Fractal Dimension of Random Attractors for Non-autonomous Fractional Stochastic Reaction-diffusion Equations

SHU Ji,BAI Qianqian,HUANG Xin and ZHANG Jian

1 School of Mathematics Sciences and V.C.&V.R.Key Lab,Sichuan Normal University,Chengdu 610068,China.

2 Department of Basic Courses,Sichuan Vocational College of Finance and Economics,Chengdu 610101,China.

3 School of Mathematical Sciences,University of Electronic Science and Technology of China,Chengdu 611731,China.

Abstract. This paper deals with non-autonomous fractional stochastic reaction-diffusion equations driven by multiplicative noise with s ∈(0,1). We first present some conditions for estimating the boundedness of fractal dimension of a random invariant set.Then we establish the existence and uniqueness of tempered pullback random attractors. Finally,the finiteness of fractal dimension of the random attractors is proved.

Key Words: Random dynamical system;random attractor;fractal dimension;fractional reactiondiffusion equation;multiplicative noise.

1 Introduction

In this paper,we will discuss the following non-autonomous fractional stochastic reactiondiffusion equation with multiplicative noise on a bounded domainU?Rn:

with boundary condition

and initial condition

wheres∈(0,1),α＞0,U?Rn(n≤3) with a smooth boundary?U,u=u(t,x) is a realvalued function onU×[τ,+∞),τ∈R,the functionsf,gsatisfy certain conditions which will be specified later.W(t) is a two-side real-valued Wiener process on a probability space. The symbol “°” means that the stochastic equation is understood in the sense of Stratonovich’s integration. Whens∈(0,1), the operator(??)sis so-called fractional Laplacian,which includes the integral and spectral fractional operators on bounded domainU,and they have different eigenfunctions and eigenvalues[1]. Whens=1, (??)sbecomes the standard Laplace operator ??.

In recent years,fractional partial differential equations have appeared in a wide range of fields,within in physics,biology,chemistry,financial mathematics,etc.,and some classical partial differential equations with fractional derivative have also been extensively studied, including fractional Schr?dinger equations [2,3], fractional Ginzburg-Landau equations [4-9], fractional Landau-Lifshitz equations [10], fractional Landau-Lifshitz-Maxwell equations[11]and fractional reaction-diffusion equations[12].

Stochastic partial differential equations(SPDEs)arise naturally in a wide variety of applications with uncertainties or random influences,called noises,are considered.During the past decades, analysis of infinite dimensional random dynamical system(RDS) has become an important field in the study of qualitative behaviour of SPDEs. The concept of pullback random attractor,which is a generalization of global attractor in deterministic systems(see[13-16]),was introduced in[17-22],and characterizes the long-time behavior of RDS perfectly. The existence of random attractors for SPDEs have been widely discussed by many authors,see,e.g.,[23-31] in the autonomous SPDEs, and[12,32-46]in the non-autonomous case.

As far as we are concerned,the finite dimensionality of an attractor is an important topic in describing the asymptotic behavior of infinite-dimensional dynamical systems.Up to now,there have several useful methods to estimate the upper bound of the Hausdorff and fractal dimensions of a random attractor. There is a fact that if a compact setAin a metric space has a bounded fractal dimension dimf(A)＜m/2 for somem∈N,thenAcan be placed in the graph of a H?lder continuous mapping which maps a compact subset of RmontoA. This indicates that the finiteness of fractal dimension of an attractor plays a very important role in the finite-dimensional reduction theory of an infinite dimensional system. However, just knowing the boundedness of Hausdorff dimension of an attractor for a system,we still have no available finite parameterization. Recently,Zhou et al. discussed the fractal dimension of random attractors of non-autonomous stochastic reaction-diffusion equation in[47],Wang discussed the asymptotic behavior of fractional reaction-diffusion equation withα∈(0,1)in[12]. However,as far as the authors are aware,there are no results for the fractal dimension of random attractors of fractional stochastic reaction-diffusion equations.

Motivated by[12,47],we are interested in bounding the fractal dimension of random attractors of non-autonomous fractional stochastic reaction-diffusion equation with multiplicative noise.The main obstacle here is that the fractional Laplacian(??)sis non-local and then deriving uniform estimates of the solutions of(1.1) is more difficulty than the standard Laplacian ??.

This paper is organized as follows. In the next section,we recall some known results and basic concepts related to the non-autonomous random dynamical system,fractional derivative,fractional Sobolev space and conditions for the fractal dimension of a random invariant set. In Section 3,we transform the stochastic equation into a random equation which solutions generate a random dynamical system. In Section 4, we derive the existence and uniqueness of a tempered pullback random attractor. Finally, we prove the boundedness of fractal dimension of the random attractor.

2 Preliminaries

In this section,we first present some basic notions about pullback random attractors for non-autonomous random dynamical systems from [38,39]. For the theory of pullback random attractors on autonomous random dynamical systems, the reader can refer to[19-22].

Let(X,‖·‖X)be a separable Hilbert space with the Borelσ-algebra B(X),(?,F,P,{θt}t∈R)be an ergodic metric dynamical system.

Definition 2.1.A mappingΦ:R+×R×?×X→X is called a continuous random dynamical system on X over(?,F,P,{θt}t∈R)if for all τ∈R,ω∈?and t,s∈R+,

(i) Φ(.,τ,.,.):R+×?×X→X is(B(R+)×F×B(X),B(X))-measurable;

(ii) Φ(0,τ,ω,·)is the identity on X;

(iii) Φ(t+s,τ,ω,·)=Φ(t,τ+s,θsω,.)°Φ(s,τ,ω,·);

(iv) Φ(t,τ,ω,·):X→X is continuous.

Definition 2.2.(1) A set-valued mapping{D(τ,ω):τ∈R,ω∈?}:? →2X,ω→D(τ,ω)issaid to be a random set if the mappingis measurable for any u∈X. If D(τ,ω)is also closed(compact)for each τ∈R,ω∈?, {D(τ,ω):τ∈R,ω∈?}is called a random closed(compact)set. A random set{D(τ,ω):τ∈R,ω∈?}is said to be bounded if there exist u0∈X and a random variable R(τ,ω)＞0such that

(2)A random set{D(τ,ω):τ∈R,ω∈?}is called tempered provided

where d(D)=sup{‖b‖X:b∈D}.

(3)A random set{B(τ,ω):τ∈R,ω∈?}is said to be a random absorbing set if for any tempered random set{D(τ,ω):τ∈R,ω∈?},there exists t0such that

(4)A random set{B1(τ,ω):τ∈R,ω∈?}is said to be a random attracting set if for any tempered random set{D(τ,ω):τ∈R,ω∈?},we have

where dH is the Hausdorff semi-distance given by dH(E,F)=supu∈Einfv∈F‖u?v‖X for any E,F?X.

(5)Dis called inclusion-closed if D={D(τ,ω):τ∈R,ω∈?}∈Dand ifR,ω∈?}is a random subset of X with

(6)LetDbe a collection of random subsets of X. ThenΦis said to beD-pullback asymptotically compact in X if for P-a.e. ω∈?,τ∈R,has a convergent subsequencein X whenever tn→∞,and xn∈B(τ?tn,θ?tnω)with{B(τ,ω):τ∈R,ω∈?}∈D.

Definition 2.3.LetDbe a collection of random subsets of X and{A(τ,ω):τ∈R,ω∈?}∈D.Then{A(τ,ω):τ∈R,ω∈?}is called aD-random attractor (orD-pullback attractor) forΦif the following conditions are satisfied

(1) {A(τ,ω):τ∈R,ω∈?}is compact,and ω→d(X,A(τ,ω))is measurable for everyX ∈X;

(2) {A(τ,ω):τ∈R,ω∈?}is strictly invariant,i.e.,?(t,τ,ω,A(τ,ω))=A(t+τ,θtω),?t≥0;

(3) {A(τ,ω):τ∈R,ω∈?}attracts all sets inD,i.e.,for all B∈D,we have

From [38], we have the existence and uniqueness theorem of random attractors for non-autonomous random dynamical system.

Theorem 2.1.LetΦbe a continuous random dynamical system on X over(?,F,P,{θt}t∈R),If there exists a closed random tempered absorbing set{B(τ,ω):τ∈R,ω∈?}ofΦandΦis asymptotically compact in X,then{A(τ,ω):τ∈R,ω∈?}is a random attractor ofΦ,where

Moreover,{A(τ,ω):τ∈R,ω∈?}is the unique random attractor ofΦ.

Next,we recall some concepts and notations of the fractional derivative and fractional Sobolev space(see[48])for details). Let S be the Schwartz space of rapidly decayingC∞functions on R,then for 0＜s＜1,the fractional Laplace operator(??)sis given by

whereC(s)is a positive constant depending onsas given by

In particular,it follows from[48]that for anyu∈S,

where F is the Fourier transform defined by

and F?1is the inverse Fourier transform. LetHs(Rn) be the fractional Sobolev space defined by

which is equipped with the norm

From now on, we write the norm ofHs(Rn) as ‖·‖sand the Gagliardo semi-norm ofi.e.

Then for allu∈Hs(Rn) we haveNote thatHs(Rn) is a Hilbert space with inner product given by

In terms of(2.2),one can verify(see[48]):

and henceis an equivalent norm ofHs(Rn). Similarly, from [48],we can defineH2s(Rn)withs∈(0,1).

In addition,we give a lemma which will be used in later sections(for details see[49]).

Lemma 2.1.If f,g∈H2s(U),then the following equation holds:

where s1and s2are nonnegative constants and satisfy s1+s2=s.

Finally, we give some sufficient conditions to bound the fractal dimension of a random invariant set for non-autonomous random dynamical system (see [25,46,47] for details).

Theorem 2.2.Let{Φ(t,τ,ω):t≥0,τ∈R,ω∈?}be a cocycle on a separable Banach space X overRand ergodic metric dynamical system(?,F,P,{θtω}t∈R). Assume that there exists a family of bounded closed random subsets{X(τ,ω):τ∈R,ω∈?}of X satisfying the following conditions:for every τ∈Rand ω∈?,

(H1)there exists a tempered random variable Rω such that the diameter‖X(τ,ω)‖X ofX(τ,ω)is bounded by Rω,i.e,supτ∈Rsupu∈X(τ,ω)‖u‖X≤Rω＜∞,and Rθtω is continuous in t for all t∈R;

(H2)invariance:X(t+τ,θtω)=Φ(t,τ,ω)X(τ,ω)for all t≥0;

(H3)there exist positive numbers λ, δ, t0, random variables C0(ω)≥0, C1(ω)≥0and mdimensional projector Pm:X→PmX(dim(PmX)=m)such that for every τ∈R,ω∈?and any u,v∈X(τ,ω),it holds that

where λ,δ,t0,m are independent of τ and ω;

(H4)λ,t0,δ,C0(ω),C1(ω)satisfy:

where“E”denotes the expectation. Then for any τ∈R, ω∈?, the fractal dimension ofX(τ,ω)has a finite bound:

where Nε(X(τ,ω))is the minimal number of balls with radius ε＞0coveringX(τ,ω)in X.

3 Random dynamical system

In this section,we will give the existence and uniqueness of solutions of problem (1.1)-(1.3)which generates a continuous random dynamical system.

The standard probability space (?,F,P) will be used in this paper where ?={ω∈C(R,R):ω(0)=0},and F is the Borelσ-algebra induced by the compact-open topology of ?,andPis the Wiener measure on(?,F). Here we will identifyW(t)withω(t),i.e.,ω(t)=W(t,ω),t∈R.

Givent∈R,defineθt:?→? by

It is known from [17,47] that there exists aθt-invariant set ?0?? such that for everyis continuous intand

where Γ is the gamma function, “E” denotes the expectation. For convenience, we will consider(1.1)forω∈?0and still write ?0as ?.

LetA=(??)s,thenAis a self-adjoint positive linear operator with positive eigenvalues:

We now convert the stochastic equation (1.1) into a pathwise deterministic one. Givenτ∈R,t≥τ,ω∈? anduτ∈L2(U), ifu=u(t,τ,ω,uτ) is a solution of (1.1)-(1.3), by the transformation

we can obtain the following equation

with boundary condition

and initial condition

Assume the functionsf,gsatisfy the following conditions:

(A1)g(x,·)∈Cb(R,L2(U))with‖g‖2=supt∈R‖g(·,t)‖2＜∞;

(A2)fis a nonlinear continuous function given by

and there exist constantsc0,c1,c2≥0 such that for allu∈R,

By the standard Galerkin method and compactness argument,as shown in[12], we can prove that,if(A1)-(A2)hold,for anys∈(0,1),τ∈R,P?a.e.ω∈? andvτ∈L2(U),the system(3.6)-(3.8)has a unique solutionv(t,τ,ω,vτ)inL2(U),which can define a continuous cocycle

given by:

where Φ1(0,τ,ω)vτ(ω)=vτ(θ?τω).

We now define a mapping Φ2:R+×R×?×L2(U)→L2(U)by

forvτ∈L2(U),t≥τ,ω∈?.Then Φ2is a continuous random dynamical system associated with(1.1)-(1.3).

Note that the two random dynamical systems Φ1and Φ2are equivalent,it is easy to check that Φ2has a random attractor provided Φ1possesses a random attractor. Therefore, we only need to consider the random attractor and estimate its fractal dimension for the cocycle Φ1in the following sections. For convenience,lettersci(i=3,4,5,···)andkj(j=1,2,3)denote different positive constants throughout this paper.

4 Random attractor

In this section, we first derive the uniform estimates of solutions for problem (3.6)-(3.8)inL2(U) andHs(U) respectively. Then, from the compactness of Sobolev embeddingwe obtain the existence and uniqueness of pullback random attractor of cocycle Φ1.

Let D=D(L2(U)) be the collection of all tempered families of nonempty subsets ofL2(U).

Lemma 4.1.Suppose(A1)-(A2)hold,then for every τ∈R,ω∈?and D={D(τ,ω):τ∈R,ω∈?}∈D,there exist T(τ,ω,D)≥0and a tempered random variable R0(τ,ω):

such that for all t≥T(τ,ω,D),the solutions v(r,τ?t,ω,vτ?t)∈L2(U)of(3.6)-(3.8)with vτ?t∈D(τ?t,θ?tω)satisfies

That is,the tempered ball

is a measurableD-pullback absorbing set in L2(U)forΦ1.

Proof.The proof can be found in[34]and we will omit the details here.

Next we derive uniform estimates of solutions inHs(U).

Lemma 4.2.Suppose(A1)-(A2)hold, then for every τ∈R,ω∈?and D={D(τ,ω):τ∈R,ω∈?}∈D,there exists T(τ,ω,D)≥1such that for all t≥T(τ,ω,D),the solutions v(r,τ?t,ω,vτ?t)∈L2(U)of problem(3.6)-(3.8)with vτ?t∈D(τ?t,θ?tω)satisfies

where

is a tempered random variable, is same as in Lemma4.1.

Proof.The proof can be found in[34]and we will omit the details here.

Lemma 4.3.Suppose(A1)-(A2)hold,then for every τ∈R,ω∈?and D={D(τ,ω):τ∈R,ω∈?}∈D,there exist T(τ,ω,D)≥1and a tempered random variable R2(τ,ω):

4.1, such that for all t≥T(τ,ω,D), the solutions v(r,τ?t,ω,vτ?t)∈L2(U)of(3.6)-(3.8)with vτ?t∈D(τ?t,θ?tω)satisfies

That is,the tempered ball

is a measurableD-pullback absorbing set in Hs(U)forΦ1.

Proof.From Lemmas 4.1 and 4.2,we can get(4.1)directly.

By the uniform estimates of the solutions of(3.6)-(3.8),we can prove the existence and uniqueness of D-pullback random attractor for the cocycle Φ1.

Theorem 4.1.Assume that(A1)-(A2)hold, then the cocycleΦ1associate with problem(3.6)-(3.8)has a compact measurableD-pullback absorbing set B1(τ,ω), thusΦ1possesses a uniqueD-pullback random attractorA={A(τ,ω):τ∈R,ω∈?}∈Din L2(U),and

where R2(τ,ω)is same as in Lemma4.3.

Proof.For anyτ∈R,ω∈?,according to Lemma 4.1,Lemma 4.3 and the compactness of embeddingwe obtain theB1(τ,ω)is a compact measurable D-pullback absorbing set for Φ1inL2(U). Therefore,it follows from Theorem 2.6 that Φ1possesses a unique D-pullback random attractor A={A(τ,ω):τ∈R,ω∈?}∈D inL2(U),and for anyτ∈R,ω∈?,A(τ,ω)?B0(τ,ω)∩B1(τ,ω). The proof is completed.

5 Fractal dimension of random attractor

Based on Theorem 2.8,in this section we prove the boundedness of fractal dimension of the random attractor A(τ,ω)for Φ1.

Firstly,from Lemmas 4.1 and 4.3,Theorem 4.4 and the invariance of random attractor,A(τ,ω)satisfies(H1)and(H2). Next,we show the Lipschitz property of Φ1on A(τ,ω).For everyτ∈R,ω∈? andvjτ(ω)∈A(τ,ω),j=1,2,let

then

By the invariance of A(τ,ω), the cocycle property of Φ1and Theorem 4.4, forr≥τ, it holds thatv1(r),v2(r)∈A(τ,θrω)?B1(θr(ω))?Hs(U)and

Lemma 5.1.Suppose(A2)holds,then for every τ∈R,ω∈?,t≥0and vjτ(ω)∈A(τ,ω),j=1,2,it holds that

Proof.Taking the inner product of(5.2)withy(r)inL2(U),we get

By(A2)and the mean value theorem,we have

whereη∈(0,1). According to(5.3)-(5.4)we obtain

Therefore,applying theGronwallinequality to(5.5)on[τ,τ+t]and replacingωbyθ?τω,we have

The proof is completed.

In the following,we show that A(τ,ω)satisfies(H3). Let{ei}i∈Nbe the eigenvectors of the operatorAcorresponding to the eigenvalues{λi}i∈NwithAei=λieifori∈N,then{ei}i∈Nform an orthonormal base ofL2(U).Write

and let

be the orthonormal projector,then

Lemma 5.2.Suppose(A1)-(A2)hold, then for every τ∈R, ω∈?and t≥1, there exist two random variables M1(ω)≥0,M2(ω)≥0and a n-dimensional orthonormal projector Pn:L2(U)→(ω)∈A(τ,ω),j=1,2,it holds that

and

Proof.Taking the inner product of(5.2)withyn=QnyinL2(U),we obtain

ByH?lderinequality,Young inequality,(A2)and(5.1),we can get

whereThus

Putting(5.11)into(5.9)we have,forr≥τ

Applying theGronwallinequality to (5.12) on[τ,τ+t] (t≥0) and replacingωbyθ?τωand from Lemma 5.1 we get

Therefore,by(5.13)and(5.14)we have

where

Thus,(5.7)is proved.From Lemma 5.1 andM2(ω)＞αz(θξω)+c2≥0,it follows that(5.8)holds. The proof is completed.

At last,we derive the finiteness of the expectations ofM1(ω)andM2(ω).

Lemma 5.3.If the coefficient α of the random term in(1.1)is small such that

then

Proof.It follows from(3.4)that

From the property of gamma function Γ,

For|β|≤1,

Thus,by(3.4)and(5.19)

Next, we prove the expectations of each term ofM2(ω) in (5.15) are bounded. From Lemma 3.1 and Lemma 3.2 we have

By(3.3)and(5.20),we get the following estimates

Thus,by(5.23)and(5.25)we obtain

Thus

The proof is completed.

As a consequence of Theorem 2.8,Lemma 5.2 and Lemma 5.3,we have the following main result in this section.

Theorem 5.1.Suppose(A1)-(A2)and(5.16)hold. Then for every τ∈R, ω∈?, the fractal dimension ofA(τ,ω)has a finite upper bound:

where

Proof.Comparing(2.5)and(5.7),we have that whenn→∞,

By(3.5)and(5.17),there exists a finite integern0∈N such that

Taket=t0＞0 in(5.7)and(5.8)satisfying

then by Theorem 2.8,for eachτ∈R,ω∈?,we have

The proof is completed.

Acknowledgement

The authors would like to thank the reviewers for their helpful comments. This work was partially supported by the National Natural Science Foundation of China(11871138),joint research project of Laurent Mathematics Center of Sichuan Normal University and National-Local Joint Engineering Laboratory of System Credibility Automatic Verification,funding of V.C.&V.R.Key Lab of Sichuan Province.