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

Abstract

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.

1. 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]. There have been some results for the classical S-H equation [2–6]. 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 [7–9]. 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 [13–19] and the references there in).

We consider the stochastic fifth-order S-H equation driven by additive white noise:with boundary conditionand the initial conditionwhere and is a bounded smooth domain.

There 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 [22–24], 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, 25–28], 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 , 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 .

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 . 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 [16–18].

2. 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 or , where is a bounded smooth domain. In the case of bounded domain, for simplicity, we use the notation to present the space . We will write the norm of as and use to denote the norm of Banach space and to denote the norm in or .

The continuous random dynamical system will be showed for the stochastic fifth-order S-H equation on (or ):with boundary conditionand with the initial conditionwhere is the smooth enough function. is a two-sided real-value Wiener process on a probability space and is given.

Let , which is a unique stationary solution of the equation [29]:

In addition, for each fixed , is pathwise continuous. And there is a tempered function :

Given a translation , one yields , where . Then, we obtain

By the Galerkin method, for all , as proved in [31], one can show that systems (9)–(11) are well-posed for every in . A continuous cocycle is defined byand a cocycle bywhere . Notice that the continuous dynamical systems and are equivalent. Similarly, is a continuous cocycle.

Let , which are a family of bound nonempty subsets of (or ) andwhere . Here, we always assume that is the collection of all tempered families of nonempty subsets of . Furthermore, we prove that there exist -pullback attractors for cocycle . In the whole paper, we assume with

3. Some Uniform Estimates

The uniform estimates are derived for solutions of systems (9)–(11) on bounded domain and , respectively.

Lemma 1. Suppose and (8) holds. Then, for every , and , there is a satisfyingfor , where is the solution of systems (9)–(11), , and and are two tempered random variables.

Proof. From (9), we deduceSinceapplying the Hölder inequality and the Young inequality, one getswhen , by the Gronwall inequality on and substituting for , one hasFrom (20), since , and is tempered, and there is satisfyingFrom (20), we can deduceFor all , we haveDenoteThen, is an absorbing set for in

Remark 1. Using the Gronwall lemma, we can obtain the following results. For every , and , there is a satisfyingwhere is the solution of systems (9)–(11), , and and are tempered random variables.

Lemma 2. Suppose and (15) holds. Then, for every , and , there is a satisfyingwhere is the solution of systems (9)–(11), , and and are tempered random variables.

Proof. By (9), one obtainsIt is easy to getIt can easily be shown thatIt is evident thatSincewe deduce thatwhere we have used the boundedness of . It shows thatSimilarly, we can getSimilarly, we have the following estimates:where .
Finally, we obtainwhereTherefore,For , choosing , by the Gronwall inequality on the interval , we obtainBy Remark 1, one can obtainNow, one can integrate (39) with respect to . Then,By Remark 1, (39)–(41), we can obtainFurthermore,

Theorem 1. Suppose and (15) holds. There exists a unique -pullback random attractor in for continuous cocycle of systems (1) and (2).
In the proof of the above lemmas, we find that all the estimates of solution do not depend on bounded domains , so these estimates are also valid for unbounded domain.

Lemma 3. Suppose and (15) holds. Then, for every , and , there are and satisfyingwhere is the solution of systems (9)–(11) and .

Proof. Let be a smooth function. When , . When , . Then, . Multiplying (21) with , we haveFor the estimate of , similar to [21], it follows thatIn addition, there areSimilarly, we getwhere we have used the boundedness of function .
It is easy to obtain the following estimates:Because and , one yieldsApplying the Gronwall lemma on , one getsNoticing and , there is a . When , we haveFrom (51), combining with Remark 1, if , there is a satisfyingAnd there is a such thatMeanwhile, there is a satisfyingBy (51)–(56), we deduce

Lemma 4. Suppose and (15) holds. Then, given and , , and , the solution satisfies the following inequality:where .

Proof. Since , then there exists a such thatBy Lemma 3, , thenBy (55) and (56), and , one gets

4. Finiteness of Fractal Dimension

Now, we are devoted to the existence of random attractor on for the random dynamical system .

Lemma 5. Suppose and (15) holds. Then, -pullback asymptotically compact holds in for continuous cocycle of (1) and (2).

Similar to the method in [30], we only give the sketch of the proof for Lemma 5.

Firstly, the weak convergencecan be given in .

Secondly, by Lemma 4, there exist enough large and satisfying

Denote the set . By the estimates of Lemma 2, the embedding is compact. It follows that the strong convergenceholds in .

According to [30], the following theorem is easily obtained. The proof is omitted.

Theorem 2. Suppose and (15) holds. Then, the continuous cocycle corresponding to problems (1) and (2) has a unique -pullback attractor in .

By Theorem 1, similar to the continuous cocycle defined in (12), the random dynamical system defined in (13) has a unique -pullback attractor, denoted by in . The boundedness of fractal dimension is proved for the random dynamical system . Because of complexity of proof on unbounded domain, we pay attention to studying the case of bounded domain . Especially, the space is denoted by .

Assume that of are a family of bounded closed random subsets, which satisfy the below conditions. For the following holds: : there is tempered random variable , which is not dependent on and satisfies the diameter of controlled by , and is continuous. : for , . : there are random variables , positive numbers , and projector satisfying for any , , where do not depend on . : satisfy the following conditions: where “E” denotes the expectation of random variables.

Once these conditions – are satisfied, the finiteness of fractal dimension can be obtained by further calculation [18, 25]. We know that conditions and hold.

Lemma 6. Let . We have

Proof. Let , when . By equation (9), we havewhere . Multiplying equation (69) with , noticing thatwe haveBy the Gronwall inequality, we obtain the estimate (68).
Let be the eigenvectors of operator corresponding to the eigenvalues with for . Let be the space spanned by and be the orthonormal projector. Let be the defined by for , with  , where . Then,

Lemma 7. For , there are a random variable and a projector satisfyingandwhere

Proof. Multiplying equation (69) with in , it follows thatThen, there exists a between and such thatandApplying Gronwall lemma to (78) and combining (68), one getsNoticing , we haveBy (79),Thus,where . Then, (73) holds. From (68), it follows that (74) holds.

Lemma 8. The expectations of are finite, that is,

Proof. Notice that satisfies for all , and satisfies the equality [25]:By ,where   are positive constants.
Since   are tempered, we obtain , and , . By the properties of , we getThen,Now, based on the conclusions of Lemmas 7 and 8, the main result is presented.

Theorem 3. The finiteness of fractal dimension for holds, that is,where .

Proof. Comparing (66) and (73), we see thatBy (67) and Lemma 8, , we deduceTake in (73) and (74) such that the following inequality is satisfied:If – holds, the following estimate is similar to [25]; here, we omit the details. Then,

Data Availability

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation, to any qualified researcher.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The paper was supported by the National Natural Science Foundation of China (nos. 11771444 and 11861013), Guangxi Natural Science Foundation (no. 2017GXNSFAA198221), Promotion of the Basic Capacity of Middle and Young Teachers in Guangxi Universities (no. 2017KY0340), and Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (no. 2018061).