# Random Attractor of Reaction-Diffusion Hopfield Neural Networks Driven by Wiener Processes

Authors: Xiao Liang; Linshan Wang; Ruili Wang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper studies the global existence and uniqueness of the mild solution for reaction-diffusion Hopfield neural networks (RDHNNs) driven by Wiener processes by applying a Schauder fixed point theorem and a priori estimate; then the random attractor for this system is also studied by constructing proper random dynamical system.


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## Body

## 1. Introduction

It is well known that the dynamics of Hopfield neural networks have been deeply investigated because they have been successfully employed in many areas such as pattern recognition, associate memory, and combinatorial optimization. The diffusion effect cannot be avoided in the neural networks when electrons are moving in asymmetric electromagnetic field. So, the dynamical behavior of reaction-diffusion Hopfield neural networks (RDHNNs, for short) has been receiving much attention, recently [1–11].But in a more realistic model, in order to describe the propagation of an electric potential in a neuron, it is sensible to include some noise in the system. In fact, a neural network can be stabilized or destabilized by certain stochastic inputs. Many scholars have been devoted to the stochastic RDHNNs such as [2, 12, 13]. Giving a deep insight into these literatures, we will find that most of these literatures consider the RDHNNs with finite dimensional Wiener processes; there are few results on the RDHNNs driven by infinite dimensional Wiener processes. However, since the neurons can be regarded as long thin cylinders, which act like electrical cables, the infinite dimensional Wiener processes are more favorable than standard Brownian motion.On the other hand, attractor plays an important role in the long time behavior for dynamical systems. The random attractor extends the concept of a strange attractor from deterministic to stochastic system. There has been great interest in random attractors for stochastic partial differential equations in recent decades. The random attractors are compact invariant random sets attracting all the orbits of attraction basin. They provide crucial geometric information about their asymptotic regime ast→∞. They can help us understand the chaotic behavior of the stochastic DRDHNNs and reduce the complexity, as well as providing the statistical properties of this system. When the global existence and uniqueness of the solution can be assured, many scholars pay much attention to the global stability, boundedness, and even synchronization of the RDHNNs [1–3, 11]. However, to the best of our knowledge, there is no result on the attractor for RDHNNs, let alone random attractor for stochastic RDHNNs. We hope this work can lay a solid foundation for the future research.So, this paper is devoted to the random attractor for the following RDHNNs driven by Wiener processes:(1)duit,x=∑j=1l∂∂xjGijx∂ui∂xj-biui+∑j=1ncijfjuj+Iidt+σidWit,x,uit,x=0,t≥t0,x∈∂O,uit0,x=ϕix,x∈O∈Rl,i=1,2,…,n,where uit,x denote the potential of the cell i at t and x∈Rl. bi are positive constants and denote the rate with which the ith unit will reset its potential to the resting state in isolation when it is disconnected from the network and external inputs at t. cij are connection weights of the neural network. fi  are the active functions of the neural network, which are continuous. σi are the intensity of the noise. Ii denote the ith component of an external input source introduced from outside the network to the ith neuron, which are constant numbers. O denotes an open bounded and connected subset of Rl with a sufficiently regular boundary ∂O. ∇ is the gradient. Initial data ϕi are F0-measurable and belong to L2O, a.e. ω∈Ω.For convenience, we rewrite system (1) in the vector form (2)du=∇·Gx∘∇u-Bu+Cf~u+Idt+σdW,ut,x=0,t≥t0,x∈∂O,ut0,x=ϕx,where C=(cij)n×n, σ=diag⁡σ1,σ2,…,σn⁡I=(I1,I2,…,In)T,  u=(u1,u2,…,un)T, ∇u=(∇u1,∇u2,…,∇un)T, W=(W1,W2,…,Wn)T, f~(u)=(f1u1,f2u2,…,fnun)T is the diagonal map, ∇ is the gradient operator, B=diag⁡(b1,b2,…,bn), ϕx=ϕ1x,ϕ2x,…,ϕnxT, G(x)=(Gij)n×l, G∘∇u=Gij∂ui/∂xjn×l is the Hadamard product between matrix G and ∇u, and ∇· is the divergence operator.We will also use the following notations in the paper.(i)
(Ω,F,P) is a complete probability space with filtration {Ft}t≥0 satisfying the usual conditions (see [1, 2, 14–18]).(ii)
Wt,x=Wt,xω is a space-time Wiener process with values in the separable Hilbert space K with EW=0 and EW,uKW,vK=t∧s(Qu,v)X, ∀s,t≥0, u,v∈X, (·,·)X denotes the inner product of X, Wt,x=∑n=1∞αnβntenx, where t∧s=min⁡t,s and the Hilbert-Schmidt operator Q is a positive definite, nuclear, symmetric, self-adjoint operator having a finite trace trQ≜∑n=1∞αn<+∞ with eigenvalues αn, en}n=1∞ is an orthogonal basis of X, and βn}n=1∞ is a sequence of mutually independent standard Brownian motions in (Ω,F,P) (see [2, 14]).(iii)
L2X0,U is the space of all Hilbert-Schmidt operators from X0≜Q1/2X into U; when equipped with the norm Φ2≜trΦQΦ∗ it becomes a Hilbert space, where Φ∈L2(X0,U); Φ∗ denotes the adjoint of Φ.


## 2. Preliminaries and Notations

In this paper, we introduce the following Hilbert spaces:U=L2On and V=H1On; according to [14, 19–21], V⊂U=U′⊂V′; U′, V′ denote the dual of the spaces U, V respectively, the injection continues, and the embedding is compact; ·, ⦀·⦀ represent the usual norm in U, V, respectively. Let us define the operator as follows: (3)A:ΠA∈U⟶UAu=∇·G∘∇u-Buand ΠA is the domain of A defined as ΠA=H2On∩H01On. A is the infinitesimal generator of an analytic semigroup Tt.Defining the Nemytskii operator as follows:(4)fux=f~ux,∀x∈O.With these notations, we rewrite system (2) in the more abstract form (5)du=Au+Cfu+Idt+σdW,ut0=ϕ.We recall that (6)WAt=σ∫-∞teAt-sdWsis the solution of the Ornstein-Uhlenbeck process (7)du=Audt+σdW.The regularity of (6) has been proved in [14, 22] and WAt has a α-Hölder continuous version with respect to t, α<1/4. Furthermore, by the law of large number(8)limt→±∞⁡WAtt=0,limt→±∞⁡ln⁡WAtt=0,a.s.(H1)
We assume thatfu≤k1+k2u, fu-fv≤k3u-v, ∀u,v∈U.(H2)
We assume that there exists a positive numberβO such that the following Poincaré inequality is valid: (9)u≤β-1⦀u⦀,∀u∈V.(H3)
We assume that there existsα>0 such that Gijx≥α/nl, i=1,2,…,n.(H4)
Letk4>2k2k5, with k4=min⁡b1,b2,…,bn and k5=max⁡cij.We also need the following propositions in the following sections.Proposition 1.
Consider the following equation:(10)ddtu=Au,t≥0,u0=ϕ.Suppose (H2), (H3) hold; let ut=Ttϕ denote the mild solution of (10); then Tt is a contraction map in U.Proof.
We recall that the solution of this linear equation isut=eAtϕ, so Tt=eAt. Now we take the inner product of (10) with ut in U, by employing the Gaussian theorem and condition (H3), we get (11)Au,u≤-α⦀u⦀2-k4u2,∀u∈V,where (·,·) is the inner product in U (see [1–3]), and we also have (12)12ddtut2+α⦀ut⦀2+k4u2≤0.By (H2), one obtains (13)ddtut2+2αβ2+k4ut2≤0.By Gronwall-Bellman inequality, we have (14)ut2≤e-2αβ2+k4tϕ2;by the definition of Tt and uniform boundedness principle, we have (15)Tt≤e-αβ2+k4t≤1.So Tt is a contraction map.Remark 2 (see [14]).
LetTt be a strongly continuous semigroup and A be the generator of Tt; then we have (16)Tt1ϕ-Tt2ϕ=∫t2t1TsAϕds=∫t2t1ATsϕds,∀ϕ∈ΠA.Proposition 3 (see [2]).
LetΨ=ψijn×n and u=u1,u2,…,unT; then ΨuRn≤nΨFuRn, where uRn represents the usual norm of Rn and ΨF represents the Frobenius norm of a matrix Ψ∈Rn×n; that is, ΨF=∑i=1n∑j=1nψij21/2.Let(X;d) be a complete separable metric space. We shall recall the notions of random dynamical system and random attractor.Definition 4 (see [22]).
Let(X,d) be a complete separable metric space. A metric dynamical system (MDS) θ≜Ω,F,P,θt,t∈R is a family of measure-preserving transformations θt:Ω→Ω, t∈R such that (i)
θ0=id, θt•θs=θt+s for all t,s∈R; • is the composition operator;(ii)
the mapt,ω→θtω is BR⊗F measurable, and θtP=P for all t∈R, where BR is the Borel σ field of R.Definition 5 (see [22–25]).
A random dynamical system (RDS) overθt is a measurable map φ, such that (1)
φ0,ωx=x, x∈X, ω∈Ω;(2)
φs+t,ωx=φs,θtω•φt,ωx, s,t∈R, x∈X, ω∈Ω;(3)
t,x→φt,ωx is continuous a.e. ω∈Ω;(4)
ω→φt,ωx is F-measurable for all t,x∈R×X.Definition 6 (see [26, 27]).
A stochastic flow is a family of mappingsSt,s;ω:X→X,-∞<s≤t<∞, parameterized by ω, such that (17)t,s,x,ω⟶St,s;ωxis BR⊗BR⊗BX⊗F-measurable and (18)St,r;ω•Sr,s;ωx=St,s;ωx,s≤r≤t,x∈X,St,s;ωx=St-s,0;θsωx,ω∈Ω.S is said to be a continuous stochastic flow, if x→St,s;ωx is continuous.Definition 7 (see [28, 29]).
A mapK:Ω→2X is said to be a closed random set if Kω is closed for a.e. ω∈Ω and the map ω→dist⁡x,Kω is a.e. measurable for all x∈X; dist denotes the Hausdorff semidistance defined as dist⁡x,A=infy∈Ad(x,y) and dist⁡A,B=supx∈Ad(x,B), A,B∈2X.Definition 8 (see [29, 30]).
A mapK:Ω→2X is said to be a compact random set if K(ω) is compact for a.e. ω∈Ω and the map ω→dist⁡x,Kω is a.s. measurable.Definition 9 (see [31, 32]).
A random setKω is called an absorbing set in X, if for all B∈X and a.e. ω∈Ω, there exists tBω>0 such that t≥tBω>0(19)φt,θ-tωB⊂Kω.Lemma 10 (Schauder fixed point theorem [33–38]).
IfB is a closed bounded convex subset of a space X and L:B→B is completely continuous, then L has a fixed point in B.Lemma 11 (see [39, 40]).
Supposeφ is a RDS on a Polish space X, and there exists a compact set K(ω) absorbing every bounded deterministic set B∈X. Then the set (20)Aω=⋃B∈X⋂T≥0⋃t≥Tϕt,θ-tωB¯¯is a random attractor for φ.


## 3. Existence and Uniqueness of the Solutions

Let(21)vt=ut-WAt,where WAt has been defined in the previous section. Then, from (5) and (6), vt satisfies the equation (22)dv=Av+Cfv+WA+Idt,vt0=ψ,where ψ=ϕ-WAt0. Let us rewrite (23) in the integral form (23)vt,ω=Ttψ+∫t0tTt-sCfv+WA+Ids.Definition 12.
Ifv satisfies (23), we say that the ut is a mild solution of (1).
LetXT∗≜Ct0,t0+T∗;U, when equipped with the norm uXT∗=supt∈t0,t0+T∗ut21/2, it becomes a Banach space. Let (24)Σm,T∗=v∈Ct0,t0+T∗;U:vt≤m∀t∈t0,t0+T∗and consider an initial data ψ which is F0-measurable and belong to U, a.e. ω∈Ω. From now on, we are going to discuss mild solution of equation (22) by Schauder fixed point method in the space XT∗ for some T∗>0.Theorem 13.
Assume thatψ is F0-measurable and belongs to U, a.e. ω∈Ω. If conditions (H1)–(H3) hold, then for any ψ<m, there exists a stopping time T∗>0, such that (1) has a mild solution in Σm,T∗.Proof.
We split our proof into the following steps.
Step 1. Σm,T∗ is a nonempty closed bounded convex set.
Letv1, v2 belong to Σm,T∗; then for 0≤λ≤1(25)λv1+1-λv2XT∗≤λv1XT∗+1-λv2XT∗≤mso λv1+1-λv2∈Σm,T∗; hence Σm,T∗ is a convex set.
Step 2. T maps Σm,T∗ into Σm,T∗.
Take anyv in Σm,T∗ and define z=Tv by (26)zt=Ttψ+∫t0tTt-sCfv+WA+Ids.By the triangle inequality (27)z≤Ttψ+∫t0tTt-sCfv+WA+Ids.By Proposition 1, it follows that (28)zt≤ψ+∫t0tCfv+WA+Ids.By using Proposition 3 and norm inequality, as well as condition (H1), we have (29)Cfv+WA≤nCFfv+WA≤nCFk1+k2v+WA≤nCFk1+k2v+WA≤nCFk1+k2vXT∗+k2WAXT∗;therefore (30)ztXT∗≤ψ+∫t0tk5WAXT∗+k6+k7vXT∗ds≤ψ+k6μ+k7+k8mt,where μ=WAXT∗, k6=nk2CF, k7=I+nCFk1, and k8=nk2CF.
Hencezt≤m for all t∈t0,t0+T∗, provided that(31)ψ+k6μ+k7+k8mt-t0<m.It is clear that, for any ψ<m, there exists a T∗ satisfying (31).
Step 3. T is a equicontinuous map of Σm,T∗.
Lett1,t2>0, then by Remark 2 and Propositions 1 and 3, we have (32)Tvt1-Tvt2=Tt1-Tt2ψ+∫t2t1Tt-sCfv+WA+Ids≤∫t2t1TsAψds+∫t2t1Tt-sCfv+WA+Ids≤∫t2t1Aψds+∫t2t1Cfv+WA+Ids≤∫t2t1Aψds+k6μ+k7+k8mt1-t2≤Aψ+k6μ+k7+k8mt1-t2,∀v∈Σm,T∗so TΣm,T∗ is an equicontinuous set.
By the Arzelá-Ascoli theorem, for any bounded setB⊂Σm,T∗, the closure of TB is compact, so T is a completely continuous map, then by Steps 1–3 and Lemma 10, T has a fixed point in Σm,T∗, which is a mild solution of (1).Theorem 14.
Suppose (H1)–(H3) hold, then for anyψ<m, (1) has at most one mild solution in t0,t0+T∗.Proof.
Letv1,v2∈Σm,T∗ be two solutions of system (23) on [0;T] with v10=v20=ψ, and set zi=Tvi, i=1,2, and z=z1-z2. Then (33)zt=Ttv10-v20+∫t0tTt-sCfv1+WA-fv2+WAds.Following the method in Step 3 of Theorem 13, we have (34)ztXT∗≤nCFk3T∗v1-v2XT∗.We take a stopping time T∗ such that (35)nCFk3T∗<1which implies that v1t=v2t and t∈t0,t0+T∗. By the combination of Theorem 13, there is one unique mild solution ut for (1).Theorem 15.
Suppose (H1)–(H3) hold. Ifv∈Ct0,t0+T;U is a solution of system (23), then (36)vt≤ψ+δfor some constant μ∞>0, where δ=eηTI2+nCFk12+2k22nCFμ∞2 and γ=2αβ2+2k4-nCF2k22+1-1.Proof.
By employing the method in [41, 42], let ψmm=1∞ be a sequence in C∞On such that (37)ψm⟶ψ∈U.Let WAm be a sequence of regular process such that (38)WAmt=∫t0tTt-sdWms⟶WAtin Ct0,t0+T×On a.e. ω∈Ω.
Letvm be the solution of (39)vmt=Ttψm+∫t0tTt-sCfvm+WAm+Ids.By using the method in the Theorem 13, it is easy to see that vm does exist on an interval of time t0,t0+Tm such that Tm→T∗ a.s. and that vm converges to v in Ct0,t0+T;U [39]. Moreover vm is regular and satisfies (40)dvm=Avm+Cfvm+WAm+Idt.
Taking the inner product of (40) with vm in U and employing the result of (11), we find that (41)12ddtvm2=vm,dvmdt=vm,Avm+Cfvm+WAm+I≤-αβ2+k4vm2+vm,Cfvm+WAm+vm,I.By the Cauchy-Schwartz inequality and Young inequality, we get (42)vm,I≤vmI≤12vm2+12I2.By using Young inequality and Proposition 3, as well as condition (H1), one obtains (43)vm,Cfvm+WAm≤vmCfvm+WAm≤nCFvmfvm+WAm≤12nCFvm2+fvm+WAm2≤12nCFvm2+2k12+2k22vm+WAm2≤12nCF4k22+1vm2+2k12+4k22WAm2.By (H2) and (H3), we deduce that (44)ddtvm2≤-2αβ2+k4vm2+nCF4k2+1vm2+2k12+4k22WAm2+vm2+I2≤-ηvm2+κ,where γ=2αβ2+2k4-nCF4k22+1-1 and κ=I2+2nCFk12+4k22nCFWAm2. By the classical Gronwall inequality, then we have (45)vmt2≤e-γtψm2+∫t0te-γsκds≤ψm2+eγTI2+2nCFk12+4k22nCFμm,∞2Twith μm,∞=supt∈t0,t0+TWAmt, for a.e. ω∈Ω.
Taking the limit asm→∞, we see that a.s. (46)vt2≤ψ2+δ.It follows that (47)vt≤ψ+δthus we complete the proof.It is easy to derive the following from Theorems13, 14, and 15.Theorem 16.
Letϕ be given which is F0-measurable and bounded in U, a.s.; then there exists a unique mild solution u of (2), with u∈Ct0,t0+T;U, ∀T<∞.


## 4. Existence of the Random Attractor

We referω=Wt and Ω=ω∈CR,Rn∣ω0=0, with P being the product measure of two Wiener measures on the negative and positive parts of Ω. In this case, there exists a Wiener shift θt defined as θtωs=ωt+s-ω(s), s,t∈R, which is an ergodic transformation.Letvt,ω;t0,ψ be the solution of (23) with v0=ψ and ut,ω;t0,ϕ is the mild solution of (1) with u0=ϕ. Then, from Theorem 15, we know that the map ψ→vt,ω;t0,ψ is continuous, by vt=ut-WAt; then one can define the maps St,s;ω and φt,ω by (48)St,t0;ωϕ=vt,ω;t0,ψ+WAt,ω,φt,ωϕ=St,0;ωϕ=vt,0;ωψ+WAt,ω.By the result of Theorem 15, S is a continuous stochastic flow and φ is a continuous RDS associated with (1).


### 4.1. Absorbing Sets inU

Let us define the operator as follows:(49)A:ΠA∈U⟶UAu=-∇·Gx∘∇uand ΠA=ΠA=H2On∩H01On. We can infer from (10) and (49) that (50)Au=-Au-Bu.The associated bilinear operator with A is defined as au,v=Au,v. By utilizing the result of [2], we have (51)au,u=∫OG:∇u∘∇udx≥α⦀u⦀2,where A:B is the Frobenius inner product of two n×m matrixes, defined as A:B≜∑in∑jmaijbij.Lett0<-1 and ψ∈U be given, and let v be the solution of equation (23). Taking the inner product of (22) with v in U we find that(52)12ddtv2=v,dvdt=v,Av+Cfv+WA+I≤-av,v-k4v2+v,Cfv+WA+v,I.By the Cauchy-Schwartz inequality and Young inequality (53)v,I≤vI≤k44v2+1k4I2.By using Young inequality and (H1) (54)v,Cfv+WA≤vCfv+WA≤k5vfv+WA≤k44v2+k52k4fv+WA2≤k44+4k22k52k4v2+2k52k4k12+2k22WA2.We deduce from (51)–(54) that (55)ddtv2+2av,v≤-ηv2+P1t,ω,where η=k4-2k22k52/k4>0 and P1t,ω=2k52/k4k1+2k2WA2+2/k4I2. By (H4), we know that η>0; using the classical Gronwall inequality, for t0≤-1, we have (56)v-12≤e-η-1-t0vt02+∫t0-1eη-1-sP1s,ωds=e-η-1-t0ut02+e-η-1-t0WAt02+∫t0-1e-η-1-sP1s,ωds.Theorem 17.
Under conditions (H1)–(H4), the RDSφ defined by (48) admits a random absorbing set in U.Proof.
Let the random variable(57)r12ω=2+∫-∞-1e-η-1-sP1s,ωds.By (8), we know that P1s,ω has at most polynomial growth for s→-∞, so rω is finite. On the other hand, WAt02 also has at most polynomial growth for t0, as t0 tending to -∞, according to (8). So, we can choose t- such that (58)e-η-1-t0ϕ2+WAt02≤2,t0≤t-.By (56), (57), and (58), for all ρ>0, there exists t(ω)≤-1, such that for all t0≤t(ω) and ϕ∈U, with ϕ≤ρ, the following inequality is satisfied:(59)v-1,ω;t0,ψ2≤r12ω.By (48) and (59), the stochastic flow of system (1) satisfies (60)S-1,t0;ωϕ≤r1ω+WAt0,ω≜r2ω.By using the relationship between stochastic flow and RDS (61)φt,θ-tωBU0,ρ=St,0;θ-tωBU0,ρ=S0,-t;ωBU0,ρ=S0,-1;ωS-1,-t;ωBU0,ρ⊂S0,-1;ωBU0,r2ω,∀ρ>0,where BU0,ρ denotes the ball of U centered at 0 with radius ρ, let K1ω=S0,-1;ωBU0,r2ω; there exists tBω>0 such that t≥tBω>0(62)φt,θ-tωBU0,ρ⊂K1ω,a.e.ω∈Ω.By Definition 9, we get the conclusion.We need the following auxiliary proposition in the following sections.Theorem 18.
There exists a random variabler3ω,r4ω>0, such that, for all ρ>0, there exist tω≤-1, such that for all t0≤t(ω) and all ϕ∈U, with ϕ≤ρ, the solution vt,ω;t0,ψ of system (22) satisfies the inequality (63)∫-10vs,ω;t0,ψ2ds≤r3ω,∫-10av,vds≤r4ω.Proof.
From (55), we can get (64)∫-10av,vds≤12v-12+12∫-10P1s,ωds.Integrating (55) over -1,0, we can also get (65)∫-10v2ds≤1ηv-12+1η∫-10P1s,ωds.Let r3ω=1/2v-12+1/2∫-10P1s,ωds, r4ω=1/ηv-12+1/η∫-10P1s,ωds, by Theorem 17, v-1 is finite, and by using (8) again, we know that r3ω, r4(ω) is finite.


### 4.2. Absorbing Set inV

We now prove the existence of an absorbing set inV. Multiplying (22) by Av and integrating over O, by (50), we have (66)Av,dvdt=Av,Av+Cfv+WA+I≤-Av2-Av,Bv+AvCfv+WA+Av,I.Using the Dirichlet boundary condition and the Green first identity (67)Av,dvdt=av,ddtv=12ddtav,v.By Gauss formula [3] (68)Av,Bv=∫OBG:∇v∘∇vdx≥k4α⦀v⦀2.We can also infer from the divergence theorem (69)Av,I=-∑i=1n∫∂OIiGi∇ui·ds=0.Using the Young inequality again, we get (70)AvCfv+WA≤Av2+14Cfv+WA2;we can deduce from Proposition 3 and condition (H1) (71)Cfv+WA2≤nCF2fv+WA2≤nCF24k22v2+2k12+4k22WA2.It follows from (66)–(71) that (72)ddtav,v≤-k9av,v+P2t,ω,where k9=2k4(73)P2t,ω=nCF22k22v2+k12+2k22WA2.Integrating (72) over s,0, we have (74)av0,v0≤avs,vs+∫s0P2s,ωds;integrating (72) again in s over -1,0 and using (51), we have (75)⦀v0⦀2≤1α∫-10avs,vsds+1α∫-10P2s,ωds.Theorem 19.
Under conditions (H1)–(H4), the RDSφ defined by (48) admits an absorbing set in V.Proof.
Put the random variable(76)r52ω≜1α∫-10avs,vsds+1α∫-10P2s,ωds.By Theorem 18, r5 is a finite number; then by employing (71), for all ρ>0, there exist tω≤-1, for all t0≤tω and all ϕ∈V, with ϕ≤ρ; the solution v0,ω;t0,ψ of system (23) satisfies the inequality (77)⦀v0,ω;t0,ϕ-WA0,ω⦀2≤r52ωwhich also means the stochastic flow of (1) satisfies (78)⦀S0,t0;ωϕ⦀≤r5ω+⦀WAt0,ω⦀≜r6ω.By using the definition of RDS and the stochastic flow, we have (79)φt,θ-tω=St,0;θ-tω=S0,-t;ω.By (78) and (79), ∀ϕ≤ρ, for sufficiently large t(80)⦀φt,θ-tωϕ⦀≤r6ω.Let K2ω=BV0,r6ω, where BV0,r6 denotes the ball of V centered at 0 with radius r6, so for all BV0,ρ∈U, there exists tBω>0 such that t≥tBω>0(81)φt,θ-tωBV0,ρ⊂K2ω,a.e.ω∈Ω.Theorem 20.
Assume that (H1)–(H4) hold; then the RDSφ generated by (1) possesses a random attractor in U.Proof.
By Theorem19, φt;ωϕ is absorbed by the random set K2ω. Since the embedding of V to U is compact, so the bounded set K2ω is compact. A combination of Lemma 11 and Theorems 17 and 19, we get the existence of random attractor for the RDS φ.If the active functionf is a bounded Lipschitz function, then we can choose a sufficiently large k1 such that k2=0, so (H4) is satisfied automatically; then we get the following theorem.Corollary 21.
Under conditions (H1)–(H3) and the fact that there exists a constantk1 such that f(u)≤k1, the RDS φ defined by (48) admits a random attractor in U.


## 4.1. Absorbing Sets inU

Let us define the operator as follows:(49)A:ΠA∈U⟶UAu=-∇·Gx∘∇uand ΠA=ΠA=H2On∩H01On. We can infer from (10) and (49) that (50)Au=-Au-Bu.The associated bilinear operator with A is defined as au,v=Au,v. By utilizing the result of [2], we have (51)au,u=∫OG:∇u∘∇udx≥α⦀u⦀2,where A:B is the Frobenius inner product of two n×m matrixes, defined as A:B≜∑in∑jmaijbij.Lett0<-1 and ψ∈U be given, and let v be the solution of equation (23). Taking the inner product of (22) with v in U we find that(52)12ddtv2=v,dvdt=v,Av+Cfv+WA+I≤-av,v-k4v2+v,Cfv+WA+v,I.By the Cauchy-Schwartz inequality and Young inequality (53)v,I≤vI≤k44v2+1k4I2.By using Young inequality and (H1) (54)v,Cfv+WA≤vCfv+WA≤k5vfv+WA≤k44v2+k52k4fv+WA2≤k44+4k22k52k4v2+2k52k4k12+2k22WA2.We deduce from (51)–(54) that (55)ddtv2+2av,v≤-ηv2+P1t,ω,where η=k4-2k22k52/k4>0 and P1t,ω=2k52/k4k1+2k2WA2+2/k4I2. By (H4), we know that η>0; using the classical Gronwall inequality, for t0≤-1, we have (56)v-12≤e-η-1-t0vt02+∫t0-1eη-1-sP1s,ωds=e-η-1-t0ut02+e-η-1-t0WAt02+∫t0-1e-η-1-sP1s,ωds.Theorem 17.
Under conditions (H1)–(H4), the RDSφ defined by (48) admits a random absorbing set in U.Proof.
Let the random variable(57)r12ω=2+∫-∞-1e-η-1-sP1s,ωds.By (8), we know that P1s,ω has at most polynomial growth for s→-∞, so rω is finite. On the other hand, WAt02 also has at most polynomial growth for t0, as t0 tending to -∞, according to (8). So, we can choose t- such that (58)e-η-1-t0ϕ2+WAt02≤2,t0≤t-.By (56), (57), and (58), for all ρ>0, there exists t(ω)≤-1, such that for all t0≤t(ω) and ϕ∈U, with ϕ≤ρ, the following inequality is satisfied:(59)v-1,ω;t0,ψ2≤r12ω.By (48) and (59), the stochastic flow of system (1) satisfies (60)S-1,t0;ωϕ≤r1ω+WAt0,ω≜r2ω.By using the relationship between stochastic flow and RDS (61)φt,θ-tωBU0,ρ=St,0;θ-tωBU0,ρ=S0,-t;ωBU0,ρ=S0,-1;ωS-1,-t;ωBU0,ρ⊂S0,-1;ωBU0,r2ω,∀ρ>0,where BU0,ρ denotes the ball of U centered at 0 with radius ρ, let K1ω=S0,-1;ωBU0,r2ω; there exists tBω>0 such that t≥tBω>0(62)φt,θ-tωBU0,ρ⊂K1ω,a.e.ω∈Ω.By Definition 9, we get the conclusion.We need the following auxiliary proposition in the following sections.Theorem 18.
There exists a random variabler3ω,r4ω>0, such that, for all ρ>0, there exist tω≤-1, such that for all t0≤t(ω) and all ϕ∈U, with ϕ≤ρ, the solution vt,ω;t0,ψ of system (22) satisfies the inequality (63)∫-10vs,ω;t0,ψ2ds≤r3ω,∫-10av,vds≤r4ω.Proof.
From (55), we can get (64)∫-10av,vds≤12v-12+12∫-10P1s,ωds.Integrating (55) over -1,0, we can also get (65)∫-10v2ds≤1ηv-12+1η∫-10P1s,ωds.Let r3ω=1/2v-12+1/2∫-10P1s,ωds, r4ω=1/ηv-12+1/η∫-10P1s,ωds, by Theorem 17, v-1 is finite, and by using (8) again, we know that r3ω, r4(ω) is finite.


## 4.2. Absorbing Set inV

We now prove the existence of an absorbing set inV. Multiplying (22) by Av and integrating over O, by (50), we have (66)Av,dvdt=Av,Av+Cfv+WA+I≤-Av2-Av,Bv+AvCfv+WA+Av,I.Using the Dirichlet boundary condition and the Green first identity (67)Av,dvdt=av,ddtv=12ddtav,v.By Gauss formula [3] (68)Av,Bv=∫OBG:∇v∘∇vdx≥k4α⦀v⦀2.We can also infer from the divergence theorem (69)Av,I=-∑i=1n∫∂OIiGi∇ui·ds=0.Using the Young inequality again, we get (70)AvCfv+WA≤Av2+14Cfv+WA2;we can deduce from Proposition 3 and condition (H1) (71)Cfv+WA2≤nCF2fv+WA2≤nCF24k22v2+2k12+4k22WA2.It follows from (66)–(71) that (72)ddtav,v≤-k9av,v+P2t,ω,where k9=2k4(73)P2t,ω=nCF22k22v2+k12+2k22WA2.Integrating (72) over s,0, we have (74)av0,v0≤avs,vs+∫s0P2s,ωds;integrating (72) again in s over -1,0 and using (51), we have (75)⦀v0⦀2≤1α∫-10avs,vsds+1α∫-10P2s,ωds.Theorem 19.
Under conditions (H1)–(H4), the RDSφ defined by (48) admits an absorbing set in V.Proof.
Put the random variable(76)r52ω≜1α∫-10avs,vsds+1α∫-10P2s,ωds.By Theorem 18, r5 is a finite number; then by employing (71), for all ρ>0, there exist tω≤-1, for all t0≤tω and all ϕ∈V, with ϕ≤ρ; the solution v0,ω;t0,ψ of system (23) satisfies the inequality (77)⦀v0,ω;t0,ϕ-WA0,ω⦀2≤r52ωwhich also means the stochastic flow of (1) satisfies (78)⦀S0,t0;ωϕ⦀≤r5ω+⦀WAt0,ω⦀≜r6ω.By using the definition of RDS and the stochastic flow, we have (79)φt,θ-tω=St,0;θ-tω=S0,-t;ω.By (78) and (79), ∀ϕ≤ρ, for sufficiently large t(80)⦀φt,θ-tωϕ⦀≤r6ω.Let K2ω=BV0,r6ω, where BV0,r6 denotes the ball of V centered at 0 with radius r6, so for all BV0,ρ∈U, there exists tBω>0 such that t≥tBω>0(81)φt,θ-tωBV0,ρ⊂K2ω,a.e.ω∈Ω.Theorem 20.
Assume that (H1)–(H4) hold; then the RDSφ generated by (1) possesses a random attractor in U.Proof.
By Theorem19, φt;ωϕ is absorbed by the random set K2ω. Since the embedding of V to U is compact, so the bounded set K2ω is compact. A combination of Lemma 11 and Theorems 17 and 19, we get the existence of random attractor for the RDS φ.If the active functionf is a bounded Lipschitz function, then we can choose a sufficiently large k1 such that k2=0, so (H4) is satisfied automatically; then we get the following theorem.Corollary 21.
Under conditions (H1)–(H3) and the fact that there exists a constantk1 such that f(u)≤k1, the RDS φ defined by (48) admits a random attractor in U.


## 5. Example and Simulation

Example 1.
(82)
u
˙
1
t
,
x
=
Δ
u
1
-
2
u
1
-
2
tanh
⁡
u
1
t
-
1
,
x
-
tanh
⁡
u
2
t
-
1
,
x
+
1.3
∂
2
W
∂
t
∂
x
,
u
˙
2
t
,
x
=
Δ
u
2
-
2
u
2
+
tanh
⁡
u
1
t
-
1
,
x
-
3
tanh
⁡
u
2
t
-
1
,
x
+
1.5
∂
2
W
∂
t
∂
x
,
u
i
t
,
0
=
u
i
t
,
20
=
0
,
t
≥
0
,
u
1
0
,
x
=
sin
⁡
0.2
π
x
,
u
2
0
,
x
=
sin
⁡
0.2
π
x
,
x
∈
O
=
0,20
.Proof.
In this caseU=L2O2 and fu is a bounded global Lipschitz continuous function with k1=20, k2=0, and k3=1, St≤1, and Δ is the Laplace operator. EW=0 and W is the cylindrical Wiener process with EWs,x∧Wt,y=t∧sx∧y, ∀s,t≥0, x,y∈0,20, E is the expectation, and a∧b=min⁡a,b. Under these assumptions, by Theorem 15, this system has a global mild solution. Meanwhile k4≥22k2k5, so according to Theorem 20, this system possesses a random attractor.
We simulate this example by using the Matlab; for detailed information see Figures1–3. A Crank-Nicolson method in time and second-order center differences in space are used to discrete this model. A Newton iterative method is used to solve the discretized nonlinear equation. For more theories about the numerical theory about SPDE, we refer to [42, 43].Figure 1
Frequency of Example1.Figure 2
Simulation ofu1 in Example 1.Figure 3
Simulation ofu2 in Example 1.The method used in this article can be extended to other systems, such as the biological systems and fluid mechanical systems [44–48]. We can also apply this method to the system driven by other types of noise, such as the G-Brown motion [49]. By the way, some control technique may be used to stabilize these systems with random attractor [50–54]. We will study these problems in the future.



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*Source: 2538658-2018-01-16.xml*