Abstract
In this paper, we first study the deterministic Swift-Hohenberg equation on a bounded domain. After obtaining some a priori estimates by the uniform Gronwall inequality, we prove the existence of an attractor by the Sobolev compact embeddings. Then, we consider the stochastic Swift-Hohenberg equation driven by additive noise on an unbounded domain and prove that the random dynamical system is asymptotically compact by uniform a priori estimates for the far-field values of the solution, which implies the existence of a random attractor for the random dynamical system associated with the stochastic Swift-Hohenberg equation.
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1 Introduction
The Swift-Hohenberg (SH) equation describes the pattern formation in fluid layers confined between horizontal well-conducting boundaries, which was proposed by Swift and Hohenberg [1] as a model for the convective instability in the Rayleigh-Bénard convection. The localized one-dimensional version of the model is as follows:
There have been some results for the local one-dimensional SH equation [2–5]. The Swift-Hohenberg equation has featured in different branches of physics, ranging from hydrodynamics to nonlinear optics, such as the Taylor-Couette flow [6, 7], study of lasers [8], and so on. The dynamical properties of the Swift-Hohenberg equation, such as the existence of a global attractor, are important for the studies of pattern formation, which ensure the stability of pattern formation and provide a mathematical foundation for the study of pattern dynamics. The authors considered the asymptotic dynamical difference between the nonlocal and local Swift-Hohenberg models in [9]. Recently, the global attractor, the stability of stationary solution, and pattern selections of the modified local Swift-Hohenberg equation have been investigated; see the references [10, 11].
After consulting the literature, we have found that there are few results about the existence of a global attractor for the local Swift-Hohenberg equation. Therefore, the existence of a global attractor for the local Swift-Hohenberg equation on a bounded domain will be given in Section 4.
In fact, when the distance from the change of stability is sufficiently small, or Rayleigh number is near thermal equilibrium, the influence of small noise or molecular noise is detected in various convection experiments [12–14]. It is difficult to stabilize the control parameters (e.g. temperature in the Rayleigh-Bénard convection) to the precision of the noise strength, which is extremely small in the case of thermal fluctuations. When the effects of thermal fluctuations on the onset of convective motion into the Bénard system are considered, the local stochastic Swift-Hohenberg equation with additive noise [1] is proposed:
Furthermore, it is also allowed to consider the effects of small possible noise from μ. So a local stochastic Swift-Hohenberg equation with multiplicative noise [15] arises:
where \(\sigma>0\), and \(\xi=\frac{dW}{dt}\) is the generalized derivative of a real-valued one-dimensional Brownian motion \(W(t)\).
There are few results on the dynamical behavior of the stochastic Swift-Hohenberg equation. Recently, some authors [16] proved the dynamics and invariant manifolds for a nonlocal stochastic Swift-Hohenberg equation. Here, the existence of a global random attractor for the stochastic Swift-Hohenberg equation with additive noise on an unbounded domain is considered. This is the main motivation of this paper. The Sobolev embeddings are no longer compact on unbounded domains. In order to overcome this difficulty, we use the method developed in [17] to prove the existence of a random attractor in the entire space. Specifically, the stochastic equation is transformed into the corresponding deterministic equation with random parameter by making use of the Ornstein-Uhlenbeck transform, and the asymptotic compactness of the random dynamical system is proved by using uniform a priori estimates for the far-field values of the solution via a truncation function.
Remark 1.1
For bounded case, “a” can be an arbitrary constant. After obtaining some a priori estimates of the solution, we can prove the existence of a global attractor by applying the compact Sobolev embeddings. In the case of unbounded domain, since the Sobolev inequality \(\|u\|_{L^{2}}\leq C\|u\|_{L^{4}}\) is invalid, we need the additional condition \(a>5\) to prove the existence of a random attractor using the current method.
In this paper, we consider the two-dimensional stochastic Swift-Hohenberg equation with additive noise
with initial condition
The paper is organized as follows. In Section 2, we recall some definitions and known results concerning global random attractors. In Section 3, we introduce the O-U transformer and transform (1.4)-(1.5) into a continuous stochastic dynamical system. In Section 4, we prove the existence of a global attractor for the corresponding deterministic dynamical system on an bounded domain. In Section 5, we obtain some uniform a priori estimates for the far-field values of the solution by the technique of a cut-off function. In Section 6, we prove the asymptotic compactness of the random dynamical system and thus deduce the existence of a global random attractor for the stochastic Swift-Hohenberg equation.
2 Preliminaries
We recall some basic concepts related to random attractors. Let \((X,\|\cdot\|_{X})\) be a separable Hilbert space with Borel σ-algebra \(\mathscr{B}(X)\) endowed with the distance d, and let \((\Omega, \mathcal{F}, \mathbb{P})\) be a probability space. We also consider the mappings \(S(t,s; \omega): X\rightarrow X\), \(-\infty< s\leq t<\infty\), parameterized by ω. There exists a group \(\theta_{t}\), \(t\in\mathbb{R}\), of measure-preserving transformations of \((\Omega, \mathcal{F}, \mathbb{P})\) such that, for all \(s< t\) and \(x\in X\),
where \(\omega(t)\) is from the two-sided Wiener space \(C_{0}(\mathbb{R};X)\) of continuous functions with values in a Banach space X, equal to 0 at \(t=0\). In this case, \(\theta_{t}\) is defined as
Definition 2.1
Let \(t\in\mathbb{R}\) and \(\omega\in\Omega\). A stochastic dynamical system with time t on a complete separable metric space \((X,d)\) with Borel σ-algebra \(\mathscr{B}\) over \(\{\theta_{t}\}\) on \((\Omega, \mathcal{F}, \mathbb{P})\) is a measurable map
such that \(S(0,0; \omega)=id\) and \(S(t,0; \omega)=S(t,s; \omega) S(s,0; \omega)\) for all \(t,s \in\mathbb{R}\) and all \(\omega\in \Omega\).
Definition 2.2
Given \(t\in\mathbb{R}\) and \(\omega\in\Omega\), \(K(t,\omega) \subset X\) is called an attracting set if for all bounded sets \(B\subset X\),
where \(d(A,B)\) is the semidistance defined by
Definition 2.3
A family \(A(\omega)\) (\(\omega\in \Omega\)) of the closed subsets of X is measurable if for all \(x\in X\), the mapping \(\omega\mapsto d(A(\omega),x)\) is measurable.
Definition 2.4
Define the random omega limit set of a bounded set \(B\subset X\) at time t as
Definition 2.5
Let \(S(t,s;\omega)_{t\geq s,\omega\in\Omega}\) be a stochastic dynamical system, and let \(A(\omega)\) be a stochastic set satisfying the following conditions:
-
(1)
It is the minimal closed set such that, for \(t\in\mathbb{R}\) and \(B\subset X\),
$$d\bigl(S(t,s;\omega)B, A(\omega)\bigr)\rightarrow0, \quad s\rightarrow- \infty. $$Then \(A(\omega)\) is said to attract B (B is a deterministic set).
-
(2)
\(A(\omega)\) is the largest compact measurable set that is invariant in sense that
$$S(t,s;\omega)A(\theta_{s}\omega)=A(\theta_{t} \omega), \quad s\leq t. $$Then \(A(\omega)\) is said to be the random attractor.
Theorem 2.6
Let \(S(t,s;\omega)_{t\geq s,\omega\in\Omega}\) be a stochastic dynamical system satisfying the following conditions:
-
(i)
\(S(t,r;\omega)S(r,s;\omega)x=S(t,s;\omega)x\) for all \(s\leq r\leq t\) and \(x\in X\),
-
(ii)
\(S(t,s;\omega)\) is continuous in X for all \(s\leq t\),
-
(iii)
for all \(s< t\) and \(x\in X\), the mapping \(\omega\mapsto S(t,s;\omega)x\) is measurable from \((\Omega, \mathcal{F})\) to \((X,\mathscr{B}(X))\),
-
(iv)
for all \(t\in\mathbb{R}, x\in X\), and \(\mathbb{P}\)-a.e. ω, the mapping \(s\mapsto S(t,s;\omega)x\) is right continuous at any point.
Assume that there exists a group \(\theta_{t}\), \(t\in\mathbb{R}\), of measure-preserving mappings such that
and for \(\mathbb{P}\)-a.e. \(\omega\in\Omega\), there exists a compact attracting set \(K(\omega)\) at time 0. We set \(\Lambda(\omega)= \overline{\bigcup_{B\subset X}A(B,\omega)}\), where the union is taken over all bounded subsets of X, and \(A(B,\omega)\) is given by
Then \(\Lambda(\omega)\) is a random attractor.
Theorem 2.7
(Uniform Gronwall lemma; see [20])
Let \(g,h,y\) be three positive locally integrable functions on \((t_{0},\infty)\) satisfying
where \(r,a_{1},a_{2},a_{3}\) are positive constants. Then
For the convenience of the following contents, we introduce some functional spaces and some notations.
\(L^{q}(D)\) is the Lebesgue space with norm \(\|\cdot\|_{L^{q}}\). The inner product on \(L^{2}(D)\) is denoted by
Particularly, \(\|u\|_{L^{\infty}}=\) \(\operatorname {ess\,sup}_{x\in D}|u(x)|\) for \(q=\infty\).
\(H^{\sigma}(D)\) is the Sobolev space \(\{u\in L^{2}(D), D^{k}u\in L^{2}(D), k\leq\sigma\}\) with norm \(\|\cdot\|_{H^{\sigma}}\). If \(D=R^{2}\), then we use the same notations. In particular, \(H_{0}^{2}(D)\) is the Sobolev space \(\{u\in L^{2}(D), D^{k}u\in L^{2}(D), k\leq2, \Delta u|_{\partial D}=0\}\).
\(\mathscr{C}(I,X)\) is the space of continuous functions from the interval I to X.
For notational simplicity, C is a generic constant and may assume various values from line to line.
3 The hydrodynamical equation with additive noise
Here we show that there is a continuous random dynamical system \((S(t,s;\omega);L^{2}(R^{2}))\) generated by the stochastic local Swift-Hohenberg equation on \(R^{2}\)
with the initial condition
where \(\Phi_{i}(x)\) is a given smooth enough function on \(R^{2}\). We need to convert the stochastic equation with random additive term into a deterministic equation with random parameter.
Now, we introduce the Ornstein-Uhlenbeck process
where \(A=\Delta^{2}\) is a positive operator. It is well known and easy to check that \(z_{i}(\cdot)\) is a stationary process with \(\mathbb{P}\)-a.e. continuous trajectories.
Putting \(z(t)=\sum_{i=1}^{m}\Phi_{i}(x)z_{i}(t)\), we have
In addition, for \(\mathbb{P}\)-a.e. \(\omega\in\Omega\), we have that
at most polynomically grows as \(t\rightarrow-\infty\), where \(p\geq2\) (see [17]).
To study (3.1)-(3.2), it is usual to translate the known \(v=u-z\) (z has the above form) and obtain the following equation:
By the Galerkin method one can show that, for all \(v_{s} \in L^{2}(R^{2})\), system (3.4)-(3.5) has a unique solution \(v\in\mathscr{C}(s,T;L^{2}(R^{2}))\cap L^{2}(s,T;H^{2}(R^{2}))\) with \(v(s)=v_{s} \) for \(\mathbb{P}\)-a.e. \(\omega\in\Omega\). It is obvious that there is a continuous stochastic dynamical system \((S(t,s; \omega); L^{2}(R^{2}))\) generated by the stochastic local Swift-Hohenberg equation with additive noise.
4 Global attractor on a bounded domain
For completeness, we first consider the following initial-boundary value problem for the deterministic local Swift-Hohenberg equation on a bounded domain:
with initial condition
and boundary conditions
where D is an open connected bounded domain in \(R^{2}\), and a is an arbitrary constant.
Theorem 4.1
For any \(u_{0}(x)\in{H}^{2}_{0}(D)\), there exists a unique, globally defined solution \(V(t)u_{0}=\sigma(u_{0},t)\) in \({H}^{2}_{0}(D)\) of system (4.1)-(4.3), and \(V(t)\) is a semigroup on \(H^{2}_{0}(D)\). Moreover, the semigroup is point dissipative in \(H^{2}_{0}(D)\) and compact in \(H^{2}_{0}(D)\) for \(t>0\). Hence, system (4.1)-(4.3) has a global attractor in \(H^{2}_{0}(D)\).
Proof
(1) Taking the inner product of (4.1) with u in H, we can obtain that
Adding the term \(\|u(t)\|^{2}_{L^{2}}\) to the above equation, we have
For the first term on the right-hand side of (4.5), using the inequality \(\|u\|_{L^{2}}\leq C\|u\|_{L^{4}}\) and ε-Young inequality, we can easily deduce that
For the second term on the right-hand side of (4.5), we have the estimate
where we applied the Gagliardo-Nirenberg inequality
and the ε-Young inequality.
Combining the above consequences, we get the inequality
By the Gronwall inequality, we have
Now we integrate with respect to s from t to \(t+1\) on the both sides of (4.6) and deduce that
By (4.7) we have
(2) Taking the inner product of (4.1) with \(\Delta^{2}u\) in H, we have
Using the Hölder inequality and ε-Young inequality, we then easily obtain
For the estimate of \((u^{3},\Delta^{2}u)\), by the Gagliardo-Nirenberg inequality
the Hölder inequality, and ε-Young inequality, we obtain the estimate
where the last inequality is owing to the boundedness of \(\|u\|_{L^{2}}\).
Plugging (4.9)-(4.10) into (4.8) yields
Letting \(\epsilon=\frac{2}{3}\), we can easily write it as
By (4.10) and Theorem 2.7 we get that
Then
Hence, (4.12) provides a uniform bound for \(\|\Delta u(t+1)\|_{L^{2}}\). Thus, the existence of an absorbing ball in the \({H}^{2}_{0}(D)\) is proved. This implies that \(V(t)\) is point dissipative in \({H}^{2}_{0}(D)\). Similarly to [10], we can obtain that \(V(t)\) is compact for \(t>0\); for details, we refer to Theorem 3.2 in [10]. Hence, Theorem 4.1 implies the existence of a global attractor for problem (4.1)-(4.3) in the space \({H}^{2}_{0}(D)\). □
5 Uniform estimates on an unbounded domain
In this section, we derive uniform estimates of a solution for system (3.4)-(3.5) on \(R^{2}\) for the purpose of proving the existence of a bounded random absorbing set and the asymptotic compactness of the stochastic dynamical system associated with the equation. In particular, we show that the tails of the solution, that is, the norms of solutions evaluated at large values of \(|x|\), are uniformly small as \(t\rightarrow0\).
Lemma 5.1
Let \(\Phi_{i}(x)\in H^{2}(R^{2})\), \(a>5\), \(v(t)\) be the solution of system (3.4)-(3.5). Then, for any given \(\eta> 0\) and \(u_{s}\in H\) satisfying \(\|u_{s}\|\leq\eta\), there exist random radii \(r_{0}(w)\), \(r_{1}(w),r_{2}(w)\) and \(s_{0}(w)\leq-1\) such that for all \(s\leq s_{0}(w)\), the following inequalities hold \(\mathbb {P}\)-a.e.:
Proof
Taking the inner product of (3.4) with \(v(t)\) in H, we can obtain that
Applying the Hölder inequality and ε-Young inequality, we have
Utilizing similar arguments, the following three estimates are also valid:
and
For the last term on the left-hand side of (5.1), applying the Gagliardo-Nirenberg inequality
the Hölder inequality, and ε-Young inequality, we obtain that
Because \((3vz^{2},v)=3\int_{\Omega}v^{2}z^{2}\geqslant0\), we drop \((3vz^{2},v)\) on the left-hand side of (5.1).
Combining the above estimates, we obtain the inequality
Let \(F(t)=C(\|\Delta z\|^{2}_{L^{2}}+\|z\|^{2}_{L^{2}}+\|z\|^{6}_{L^{6}}+\|z\| ^{6}_{L^{2}})\). Then
By the Gronwall inequality for \(s\leq-1\) and \(t\in[-1,0]\) we have
where \(F(\sigma)\) grows at most polynomially as \(\sigma\rightarrow -\infty\) \(\mathbb{P}\)-a.e. Because \(F(\sigma)\) is multiplied by a function that decays exponentially, the integral in (5.3) converges. Then, given \(\eta > 0\), we can choose \(s_{0}(\omega)\leq-1\) depending only on ω such that
We can deduce from (5.3) that, for all \(s\leq s_{0}(\omega)\),
Similarly, because \(z(s)\) grows at polynomially as \(s\rightarrow -\infty\) and \(z(s)\) is multiplied by a function that decays exponentially, the term
is bounded. Now we integrate with respect to t from −1 to 0 on the both sides of (5.2) and deduce that
On the other hand, we can obtain
The proof is complete. □
Lemma 5.2
Let \(\Phi_{i}(x)\in H^{2}(R^{2})\), \(a>5\), \(v(t)\) be the solution of system (3.4)-(3.5). Then, for any given \(\eta> 0\) and \(u_{s}\in H\) satisfying \(\|u_{s}\|\leq\eta\), there exist random radii \(r_{3}(\omega)\), \(r_{4}(\omega)\), \(r_{5}(\omega)\) such that the following inequalities hold \(\mathbb{P}\)-a.e.:
Proof
Integrating from s (\(s\leq-1\)) to 0 on the both sides of (5.2), we have
For the first term of the right-hand side of (5.4), since \(F(\tau )\) grows at most polynomially as \(\tau\rightarrow-\infty\) \(\mathbb {P}\)-a.e., and it is multiplied by a function that decays exponentially, the integral in (5.4) converges. Then, for the second term of the right-hand side of (5.4), there exists a \(s_{2}(\omega)\leq s_{1}(\omega)\) satisfying
The proof is complete. □
Lemma 5.3
Let \(\Phi_{i}(x)\in H^{2}(R^{2})\), \(a>5\), \(v(t)\) be the solution of system (3.4)-(3.5).Then, for any given \(\eta> 0\) and \(u_{s}\in H\) satisfying \(\Vert u_{s}\Vert \leq\eta\), there exist random radii \(r_{6}(\omega),r_{7}(\omega)\), and \(s_{0}(\omega)\leq-1\) such that for all \(s\leq s_{0}(\omega)\), the following inequality holds \(\mathbb{P}\)-a.e.:
Proof
Taking the inner product of (3.4) with \(\Delta^{2}v\) in H, we can obtain
Applying the Hölder inequality and ε-Young equality, we have
It can easily be shown that
For the estimate of \(a(z,\Delta^{2}v)\), it is evident that
According to the Gagliardo-Nirenberg inequality
the Hölder inequality, and ε-Young inequality, we deduce that
where the last inequality is owing to the boundedness of \(\|v\|_{L^{2}}\). It is obvious that
By the Gagliardo-Nirenberg inequality
Hölder inequality, and ε-Young inequality, we have the estimate
where the last inequality is owing to the boundedness of \(\|v\| _{L^{2}}\).
On account of the boundedness of \(\|v(t)\|_{L^{2}}\), it is easy to check that
Combining the above estimates, we obtain the inequality
Let \(G(t)=C(\|\Delta z\|_{L^{2}}^{2}+\|z\|_{L^{6}}^{6}+\|z\|_{L^{\infty}}^{\frac {8}{3}}+\|z\|_{L^{\infty}}^{4}+\|z\|_{L^{2}}^{2})\), which grows at most polynomially as \(t\rightarrow-\infty\) \(\mathbb{P}\)-a.e. Then
Integrating from θ to t for any \(-1\leq\theta\leq t\leq0\), we have
Now integrating with respect to θ on \([-1,0]\) on both sides of (5.6), as Lemma 5.1, satisfies for all \(s< s_{0}(\omega)\), there exists \(s_{0}(\omega)\) such that, for all \(s< s_{0}(\omega)\),
On the other hand, we can obtain
The proof is complete. □
Lemma 5.4
Let \(\Phi_{i}(x) \in H^{2}(R^{2}), a>5, \eta> 0\) be given, and \(u_{s} \in H\) satisfy \(\|u_{s}\|\leq\eta\). Then, for every \(\epsilon> 0\) and \(\mathbb{P}\)-a.e. \(\omega\in\Omega\), there exist \(\bar {s}(\omega)\leq-1\) and \(\bar{k}(\epsilon) > 0\) such that, for all \(s \leq\bar{s}(\omega)\) and \(k > \bar{k}(\epsilon)\), the solution \(v(t)\) of system (3.4)-(3.5) with \(v_{s}=u_{s}-z(s)\) satisfies
Proof
Let \(\theta(s)\) be a smooth function defined on \(R^{+}\) such that \(0 \leq\theta(s) \leq1\) for all \(s\in R^{+}\) and
Then there exists a positive constant C such that \(|\theta '(s)|+|\theta''(s)|< C\) for all \(s\in R^{+}\).
Multiplying (3.4) with \(\theta(\frac{|x|^{2}}{k^{2}})v\) and then integrating the resulting identity, we find
For the estimate of \(\int_{R^{2}}\theta(\frac{|x|^{2}}{k^{2}})v\Delta^{2}v\,dx\),
For the first term on the right-hand side of (5.9), we have
Similarly, for the second and third terms on the right-hand side of (5.9), we have
and
Substituting estimates (5.10)-(5.12), it follows that
By the Hölder inequality and ε-Young inequality we have
Similarly, we get
where the second inequality is owing to the boundedness of function θ.
For the estimate of \(\int_{R^{2}}\theta(\frac{|x|^{2}}{k^{2}})vz\,dx\), we obtain
For the estimate of \(\int_{R^{2}}\theta(\frac{|x|^{2}}{k^{2}})vz^{3}\,dx\), we get
For the estimate of\(\int_{R^{2}}\theta(\frac{|x|^{2}}{k^{2}})v^{3}z\,dx\), we have
Because \(\int_{R^{2}}\theta(\frac{|x|^{2}}{k^{2}})v^{2}z^{2}\,dx\geq0\), we drop it on the left-hand side of (5.8).
From the above estimates we can obtain the inequality
By the Gronwall inequality for \(s\leq-1\) and \(t\in[-1,0]\) we have
For the first term of the right-hand side of (5.14), there exists \(s_{1}(\omega)\) such that, for all \(s\leq s_{1}(\omega)\), \(e^{(a-5)(s+1)}\) decays exponentially as \(s\rightarrow-\infty\). Then there exists \(\bar {s}< s_{1}(\omega)\) such that, for all \(s\leq\bar{s}\), we have
For the second term of the right-hand side of (5.14), by the Gagliardo-Nirenberg inequality and ε-Young inequality we have
According to Lemma 5.2, there exists \(k_{1}>0\) such that, for all \(k>k_{1}\), we have
It follows from (5.15)-(5.16) that
Let \(E(t)=C(\|z\|^{2}_{H^{2}}+\|z\|^{3}_{H^{2}}+\|z\|^{4}_{H^{2}})\). When k is large enough, we have
because \(E(t)=C(\|z\|^{2}_{H^{2}}+\|z\|^{3}_{H^{2}}+\|z\|^{4}_{H^{2}})\leq\epsilon \sum_{k=1}^{m}(|z_{k}|^{2}+|z_{k}|^{3}+|z_{k}|^{4})\) when \(|x|\geq k\) and \(\sum_{k=1}^{m}(|z_{k}|^{2}+|z_{k}|^{3}+|z_{k}|^{4})\) grows at most polynomially.
Then we can obtain that, for \(\epsilon>0\), there exist \(\bar{s},\bar {k}=\max\{k_{1},k_{2}\}\) such that, for all \(s\leq\bar{s}\) and \(k\geq\bar{k}\),
□
Lemma 5.5
Let \(\Phi_{i}(x) \in H^{2}(R^{2}), a>5, \eta> 0\) be given, and \(u_{s} \in H\) satisfy \(\|u_{s}\|\leq\eta\). Then, for every \(\epsilon> 0\) and \(\mathbb{P}\)-a.e. \(\omega\in\Omega\), there exist \(s'(\omega )\leq-1\) and \(k'(\epsilon) > 0\) such that, for all \(s \leq s'(\omega)\) and \(k >k'(\epsilon)\), the solution \(u(t)\) satisfies the inequality
Proof
Since \(z(t)\in H^{2}\), we have
By Lemma 5.4 we know that
Let \(\bar{s}(\omega)\) and k̄ be the constants in Lemma 5.1. Then choosing \(s' >\bar{s}\) and \(k'>\bar{k}\), for all \(s\leq s',k>k'\), by (5.17) and (5.18) we have
□
6 Random attractors
Motivated by these previous works, in this section, we are interested in the existence of a random attractor for the random dynamical system \(S(t,s;\omega)\) associated with the stochastic Swift-Hohenberg equation on \(R^{2}\).
Lemma 6.1
Assume that \(\Phi_{i}(x)\in H^{2}(R^{2})\). Then the random dynamical system \(S(t,s;\omega)\) is asymptotically compact in \(L^{2}(R^{2})\); that is, for \(\mathbb{P}\)-a.e. \(\omega\in\Omega\), the sequence \(u(0,s_{n};\omega)\) has a convergent subsequence in \(L^{2}(R^{2})\), provided that \(s_{n}\rightarrow-\infty\).
Proof
Let \(s_{n}\rightarrow-\infty\). Then by Lemma 5.1, for \(\mathbb{P}\)-a.e. \(\omega\in\Omega\), we obtain
Hence, there is \(\xi\in L^{2}(R^{2})\) such that, up to a subsequence,
Next, we prove that the weak convergence of (6.1) is in fact the strong convergence.
Given \(\epsilon>0\), by Lemma 5.5 there are \(T_{1}(\eta,\omega,\epsilon)\) and \(k(\omega,\epsilon)\) such that, for all \(s< T_{1}\), we have
Since \(s_{n}\rightarrow-\infty\), there is \(N_{1}(\eta,\omega,\epsilon ,)\) such that \(s_{n}< N_{1}\) for every \(n>N_{1}\). Therefore, it follows from (6.2) that, for all \(n>N_{1}\), we have
On the other hand, by Lemma 5.3 there are \(T_{2}(\eta,\omega)\) and \(r(\omega)\) such that, for all \(s< T_{2}\), we have
Denote \(Q_{R}=\{x\in R^{2}:|x|\leq R\}\). By the compactness of embedding \(H^{2}(Q_{R})\hookrightarrow L^{2}(Q_{R})\) it follows from (6.4) that there is a subsequence
which shows that, for given \(\epsilon>0\), there exists \(N_{2}(\eta ,\omega,\epsilon)\) such that, for all \(n>N_{2}\),
Note that \(\xi\in L^{2}(R^{2})\), so there exists \(R'(\epsilon)>R\) such that
and
Let \(N_{3}=\max\{N_{1},N_{2}\}\). By (6.3), (6.6), and (6.7) we find that, for all \(n\geq N_{3}\), we have
which shows that
as desired. The proof is complete. □
We are now in a position to present our main result, the existence of a random attractor for \(S(t,s;\omega)\) in \(L^{2}(R^{2})\).
Theorem 6.2
Let \(\Phi(x)\in H^{2}(R^{2})\) and \(a>5\). Then the random dynamical system \(S(t,s;\omega)\) has a unique random attractor in \(L^{2}(R^{2})\).
Proof
Notice that \(S(t,s;\omega)\) has a closed random absorbing set in \(H^{1}(R^{2})\) by Lemmas 5.1 and 5.3 and is asymptotically compact in \(L^{2}(R^{2})\) by Lemma 6.1. Hence, the existence of a unique random attractor for \(S(t,s;\omega)\) follows from Theorem 2.6 immediately. □
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Acknowledgements
This paper was supported by NSFC Grant Nos. 11201475, 11371183, 11301097, GXNSF Grant No. 2014GXNSFAA118016, the project of outstanding young teachers’ training in higher education institutions of Guangxi and the Doctoral Grant of Guangxi University of Science and Technology No. 03081587-8.
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Guo, C., Chen, Y. & Guo, Y. Dynamical behaviors of stochastic local Swift-Hohenberg equation on unbounded domain. J Inequal Appl 2016, 228 (2016). https://doi.org/10.1186/s13660-016-1166-1
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DOI: https://doi.org/10.1186/s13660-016-1166-1