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

Authors: Yanfeng Guo; Chunxiao Guo; Yongping Xi
Journal: Complexity
 (2020)

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## 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.


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## Body

## 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:(1)du+Δ2u+2Δu+au+u5−gx,tdt=ϕxdWt,x,t∈D×κ,+∞,κ∈ℝ,with boundary condition(2)ut,x=0,x∈∂D,and the initial condition(3)ux,κ=uκx,x∈D,where a>0 and D⊂ℝ2 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 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 ℝ2.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 [16–18].


## 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 L2ℝ2 or L2D, where D is a bounded smooth domain. In the case of bounded domain, for simplicity, we use the notation H to present the space L2D. We will write the norm of Lpℝ2LpD as ⋅Lp and use ⋅X to denote the norm of Banach space X and ⋅s to denote the norm in HsD or Hsℝ2.The continuous random dynamical system will be showed for the stochastic fifth-order S-H equation onL2ℝ2 (or L2D):(4)du+Δ2u+2Δu+au+u5−gx,tdt=∑i=1mϕixdWit,with boundary condition(5)ut,x=0,x∈∂D,and with the initial condition(6)ux,κ=uκx,x∈D,where ϕix∈H4ℝ2i=1,…,m is the smooth enough function. W is a two-sided real-value Wiener process on a probability space and gx,⋅∈Cbℝ,H is given.Letz˜ω=−a∫−∞0eaκωκdκ,ω∈Ω, which is a unique stationary solution of the equation [29]:(7)dz˜+az˜dt=dW.In addition, for each fixedω∈Ω, z˜θtω is pathwise continuous. And there is a tempered function rω>0:(8)z˜θtω2+z˜θtωp≤ea−6/2|trω,t∈ℝ,P−a.e.ω∈Ω.Given a translationvt,κ,ω,vκ=ut,κ,ω,uκ−zθtω, one yields vκ=uκ−zω, where zθtω=ϕxz˜θtω. Then, we obtain(9)dvdt+Δ2v+2Δv+2Δzθtω+av+Δ2zθtω+v5+z5θtω+5v4zθtω+10v3z2θtω+10v2z3θtω+5vz4θtω=gx,t,(10)vt,x=0,x∈∂D,(11)vκ,κ,ω,x=vκω=uκx−zω.By the Galerkin method, for allvκ∈H, as proved in [31], one can show that systems (9)–(11) are well-posed for every ω∈Ω in H. A continuous cocycle Φ:ℝ+×ℝ×Ω×H⟶H is defined by(12)Φt,κ,ω,uκ=Φt,κ,ωuκ=ut+κ,κ,θ−κω,uκ,and a cocycle Ψ:ℝ+×ℝ×Ω×H⟶H by(13)Ψt,κ,ω,vκ=Ψt,κ,ωvκ=vt+κ,κ,θ−κω,vκ,where vκ=uκ−zω. Notice that the continuous dynamical systems Φ and Ψ are equivalent. Similarly, Φ:ℝ+×ℝ×Ω×L2ℝ2⟶L2ℝ2 is a continuous cocycle.LetB=Bκ,ω:κ∈ℝ,ω∈Ω, which are a family of bound nonempty subsets of H (or L2ℝ2) and(14)limt⟶+∞e−ctBκ+t,θtω2=0,∀c>0,κ∈ℝ,ω∈Ω,where B=supu∈Bu. 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 gx,⋅∈Cbℝ,H with(15)g2=supt∈ℝg⋅,t2<+∞.


## 3. Some Uniform Estimates

The uniform estimates are derived for solutions of systems (9)–(11) on bounded domain D and ℝ2, respectively.Lemma 1.
Supposea>6 and (8) holds. Then, for every κ∈ℝ,ω∈Ω, and B=Bκ,ω:κ∈ℝ,ω∈Ω∈D, there is a T0κ,ω,B≥1 satisfying(16)vκ,κ−t,θ−κω,vκ−t2≤r0κ,w,uκ,κ−t,θ−κω,uκ−t2≤r1κ,w,for ∀t≥T0κ,ω,B, where v is the solution of systems (9)–(11), vκ−t+zθ−tω∈Bκ−t,θ−tω, and r0κ,w and r1κ,w are two tempered random variables.Proof.
From (9), we deduce(17)12ddtv2+Δv2=−av2−vL66−2Δv+2Δzθtω+Δ2zθtω,v−z5θtω+5v4zθtω+10v3z2θtω,v−52v2z3θtω+vz4θtω,v+gt,v.
Since(18)10v3z2θtω,v=10∫Dv4z2θtωdx≥0,5vz4θtω,v=5∫Dv2z4θtωdx≥0,applying the Hölder inequality and the ε−Young inequality, one gets(19)ddtv2+vL66+a−6v2+Δv2≤Cz˜θtω10+z˜θtω6+z˜θtω2+1agt2≐Fθtω+1agt2,when r≥κ−t, by the Gronwall inequality on κ−t,r and substituting θ−κω for ω, one has(20)vr,κ−t,θ−κω,vκ−t2≤e−a−6r+t−κvκ−t2+∫κ−tre6−ar−sFθs−κω+e6−ar−sags2ds≤e−a−6tvκ−t2+∫−∞0e−6−asFθsω+e−6−asags+κ2ds.
From (20), since vκ−t+zθ−tω∈Bκ−t,θ−tω, and B=Bκ,ω:κ∈ℝ,ω∈Ω is tempered, and there is T0=T0κ,ω,B>1 satisfying(21)e−a−6tvκ−t2≤1,∀t≥T0.
From (20), we can deduce(22)vκ,κ−t,θ−κω,vκ−t2≤1+C∫−∞0ea−6s2rωds+1a∫−∞0‖s+κ2ea−6sds≤e−a−6tvκ−t2+2Ca−6rω+1a∫−∞0ea−6sgs+κ2ds≐r0κ,ω,∀t≥T0κ,ω,B.
For allvκ−t+zθ−tω∈Bκ−t,θ−tω, we have(23)uκ,κ−t,θ−κω,uκ−t2≤2zω2+2+C∫−∞0ea−6s/2rωds+∫−∞0‖gs+κ‖2ea−6sds≐r1κ,ω,∀κ∈ℝ,ω∈Ω.
Denote(24)Kκ,ω=u∈H:u2≤r1κ,ω,ω∈Ωκ∈ℝ∈D.
Then,Kκ,ω is an absorbing set for Φ in DRemark 1.
Using the Gronwall lemma, we can obtain the following results. For everyκ∈ℝ,ω∈Ω, and B=Bκ,ω:κ∈ℝ,ω∈Ω∈D, there is a T1κ,ω,B≥T0≥1 satisfying(25)∫κ−1κvs,κ−t,θ−κω,vκ−t2ds≤r2κ,w,∀t≥T1κ,ω,B,∫−t0ea−6sΔvs+κ,κ−t,θ−κω,vκ−t2ds+∫−t0ea−6svs+κ,κ−t,θ−κω,vκ−tL66ds≤r3κ,ω,∀t≥T1κ,ω,B,where v is the solution of systems (9)–(11), vκ−t+zθ−tω∈Bκ−t,θ−tω, and r2κ,w and r2κ,w are tempered random variables.Lemma 2.
Supposea>6 and (15) holds. Then, for every κ∈ℝ,ω∈Ω, and B=Bκ,ω:κ∈ℝ,ω∈Ω∈D, there is a T1κ,ω,B≥T0≥1 satisfying(26)Δvκ,κ−t,θ−κω,vκ−t2≤r4κ,ω,∀t≥T1κ,ω,B,where v is the solution of systems (9)–(11), vκ−t+zθ−tω∈Bκ−t,θ−tω, and r4κ,w and r5κ,w are tempered random variables.Proof.
By (9), one obtains(27)12ddtΔv2+2Δv,Δ2v+2Δzθtω+Δ2zθtω,Δ2v+aΔv2+Δv2+v5,Δ2v+z5θtω,Δ2v+v4zθtω,Δ2v+10v3z2θtω,Δ2v+10v2z3θtω,Δ2v+5vz4θtωΔ2v=gx,t,Δ2v.
It is easy to get(28)2Δv,Δ2v≤14Δ2v2+4Δv2.
It can easily be shown that(29)2Δzθtω,Δ2v≤112Δ2v2+CΔzθtω2,gx,t,Δ2v≤112Δ2v2+3‖gt‖2.
It is evident that(30)Δ2zθtω,Δ2v≤112Δ2v2+Czθtω2.
Since(31)vL8≤CΔ2v3/16v13/16,vL4≤CΔ2v1/8v7/8,we deduce that(32)5v4zθtω,Δ2v≤5vL84Δ2vzθtωL∞CΔ2v7/4v13/4zθtωL∞≤112Δ2v2+CzθtωL∞8,5vz4θtω,Δ2v≤5vL4Δ2vzθtωL164≤CΔ2v9/8zθtωL164≤112Δ2v2+CzθtωL1664/7,where we have used the boundedness of v. It shows that(33)z5θtω,Δ2v≤112Δ2v2+CzθtωL1010.
Similarly, we can get(34)10v3z2θtω,Δ2v≤10vL123Δ2vzθtωL82≤CΔ2v13/8zθtωL82v19/8≤CΔ2v13/8zθtωL82≤112Δ2v2+CzθtωL832/3,10v2z3θtω,Δ2v≤10vL82Δ2vzθtωL123≤CΔ2v3/8v13/8Δ2vzθtωL123≤CΔ2v11/8zθtωL123≤112Δ2v2+CzθtωL1248/5.
Similarly, we have the following estimates:(35)∫D∇v2v3∇2vdx≤∇vL∞2∇2vvL63≤CvΔ2vΔvvL63≤CΔ2vΔvvL63≤1240Δ2v2+CΔv2vL66,v5,Δ2v=−5∫Dv4∇v∇3vdx=5∫D∇∇vv4∇2vdx=5∫D∇2vv4+4∇v2v3∇2vdx=5∫D∇2v2v4dx+20∫D∇v2v3∇2vdx,where ∫D∇2v2v4dx≥0.
Finally, we obtain(36)ddtΔv2+2a−8Δv2≤CvL66Δv2+Hθtω+3gt2,
where(37)Hθtω=Cz˜θtω2+z˜θtω10+z˜θtω32/3+z˜θtω8+z˜θtω64/7+z˜θtω48/5.
Therefore,(38)ddtΔv2≤CvL66Δv2+Hθtω+3gt2.
Fors∈κ−1,κ, choosing t≥T1κ,ω,B≥1, by the Gronwall inequality on the interval s,κ, we obtain(39)Δvκ,κ−t,θ−κω,vκ−t2≤e∫κ−1κCvL66dσΔvs,κ−t,θ−κω,vκ−t2+∫κ−1κHθr−κωdr+3∫κ−1κgr2dr.
By Remark1, one can obtain(40)∫κ−1κvs,κ−t,θ−κω,vκ−tL66ds≤Cr3κ,ω.
Now, one can integrate (39) with respect to s. Then,(41)Δvκ,κ−t,θ−κω,vκ−t2≤e∫κ−1κCvL66dσ∫κ−1κΔvs,κ−t,θ−κω,vκ−t2ds+∫−10Hθrω+3gr+κ2dr≤eCr3κ,ωr2κ,ω+2Ca−6rω+3∫−10gr+κ2dr.
By Remark1, (39)–(41), we can obtain(42)Δvκ,κ−t,θ−κω,vκ−t2≤r4κ,ω.
Furthermore,(43)Δuκ,κ−t,θ−κω,uκ−t2≤r5κ,ω.Theorem 1.
Supposea>6 and (15) holds. There exists a unique D-pullback random attractor A=Aκ,ω:κ∈ℝ,ω∈Ω∈D in H=L2D for continuous cocycle Φ 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 domainsD, so these estimates are also valid for unbounded domain.Lemma 3.
Supposea>6 and (15) holds. Then, for every κ∈ℝ,ω∈Ω, and B=Bκ,ω:κ∈ℝ,ω∈Ω∈D, there are T2κ,ω,B,ϵ≥T1≥1 and k3κ,ω,ϵ≥1 satisfying(44)∫x|≥kvκ,κ−t,θ−κω,vκ−t2dx≤ϵ,∀t≥T2κ,ω,B,ϵ,k≥k3,where v is the solution of systems (9)–(11) and vκ−t+zθ−tω∈Bκ−t,θ−tω.Proof.
Let0≤θs≤1s∈ℝ+ be a smooth function. When 0≤s≤1, θs=0. When s≥2, θs=1. Then, θ′s+θ″s<C. Multiplying (21) with θx|2/k2v, we have(45)12ddt∫ℝ2θx2k2v2dx+∫ℝ2θx2k2vΔ2v+2Δvdx+∫ℝ2θx2k2v2Δzθtω+av+v5+Δ2zθtωdx+5∫ℝ2θx2k2vzv4+2v3z+2v2z2+vz3+15z4dx=∫ℝ2θx2k2vgx,tdx.
For the estimate of∫ℝ2θx2/k2vΔ2vdx, similar to [21], it follows that(46)∫ℝ2θx2k2vΔ2vdx≥−Ckv2+∇v2+Δv2+∫ℝ2θx2k2Δv2dx.
In addition, there are(47)∫ℝ2θx2k2vΔvdx≤12∫ℝ2θx2k2Δv2dx+12∫ℝ2θx2k2v2dx,∫ℝ2θx2k2vΔ2zθtωdx≤a2∫ℝ2θx2k2v2dx+C∫ℝ2θx2k2Δ2zθtω2dx,∫ℝ2θx2k2vgx,tdx≤12∫ℝ2θx2k2v2dx+C∫ℝ2θx2k2gt2dx.
Similarly, we get(48)∫ℝ2θx2k2vΔzθtωdx≤12∫ℝ2θx2k2v2dx+C∫ℝ2θx2k2Δzθtω2dx,where we have used the boundedness of function θ.
It is easy to obtain the following estimates:(49)∫ℝ2θx2k2v5zθtωdx≤110∫ℝ2θx2k2v6dx+C∫ℝ2θx2k2zθtω6dx,∫ℝ2θx2k2vz5θtωdx≤12∫ℝ2θx2k2v2dx+C∫ℝ2θx2k2zθtω10dx,(50)∫ℝ2θx2k2v3z3θtωdx≤120∫ℝ2θx2k2v6dx+C∫ℝ2θx2k2zθtω6dx.
Because∫ℝ2θx2/k2v4z2θtωdx≥0 and ∫ℝ2θx2/k2v2z4θtωdx≥0, one yields(51)ddt∫ℝ2θx|2k2v2dx+a−6∫ℝ2θx|2k2v2dx≤Ckv2+∇v2+Δv2+C∫ℝ2θx|2k2Δz2+z6+Δ2z2+z10dx+C∫ℝ2θx|2k2gt2dx.
Applying the Gronwall lemma onκ−t,κ, one gets(52)∫ℝ2θx2k2vκ,κ−t,θ−κω,vκ−t2dx≤e−a−6tvκ−t2+Ck∫−t0ea−6svs+κ,κ−t,θ−κω,vκ−t2+∇vs+κ,κ−t,θ−κω,vκ−t2+Δvs+κ,κ−t,θ−κω,vκ−t2ds+C∫−∞0ea−6s∫ℝ2θx2k2Δzθsω2+Δ2zθsω2+zθsω6+zθsω10dxds+C∫−∞0ea−6s∫ℝ2θx2k2gs+κ2dxds.
Noticingvκ−t+zθ−tω∈Bκ−t,θ−tω and B∈D, there is a T2≥T1t,ω,B,ϵ. When t≥T2, we have(53)e−a−6tvκ−t2≤ϵ4.
From (51), combining with Remark 1, if ϵ>0, there is a k1κ,ϵ≥1 satisfying(54)Ck∫−t0ea−6sv2+Δv2ds≤ϵ4,∀k≥k1κ,ϵ.
And there is ak2=k2ϵ≥k1κ,ϵ such that(55)C∫−∞0ea−6s∫ℝ2θx2k2Δzθsω2+Δ2zθsω2+zθsω6+zθsω10dxds≤Cϵ∫−∞0ea−6sz˜θsω2+z˜θsω6+z˜θsω10ds≤ϵ4,k≥k2.
Meanwhile, there is ak3κ,ϵ≥k2 satisfying(56)C∫−∞0ea−6s∫ℝ2θx2k2gs+κ2dxds≤ϵ4,k≥k3κ,ϵ.
By (51)–(56), we deduce(57)∫x≥kvκ,κ−t,θ−κω,vκ−t2dx≤ϵ,∀k≥k3κ,ϵ.Lemma 4.
Supposea>6 and (15) holds. Then, given ϵ>0 and B=B,ω:κ∈ℝ,ω∈Ω∈D, ∃T2=T2κ,ω,B,ϵ≥T1≥1, and k4=k4ω,ϵ>0, the solution uκ,κ−t,θκω,uκ−t=vκ,κ−t,θκω,vκ−t+zθtω satisfies the following inequality:(58)∫x≥k′uκ,κ−t,θ−κω,uκ−t2dx≤ϵ,∀t≥T2,k′≥k4,where vκ−t+zθ−tω∈Bκ−t,θ−tω.Proof.
Sincezθtω∈H4ℝ2, then there exists a k1′>0 such that(59)∫x≥k1′zθtω2dx≤ϵz˜θtω2.
By Lemma3, ∃k2′>k3>0, then(60)∫x≥k2′vκ,κ−t,θ−κω,vκ−t2dx≤ϵ4,∀t≥T2κ,ω,B,ϵ.
By (55) and (56), ∀t≥T2 and k′≥k4=maxk1′,k2′, one gets(61)∫x≥k′uκ,κ−t,θ−κω,uκ−t2dx≤ϵ.


## 4. Finiteness of Fractal Dimension

Now, we are devoted to the existence of random attractor onℝ2 for the random dynamical system Φ.Lemma 5.
Supposea>6 and (15) holds. Then, D-pullback asymptotically compact holds in L2ℝ2 for continuous cocycle Φ of (1) and (2).Similar to the method in [30], we only give the sketch of the proof for Lemma 5.Firstly, the weak convergence(62)Φtn,κ−tn,θ−tnω,u0,n⇀ξcan be given in L2ℝ2.Secondly, by Lemma4, there exist enough large t and M satisfying(63)∫x≥MΦt,κ−t,θ−tω,u02dx≤ϵ.Denote the setQM=x∈ℝ2:x≤M. By the estimates of Lemma 2, the embedding H2QM↦L2QM is compact. It follows that the strong convergence(64)Φtn,κ−tn,θ−tnω,u0,n⟶ξholds in L2ℝ2.According to [30], the following theorem is easily obtained. The proof is omitted.Theorem 2.
Supposea>6 and (15) holds. Then, the continuous cocycle Φ corresponding to problems (1) and (2) has a unique D-pullback attractor A=Aκ,ω:κ∈ℝ,ω∈Ω∈D in L2ℝ2.By Theorem1, 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 A˜=A˜κ,ω:κ∈ℝ,ω∈Ω in H=L2D∈D. The 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 L2D is denoted by H.Assume thatχκ,ωκ∈ℝ,ω∈Ω of X are a family of bounded closed random subsets, which satisfy the below conditions. For ∀ω∈Ω,κ∈ℝ, the following holds:H1: there is tempered random variable Rω, which is not dependent on κ and satisfies the diameter χκ,ωX of χκ,ω controlled by Rω, and Rθtω∀t∈ℝ is continuous.H2: for ∀t>0, χt+κ,θtω=Ψt,κ,ωχκ,ω.H3: there are random variables C0ω≥0, positive numbers λ,δ,t0, and projector Pm:X⟶PmX satisfying(65)PmΨt0,κ,ωv−PmΨt0,κ,ωuX≤e∫0t0C0θsωdsv−uX,(66)I−PmΨt0,κ,ωv−I−PmΨt0,κ,ωuX≤e−λt0+δe∫0t0C0θsωdsv−uX,for anyu,v∈χκ,ω, κ∈ℝ,andω∈Ω, where λ,δ,t0,m do not depend on κ,ω.H4: λ,δ,t0,C0ω satisfy the following conditions:(67)0≤EC0ω<∞,λ≥32EC0ω,0<δ≤132,4ln2≤λt0≤8ln2,where “E” denotes the expectation of random variables.Once these conditionsH1–H4 are satisfied, the finiteness of fractal dimension can be obtained by further calculation [18, 25]. We know that conditions H1 and H2 hold.Lemma 6.
Letω∈Ω,κ∈ℝ,t≥0. We have(68)φ1t+κ,κ,θ−κω,φ1κθ−κω−φ2t+κ,κ,θ−κω,φ2κθ−κω≤e∫0t1−adsφ1κ−φ2κ,φ1κ,φ2κ∈A˜.Proof.
Letφjr=φjr,κ,ω,φjκω,yr=φ1r−φ2r,j=1,2, when r≥κ. By equation (9), we have(69)dydt+Δ2y+2Δy+ay+fφ1−fφ2=0,r≥κ,yr,ω,x=0,x∈∂D,yκ,ω,x=yκω,x,κ∈ℝ,x∈D,where fζ=ζ+z5. Multiplying equation (69) with yr, noticing that(70)∫Dyfφ1−fφ2dx≥0,we have(71)ddryr2≤21−ayr2,∀r≥κ.
By the Gronwall inequality, we obtain the estimate (68).
Letejj∈ℕ be the eigenvectors of operator −Δ corresponding to the eigenvalues λjj∈ℕ with −Δej=λjej for j∈ℕ. Let Hn be the space spanned by ejj=1n and Pn:H⟶Hn be the orthonormal projector. Let Qn:H⟶Hn⊥ be the defined by Pnv=vn for v∈H, with  v=vn+vn⊥, where vn∈Hn,vn⊥∈Hn⊥. Then,(72)λn+1Qnv2≤v12.Lemma 7.
Forφ1κ,φ2κ∈A˜κ,ω, there are a random variable C0ω≥0 and a projector Pn:H⟶Hn satisfying(73)I−PnΨt,κ,θ−κωφ1κθ−κω−I−PnΨt,κ,θ−κωφ2κθ−κω,≤e4−a−1/4λn+12t+δne∫0tC0θsωdsφ1κ−φ2κ,and(74)PnΨt,κ,θ−κωφ1κθ−κω−PnΨt,κ,θ−κωφ2κθ−κω≤e∫0tC0θsωdsφ1κ−φ2κ,∀κ∈ℝ,ω∈Ω,t≥0,where(75)δn=1λn+1.Proof.
Multiplying equation (69) with yn=Qny in H, it follows that(76)2Δy,yn≤14Δy2+4yn2,∇v2≤CΔvv≤r1ω+Cr8ω=M02θrω.
Then, there exists aϑ between φ1r and φ2r such that(77)fφ1−fφ2,yn≤5∫Dϑ+zθrω12dx1/3⋅∫Dy2dx1/2∫Dyn6dx1/6≤CϑH014+zθrωL124yynL6≤CM04θrω+zθrωL1242y2+12Δyn2and(78)ddtyn2+12λn+12+2a−8yn2≤CM04θrω+zθrωL1242y2.
Applying Gronwall lemma to (78) and combining (68), one gets(79)ynκ+t,κ,θ−κω,yκθ−κω2≤ynκ2e∫κκ+t8−2a−1/2λn+12ds+∫κκ+tCM04θr−κω+zθr−κωL1242φ1r−φ2r2e∫rκ+t8−2a−1/2λn+12dsdr≤ynκ2e∫0t8−2a−1/2λn+12ds+φ1κ−φ2κ2∫κκ+tCM04θr−κω+zθr−κωL1242e∫κr21−adse∫rκ+t8−2a−1/2λn+12dsdr≤ynκ2e∫0t8−2a−1/2λn+12ds+φ1κ−φ2κ2e∫0t8−2ads∫0tCM04θr−κω+zθr−κωL1242e−1/2λn+12t−rdr.
Noticingx<exx≥0, we have(80)∫0tCM04θrω+‖zθrω‖L1242⋅e−1/2λn+12t−rdr≤∫0tC2⋅M04θrω+‖zθrω‖L1244dr12∫0te−λn+12t−rdr1/2≤1λn+1e∫0tC2M04θrω+‖zθrω‖L1244dr.
By (79),(81)ynκ+t,κ,θ−κω,yκθ−κω2≤e∫0t8−2a−1/2λn+12ds+1λn+1e∫0t8−2a+C2M04θsω+zθsωL1244dsφ1κ−φ2κ2/
Thus,(82)ynκ+t,κ,θ−κω,yκθ−κω≤e∫0t4−a−1/4λn+12ds+1λn+1e∫0t4−a+C2/2M04θsω+zθsωL1244dsφ1κ−φ2κ,where C0ω=4−a+C2/2M04ω+zωL1244. Then, (73) holds. From (68), it follows that (74) holds.Lemma 8.
The expectations ofC0ω,C02ω are finite, that is,(83)0≤EC0ω,EC02ω<∞.Proof.
Notice thatz˜θtω satisfies Ez˜θtωr=Γ1+r/2/π, for all r>0,t∈ℝ, and Γ satisfies the equality [25]:(84)Γ1+2k2=2k−12k−3⋯3⋅12kπ,Γ1+2k+12=k!,∀k∈ℕ.
ByM02ω=r1ω+C⋅r8ω,(85)C0ω=C¯1+r18ω+r88ω+zθtωL1216,C02ω=C1+C2r116ω+C3r816ω+C4zθtωL1232,where C¯,C1,C2,C3,C4   are positive constants.
Sincer1ω,r8ω  are tempered, we obtain Er18ω<∞, Er88ω<∞ and Er116ω<∞, Er816ω<∞. By the properties of z˜θtω, we get(86)EzθtωL1216=ϕxL1216⋅Ez˜θtω16=ϕxL121615⋅13⋅11⋯3⋅128<∞,EzθtωL1232=ϕxL1232⋅Ez˜θtω32=ϕxL123231⋅29⋅27⋯3⋅1216<∞.
Then,(87)EC0ω=C¯1+Er18ω+Er88ω+EzθtωL1216<∞,EC02ω=C1+C2Er116ω+C3Er816ω+C4EzθtωL1232<∞.
Now, based on the conclusions of Lemmas7 and 8, the main result is presented.Theorem 3.
The finiteness of fractal dimension forA˜κ,ω holds, that is,(88)dimfA˜κ,ω≤8n0lnn0λn0+11/2+1ln2<∞,κ∈ℝ,ω∈Ω,where n0=minn:λn+1≥max128EC0ω+16−4a1/2,1024<∞.Proof.
Comparing (66) and (73), we see that(89)λ=14λn+12+a−4⟶∞,δ=1λn+1⟶0,asn⟶∞.
By (67) and Lemma 8, ∃n0∈ℕ, we deduce(90)14λn+12+a−4≥32EC0ω,0<1λn+1<132.
Taket=t0>0 in (73) and (74) such that the following inequality is satisfied:(91)0<16ln2λn+12+4a−16≤t0≤32ln2λn+12+4a−16<∞.
IfH1–H4 holds, the following estimate is similar to [25]; here, we omit the details. Then,(92)dimfA˜κ,ω≤8n0lnn0/δ+1ln2=8n0lnn0λn0+11/2+1ln2<∞,∀κ∈ℝ,ω∈Ω.



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