Random A ttractor for the Nonclassical D iffusion Equation w ith Fad ing M em ory

CHENG Shuilin

School ofStatisticsand M athematics,Zhongnan University ofEconom icsand Law, Wuhan 430073,China.

Received 22M ay 2015;A ccep ted 5 Ju ly 2015

Random A ttractor for the Nonclassical D iffusion Equation w ith Fad ing M em ory

CHENG Shuilin?

School ofStatisticsand M athematics,Zhongnan University ofEconom icsand Law, Wuhan 430073,China.

Received 22M ay 2015;A ccep ted 5 Ju ly 2015

.In thispaper,w e consider thestochasticnonclassicald iffusion equationw ith fad ingm em ory on a bounded dom ain.By decom position of the solu tion operator,w e give the necessary cond ition of asym p totic sm oothness of the solu tion to the initial boundary value p rob lem,and then w e p rove the existence of a random attracto r in the space M1=D(A12)× L2μ(R+,D(A12)),w here A= ? Δ w ith Dirich let boundary cond ition.

Stochastic nonclassicald iffusion equations;fadingm em ory;random attractor.

1 In troduction

In this paper,w e use them ethod described in[1–3]to consider a stochastic nonclassical d iffusion equation w ith fad ingm em ory on a bounded dom ain:

w here D is a bounded dom ain in Rn(n ≥ 3),h(x)∈ H10(D)TH2(D),W(t)is a tw o-sided real valued Wiener p rocess on the p robability space(?,F,P).Abou t the forcing term g and the nonlinearity f,w e assum e that g(x) ∈ H?1(D),f is a Lipschitz con tinuous function and satisfi es thatfor s∈R and 2≤ p≤2n/(n?2).

This problem has its origin in them athem atical descrip tion of viscoelasticm aterials. It isw ell know n that the viscoelasticm aterials exhibit natural dam p ing,w hich is due to the specialproperty of thesem aterials to retain am em ory of their pasthistory.And from them aterials pointof view,the p roperty ofm em ory com es from them em ory kernelκ(s), w hich decays to zero w ith exponential rate,so it is the fad ingm em ory of the far history in them odel.

Many au thors have stud ied the classical reaction d iffusion equations[3–8],but it doesn’t contain all the aspects of the reaction d iffusion problem.K.Ku ttler and E.C. Aifantis[9]proved the existenceand uniqueness of the solu tion for the nonclassical d iffusion equations ut?Δut?Δu=f(u)+g(x),and Q.Z.Ma[10]study the existence of global attractors for nonclassical d iffusion equations in H1(RN)w ith the non linearity satisfies thearbitrary order polynom ialgrow th cond itions.In[3,11–14],attractors for the w ave equations or nonclassical d iffusion equationsw ith them em ory kernelhave been proved.N.Tatar[1]and Wu,Liu etal[15,16]considered the exponen tialstability for the w ave equation w ith a tem poral non-local term and asym p totic behavior o f so lu tions for thew aveequationsw ithm em ory.

In recent years,the existence and uniqueness of the solu tions and the long tim e dynam ics for the stochastic reaction d iffusion equation have been p roved,w hich contain bounded and unbounded dom ains[17–20].

In this paper,suppose that them em ory kernel and assum e that thereexistsa positive constan tδ ＞0 such that the function μ(s)=?κ′(s) satisfi es

To ou r know ledge,Eq.(1.1)has not been considered by p redecessors and is stud ied fi rstly as a new m odel in this paper.Since Eq.(1.1)con tains m em ory term,w e fi rst construct relatively com p licate solu tion space and m ake a p rioriestim ate in the space. Sim u ltaneously,it isalso d iffi cu lt to testand verify thep roperty of thesolution sem igroup such as con tinuity,com pactness,or asym p totic com pactness.These obstacles possibly lead to variousadd itional cond itionsw hich seem unnecessary on term sof(1.1)(cf.[21]).

This paper is organized as follow s.In the next section,w e recall som e p relim inaries, includ ing the notation thatw ew illuse,som e p relim inary resu lts related to the solu tion of the stochastic equation and the random attractor for random dynam ical system s.In Section 3,fi rstly,w e define a continuous random dynam ical system for p rove the existence and uniqueness of the solution for the stochastic nthe nonclassical d iffusion equation w ith fad ingm em ory,then p rove exist a closed random absorbing and establish the asym p totic com pactness of the random dynam ical system and prove the existence of a random attractor.

Throughou t th is paper,w e denote C is a constant from line to line.

2 Prelim inaries

In this section,w e recall the fundam en tal resu lts related to som e basic function spaces, inequalitiesand characterization sim ilar to[3].

Let A=?Δ,w ith the dom ain D(A)=H10(D)∩H2(D),and the fractionalpow ers space D(Aα2),α ∈ R,the(·,·)D(Aα2),‖ ·‖D(Aα2)is the inner p roduct and norm,respectively.For convenience,w e use Hα=D(Aα2),the norm ‖·‖Hα= ‖ ·‖D(Aα2),and H0=L2(D),H1= H10(D).

Sim ilar to[19],for them em ory kernel μ(·),w e denote L2μ(R+;Hα)the H ilbert space of functions φ :R+→ Hα,endow ed w ith the inner p roductand norm respectively,

Define the space

w ith the inner p roduct

and the norm

We also introduce the fam ily of H ilbert spaces Mα=Hα× L2μ(R+;Hα),and endow ed norm: In the follow ing of thisarticle,w e denote ‖·‖2Hα,μ:= ‖·‖2α,μ.Form ore details,see[3,13].

Next,w e recall som e basic concepts related to random attractors for stochastic dynam icalsystem s[17].

Let ? ={ω ∈ C(R,R):ω(0)=0},F is the Borel σ-algebra on ?,and P is the correspond ingW ienerm easu re.Define

Then θ=(θt)t∈Ris them easurablem ap and θ0is the identity on ?, θt+s= θt?θsfor all s,t∈ R.That is,(?,F,P,(θt)t∈R)is called am etric dynam ical system.

Definition 2.1.(?,F,P,(θt)t∈R)iscalled ametricdynam icalsystem if θ:R × ? → ? is(B(R)× F,F)-measurable,θ0is the identity on ?,θs+t= θs?θtforalls,t∈ R and θtP=P forall t∈ R.Definition 2.2.A continuous random dynamical system(RDS)on E over ametric dynamical system(?,F,P,(θt)t∈R)isamapping

which is(B(R+)× F × B(E),B(E))-measurable and satisfies,for P ? a.e.ω ∈ ?,

(i) Φ(0,ω,·)is the identity on E;

(ii) Φ(t+s,ω,·)= Φ(t,ωs,·)?Φ(s,ω,·)forall t,s∈ R+;

(iii) Φ(t,ω,·):E → E iscontinuousforall t∈ R+.

Defi n ition 2.3.A random bounded set{B(ω)}ω∈?of E is called tempered w ith respect to (θt)t∈Riffor P?a.e.ω ∈ ?,forallβ ＞0,

where d(B)=supx∈B‖x‖E.

Definition 2.4.Let D be a collection of random subsets of E and{K(ω )}ω∈?∈ D.Then {K(ω)}ω∈?is called a random absorbing set for Φ in D iffor every B ∈ D and P?a.e.ω ∈ ?, there exists tB(ω)＞0 such that forall t≥ tB(ω)

Definition 2.5.Let D bea collection ofrandom subsetsofE.Then Φ is said to beasymptotically compact in E iffor P ? a.e.ω ∈ ?,{Φ(tn,θ?tnω,xn)}∞n=1has a convergent subsequence in E whenever tn→ ∞,and xn∈ B(θ?tnω)w ith{B(ω)}ω∈?∈ D.

Defi n ition 2.6.Let D bea collection ofrandom subsetsofE and A(→)ω∈?∈D.Then A(→)ω∈?iscalled a D-random attractorfor Φ ifthefollowing conditionsare satisfied,for P ? a.e.ω ∈ ?,

(i)A(→)is compact,and ω ■→ d(x,A(→))ismeasurableforevery x ∈ E;

(ii)A(→)ω∈?is invariant,that is,Φ(t,ω,A(→))=A(θtω),?t≥ 0;

(iii)A(→)ω∈?attracts every set in D,that is,for every

where d is the Hausdorffsem i-metric given by

forany X ? E,and Y ?E.

Theorem 2.1.([17,18,20])Let Φ be a continuous random dynam ical system with state space E over(?,F,P,(θt)t∈R).Ifthere isa closed random absorbing set B(ω)ofΦ and Φ isasymptotically compact in E,then{A(→)}is a random attractor ofΦ,w here

M oreover,{A(→)}is theuniquerandom attractor ofΦ.

As in[3],w e define a new variab le to refl ect the fad ing m em ory o f(1.1):

Hence,

Therefore,w e can rew rite(1.1)as follow s.

w here u0(x,s)satisfies that there exist tw o positive constants C and k,such that

Lemm a 2.1.([3,19])Assume that μ ∈ C1(R+)∩ L1(R+)is a nonnegative function,and there exists s0∈ R+,such that μ(s0)=0,then μ(s)=0 holdsforall s≥ 0.M oreover,for three Banach spaces B0,B1and B2,B0and B1are reflexiveand where,the embedding B0■→ B1is compact.Let K ? L2μ(R+;B1)satisfy

Then K isrelatively compact in L2μ(R+;B1).

3 The random attractor

In th is section,w e p rove that the stochastic nonclassical d iffusion p rob lem(2.4)has a random attractor.First,w e convert the p roblem(2.4)into a determ inistic system w ith a random param eter.

Wenow introduce an Ornstein-Uh lenbeck p rocess:

w hich is the solu tion of the Ornstein-Uh lenbeck equation

It is know n that there existsa θt-invariant set ?? ? ? o f fu ll P m easu re such that t→ X(θt) is con tinuous for every ω ∈ ??,and the random variable|X(θt)|is tem pered,see,e.g., [1,8,19].

Set Z(θtω)=(I?Δ)?1hX(θtω),w here Δ is the Lap lacian w ith dom ain H10(D)∩H2(D), w e fi nd that

Let v(x,t)(ω)=u(x,t)(ω)? Z(θtω),w here u(x,t)(ω)satisfies(2.4).Then w e have By the Galerkin m ethod as in[13],under assum p tions(1.2)-(1.3),for P?a.e.ω ∈ ?,and for all z0=(v0,η0)∈ M1,p rob lem(3.2)has a un ique so lu tion z=(v,ηt)in M1,satisfying z(·,z0,ω)∈ C([0,∞);M1)∩ L∞([0,∞);M1).

Throughou t this paper,w ealw aysw rite

If u is a solu tion of problem(1.1)in som e sense,w e can define a continuous dynam ical system Ψ on M1w ith

Form oreabou t the random continuous dynam ical system,see[1].

In order to prove the asym ptotic com pactness and the existence of global attractors, w e give the fo llow ing resu lts.

Lemm a 3.1.([3])Set I=[0,T],? T ＞ 0.Letmemory kernel μ(s)satisfy(1.3),then for any ηt∈ C(I;L2μ(R+;Hr)),0 ＜ r＜ 3,there existsa constant δ ＞ 0,such that

We fi rst show that the random dynam ical system Ψ has a closed random absorbing set in D,and then p rove thatΨ isasym p totically com pact.

Lemm a3.2.Assumethath(x)∈H10(D)TH2(D)and(1.2)-(1.3)hold.Let B={B(ω)}ω∈?∈D. Then for P?a.e.ω ∈ ?,there isapositive random function r1(ω)and a constant T=T(B,ω)＞0

such that forall t≥ T,

the solution of(3.2)has the follow ing uniform estimate

Proof.Taking the inner p roducto f the fi rstequation of(3.2)w ith v in L2(D),w e obtain

From(2.2)and(2.3),w e get

Hence,w e can rew rite(3.6)as

By Young inequality,Lemm a 3.1 and(1.2),w e have

and

It follow s from(3.9)-(3.12)that

Then,choose the right ε1and ε2such that1?Cε2＞ 0 and δ/2?Cε1＞ 0.

By theem bedd ing theorem,w e have

Denote

Then(3.13)-(3.14)im p ly that

Accord ing to Gronw all’s lemm a,

Substitu ting ω by θ?tω,then w ehave from(3.16)that

Note that|X(θsω)|is the tem pered,and Z(θtω)=(I?Δ)?1hX(θtω),h(x)∈H10(D)TH2(D), w e can choose

Then r1(ω)is the tem pered since|X(θsω)|has atm ost linear grow th rate at infinity,so the proof is com p leted.

To prove the asym ptotic com pactness of the solution sem igroup,w e decom pose the solu tion y(t)=(u(t),ηt)of(3.2)as follow s[3]:

w here y1(t)=(u1(t),η1t),y2(t)=(u2(t),η2t)satisfy the fo llow ing p rob lem s,respectively

and

here the nonlinearity f=f1+f2are all satisfies(1.2).The forcing term g(x) ∈ H?1(D) and d rifting term h(x) ∈ H10(D)TH2(D),there exist functions g1(x) ∈ L2(D),h1(x) ∈H10(D)T H2(D),for any ∈＞ 0,such that

Set Z1(θtω)=(I? Δ)?1h1X(θtω),w e find that

Let v1(t,ω)=u1(t,ω)? Z(θtω)+Z1(θtω),w here u1(t,ω)satisfies(3.20),and v2(t,ω)= u2(t,ω)?Z1(θtω),u2(t,ω)is the solu tion of(3.21).Then for v1(t,ω)and v2(t,ω)w e have that

Sim ilar to thep roblem(3.2),w ealso have the correspond ingexistenceand uniqueness of solutions for(3.24)and(3.25).For the convenience,w e denote the solution operators of(3.24)and(3.25)by{S1(t)}t≥0and{S2(t)}t≥0,respectively.Then,for every z0∈ M1, w e have

Next,w e give som e lemm as to p rove the asym p totic sm oothness.

Lemm a 3.3.Assume that the conditions on f,f1,g,g1hold.Let B={B(ω)}ω∈?∈ D.Then for P?a.e.ω ∈ ?,there isa constant T2=T2(B,ω)＞ 0,? ε＞ 0,if

then forall t≥ T2,the solution of(3.24)satisfies thefollow ing uniform estimate

where thepositive random function r1(ω)is defined in Lemma3.2.

Proof.From(3.8)w e substitu ting f,g,Z(θtω)by f1,g?g1,Z(θtω)? Z1(θtω),respectively. Sim ilar to the p roof the Lemm a 3.2,w e have

Lemm a 3.4.Assume that the conditionson f,f1,g,g1,h,h1hold.Let B={B(ω)}ω∈?∈D.Then for P?a.e.ω ∈ ?,there isa positive random function r2(ω)and

such thatforevery given T ≥ 0,the solution of(3.25)has thefollow ing uniform estimates

where l=m in{1,(2n? p(n?2))/2}.

Proof.M ultip lying(3.25)by Alv2and integrating over D,w e can get

From(2.2)and(3.25),w e can obtain that

and

By(1.2)and them ean value theorem,w e have

w here ξisbetw een v+Z(θtω)and v1+Z(θtω)?Z1(θtω),and

Using theem bedd ing theorem,w e get

and

w herew e have used the inequality and the em bedd ing theorem

Note that

and

Thanks to Lemm a 3.1,the p roperty of solution of(3.2)and(3.20),and(3.30)-(3.34),w e can deduce that

w here β =m ax{C,δ2+ ε}.

Then app lying Gronw all lemm a,w e get that

Thus,for every given T ＞ 0,w ehave

w here r2(ω)=RT0e2β(T?s)(1+|X(θsω)|2))d s is a random function.

We com p lete the p roof.

Since ηt(x,s)=Rs0u(x,t?r)d r,s≥ 0 and(3.25),w ehave

form ore in form ation on ηt(x,s),see[3],w e have

Lemm a 3.5.Let Π :H1× L2μ(R+;H1) → L2μ(R+;H1)is a projection operator,setting ΓT2:= ΠS2(T,B0(ω)),B0(ω)is a random bounded absorbing set from Lemma 3.4,S2(T,·)is the solution operators of(3.25),and under the assumption of Lemma 3.4,there is a positive random function r3(ω)depend on T,such that

(i) ΓT2isbounded in L2μ(R+;H1+l)∩ H1μ(R+;H1);

Proof.By the random translation,(3.38)and Lemm a 3.4,w e can p rove this lemm a.

Therefore,Lemm a 2.1 im p lies that ΓT2is relatively com pact in L2μ(R+;H1).And m aking use of the com pactem bedd ing H1+l■→ Hl,w e obtain

Lemm a 3.6.Let S2(t,·)be the corresponding solution operator of(3.25),and the assumption of Lemma3.4 and 3.5 hold,then forany T ＞ 0,S2(T,B0(ω))isrelatively compact in M1.

We are now in a position to p rove existence of a random attractor for the stochastic nonclassicald iffusion equation w ith fad ingm em ory.

Theorem 3.1.Let{S(t)}t≥0be thesolution operatorofequations(3.2),and the conditionsofthe Lemma3.6 hold,then therandom dynam icalsystem Ψ has aunique random attractor in M1.

Proof.Notice thatΨ has a closed absorbing set B={B(ω)}ω∈?∈ D by Lemm a 3.2,and is relatively com pact in M1by Lemm a3.3 and Lemm a3.6.Hence theexistenceofa unique D-random attractor follow s from Theorem 2.1 imm ed iately.

A cknow ledgem en t

Thisw ork w as supported by Foundation of Young Teachers of Zhongnan University of Econom icsand Law 31541411210.

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?Correspond ing au tho r.Emailaddress:chengsl zncd@znufe.edu.cn,chengslhust@sina.com(S.L.Cheng)

AMSSub ject Classifi cations:35B40,35B41,35K55

Chinese Library Classifi cations:O175.26