Fractal Dimension for the Nonautonomous Stochastic Fifth-Order Swift–Hohenberg Equation

Some dynamics behaviors for the nonautonomous stochastic ﬁfth-order Swift–Hohenberg equation with additive white noise are considered. The existence of pullback random attractors for the nonautonomous stochastic ﬁfth-order Swift–Hohenberg equation with some properties is mainly investigated on the bounded domain and unbounded domain, through the Ornstein–Uhlenbeck transformation and tail-term estimates. Furthermore, on the basis of some conditions, the ﬁniteness of fractal dimension of random attractor is proved.


Introduction
Swift and Hohenberg proposed the Swift-Hohenberg (S-H) equation as a model for the convective instability in the Rayleigh-Bénard convection in 1997 [1]. ere have been some results for the classical S-H equation [2][3][4][5][6]. Peletier and his collaborators have studied the S-H equation from different aspects, such as the stability of stationary solutions and pattern selections of solutions [7][8][9]. Recently, some results about pullback attractor [10,11] and uniform attractor [12] of S-H equation are investigated. As we know, more and more authors investigated the random attractors and have obtained many important results. Meanwhile, there has been tremendous interest in developing the fractal dimension estimate of random attractors in recent years (see [13][14][15][16][17][18][19] and the references there in).
We consider the stochastic fifth-order S-H equation driven by additive white noise: du + Δ 2 u + 2Δu + au + u 5 − g(x, t) dt � ϕ(x)dW(t), ((x, t) ∈ D ×(κ, +∞), κ ∈ R, (1) with boundary condition u(t, x) � 0, x ∈zD , (2) and the initial condition where a > 0 and D ⊂ R 2 is a bounded smooth domain. ere are some results about dynamics behaviors for the classical autonomous stochastic S-H equation [20,21]. Except for the study of the existence of the random attractor [22][23][24], Zhou et al. have established an efficient theory about the finite fractal dimensions of random attractor [25,26]. To our knowledge, the fractal dimension estimate has been barely studied for the random attractors of the stochastic fifth-order S-H equation yet. According to the ideas in [21,[25][26][27][28], stochastic dynamics behaviors of the random attractor are considered for stochastic equations (1)-(3) in two cases.
Firstly, we mainly give the existence of random attractor for the fifth-order S-H equation corresponding to (1)- (3). A few results about the dynamics behaviors of the fifth-order S-H equation with additive noise have been given when the nonlinear term is five ordered. Due to the increasing order, more difficult terms can be produced to derive uniform estimates for the solution of equations (1)- (3). In order to overcome these difficulties aroused by the fifth-order term, we mainly use the method of integration by parts after Ornstein-Uhlenbeck transformation. Notice that the uniform estimates are independent of bounded region D, and we can obtain the existence of random attractor by proving that the random dynamical system is asymptotically compact through tail-term estimates on unbounded domain R 2 .
Secondly, we are devoted to the finiteness of fractal dimension for the random attractor of (1)- (3). Because of the complexity of proving the boundedness of fractal dimension on unbounded domain, we pay attention to studying the fifth-order S-H equation on bounded domain D. Furthermore, we will discuss the case of unbounded domain. Here, in order to obtain the boundedness of the fractal dimension, some sufficient conditions are proposed [16][17][18].

Preliminaries
We give the theorem related to random attractors. Since these conclusions are classic, we will not discuss them in detail, and readers can check the relevant literature [22-24, 29, 30].
In this paper, we will use ‖ · ‖ to denote the norm and (·, ·) to denote the inner product in L 2 (R 2 ) or L 2 (D), where D is a bounded smooth domain. In the case of bounded domain, for simplicity, we use the notation H to present the space L 2 (D). We will write the norm of L p (R 2 )(L p (D)) as ‖ · ‖ L p and use ‖ · ‖ X to denote the norm of Banach space X and ‖ · ‖ s to denote the norm in H s (D) or H s (R 2 ). e continuous random dynamical system will be showed for the stochastic fifth-order S-H equation on L 2 (R 2 ) (or L 2 (D)): with boundary condition and with the initial condition where is the smooth enough function. W is a two-sided real-value Wiener process on a probability space and g(x, − ∞ e aκ ω(κ)dκ, ω ∈ Ω, which is a unique stationary solution of the equation [29]: In addition, for each fixed ω ∈ Ω, z(θ t ω) is pathwise continuous. And there is a tempered function r(ω) > 0: By the Galerkin method, for all v κ ∈ H, as proved in [31], one can show that systems (9)-(11) are well-posed for every (12) and a cocycle Ψ: (13) where v κ � u κ − z(ω). Notice that the continuous dynamical systems Φ and Ψ are equivalent. Similarly, Φ: which are a family of bound nonempty subsets of H (or L 2 (R 2 )) and where ‖B‖ � sup u∈B ‖u‖. Here, we always assume that D is the collection of all tempered families of nonempty subsets of H. Furthermore, we prove that there exist D-pullback attractors for cocycle Φ. In the whole paper, we assume g(x, ·) ∈ C b (R, H) with
Proof. From (9), we deduce Since applying the Hölder inequality and the ε− Young inequality, one gets when r ≥ κ − t, by the Gronwall inequality on [κ − t, r] and substituting θ − κ ω for ω, one has v r, From (20), Using the Gronwall lemma, we can obtain the following results. For every κ ∈ R, ω ∈ Ω, and where v is the solution of systems (9)- (11), , and r 2 (κ, w) and r 2 (κ, w) are tempered random variables.
Proof. By (9), one obtains It is easy to get It can easily be shown that It is evident that Since we deduce that where we have used the boundedness of ‖v‖. It shows that Similarly, we can get Similarly, we have the following estimates: where Finally, we obtain d dt where erefore, For s ∈ [κ − 1, κ], choosing t ≥ T 1 (κ, ω, B) ≥ 1, by the Gronwall inequality on the interval [s, κ], we obtain By Remark 1, one can obtain Now, one can integrate (39) with respect to s. en, By Remark 1, (39)-(41), we can obtain Furthermore,  (1) and (2).
In the proof of the above lemmas, we find that all the estimates of solution do not depend on bounded domains D, so these estimates are also valid for unbounded domain. en, for every κ ∈ R, ω ∈ Ω, and B � B(κ, ω): where v is the solution of systems (9)- (11) and For the estimate of R 2 θ(|x| 2 /k 2 )vΔ 2 vdx, similar to [21], it follows that In addition, there are Similarly, we get Complexity 5 where we have used the boundedness of function θ.

Finiteness of Fractal Dimension
Now, we are devoted to the existence of random attractor on R 2 for the random dynamical system Φ. (15) holds. en, D-pullback asymptotically compact holds in L 2 (R 2 ) for continuous cocycle Φ of (1) and (2).

Lemma 5. Suppose a > 6 and
Similar to the method in [30], we only give the sketch of the proof for Lemma 5.
Firstly, the weak convergence Φ t n , κ − t n , θ − t n ω, u 0,n ⇀ξ (62) can be given in L 2 (R 2 ). Secondly, by Lemma 4, there exist enough large t and M satisfying Denote the set Q M � x ∈ R 2 : |x| ≤ M . By the estimates of Lemma 2, the embedding H 2 (Q M )↦L 2 (Q M ) is compact. It follows that the strong convergence According to [30], the following theorem is easily obtained. e proof is omitted. > 6 and (15) holds. en, the continuous cocycle Φ corresponding to problems (1) and (2) has a unique D-pullback attractor

Theorem 2. Suppose a
By eorem 1, similar to the continuous cocycle Φ defined in (12), the random dynamical system Ψ defined in (13) has a unique D-pullback attractor, denoted by e boundedness of fractal dimension is proved for the random dynamical system Ψ. Because of complexity of proof on unbounded domain, we pay attention to studying the case of bounded domain D. Especially, the space L 2 (D) is denoted by H.
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Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.