} \omega )\),
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 \(u_{\delta ,\tau -t}\in \mathcal{D}_{\delta }(\tau -t, \theta _{-t} \omega )\),
By (3.6) and the Assumption F, we have
We substitute (3.7)–(3.9) into (3.4) to find that, for all \(\delta \in (0,\delta _{4}]\), \(t\geq T_{\delta }\) and \(u_{\delta ,\tau -t}\in \mathcal{D}_{\delta }(\tau -t, \theta _{-t} \omega )\),
By using the relationship
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 \(n\geq N\),
where, by the continuity of \(\mathcal{G}_{\delta }\),
Therefore, the sequence
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
We obtain a random equation:
with the initial-boundary conditions
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
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
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
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
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
Proof
For each \(\delta \in (0,\delta _{0}]\) with the positive number \(\delta _{0}\) in Lemma 3.1, we define
By the difference between Eqs. (2.8) and (3.13), we obtain
Multiplying (4.3) with \(\overline{\xi _{\delta }}\) and taking the real part, we obtain
We split the last term of (4.4) to obtain
By the Gagliardo–Nirenberg inequality, \(\|w\|_{4}^{4}\leq c\|w\|^{2} \|\nabla w\|^{2}\), we have
By Lemma 2.1 or (2.9), we have, as \(\delta \to 0\),
which further implies
Hence, by (4.6) and \(\beta \in C_{b}(\mathbb{R},\mathbb{R})\),
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
Hence, for all \(t\in [\tau ,\tau +T]\),
By (4.5), (4.7) and (4.9), we have
On the other hand, the Young inequality gives
We substitute (4.10) and (4.11) into (4.4) to obtain
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 \(t\in [\tau ,\tau +T]\),
By (2.16)–(2.17) in Lemma 2.2, there is a \(\delta _{0}>0\) such that
Since \(v\in L^{2}(\tau , \tau +T, \mathbb{H}_{0}^{1}(\mathcal{I})\), we have
Noticing that f is locally integrable, we have, for all \(\delta \in (0,\delta _{0}]\),
By Lemma 2.1 and \(\|u_{\delta ,\tau }-u_{\tau }\|\rightarrow 0\) as \(\delta \rightarrow 0\), we have
as \(\delta \rightarrow 0\). On the other hand,
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
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
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,
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
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
By the compactness of the Sobolev embedding, the sequence \(\{y_{n} ^{1}\}\) has a convergent subsequence \(\{y_{n1}^{1}\}\) such that
Repeating this process, there are \(y_{n,m-1}^{m}\in \mathcal{A}_{ \delta _{n, m-1}}(\tau -m, \theta _{-m}\omega )\) such that
and, for an index subsequence \(\{nm\}\) of \(\{n,m-1\}\),
We consider the diagonal subsequence \(\{nn\}\) of \(\{n\}\) to obtain
By the convergence (4.1) in Lemma 4.1, we have
On the other hand, by Proposition 2.3,
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
as \(m\to \infty \). Hence, \(z_{0}\in \mathcal{A}_{0}(\tau ,\omega )\) as desired. □
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Wang, F., Li, J. & Li, Y. Random attractors for Ginzburg–Landau equations driven by difference noise of a Wiener-like process. Adv Differ Equ 2019, 224 (2019). https://doi.org/10.1186/s13662-019-2165-6
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DOI: https://doi.org/10.1186/s13662-019-2165-6