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Research article

Completely monotonic integer degrees for a class of special functions

  • Received: 14 January 2020 Accepted: 26 March 2020 Published: 07 April 2020
  • MSC : 26A48, 33B15, 44A10

  • Let fn(x) (n=0,1,) be the remainders for the asymptotic formula of lnΓ(x) and Rn(x)=(1)nfn(x). This paper introduced the concept of completely monotonic integer degree and discussed the ones for the functions (1)mR(m)n(x), then demonstrated the correctness of the existing conjectures by using a elementary simple method. Finally, we propose some operational conjectures which involve the completely monotonic integer degrees for the functions (1)mR(m)n(x) for m=0,1,2,.

    Citation: Ling Zhu. Completely monotonic integer degrees for a class of special functions[J]. AIMS Mathematics, 2020, 5(4): 3456-3471. doi: 10.3934/math.2020224

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  • Let fn(x) (n=0,1,) be the remainders for the asymptotic formula of lnΓ(x) and Rn(x)=(1)nfn(x). This paper introduced the concept of completely monotonic integer degree and discussed the ones for the functions (1)mR(m)n(x), then demonstrated the correctness of the existing conjectures by using a elementary simple method. Finally, we propose some operational conjectures which involve the completely monotonic integer degrees for the functions (1)mR(m)n(x) for m=0,1,2,.


    Let QRn be a bounded C2-domain and OεRn+1 be the domain

    Oε={x=(x,xn+1)|x=(x1,,xn)Qand0<xn+1<εg(x)},

    where gC2(¯Q,(0,+)) and 0<ε1. Since gC2(¯Q,(0,+)), there exist two positive constants γ1 and γ2 such that

    γ1g(x)γ2,x¯Q. (1)

    Denote O=Q×(0,1) and ˜O=Q×(0,γ2) which contains Oε for 0<ε1. Given τR, we will study the limit of asymptotical behavior of the following stochastic reaction-diffusions equation with multiplicative noise defined on the thin domain Oε as ε tends to 0:

    {dˆuεΔˆuεdt=(H(t,x,ˆuε(t))+G(t,x))dt+mj=1cjˆuεdwj,xOε,t>τ,ˆuενε=0,xOε, (2)

    with the initial condition

    ˆuε(τ,x)=ˆϕε(x),xOε, (3)

    where νε is the unit outward normal vector to Oε, H is a superlinear source term, G is a function defined on RטO, cjR for j=1,2,,m, wj, j=1,2,,m, are independent two-sided real-valued Wiener processes on a probability space, and the symbol indicates that the equation is understood in the sense of Stratonovich integration.

    As ε0, we will show in certain sense that the limiting behavior of (2) is governed by the following equation:

    {du01gni=1(gu0yi)yidt=(H(t,(y,0),u0(t))+G(t,(y,0)))dt+mj=1cju0dwj,y=(y1,,yn)Q,t>τ,u0ν0=0,yQ, (4)

    with the initial condition

    u0(τ,y)=ϕ0(y),yQ, (5)

    where ν0 is the unit outward normal to Q.

    Random attractors have been investigated in [2,5,10,19,9] in the autonomous stochastic case, and in [3,21,22,23] in the non-autonomous stochastic case. Recently, the limiting dynamical behavior of stochastic partial differential equations on thin domain was studied in [16,20,13,14,11,12,17,4]. However, in [17,13], we only investigated the limiting behavior of random attractors in L2(O) of stochastic evolution equations on thin domain. In this paper, we will prove the existence and uniqueness of bi-spatial pullback attractor for the systems defined on fixed domain O converted from (2)-(3) when the initial space is L2(O) and the terminate space is H1(O) and establish upper semicontinuity result for the corresponding family of random attractors in H1(O) as ε approaches 0.

    Let X be a Banach space. The norm of X is written as X. Let M=L2(Q) and N=L2(O). We denote by (,)Y the inner product in a Hilbert space Y. The letter c and ci, iN, are generic positive constants which may change its values from line to line.

    We organize the paper as follows. In the next section, we establish the existence of a continuous cocycle in N for the stochastic equation defined on the fixed domain O converted from (2)-(3). We also describe the existence of a continuous cocycle in M for the stochastic equation (4)-(5). Section 3 contains all necessary uniform estimates of the solutions. We then prove the existence and uniqueness of regular random attractors for the stochastic equations in section 4, and analyze convergence properties of the solutions as well as the random attractors in H1(O) in section 5.

    Here we show that there is a continuous cocycle generated by the reaction-diffusion equation defined on Oε with multiplicative noise and deterministic non-autonomous forcing:

    {dˆuεΔˆuεdt=(H(t,x,ˆuε(t))+G(t,x))dt+mj=1cjˆuεdwj,x=(x,xn+1)Oε,t>τ,ˆuενε=0,xOε, (6)

    with the initial condition

    ˆuετ(x)=ˆϕε(x),xOε, (7)

    where νε is the unit outward normal to Oε, G:RטOR belongs to L2loc(R,L(˜O)), cjR, wj (j=1,2,,m) are independent two-sided real-valued Wiener processes on a probability space which will be specified later, and H is a nonlinear function satisfying the following conditions: for all x˜O and t,sR,

    H(t,x,s)sλ1|s|p+φ1(t,x), (8)
    |H(t,x,s)|λ2|s|p1+φ2(t,x), (9)
    H(t,x,s)sλ3, (10)
    |H(t,x,s)x|ψ3(t,x), (11)

    where p>2, λ1 λ2 and λ3 are positive constants, φ1Lloc(R,L(˜O)) and φ2,ψ3L2loc(R,L(˜O)).

    Throughout this paper, we fix a positive number λ(0,λ1) and write

    h(t,x,s)=H(t,x,s)+λs (12)

    for all x˜O and t,sR. Then it follows from (8)-(11) that there exist positive numbers α1, α2, β, b1 and b2 such that

    h(t,x,s)sα1|s|p+ψ1(t,x), (13)
    |h(t,x,s)|α2|s|p1+ψ2(t,x), (14)
    h(t,x,s)sβ, (15)
    |h(t,x,s)x|ψ3(t,x), (16)

    where ψ1(t,x)=φ1(t,x)+b1 and ψ2(t,x)=φ2(t,x)+b2 for x˜O and t,sR.

    Substituting (12) into (6) we get for t>τ,

    {dˆuε(Δˆuελˆuε)dt=(h(t,x,ˆuε(t))+G(t,x))dt+mj=1cjˆuεdwj,x=(x,xn+1)Oε,ˆuενε=0,xOε, (17)

    with the initial condition

    ˆuετ(x)=ˆϕε(x),xOε. (18)

    We now transfer problem (17)-(18) into an initial boundary value problem on the fixed domain O. To that end, we introduce a transformation Tε:OεO by Tε(x,xn+1)=(x,xn+1εg(x)) for x=(x,xn+1)Oε. Let y=(y,yn+1)=Tε(x,xn+1). Then we have

    x=y,xn+1=εg(y)yn+1.

    It follows from [18] that the Laplace operator in the original variable xOε and in the new variable yO are related by

    Δxˆu(x)=|J|divy(|J|1JJyu(y))=1gdivy(Pεu(y)),

    where we denote by u(y)=ˆu(x) and Pε is the operator given by

    Pεu(y)=(guy1gy1yn+1uyn+1guyngynyn+1uyn+1ni=1yn+1gyiuyi+1ε2g(1+ni=1(εyn+1gyi)2)uyn+1).

    In the sequel, we abuse the notation a little bit by writing h(t,x,s) and G(t,x) as h(t,x,xn+1,s) and G(t,x,xn+1) for x=(x,xn+1), respectively. With this agreement, for any function F(t,y,s), we introduce

    Fε(t,y,yn+1,s)=F(t,y,εg(y)yn+1,s),F0(t,y,s)=F(t,y,0,s),

    where y=(y,yn+1)O and t,sR, Then problem (17)-(18) is equivalent to the following system for t>τ,

    {duε(1gdivy(Pεuε)λuε)dt=(hε(t,y,uε(t))+Gε(t,y))dt+mj=1cjuεdwj,y=(y,yn+1)O,Pεuεν=0,yO, (19)

    with the initial condition

    uετ(y)=ϕε(y)=ˆϕεT1ε(y),yO, (20)

    where ν is the unit outward normal to O.

    Given tR, define a translation θ1,t on R by

    θ1,t(τ)=τ+t,for allτR. (21)

    Then {θ1,t}tR is a group acting on R. We now specify the probability space. Denote by

    Ω={ωC(R,R):ω(0)=0}.

    Let F is the Borel σ-algebra induced by the compact-open topology of Ω, and P the corresponding Wiener measure on (Ω,F). There is a classical group {θt}tR acting on (Ω,F,P), which is defined by

    θtω()=ω(+t)ω(t),ωΩ,tR. (22)

    Then (Ω,F,P,{θt}tR) is a metric dynamical system (see [1]). On the other hand, let us consider the one-dimensional stochastic differential equation

    dz+αzdt=dw(t), (23)

    for α>0. This equation has a random fixed point in the sense of random dynamical systems generating a stationary solution known as the stationary Ornstein-Uhlenbeck process (see [6] for more details). In fact, we have

    Lemma 2.1. There exists a {θt}tR-invariant subset ΩF of full measure such that

    limt±|ω(t)|t=0for allωΩ,

    and, for such ω, the random variable given by

    z(ω)=α0eαsω(s)ds

    is well defined. Moreover, for ωΩ, the mapping

    (t,ω)z(θtω)=α0eαsθtω(s)ds=α0eαsω(t+s)ds+ω(t)

    is a stationary solution of (23) with continuous trajectories. In addition, for ωΩ

    limt±|z(θtω)|t=0,limt±1tt0z(θsω)ds=0, (24)
    limt±1tt0|z(θsω)|ds=E|z|<. (25)

    Denote by zj the associated Ornstein-Uhlenbeck process corresponding to (23) with α=1 and w replaced by wj for j=1,,m. Then for any j=1,,m, we have a stationary Ornstein-Uhlenbeck process generated by a random variable zj(ω) on Ωj with properties formulated in Lemma 2.1 defined on a metric dynamical system (Ωj,Fj,Pj,{θt}tR). We set

    ˜Ω=Ω1××Ωmand  F=mj=1Fj,

    Then (˜Ω,F,P,{θt}tR) is a metric dynamical system.

    Denote by

    SCj(t)u=ecjtu,foruL2(O),

    and

    T(ω):=SC1(z1(ω))SCm(zm(ω))=emj=1cjzj(ω)IdL2(O),ωΩ.

    Then for every ωΩ, T(ω) is a homeomorphism on L2(O), and its inverse operator is given by

    T1(ω):=SCm(zm(ω))SC1(z1(ω))=emj=1cjzj(ω)IdL2(O).

    It follows that T1(θtω) has sub-exponential growth as t± for any ω˜Ω. Hence T1 is tempered. Analogously, T is also tempered. Obviously, sups[s0a,s0+a]T(θsω) is still tempered for every s0R and aR+.

    On the other hand, since zj,j=1,,m, are independent Gaussian random variables, by the ergodic theorem we still have a {θt}tR-invariant set ˆΩF of full measure such that

    limt±1tt0T(θτω)2dτ=ET2=mj=1E(e2cjzj)<,

    and

    limt±1tt0T1(θτω)2dτ=ET12=mj=1E(e2cjzj)<.

    Remark 1. We now consider θ defined in (22) on ˜ΩˆΩ instead of Ω. This mapping possesses the same properties as the original one if we choose F as the trace σ-algebra with respect to ˜ΩˆΩ. The corresponding metric dynamical system is still denoted by (Ω,F,P,{θt}tR) throughout this paper.

    Next, we define a continuous cocycle for system (19)-(20) in N. This can be achieved by transferring the stochastic system into a deterministic one with random parameters in a standard manner. Let uε be a solution to (19)-(20) and denote by vε(t)=T1(θtω)uε(t) and δ(ω)=mj=1cjzj(ω). Then vε satisfies

    {dvεdt1gdivy(Pεvε)=(λ+δ(θtω))vε+T1(θtω)hε(t,y,T(θtω)vε(t))+T1(θtω)Gε(t,y),yO,t>τ,Pεvεν=0,yO, (26)

    with the initial conditions

    vετ(y)=ψε(y),yO, (27)

    where ψε=(T1(θτω))ϕε.

    Since (26) is a deterministic equation, by the Galerkin method, one can show that if H satisfies (8)-(11), then for every ωΩ, τR and ψεN, (26)-(27) has a unique solution vε(t,τ,ω,ψε)C([τ,τ+T),L2(O))L2((τ,τ+T),H1(O))C([τ+ϵ,τ+T),H1(O)) with vτ(,τ,ω,ψε)=ψε for every T>0 and 0<ϵ<T. Furthermore, one may show that vε(t,τ,ω,ψε) is (F,B(N))-measurable in ωΩ and continuous with respect to ψε in N for all tτ. Since uε(t,τ,ω,ϕε)=T(θtω)vε(t,τ,ω,ψε) with ϕε=(T(θτω))ψε, we find that uε(t) is continuous in both tτ and ϕεN and is (F,B(N))-measurable in ωΩ. In addition, it follows from (26) that uε is a solution of problem (19)-(20). We now define Φε:R+×R×Ω×NN by

    Φε(t,τ,ω,ϕε)=uε(t+τ,τ,θτω,ϕε)=T(θt+τω)vε(t+τ,τ,θτω,ψε),for all(t,τ,ω,ϕε)R+×R×Ω×N. (28)

    By the properties of uε, we find that Φε is a continuous cocycle on N over (R,{θ1,t}tR) and (Ω,F,P,{θt}tR), where {θ1,t}tR and {θt}tR are given by (21) and (22), respectively. In this paper, we will first prove the asymptotic compactness of solutions in H1(O) and then establish the existence and upper semicontinuity in H1(O) of (N,H1(O))-random attractors.

    Let Rε:L2(Oε)L2(O) be an affine mapping of the form

    (Rεˆϕε)(y)=ˆϕε(T1εy),ˆϕεL2(Oε).

    Given tR+, τR, ωΩ and ˆϕεL2(Oε), we can define a continuous cocycle ˆΦε for problem (6)-(7) by the formula

    ˆΦε(t,τ,ω,ˆϕε)=R1εΦε(t,τ,ω,Rεˆϕε).

    The same change of unknown variable v0(t)=T1(θtω)u0(t) transforms equation (4) into the following random partial differential equation on Q:

    {dv0dtni=11g(gv0yi)yi=(λ+δ(θtω))v0+T1(θtω)h0(t,y,T(θtω)v0(t))+T1(θtω)G0(t,y),yQ,t>τ,v0ν0=0,yQ, (29)

    with the initial conditions

    v0τ(y)=ψ0(y),yQ, (30)

    where ψ0=(˜T1(θτω))ϕ0.

    The same argument as above allows us to prove that problem (4) and (5) generates a continuous cocycle Φ0(t,τ,ω,ϕ0) in the space M.

    Now we want to write equation (26)-(27) as an abstract evolutionary equation. We introduce the inner product (,)Hg(O) on N defined by

    (u,v)Hg(O)=Oguvdy,for allu,vN

    and denote by Hg(O) the space equipped with this inner product. Since g is a continuous function on ¯Q and satisfies (1), one easily shows that Hg(O) is a Hilbert space with norm equivalent to the natural norm of N.

    For 0<ε1, we introduce a bilinear form aε(,):H1(O)×H1(O)R, given by

    aε(u,v)=(Jyu,Jyv)Hg(O), (31)

    where

    Jyu=(uy1gy1gyn+1uyn+1,,uyngyngyn+1uyn+1,1εguyn+1).

    By introducing on H1(O) the equivalent norm, for every 0<ε1,

    uH1ε(O)=(O(|yu|2+|u|2+1ε2u2yn+1)dy)12, (32)

    we see that there exist positive constants ε0, η1 and η2 such that for all 0<εε0 and uH1(O),

    η1O(|yu|2+1ε2u2yn+1)dyaε(u,u)η2O(yu|2+1ε2u2yn+1)dy (33)

    and

    η1u2H1ε(O)aε(u,u)+u2L2(O)η2u2H1ε(O). (34)

    Denote by Aε an unbounded operator on Hg(O) with domain

    D(Aε)={vH2(O),Pεvν=0onO}

    as defined by

    Aεv=1gdivPεv,vD(Aε).

    Then we have

    aε(u,v)=(Aεu,v)Hg(O),uD(Aε),vH1(O). (35)

    Using Aε, (26)-(27) can be written as

    {dvεdt+Aεvε=(λ+δ(θtω))vε+T1(θtω)hε(t,y,T(θtω)vε(t))+T1(θtω)Gε(t,y),yO,t>τ,vετ=ψε. (36)

    To reformulate system (29)-(30), we introduce the inner product (,)Hg(Q) on M defined by

    (u,v)Hg(Q)=Qguvdy,for allu,vM,

    and denote by Hg(Q) the space equipped with this inner product. Let a0(,):H1(Q)×H1(Q)R be a bilinear form given by

    a0(u,v)=Qgyuyvdy.

    Denote by A0 an unbounded operator on Hg(Q) with domain

    D(A0)={vH2(Q),vν0=0onQ}

    as defined by

    A0v=1gni=1(gvyi)yivD(A0).

    Then we have

    a0(u,v)=(A0u,v)Hg(Q),uD(A0),vH1(Q).

    Using A0, (29)-(30) can be written as

    {dv0dt+A0v0=(λ+δ(θtω))v0+T1(θtω)h0(t,y,T(θtω)v0(t))+T1(θtω)G0(t,y),yQ,t>τ,v0τ(s)=ψ0(s),s[ρ,0]. (37)

    Hereafter, we set X0=M, Xε=L2(Oε) and X1=N. For every i=ε,0 or 1, a family Bi={Bi(τ,ω):τR,ωΩ} of nonempty subsets of Xi is called tempered if for every c>0, we have:

    limtectBi(τ+t,θtω)Xi=0,

    where BiXi=supxBixXi. The collection of all families of tempered nonempty subsets of Xi is denoted by Di, i.e.,

    Di={Bi={Bi(τ,ω):τR,ωΩ}:Bi is tempered in Xi}.

    Our main purpose of the paper is to prove that the cocycle ˆΦε and Φ0 possess a unique (L2(Oε),H1(Oε))-random attractor ˆAε and (M,J)-random attractor A0, respectively. Furthermore ˆAε is upper-semicontinuous at ε=0, that is, for every τR and ωΩ,

    limε0supuεˆAεinfu0A0ε1uεu02H1(Oε)=0. (38)

    To prove (38), we only need to show that the cocycle Φε has a unique (N,H)-random attractor Aε and it is upper-semicontinuous at ε=0 in the sense that for every τR and ωΩ,

    limε0distH(Aε(τ,ω),A0(τ,ω))=0,

    which will be established in the last section of the paper.

    Furthermore, we suppose that there exists λ0>0 such that

    ¯γΔ=λ02E(|δ(ω)|)>0. (39)

    Let us consider the mapping

    γ(ω)=λ02|δ(ω)|. (40)

    By the ergodic theory and (39) we have

    limt±1tt0γ(θlω)dl=Eγ=¯γ>0. (41)

    The following condition will be needed when deriving uniform estimates of solutions:

    τe12¯γs(G(s,)2L(˜O)+φ1(s,)2L(˜O)+ψ3(s,)2L(˜O))ds<,τR. (42)

    When constructing tempered pullback attractors, we will assume

    limreσr0e12¯γs(G(s+r,)2L(˜O)+φ1(s+r,)2L(˜O)+ψ3(s+r,)2L(˜O))ds=0,σ>0. (43)

    Since ψ1=φ1+b1 for some positive constant b1, it is evident that (42) and (43) imply

    τe12¯γs(G(s,)2L(˜O)+ψ1(s,)L(˜O)+ψ3(s,)2L(˜O))ds<,τR (44)

    and

    limreσr0e12¯γs(G(s+r,)2L(˜O)+ψ1(s+r,)2L(˜O)+ψ3(s+r,)2L(˜O))ds=0, (45)

    for any σ>0.

    In this section, we recall and generalize some results in [17] and derive some new uniform estimates of solutions of problem (36) or (19)-(20) which are needed for proving the existence of D1-pullback absorbing sets and the D1-pullback asymptotic compactness in H1(O) of the cocycle Φε.

    Lemma 3.1. Assume that (8)-(11), (39) and (42) hold. Then for every 0<εε0, τR, ωΩ, and D1={D1(τ,ω):τR, ωΩ}D1, there exists T=T(τ,ω,D1)2, independent of ε, such that for all tT, λ1>λ0 and ψεD1(τt,θtω), the solution vε of (36) with ω replaced by θτω satisfies

    sup1s0vε(τ+s,τt,θτω,ψε)2H1ε(O)R2(τ,ω), (46)

    where R2(τ,ω) is determined by

    R2(τ,ω)=r1(ω)R1(τ,ω)+c0e¯γrT1(θrω)2(G(r+τ,)2L(˜O)+ψ3(r+τ,)2L(˜O))dr, (47)

    where R1(τ,ω) is determined by

    R1(τ,ω)=c0er0γ(θlω)dlT1(θrω)2G(r+τ,)2L(˜O)dr+c0er0γ(θlω)dlT1(θrω)2ψ1(r+τ,)2L(˜O)dr, (48)

    and r1(ω) is a tempered function, and c is independent of ε.

    Proof. The proof is similar as that of Lemma 3.4 in [17], so we only sketch the proof here. Taking the inner product of (36) with vε in Hg(O), we find that

    12ddtvε2Hg(O)aε(vε,vε)+(λ0+δ(θtω))vε2Hg(O)+(T1(θtω)hε(t,y,T(θtω)vε(t)),vε)Hg(O)+(T1(θtω)Gε(t,y),vε)Hg(O). (49)

    By (13), we have

    ddtvε2Hg(O)+2aε(vε,vε)+λ02vε2Hg(O)+2α1γ1T1(θtω)2uεpLp(O)(λ0+2δ(θtω))vε2Hg(O)+2λ0γ2|˜O|T1(θtω)2G(t,)2L(˜O)+2γ2|˜O|T1(θtω)2ψ1(t,)L(˜O). (50)

    Then, we have for any στ,

    eστγ(θlω)dlvε(σ)2Hg(O)+2στerτγ(θlω)dlaε(vε(r),vε(r))dr+λ02στerτγ(θlω)dlvε(r)2Hg(O)dr+2α1γ1στT1(θrω)2erτγ(θlω)dluε(r)pLp(O)drvε(τ)2Hg(O)+2λ0γ2|˜O|στerτγ(θlω)dlT1(θrω)2G(r,)2L(˜O)dr+2γ2|˜O|στerτγ(θlω)dlT1(θrω)2ψ1(r,)2L(˜O)dr, (51)

    where γ(θtω)=λ0+δ(θtω).

    Thus by the similar arguments as Lemma 3.1 in [17] we get for every τR, ωΩ, and D1D1, there exists T=T(τ,ω,D1)>0 such that for all tT,

    vετ(,τt,θτω,ψ)2L2(O)c0er0γ(θlω)dlψ1(r+τ,)2L(˜O)dr+c0er0γ(θlω)dlT1(θrω)2G(r+τ,)2L(˜O)dr+c0er0γ(θlω)dlT1(θrω)2ψ1(r+τ,)2L(˜O)dr. (52)

    Moreover, taking the inner product of (36) with Aεvε in Hg(O), we find that

    12ddtaε(vε,vε)+Aεvε2Hg(O)(λ0+δ(θtω))aε(vε,vε)+(T1(θtω)hε(t,y,T(θtω)vε(t)),Aεvε)Hg(O)+(T1(θtω)Gε(t,y),Aεvε)Hg(O). (53)

    By (15)-(16) we have

    ddtaε(vε,vε)+Aεvε2Hg(O)(c+2δ(θtω))aε(vε,vε)+cT1(θtω)2(G(t,)2L(˜O)+ψ3(t,)2L(˜O)), (54)

    The left proof is similar of that Lemma 3.4 in [17], so we omit it here.

    We are now in a position to establish the uniform estimates for the solution uε of the stochastic equation (19)-(20) by using those estimates for the solution vε of (36) and the relation between vε and uε.

    Lemma 3.2. Assume that (8)-(11), (39) and (42) hold. Then for every 0<εε0, τR, ωΩ, and D1={D1(τ,ω):τR,ωΩ}D1, there exists T=T(τ,ω,D1)2, independent of ε, such that for all tT, λ1>λ0 and ϕεD1(τt,θtω), the solution uε of (19)-(20) with ω replaced by θτω satisfies

    sup1s0uε(τ+s,τt,θτω,ϕε)2H1ε(O)r2(ω)R2(τ,ω), (55)

    where r2(ω) is a tempered function and R2(τ,ω) is given by (47).

    Lemma 3.3. Assume that (8)-(11), (39) and (42) hold. Then for every 0<εε0, τR, ωΩ, and D1={D1(τ,ω):τR, ωΩ}D1, there exists T=T(τ,ω,D1)2, independent of ε, such that for all tT, λ1>λ0 and ψεD1(τt,θtω), the solution vε of (36) with ω replaced by θτω satisfies

    sup1s0vε(τ+s,τt,θτω,ψε)pLp(O)+ττρvε(r,τt,θτω,ψε)2p2L2p2(O)drR3(τ,ω), (56)

    where R3(τ,ω)< for every τR and ωΩ.

    Proof. The proof is similar as that of Lemma 3.6 in [14], so we omit it here.

    Lemma 3.4. Assume that (8)-(11), (39) and (42) hold. Then for every η>0, τR, ωΩ, and D1={D1(τ,ω):τR,ωΩ}D1, there exist T=T(τ,ω,D1)2, γ=γ(ω)>0, a large M=M(τ,ω,η)>0 and 0<ε1<ε0 such that for all tT, λ1>λ0, 0<ε<ε1 and ψεD1(τt,θtω), the solution vε of (36) with ω replaced by θτω satisfies

    01eγMp2s{yO: vε(s+τ,τt,θτω,ψε)2M}|vε(s+τ,τt,θτω,ψε)|2p2dydsη, (57)
    01eγMp2s{yO: vε(s+τ,τt,θτω,ψε)2M}|vε(s+τ,τt,θτω,ψε)|2p2dydsη. (58)

    Proof. Let M be a positive number to be specified later. Taking the scalar product of (36) with (vεM)p1+, where (vεM)+=max{vεM,0}, we have

    1pddt(vεM)+pLp(O)+(p1)vεM(vεM)p2aε(vε,vε)dx(δ(θtω)vε,(vεM)p1+)+(T1(θtω)hε(t,y,T(θtω)vε),(vεM)p1+)+(T1(θtω)Gε(t,y),(vεM)p1+). (59)

    For the first term on the right side of (59) we have

    |(δ(θtω)vε,(vεM)p1+)|1p|δ(θrω)|pO|vε|pdx+p1pO(vεM)p+dx. (60)

    For the second term on the right-hand side of (59), by (8), we obtain, for vε>M,

    hε(t,y,T(θtω)vε) (vεM)p1+α1T(θtω)p1(vε)p1(vM)p1+
    +T(θtω)1ψ1(t,y,εg(y)yn+1)(vε)1(vεM)p1+
    12α1Mp2T(θtω)p1(vεM)p+12α1T(θtω)p1(vεM)2p2+
    +T1(θtω)1|ψ1(t,y,εg(y)yn+1)|(vεM)p2+

    which implies

    (T1(θtω)hε(t,y,T(θtω)vε), (vεM)p1+)
    12α1Mp2T(θtω)p2O(vεM)p+dx12α1T1(θtω)p2O(vεM)2p2+dx
    +T(θtω)2O|ψ1(t,y,εg(y)yn+1)|(vεM)p2+dx
    12α1Mp2T(θtω)p2O(vεM)p+dx12α1T(θtω)p2O(vεM)2p2+dx
    +p2pO(vεM)p+dx+2pT(θtω)pO|ψ1(t,y,εg(y)yn+1)|p2dy. (61)

    The last term in (59) is bounded by

    \begin{align} \left( {{\mathcal T}^{ - 1} \left( {\theta _{t} \omega } \right)G_\varepsilon\left( {t,y} \right),\left ( {v^\varepsilon - M} \right)_ + ^{p - 1} } \right) &\le {\frac 18} \alpha_1 \|{\mathcal T} \left( {\theta _{t}\omega } \right)\|^{p-2} \int_{\mathcal O} {\left( {v^\varepsilon - M} \right)_ + ^{2p - 2} } dx \\ &\quad + \frac{2}{\alpha_1}\|{\mathcal T} \left( {\theta _{t}\omega } \right)\|^{-p} \int_{v^\varepsilon \ge M} {\left| {G_\varepsilon\left( {t,y} \right)} \right|^2 } dy. \end{align} (62)

    All above estimates yield

    \begin{align} & \frac{d}{{dt}}\left\| {\left( {v^\varepsilon - M} \right)_ + } \right\|_{L^p(\mathcal O)}^p -(2p-3-{\frac 12}p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{t}\omega } \right)\|^{p-2}) \int_{\mathcal O}(v^\varepsilon-M)_+^{p} dx\\ &\quad +{\frac 14}p \alpha_1 \|{\mathcal T} \left( {\theta _{t}\omega } \right)\|^{p-2} \int_{\mathcal O} (v^\varepsilon-M)_+^{2p-2} dx \\ &\le | {\delta\left( {\theta _r \omega } \right)} |^{p} \int_{\mathcal O} {|v^\varepsilon| ^{p} } dx+2 \| {\mathcal T} \left( {\theta _{t}\omega } \right)\|^{ - p} \int_{\mathcal O} | \psi_1(t,y^{*},\varepsilon g(y^{*})y_{n+1}) |^ {\frac p2} dy\\ &\quad + \frac{2p}{\alpha_1}\|{\mathcal T} \left( {\theta _{t}\omega } \right)\|^{-p} \int_{\mathcal O} {\left| {G_\varepsilon\left( {t,y} \right)} \right|^2 } dy. \end{align} (63)

    Multiplying (63) by e^{-\int_0^t( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } , then integrating on (\tau -1, \tau) we obtain

    \| \left ( v^\varepsilon(\tau, \tau-t, \omega, \psi^\varepsilon) -M \right )_+\|^p_{L^p(\mathcal O)}
    +{\frac 14} p \alpha_1 \int_{\tau -1}^{\tau} \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{p-2} e^{-\int_{\tau}^\zeta ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr }
    \times \int_{\mathcal O} ( v^\varepsilon(\zeta, \tau-t, \omega, \psi^\varepsilon) -M) _+^{2p-2} dx d\zeta
    \le e^{-\int_{\tau}^{\tau -1} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2})dr } \| \left ( v^\varepsilon(\tau-1, \tau-t, \omega, \psi^\varepsilon) -M \right )_+\|^p_{L^p(\mathcal O)}
    + \int_{\tau -1}^{\tau} | \delta (\theta_\zeta \omega)|^p e^{-\int_{\tau}^\zeta ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \|v^\varepsilon(\zeta, \tau-t, \omega, \psi^\varepsilon)\|^p_{L^p(\mathcal O)} d\zeta
    +2|\mathcal O| \int_{\tau -1}^{\tau} \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{-p} e^{-\int_{\tau+\xi}^\zeta ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \| \psi_1(\zeta,\cdot) \|_{L^\infty(\widetilde {\mathcal O})}^ {\frac p2} d\zeta.
    \begin{equation} +\frac{2p|\mathcal O|}{\alpha_1} \int_{\tau -1}^{\tau} \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{-p} e^{-\int_{\tau+\xi}^\zeta ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \| G(\zeta,\cdot) \|^2_{L^\infty(\widetilde {\mathcal O})} d\zeta, \end{equation} (64)

    where |\mathcal O| stands for the Lebesgue measure of \mathcal O . Replacing \omega by \theta_{-\tau} \omega in (64) we get

    {\frac 14} p \alpha_1 \int_{-1}^0 \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{p-2} e^{-\int_0^{\zeta} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr }
    \times\int_{\mathcal O} ( v^\varepsilon(\zeta+\tau, \tau-t,\theta_{-\tau} \omega, \psi^\varepsilon) -M) _+^{2p-2} dx d\zeta
    \le e^{-\int_0^{ -1} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2})dr } \| \left ( v^\varepsilon(\tau-1, \tau-t,\theta_{-\tau} \omega, \psi^\varepsilon) -M \right )_+\|^p_{L^p(\mathcal O)}
    + \int_{ -1}^0 | \delta (\theta_{\zeta+\xi} \omega)|^p e^{-\int_0^{\zeta} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \|v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon)\|^p_{L^p(\mathcal O)} d\zeta
    +2|\mathcal O| \int_{ -1}^0 \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{-p} e^{-\int_0^{\zeta} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \| \psi_1(\zeta+\tau,\cdot) \|_{L^\infty(\widetilde {\mathcal O})}^ {\frac p2} d\zeta.
    \begin{equation} +\frac{2p|\mathcal O|}{\alpha_1} \int_{ -1}^0 \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{-p} e^{-\int_0^{\zeta} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \| G(\zeta+\tau,\cdot) \|^2_{L^\infty(\widetilde {\mathcal O})} d\zeta. \end{equation} (65)

    Since \omega is continuous on [-1,0] , there exist c_1 = c_1(\omega, p, \alpha_1)>0 and c_2 = c_2(\omega, p, \alpha_1)>0 such that

    \begin{equation} c_1 \le {\frac 12} p \alpha_1 \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2} \le c_2 \quad \text{ for all } \ r\in [-\rho-1,0]. \end{equation} (66)

    By (66) we obtain

    \begin{equation} e^{c_2 M^{p-2} \zeta } \le e^{\int_\xi^{\zeta+\xi} {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2} dr } \le e^{c_1M^{p-2} \zeta } \quad \text{ for all } \ \zeta \in [-1,0]\,\,\text{and}\,\,\xi\in [-\rho,0]. \end{equation} (67)

    For the left-hand side of (65), by (67) we find that there exists c_3 = c_3(\omega)>0 such that

    {\frac 14} p \alpha_1 \int_{-1}^0 \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{p-2} e^{-\int_0^{\zeta} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr }
    \int_{\mathcal O} ( v^\varepsilon(\zeta+\tau, \tau-t,\theta_{-\tau} \omega, \psi^\varepsilon) -M) _+^{2p-2} dx d\zeta
    \begin{equation} \ge c_3 \int_{ -1}^0 e^{c_2 M^{p-2} \zeta } \int_{\mathcal O} ( v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon) -M) _+^{2p-2} dx d\zeta. \end{equation} (68)

    For the first term on the right-hand side of (65), by (67) we obtain

    e^{-\int_0^{ -1} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2})dr } \| \left ( v^\varepsilon(\tau-1, \tau-t,\theta_{-\tau} \omega, \psi^\varepsilon) -M \right )_+\|^p_{L^p(\mathcal O)}
    \le e^{2p-3} e^{-c_1M^{p-2} } \| \left ( v^\varepsilon(\tau-1, \tau-t,\theta_{-\tau} \omega, \psi^\varepsilon) -M \right )_+\|^p_{L^p(\mathcal O)}
    \begin{equation} \le e^{2p-3} e^{-c_1M^{p-2} } \| v^\varepsilon(\tau-1, \tau-t,\theta_{-\tau} \omega, \psi^\varepsilon)\|^p_{L^p(\mathcal O)}. \end{equation} (69)

    Similarly, for the second terms on the right-hand side of (65), we have from (67) there exists c_4 = c_4(\omega)>0 such that

    \int_{ -1}^0 | \delta (\theta_{\zeta} \omega)|^p e^{-\int_0^{\zeta} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \|v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon)\|^p_{L^p(\mathcal O)} d\zeta
    \begin{equation} \leq c_4 \int_{-1}^0 e^{ c_1M^{p-2} \zeta } \|v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon)\|^p_{L^p(\mathcal O)} d\zeta \end{equation} (70)

    Since \varphi_1 \in L^\infty_{loc}( \mathbb{R}, {L^\infty(\widetilde {\mathcal O})}) and G\in L^2_{loc} ( \mathbb{R}, {L^\infty(\widetilde {\mathcal O})}) , for the two three terms on the right-hand side of (65), by (67) we obtain there exists c_5 = c_5(\tau,\omega)>0 such that

    2|\mathcal O| \int_{ -1}^0 \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{-p} e^{-\int_0^{\zeta} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \| \psi_1(\zeta+\tau,\cdot) \|_{L^\infty(\widetilde {\mathcal O})}^ {\frac p2} d\zeta.
    +\frac{2p|\mathcal O|}{\alpha_1} \int_{ -1}^0 \|{\mathcal T} \left( {\theta _{\zeta}\omega } \right)\|^{-p} e^{-\int_0^{\zeta} ( 2p-3 - {\frac 12} p \alpha_1 M^{p-2} \|{\mathcal T} \left( {\theta _{r}\omega } \right)\|^{p-2}) dr } \| G(\zeta+\tau,\cdot) \|^2_{L^\infty(\widetilde {\mathcal O})} d\zeta
    \begin{equation} \le c_5 \int_{-1}^0 e^{ c_1M^{p-2} \zeta } d\zeta \le c_1^{-1}c_5 M^{2-p}. \end{equation} (71)

    By (68)-(71) we get from (65) that

    c_3 \int_{ -1}^0 e^{c_2 M^{p-2} \zeta } \int_{\mathcal O} ( v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon) -M) _+^{2p-2} dy d\zeta
    \le e^{2p-3} e^{-c_1M^{p-2} } \|v^\varepsilon({\tau -1}, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon) \|^p_{L^p(\mathcal O)}
    + c_4 \int_{-1}^0 e^{ c_1M^{p-2} \zeta } \|v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon)\|^p_{L^p(\mathcal O)} d\zeta + c_1^{-1}c_5 M^{2-p},

    which together with Lemma 3.2 and Lemma 3.3 implies that there exist c_6 = c_6(\tau, \omega)>0 and T = T(\tau, \omega, D_1) \ge 2 such that for all t \ge T ,

    c_3 \int_{ -1}^0 e^{c_2 M^{p-2} \zeta } \int_{\mathcal O} ( v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon) -M) _+^{2p-2} dx d\zeta
    \begin{equation} \le c_6 e^{-c_1M^{p-2} } + c_6 \int_{-1}^0 e^{ c_1M^{p-2} \zeta } d\zeta + c_1^{-1}c_5M^{2-p} \le c_6 e^{-c_1M^{p-2} } + c_1^{-1}(c_5+c_6) M^{2-p}. \end{equation} (72)

    Since p>2 , we find that for every \eta>0 , there exists M_0 = M_0(\tau, \omega, \eta)>0 such that for all M\ge M_0 and t\ge T ,

    \begin{equation} \int_{ -1}^0 e^{c_2 M^{p-2} \zeta } \int_{\mathcal O} ( v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon) -M) _+^{2p-2} dy d\zeta \le \eta. \end{equation} (73)

    Note that |v| \le 2 (v-M)_+ for v\ge2 M , which together with (73) yields that for all M \ge M_0 and t\ge T ,

    \int_{ -1}^0 e^{c_2 M^{p-2} \zeta } \int_{ \{ y\in \mathcal O: \ v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon ) \ge 2M\} } | v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon )|^{2p-2} dy d\zeta\\\le 2^{2p-2} \int_{ -1}^0 e^{c_2 M^{p-2} \zeta } \int_{\mathcal O} ( v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon) -M) _+^{2p-2} dx d\zeta \le 2^{2p-2} \eta. (74)

    Similarly, one can verify that there exist M_1 = M_1(\tau, \omega, \eta)>0 and T_1 = (\tau, \omega, D) \ge 2 such that for all M\ge M_1 and t\ge T_1 ,

    \begin{equation} \int_{ -1}^0 e^{c_2 M^{p-2} \zeta } \int_{ \{ y\in \mathcal O: \ v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon ) \le - 2M \} } | v^\varepsilon(\zeta+\tau, \tau-t, \theta_{-\tau}\omega, \psi^\varepsilon )|^{2p-2} dy d\zeta \le 2^{2p-2} \eta. \end{equation} (75)

    Then Lemma 3.4 follows from (3) and (75) immediately.

    Note that A_\varepsilon is a family of linear operators in Y_\varepsilon , for 0\leq\varepsilon\leq \varepsilon_0 , where Y_\varepsilon = H_g(\mathcal O) , for 0<\varepsilon\leq \varepsilon_0 , and Y_0 = H_g(\mathcal Q) , is self-adjoint and has a compact resolvent. Then, \sigma(A_\varepsilon) consists of only eigenvalues \{\lambda_n^\varepsilon\}_{n = 1}^\infty with finite multiplicity:

    \begin{equation*} 0\leq \lambda_1^\varepsilon \leq\lambda_2^\varepsilon\leq\ldots\leq\lambda_n^\varepsilon\leq\cdots \to +\infty, \end{equation*}

    and their associated eigenfunctions \{\varpi_n^\varepsilon\}_{n = 1}^{\infty} form an orthonormal basis of Y_\varepsilon .

    It follows from Corollary 9.7 in [8] that the eigenvalues and the eigenfunctions of A_\varepsilon are convergent with respect to \varepsilon .

    Next, we introduce the spectral projections. We use P^\varepsilon_m to denote the projection from Y_\varepsilon onto the eigenspace \text{span}\{\varpi_i^\varepsilon\}_{i = 1}^m given by

    \begin{equation*} P^\varepsilon_n (u) = \sum\limits_{i = 1}^m(u,\varpi_i^\varepsilon)_{Y_\varepsilon}\varpi_i^\varepsilon \quad\text{for}\; u\in Y_\varepsilon. \end{equation*}

    We use Q^\varepsilon_m to denote its orthogonal complement projection, i.e., P^\varepsilon_m+Q^\varepsilon_m = I_\varepsilon , where I_\varepsilon is the identity operators on Y_\varepsilon . It is clear that

    \begin{equation} a_\varepsilon \left( {u,u} \right) = \left( {A_\varepsilon u,u} \right)_{H_g \left(\mathcal O \right)} \le \lambda ^\varepsilon _n \left( {u,u} \right)_{H_g \left( \mathcal O \right)}, \quad \forall u \in P_n^\varepsilon D\left( {A_\varepsilon ^{1/2} } \right). \end{equation} (76)

    and

    \begin{equation} a_\varepsilon \left( {u,u} \right) = \left( {A_\varepsilon u,u} \right)_{H_g \left(\mathcal O \right)} \ge \lambda _{m + 1}^\varepsilon \left( {u,u} \right)_{H_g \left(\mathcal O \right)} ,\quad u \in Q_m^\varepsilon D\left( {A_\varepsilon ^{1/2} } \right). \end{equation} (77)

    Let u^\varepsilon = u_1^\varepsilon + u_2^\varepsilon and v^\varepsilon = v_1^\varepsilon + v_2^\varepsilon , where u_1^\varepsilon = P_m^\varepsilon u^\varepsilon , u_2^\varepsilon = Q_m^\varepsilon u^\varepsilon , v_1^\varepsilon = P_m^\varepsilon v^\varepsilon , and v_2^\varepsilon = Q_m^\varepsilon v^\varepsilon , respectively.

    Lemma 3.5. Assume that (8)-(11), (39) and (42) hold. Then for every \tau\in \mathbb R , \omega\in \Omega , \eta>0 and D_1 = \left\{ {D_1\left( {\tau ,\omega } \right):\tau \in \mathbb R,\omega \in \Omega } \right\}\in \mathcal{D}_1 , there exists T = T(\tau,\omega,D_1,\eta)\geq 2 , m = m(\tau,\omega,D,\eta)\in \mathbb N and 0<\varepsilon_1 = \varepsilon_1(n)<\varepsilon_0 such that for all t\geq T , 0<\varepsilon<\varepsilon_1 and \phi^\varepsilon \in D_1 \left( {\tau - t,\theta _{ - t} \omega } \right) , the solution u^\varepsilon of (19)-(20) with \omega replaced by \theta_{-\tau}\omega satisfies

    \left\| { u^\varepsilon_2\left( {\tau,\tau-t,\theta _{ - \tau} \omega , \phi^\varepsilon} \right)} \right\|_{H^1(\mathcal O)} \le \eta.

    Proof. Taking the inner product (36) with A_\varepsilon v_{2}^\varepsilon in H_g \left( \mathcal O \right) , we get

    \begin{align} & \frac{1}{2}\frac{d}{{dt}}a_\varepsilon(v_2^\varepsilon,v_2^\varepsilon) + \left\| {A_\varepsilon v_2^\varepsilon } \right\|^2 \leq \left( {\delta \left( {\theta _{t } \omega } \right)v_2^\varepsilon , A_\varepsilon v_2^\varepsilon } \right) \\ &\quad + \left( {Q_n^\varepsilon{\mathcal T}^{ - 1} \left( {\theta _{t } \omega } \right)h_\varepsilon\left(t, {y,{\mathcal T}\left( {\theta _{t } \omega } \right)v^\varepsilon} \right), A_\varepsilon v_2^\varepsilon } \right) \\ &\quad+\left(Q_n^\varepsilon {{\mathcal T}^{ - 1} \left( {\theta _{t } \omega } \right)G_\varepsilon\left( {t,y} \right), A_\varepsilon v_2^\varepsilon } \right). \end{align} (78)

    For the first term on the right-hand side of (78), we have

    \begin{equation} \left( {\delta \left( {\theta _{t } \omega } \right)v_2^\varepsilon ,A_\varepsilon v_2^\varepsilon } \right) \le \frac{1}{8}\left\| {A_\varepsilon v_2^\varepsilon } \right\|^2 + 2| {\delta \left( {\theta _{t } \omega } \right)} |^2 \left\| {v_2^\varepsilon } \right\|^2. \end{equation} (79)

    For the superlinear term, we have from (9) that

    \begin{align} &\left( {Q_n^\varepsilon{\mathcal T}^{ - 1} \left( {\theta _{t } \omega } \right) h_\varepsilon\left(t, {y,{\mathcal T}\left( {\theta _{t } \omega } \right)v^\varepsilon} \right), A_\varepsilon v_2^\varepsilon } \right) \\ & \le \frac{1}{8}\left\| {A_\varepsilon v_2^\varepsilon } \right\|^2 + 2\left\| {{\mathcal T}^{ - 1} \left( {\theta _{t } \omega } \right)} \right\|^2 \int_{\mathcal O} {\left| {h_\varepsilon\left(t, {y,{\mathcal T}\left( {\theta _{t} \omega } \right)v^\varepsilon} \right)} \right|} ^2 dy \\ &\le \frac{1}{8}\left\| {A_\varepsilon v_2^\varepsilon } \right\|^2 + 2\alpha_2\left\| {{\mathcal T}^{ - 1} \left( {\theta _{t } \omega } \right)} \right\|^2 \int_{\mathcal O} {\left( {\left| {{\mathcal T}\left( {\theta _{t } \omega } \right)v^\varepsilon} \right|^{p - 1} + \psi _2 \left( t,y^{*},\varepsilon g(y^{*})y_{n+1} \right)} \right)} ^2 dy \\ &\le \frac{1}{8}\left\| {A_\varepsilon v_2^\varepsilon } \right\|^2 + 4\alpha_2\left\| {{\mathcal T}\left( {\theta _{t } \omega } \right)} \right\|^{2p - 4} \left\| v \right\|_{2p - 2}^{2p - 2} + 4\alpha_2|\mathcal O|\left\| {{\mathcal T}^{ - 1} \left( {\theta _{t } \omega } \right)} \right\|^2\| \psi_2(t,\cdot) \|^2_{L^\infty(\widetilde {\mathcal O})} . \end{align} (80)

    For the last term on the right-hand side of (78), we have

    \begin{align} \left( {Q_n^\varepsilon T^{ - 1} \left( {\theta _t \omega } \right) G_\varepsilon \left( {t,y} \right),A_\varepsilon v_2^\varepsilon } \right) \le \frac{1}{8}\left\| {A_\varepsilon v_2^\varepsilon } \right\|^2 + 2|\mathcal O|\left\| {T^{ - 1} \left( {\theta _t \omega } \right)} \right\|^2 \left\| {G\left( {t, \cdot } \right)} \right\|_{L^\infty \left( \widetilde {\mathcal O} \right)}^2 \end{align} (81)

    Noting that \left\| {A_\varepsilon v_2^\varepsilon } \right\|^2 \ge \lambda _{n + 1}^\varepsilon a_\varepsilon(v_2^\varepsilon,v_2^\varepsilon) , we obtain from all above estimates that

    \begin{align} & \frac{d}{{dt}}a_\varepsilon(v_2^\varepsilon,v_2^\varepsilon) + \lambda _{n + 1}^\varepsilon a_\varepsilon(v_2^\varepsilon,v_2^\varepsilon) \le 4 {\delta^2 \left( {\theta _{t } \omega } \right)} \left\| {v_2^\varepsilon } \right\|^2 \\ &\quad + 8\alpha_2\left\| {{\mathcal T}\left( {\theta _{t } \omega } \right)} \right\|^{2p - 4} \left\| v^\varepsilon \right\|_{2p - 2}^{2p - 2} \\ &\quad + c\left\| {{\mathcal T}^{ - 1} \left( {\theta _{t } \omega } \right)} \right\|^2(\| \psi_2(t,\cdot) \|^2_{L^\infty(\widetilde {\mathcal O})} +\left\| {G\left( {t,\cdot} \right)} \right\|^2_{L^\infty(\widetilde {\mathcal O})}). \end{align} (82)

    Taking \xi\in (\tau-1,\tau) , multiplying (82) by e^{ \lambda _{n + 1}^\varepsilon}t , first integrating with respect to t on (\xi, \tau) , integrating with respect to \xi on (\tau-1,\tau) , and then replacing \omega by \theta_{-\tau}\omega , we get

    \begin{align} & a_\varepsilon( v_2^\varepsilon \left( \tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right),v_2^\varepsilon \left( \tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)) \\ &\le\int_{\tau - 1 }^{\tau} {e^{\lambda _{n + 1}^\varepsilon \left( {r - \tau } \right)} } a_\varepsilon( v_2^\varepsilon \left( r,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right),v_2^\varepsilon \left( r,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)) dr\\ & + 4\delta^2\int_{{\tau} - 1}^{\tau} {e^{\lambda _{n + 1}^\varepsilon \left( {r - \tau } \right)} \delta ^2 \left( {\theta _{r - \tau } \omega } \right)} a_\varepsilon( v_2^\varepsilon \left( r,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right),v_2^\varepsilon \left( r,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)) dr \\ & + 8\alpha_2\int_{{\tau} - 1 }^{\tau} {e^{\lambda _{n + 1}^\varepsilon \left( {r - \tau } \right)} \left\| {{\mathcal T}\left( {\theta _{r - \tau } \omega } \right)} \right\|^{2p - 4} \left\| {v^\varepsilon\left( r,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)} \right\|_{2p - 2}^{2p - 2} } dr \\ &+c\int_{{\tau} - 1 }^{\tau}e^{\lambda _{n + 1}^\varepsilon \left( {r - \tau } \right)} \left\| {{\mathcal T}^{ - 1} \left( {\theta _{r - \tau } \omega } \right)} \right\|^2 (\| \psi_2(r,\cdot) \|^2_{L^\infty(\widetilde {\mathcal O})}) dr \\ & + c\int_{{\tau} - 1 }^{\tau} {e^{\lambda _{n + 1}^\varepsilon \left( {r - \tau } \right)} \left\| {{\mathcal T}^{ - 1} \left( {\theta _{r - \tau } \omega } \right)} \right\|^2 \left\| {G\left( {r,\cdot} \right)} \right\|^2_{L^\infty(\widetilde {\mathcal O})} } dr. \end{align} (83)

    Since \varphi_2,G \in L^2_{loc}( \mathbb{R}, {L^\infty(\widetilde {\mathcal O})}) , \|\mathcal T(\theta_t\omega)\| is continuous on [-1,0] and \lambda^\varepsilon_{n+1} approaches \lambda^0_{n+1} as \varepsilon\rightarrow 0 , we find that there exists c = c(\omega) >0 and 0<\varepsilon^{*}<\varepsilon_0 such that for 0<\varepsilon<\varepsilon^{*} ,

    \begin{align} &a_\varepsilon( v_2^\varepsilon \left( \tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right),v_2^\varepsilon \left( \tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right))\\ & \le c\int_{ - 1 }^0 {e^{\lambda _{n + 1}^\varepsilon r} \left\| {v^\varepsilon\left( r+\tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)} \right\|_{2p - 2}^{2p - 2} } dr \\ &\quad + c\int_{ - 1 }^{0} {e^{\lambda _{n + 1}^\varepsilon r} } a_\varepsilon( v^\varepsilon \left(r+ \tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right),v^\varepsilon \left( r+\tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)) dr \end{align}
    \begin{align} &\quad + c\int_{ - 1 }^{0} {e^{\lambda _{n + 1}^\varepsilon r} } dr\\ & \le c\int_{ - 1 }^0 {e^{(\lambda _{n + 1}^0-1) r} \left\| {v^\varepsilon\left( r+\tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)} \right\|_{2p - 2}^{2p - 2} } dr \\ &\quad + c\int_{ - 1 }^{0} {e^{(\lambda _{n + 1}^0-1) r} } a_\varepsilon( v^\varepsilon \left( r+\tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right),v^\varepsilon \left( r+\tau+s,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)) dr\\ &\quad + c\int_{ - 1 }^0 {e^{(\lambda _{n + 1}^0-1) r} \left\| {v^\varepsilon\left( {r+\tau - \rho_0(r+\tau+s),\tau-t, \theta_{-\tau}\omega,\psi^\varepsilon } \right)} \right\|^2 } dr\\ &\quad + c\int_{ - 1 }^{0} {e^{(\lambda _{n + 1}^0-1) r} } dr. \end{align} (84)

    Given \eta>0 , let T = T (\tau, \omega, D_1) \geq 2, \gamma = \gamma(\omega) > 0 , M = M(\tau, \omega, \eta) \geq 1 and 0<\varepsilon_1<\varepsilon^{*} be the constants in Lemma 3.4. Choose N_1 = N_1(\tau, \omega, \eta) \geq1 large enough such that \lambda_{n+1}^0-1\geq \gamma M^{p-2} for all n \geq N_1 . Then, by Lemma 3.4, we obtain, for all t \geq T , n \geq N_1 and 0<\varepsilon<\varepsilon_1 ,

    \begin{align} & c\int_{ - 1 }^0 {e^{(\lambda _{n + 1}^0-1) r} \left\| {v^\varepsilon\left( r+\tau,\tau-t,\theta_{-\tau}\omega, \psi^\varepsilon \right)} \right\|_{2p - 2}^{2p - 2} } dr \\ &\leq c\int_{ - 1 }^0 {e^{(\lambda _{n + 1}^0-1) r} \int_{\{y\in \mathcal O:|v^\varepsilon|\geq 2M\}}| {v^\varepsilon\left( r+\tau, \tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)} |^{2p - 2}dy } dr \\ &\quad+ c\int_{ - 1 }^0 {e^{(\lambda _{n + 1}^0-1) r} \int_{\{y\in \mathcal O:|v^\varepsilon| < 2M\}} | {v^\varepsilon\left( r+\tau,\tau-t, \theta_{-\tau}\omega,\psi^\varepsilon \right)} |^{2p - 2}dy } dr \\ &\leq c\int_{ - 1 }^0 {e^{\gamma M^{p-2} r} \int_{\{y\in \mathcal O:|v^\varepsilon|\geq 2M\}}| {v^\varepsilon\left( r+\tau, \tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)} |^{2p - 2}dy } dr \\ &\quad+ c\int_{ - 1 }^0 {e^{(\lambda _{n + 1}^0-1) r} \int_{\{y\in \mathcal O:|v^\varepsilon| < 2M\}} | {v^\varepsilon\left( r+\tau,\tau-t, \theta_{-\tau}\omega,\psi^\varepsilon \right)} |^{2p - 2}dy } dr \\ &\leq \eta+c2^{2p-2} M^{2p-2}|\mathcal O| \int_{ - 1 }^0 {e^{(\lambda _{n + 1}^0-1) r} } dr \leq \eta+c2^{2p-2} M^{2p-2}|\mathcal O| \frac{1}{\lambda _{n + 1}^0-1} . \end{align} (85)

    For the last three terms on the right-hand side of (84), by Lemma 3.1, we find that there exist c_1 = c_1(\tau, \omega)> 0 and T_1 = T_1(\tau, \omega, D_1) \geq T such that for all t \geq T_1 ,

    \begin{align} & c\int_{ - 1 }^{0} {e^{(\lambda _{n + 1}^0-1) r} } a_\varepsilon( v^\varepsilon \left( r+\tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right),v^\varepsilon \left( r+\tau,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)) dr\\ &\quad+ c\int_{ - 1 }^{0} {e^{(\lambda _{n + 1}^0-1) r} } dr \leq c_1 \int_{ - 1 }^{0} {e^{(\lambda _{n + 1}^0-1) r} } dr \leq c_1 \frac{1}{\lambda _{n + 1}^0-1} . \end{align} (86)

    Since \lambda_{n+ 1}^0\rightarrow \infty as n\rightarrow \infty , we obtain from (84)-(86) that there exists N_2 = N_2(\tau, \omega, \eta) \geq N_1 such that for all n \geq N_2 , t \geq T_1 and 0<\varepsilon<\varepsilon_1 ,

    a_\varepsilon( v_2^\varepsilon \left( \tau+s,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right),v_2^\varepsilon \left( \tau+s,\tau-t,\theta_{-\tau}\omega,\psi^\varepsilon \right)) \le 2\eta,

    which together v^\varepsilon(t) = {\mathcal T}^{ - 1} \left( {\theta _t \omega } \right) u^\varepsilon(t) and (77) completes the proof.

    In this subsection, we establish the existence of \mathcal D_1 -pullback attractor for the cocycle \Phi_\varepsilon associated with the stochastic problem (19)-(20). We first show that problem (19)-(20) has a tempered pullback absorbing set as stated below.

    Lemma 4.1. Suppose (8)-(11), (39) and (43) hold. Then the cocycle \Phi_\varepsilon associated with problem (19)-(20) has a closed measurable \mathcal D_1 -pullback absorbing set K = \{K \left( {\tau ,\omega } \right):\tau\in \mathbb R,\omega\in \Omega\}\in \mathcal D_1 .

    Proof. We first notice that, by Lemma 3.2, \Phi_\varepsilon has a closed \mathcal D_1 -pullback absorbing set K in H^1(\mathcal O). More precisely, given \tau\in \mathbb R and \omega\in \Omega , let

    \begin{equation} K \left( {\tau ,\omega } \right) = \left\{ {u \in H^1(\mathcal O) :\left\| u \right\|_{H^1(\mathcal O) }^2 \le L\left( {\tau ,\omega } \right)} \right\}, \end{equation} (87)

    where L\left( {\tau ,\omega } \right) is the constant given by the right-hand side of (55). It is evident that, for each \tau\in \mathbb R, L(\tau,\cdot ) : \Omega \rightarrow \mathbb R is ( \mathcal{F}, \mathcal{B} ( \mathbb R )) -measurable. In addition, for every \tau\in \mathbb R , \omega\in \Omega , and D\in \mathcal{D}_1 , there exists T = T(\tau,\omega,D)\geq 2 such that for all t\geq T ,

    \Phi_\varepsilon \left( {t,\tau - t,\theta _{ - t} \omega ,D \left( {\tau - t,\theta _{ - t} \omega } \right)} \right) \subseteq K\left( {\tau ,\omega } \right).

    Thus we find that K = \left\{ {K \left( {\tau ,\omega } \right):\tau \in \mathbb R,\omega \in \Omega } \right\} is a closed measurable set and pullback-attracts all elements in \mathcal D_1 . By the similar argument as in [15] we can obtain easily from (43) that K = \{K (\tau, \omega) : \tau\in \mathbb R, \omega \in \Omega\} is tempered. Consequently, K is a closed measurable D_1 -pullback absorbing set for \Phi_\varepsilon in D_1 .

    Lemma 4.2. Assume that (8)-(11), (39) and (43) hold. Then, the cocycle \Phi_\varepsilon is {\mathcal{D}}_1 -pullback asymptotically compact in H^1(\mathcal O) ; that is, for all \tau\in \mathbb R and \omega\in \Omega , \left\{ {\Phi_\varepsilon \left( {{t_n},\tau-t_n ,{\theta _{ - {t_n}}}\omega ,{x_n}} \right)} \right\}_{n = 1}^\infty has a convergent subsequence in H^1(\mathcal O) whenever t_n\rightarrow \infty and {x_n} \in D_1\left( {\tau-t_n,{\theta _{- {t_n}}}\omega } \right) with \{ D_1\left( {\tau ,\omega } \right):\tau \in \mathbb R , \omega \in \Omega \}\in {\mathcal{D}}_1 .

    Proof. We will show that for every \eta>0 , the sequence \{u^\varepsilon\left( {\tau,\tau-t_n,\theta _{ - \tau} \omega , \phi^\varepsilon} \right) \}_{n = 1}^\infty has a finite open cover of balls with radii less than \eta . By Lemma 3.5, we infer that there exists N_1 = N_1(\tau,\omega,D_1,\eta)\geq 1 , m_0 = m_0(\tau,\omega,D_1,\eta)\in \mathbb N and 0<\varepsilon_1 = \varepsilon_1(m_0)<\varepsilon_0 such that for all n \ge N_1 and 0<\varepsilon<\varepsilon_1 ,

    \begin{equation} \left\| { u^\varepsilon_2\left( {\tau,\tau-t_n,\theta _{ - \tau} \omega , \phi^\varepsilon} \right)} \right\|_{H^1(\mathcal O)} = \left\| { Q_{m_0}u^\varepsilon\left( {\tau,\tau-t_n,\theta _{ - \tau} \omega , \phi^\varepsilon} \right)} \right\|_{H^1(\mathcal O)} < {\frac {\eta}{4}}. \end{equation} (88)

    On the other hand, by Lemma 3.2 we find that the sequence \{P_{m_0} u^\varepsilon\left( {\tau,\tau-t_n,\theta _{ - \tau} \omega , \phi^\varepsilon} \right) \}_{n = 1}^\infty is bounded in the finite-dimensional space P_{m_0} H^1 (\mathcal O) and hence is precompact, which together with (88) shows that the sequence u^\varepsilon\left( {\tau,\tau-t_n,\theta _{ - \tau} \omega , \phi^\varepsilon} \right) has a finite open cover of balls with radii less than \eta in H^1(\mathcal O) , as desired.

    Theorem 4.3. Assume that (8)-(11), (39) and (43) hold. Then, the cocycle \Phi_\varepsilon has a unique \mathcal{D}_1 -pullback (\mathcal N, H^1(\mathcal O)) -attractor \mathcal A_\varepsilon = \{\mathcal A_\varepsilon(\tau,\omega):\tau\in \mathbb R,\omega\in\Omega\} .

    Proof. First, we know from Lemma 4.1 that \Phi_\varepsilon has a a closed measurable \mathcal D_1 -pullback absorbing set {K \left( {\tau ,\omega } \right)} . Second, it follows from Lemma 4.2 that \Phi_\varepsilon is \mathcal{D}_1 -pullback asymptotically compact from \mathcal N to H^1(\mathcal O) . Hence, the existence of a unique \mathcal{D}_1 -pullback (\mathcal N, H^1(\mathcal O)) -attractor for the cocycle \Phi_\varepsilon follows from Proposition 2.5 in [7].

    Analogous results also hold for the solution of (4)-(5). In particular, we have:

    Theorem 4.4. Assume that (8)-(11), (39) and (43) hold. Then, the cocycle \Phi_0 has a unique \mathcal{D}_0 -pullback ( \mathcal M , H^1(\mathcal Q) )-attractor \mathcal A_0 = \{\mathcal A_0(\tau,\omega):\tau\in \mathbb R,\omega\in\Omega\} .

    The following estimates are needed when we derive the convergence of pullback attractors. By the similar proof of that of Theorem 5.1 in [14], we get the following lemma.

    Lemma 5.1. Assume that (8)-(11) and (39) hold. Then for every 0<\varepsilon\leq \varepsilon_0 , \tau\in \mathbb R , \omega\in \Omega , T>0 , and \lambda_1> \lambda_0 , the solution v^\varepsilon of (36) satisfies, for all t\in [\tau, \tau+T] ,

    \begin{eqnarray} &&\int_{\tau}^{t} {{\left\| {v^\varepsilon \left( {r,\tau , \omega ,\psi^\varepsilon } \right)} \right\|_{H_\varepsilon ^1(\mathcal O) }^2 }} dr \le c \left\| \psi^\varepsilon \right\|_{\mathcal N}^2 \\ &&\mathit{\mbox{}} +c\int_{ \tau }^{\tau+T} \left( {\left\| {G\left( { r,\cdot} \right)} \right\|_{L^\infty( { {\widetilde {\mathcal O}} } )}^2 + \left\| {\psi_1 \left( { r,\cdot } \right)} \right\|_{L^\infty( { {\widetilde {\mathcal O}} } )}^2 } \right)dr, \end{eqnarray}

    where c is a positive constant depending on \tau , \omega , \lambda_0 and T , but independent of \varepsilon .

    Similarly, one can prove

    Lemma 5.2. Assume that (8)-(11) and (39) hold. Then for every \tau\in \mathbb R , \omega\in \Omega , T>0 , and \lambda_1>\lambda_0 , the solution v^0 of (37) satisfies, for all t\in [\tau, \tau+T] ,

    \begin{eqnarray} &&\int_{\tau}^{t} {{\left\| {v^0 \left( {r,\tau , \omega ,\psi^0 } \right)} \right\|_{H ^1(\mathcal Q) }^2 }} dr \\ &\le& c \left\| \psi^0 \right\|_{\mathcal M}^2+c \int_{ \tau }^{\tau+T} \left( {\left\| {G\left( { r,\cdot} \right)} \right\|_{L^\infty( { {\widetilde {\mathcal O}} } )}^2 + \left\| {\psi_1 \left( { r,\cdot } \right)} \right\|_{L^\infty( { {\widetilde {\mathcal O}} } )}^2 } \right)dr, \end{eqnarray}

    where c is a positive constant depending on \tau , \omega , \lambda_0 and T , but independent of \varepsilon .

    In the sequel, we further assume the functions G and H satisfy that for all t,s\in {\mathbb R} ,

    \begin{equation} \left\| {G_\varepsilon(t,\cdot) - G_0(t,\cdot) } \right\|_{L^2 \left( \mathcal O \right)} \le \kappa_1(t)\varepsilon \end{equation} (89)

    and

    \begin{equation} \left\| {H_\varepsilon(t,\cdot,s) - H_0(t,\cdot,s) } \right\|_{L^2 \left( \mathcal O \right)} \le \kappa_2(t)\varepsilon , \end{equation} (90)

    where \kappa_1(t), \kappa_2(t)\in L^2_{loc}({\mathbb R}) .

    By (12) and (90) we have, for all x\in \widetilde{\mathcal O} and t,s\in {\mathbb R} ,

    \begin{equation} \left\| {h_\varepsilon(t,\cdot,s) - h_0(t,\cdot,s) } \right\|_{L^2 \left( \mathcal O \right)} \le \kappa_2(t)\varepsilon. \end{equation} (91)

    Since \mathcal M can be embedded naturally into \mathcal N as the subspace of functions independent of y_{n+1} , we can consider the cocycle \Phi_0 as a mapping from \mathcal M into \mathcal N . Therefore we can compare \Phi_0 with \Phi_\varepsilon .

    Theorem 5.3. Suppose (8)-(11), (39), and (89)-(90) hold. Given \tau\in {\mathbb{R}} , \omega\in \Omega , \varepsilon_n\rightarrow 0 and a positive number L(\tau, \omega) , if \phi{^{\varepsilon_n}}\in H^1_{\varepsilon_n}(\mathcal O)) such that \left\| {\phi{^{\varepsilon_n}} } \right\|_{H^1_{\varepsilon_n}(\mathcal O))} \le L(\tau, \omega) , then there exists \phi^0\in \mathcal M such that, up to a subsequence, for t\geq 0 ,

    \mathop {\lim }\limits_{n \to \infty} \left\| {\Phi _{\varepsilon_n} \left( {t,\tau,\omega ,\phi^{\varepsilon_n} } \right) - \Phi _{0 } \left( {t,\tau,\omega ,\phi^0} \right)} \right\|_{\mathcal N} = 0.

    Proof. Since \phi^{\varepsilon_n}\in H^1_{\varepsilon_n}(\mathcal O) , there exists \phi^0\in \mathcal M such that \phi^{\varepsilon_n} \rightarrow \phi^0 in \mathcal N . By the similar proof of that of Theorem 5.4 in [14], for any T>0 , we have for t\in[\tau,\tau+T]

    \begin{eqnarray} && \left\| {v^{\varepsilon_n} \left( t \right)}-{v^0 \left( t \right)} \right\|_{\mathcal N}^2 \le c \left\| \phi^{\varepsilon_n}-\phi^0 \right\|_{{\mathcal N}}^2 +c\mathop {\max }\limits_{\nu \in \left[ {\tau ,t} \right]} \xi \left( {\theta _\nu \omega } \right) \int_\tau ^t { \left\| {{v^{\varepsilon_n} \left( s \right)}- {v^0 \left( s \right)}} \right\|_{{\mathcal N}}^2 } ds \end{eqnarray}
    \begin{eqnarray} &&\mbox{} + c{\varepsilon_n} \mathop {\max }\limits_{\nu \in \left[ {\tau ,t} \right]} \left\| {{\mathcal T}^{ - 1} \left( {\theta _{\nu} \omega } \right)} \right\| \int_\tau ^t {\left( \left\| {v^{\varepsilon_n}(s) } \right\|_{H_{\varepsilon_n} ^1 \left( {\mathcal O} \right)}^2 + \left\| v^0(s) \right\|_{H ^1 \left( {\mathcal Q} \right)}^2 \right)} ds \\ &&\mbox{} + c{\varepsilon_n} \mathop {\max }\limits_{\nu \in \left[ {\tau ,t} \right]} \left\| {{\mathcal T}^{ - 1} \left( {\theta _{\nu} \omega } \right)} \right\| \int_\tau ^t {\left(\kappa^2_1(s)+\kappa^2_2(s) \right)} ds \\ &&\mbox{}+ c{\varepsilon_n} \int_\tau ^t {\left( {\left\| {v^{\varepsilon_n}(s) } \right\|_{H_{\varepsilon_n} ^1 \left( \mathcal O \right)}^2 + \left\| {v^0(s) } \right\|_{H^1 \left( \mathcal Q \right)}^2 } \right)} ds, \end{eqnarray} (92)

    where \xi(\theta_t\omega) = \beta+|\delta(\theta_t\omega)|. By Lemma 5.1 and Lemma 5.2 we find that there exists a positive constant \varrho = \varrho(\tau,\omega,T) , independent of {\varepsilon_n} , such that for all t\in[\tau,\tau+T] ,

    \begin{eqnarray} \left\| {{v^{\varepsilon_n} \left( t \right)}-{v^0 \left( t \right)}} \right\|_{\mathcal N }^2 & \le& e^{c(1+ \mathop {\max }\limits_{\nu \in \left[ {\tau ,\tau+T} \right]} \xi \left( {\theta _\nu \omega } \right)) T} \left\|\phi^{\varepsilon_n}-\phi^0 \right\|_{\mathcal N }^2 \\ &&\mbox{} +\varrho{\varepsilon_n} e^{c(1+ \mathop {\max }\limits_{\nu \in \left[ {\tau ,\tau+T} \right]} \xi \left( {\theta _\nu \omega } \right)) T} [ \| {\psi ^0 } \|_{\mathcal M}^2+\| {\psi ^{\varepsilon_n} } \|_{\mathcal N}^2 \\ &&\mbox{}+\int_\tau ^{\tau+T} {\left(\kappa^2_1(s)+\kappa^2_2(s) \right)} ds \\ &&\mbox{}+ {\int_{ \tau }^{\tau+T} {( {\| {G( { s,\cdot} )} \|_{L^\infty ( {\widetilde{\mathcal O}} )}^2 + \| {\psi_1 ( { s,\cdot} )} \|_{L^\infty ( {\widetilde{\mathcal O} } )}^2 } ) }ds} ]. \end{eqnarray} (93)

    Notice that, for all t\in[\tau,\tau+T] ,

    \begin{eqnarray} &&\| {u^{\varepsilon_n} ( {t,\tau,\omega ,\phi^\varepsilon } ) - u^0( {t,\tau,\omega ,\phi^0 } )} \|_{\mathcal N}^2 \\ &\leq& \mathop {\max }\limits_{\nu \in [ {\tau ,\tau + T} ]} \| {{\mathcal T}( {\theta _\nu \omega } )} \|^2 \| {v^{\varepsilon_n} ( {t,\tau,\omega ,{ {\mathcal T}}^{ - 1} ( {\theta _\tau \omega } )\phi^\varepsilon } ) - v^0( {t,\tau,\omega ,{ {\mathcal T}}^{ - 1} ( {\theta _\tau \omega } )\phi^0 } }\|_{\mathcal N }^2, \end{eqnarray}

    which together with (93) implies the desired results.

    The next result is concerned with uniform compactness of attractors with respect to \varepsilon .

    Lemma 5.4. Assume that (8)-(11), (39) and (43) hold. If \varepsilon_n\rightarrow 0 and u^{\varepsilon_n} \in \mathcal A_{\varepsilon _n } \left( {\tau ,\omega } \right) , then there exist a subsequence of ({u^{\varepsilon_n} })_{n\in \mathbb{N}} , again denoted by (u^{\varepsilon_n} )_{n\in \mathbb{N}} , and u\in H^1(\mathcal Q) such that

    \mathop {\lim }\limits_{n \to \infty } \left\| {u^{\varepsilon_n} - u} \right\|_{H^1(\mathcal O)} = 0.

    Proof. Take a sequence t_n \to \infty . By the invariance of \mathcal A_{\varepsilon _n } there exists \phi^{\varepsilon_n} \in \mathcal A_{\varepsilon_n } \left( {\tau - t_n ,\theta _{ - t_n } \omega } \right) such that

    \begin{equation} u^{\varepsilon_n} = \Phi _{\varepsilon _n } \left( {t_n ,\tau - t_n ,\theta _{ - t_n } \omega ,\phi^{\varepsilon_n} } \right). \end{equation} (94)

    By Lemma 4.1, we have \phi^{\varepsilon_n} \in K \left( {\tau - t_n ,\theta _{ - t_n } \omega } \right)\in D_1. Since \varepsilon_n\rightarrow 0 and t_n \to \infty , By Lemma 3.5, for any \eta>0 , there exists a large enough N_1\in \mathbb N such that for all n\geq N_1 ,

    \begin{equation} \left\| { {Q_{N_1}^{\varepsilon_n} } u^{\varepsilon_n}\left( {\tau,\tau-t_n,\theta _{ - \tau} \omega , \phi^{\varepsilon_n}} \right)} \right\|_{H^1(\mathcal O)} \le \eta. \end{equation} (95)

    By Lemma 3.2, we have

    \begin{equation} \|{ {P_{N_1}^{\varepsilon_n} } u^{\varepsilon_n}\left( {\tau,\tau-t_n,\theta _{ - \tau} \omega , \phi^{\varepsilon_n}} \right)}\|_{H^1(\mathcal O)} < M. \end{equation} (96)

    It follows from (95) and (96) that (u^{\varepsilon_n}\left( {\tau,\tau-t_n,\theta _{ - \tau} \omega , \phi^{\varepsilon_n}} \right))_{n\in \mathbb{N}} is precompact in H^1(\mathcal O) . Since the estimate (55) holds, there exists u in H^1(\mathcal Q) and a subsequence of ({u^{\varepsilon_n}})_{n\in \mathbb{N}} , again denoted by (u^{\varepsilon_n})_{n\in \mathbb{N}} , such that

    \begin{equation} \mathop {\lim }\limits_{n \to \infty } \left\| {u^{\varepsilon_n} - u } \right\|_{H^1(\mathcal O)} = 0. \end{equation} (97)

    This completes the proof.

    Now we are in a position to prove the main result of this paper.

    Theorem 5.5. Assume that (8)-(11), (39), (43), and (89)-(90) hold. The attractors \mathcal{A}_\varepsilon are upper-semicontinuous at \varepsilon = 0 , that is, for every \tau\in \mathbb R and \omega\in \Omega ,

    \mathop {\lim }\limits_{\varepsilon \to 0} \mathit{\text{dist}}_{H^1(\mathcal O)} \left( {\mathcal{A}_\varepsilon \left(\tau, \omega \right),\mathcal{A}_0 \left(\tau, \omega \right)} \right) = 0.

    Proof. Given \tau\in {\mathbb{R}} and \omega \in \Omega , by the invariance of \mathcal{A}_\varepsilon and (55) we find that there exists \varepsilon_0>0 such that

    \begin{equation} \| u\|^2_{H^1_\varepsilon ( {\mathcal{O}})} \le L(\tau, \omega) \quad \mbox{for all } \ 0 < \varepsilon < \varepsilon_0 \ \ \mbox{and} \ u \in \mathcal{A}_\varepsilon (\tau, \omega), \end{equation} (98)

    where L(\tau, \omega) is the positive constant given by the right-hand side of (55) which is independent of \varepsilon . If the theorem is not true, there exist \delta>0 , a sequence (\varepsilon_n)_{n\in \mathbb{N}} of positives constants, \varepsilon_n\rightarrow 0 , and a sequence (z_n)_{n\in \mathbb{N}} , z_n\in \mathcal{A}_{\varepsilon_n} (\tau,\omega) for all {n\in \mathbb{N}} , such that

    \begin{equation} \text{dist}_{H^1(\mathcal O)} \left( {z_n ,\mathcal{A}_0 \left(\tau, \omega \right)} \right) \ge \delta\quad \text{for all}\quad n\in \mathbb{N}. \end{equation} (99)

    By Lemma 5.4 there exists z_{\ast} in H^1(\mathcal Q) and a subsequence of ({z_n})_{n\in \mathbb{N}} , again denoted by (z_n)_{n\in \mathbb{N}} , such that

    \begin{equation} \mathop {\lim }\limits_{n \to \infty } \left\| {z_n - z_ * } \right\|_{H^1(\mathcal O)} = 0. \end{equation} (100)

    By the invariance property of the attractor \mathcal{A}_{\varepsilon_n} (\tau,\omega) , for every t>0 there exists y_n^t \in \mathcal{A}_{\varepsilon_n} \left( \tau-t,{\theta _{ - t} \omega } \right) such that

    \begin{equation} z_n = \Phi _{\varepsilon _n } \left( {t,\tau-t,\theta _{- t} \omega ,y_n^t } \right). \end{equation} (101)

    By Lemma 5.4 again there exists y_{\ast}^t in H^1(\mathcal Q) and a subsequence of ({y_n^t})_{n\in \mathbb{N}} , again denoted by (y_n^t)_{n\in \mathbb{N}} , such that

    \begin{equation} \mathop {\lim }\limits_{n \to \infty } \left\| {y_n^t - y_*^t } \right\|_{H^1(\mathcal O)} = 0. \end{equation} (102)

    It follows from Theorem 5.3 that for every t>0 ,

    \begin{eqnarray} \mathop {\lim }\limits_{n \to \infty } \Phi _{\varepsilon _n } \left( {t,\tau-t,\theta _{- t} \omega ,y_n^t } \right) = \Phi _0 \left( {t,\tau-t,\theta _{ - t} \omega ,y_ * ^t } \right)\quad \text{in}\quad \mathcal N. \end{eqnarray} (103)

    By (100), (101), (103) and uniqueness of limits we obtain

    \begin{equation} z_ * = \Phi _0 \left( {t,\tau-t,\theta _{ - t} \omega ,y_ * ^t } \right)\quad \text{in}\quad H^1(\mathcal O). \end{equation} (104)

    Notice that \mathcal{A}_{\varepsilon_n} (\tau-t,\theta_{-t}\omega)\subseteq K(\tau-t,\theta_{-t}\omega) and y^t_n\in \mathcal{A}_{\varepsilon_n} (\tau-t,\theta_{-t}\omega) for all n\in \mathbb{N}. Thus by (98) we have

    \begin{equation} \mathop {\lim \sup }\limits_{n \to \infty } \left\| {y_n^t } \right\|_{H^1(\mathcal O)} \le \left\| {K\left( {\tau - t,\theta _{- t} \omega } \right)} \right\|_{H^1(\mathcal O)} \le L\left( {\tau - t,\theta _{ - t} \omega } \right). \end{equation} (105)

    By (102) and (105) we get, for every t>0 ,

    \begin{equation} \left\| {y_ * ^t } \right\|_{H^1(\mathcal Q)} \le L\left( {\tau - t,\theta _{- t} \omega } \right). \end{equation} (106)

    By K_0\in \mathcal D_0 and the attraction property of \mathcal A_0 in \mathcal D_0 , we obtain from (104) and (106) that

    \begin{eqnarray} && \text{dist}_{H^1(\mathcal Q)} \left( {z_ * ,\mathcal A_0 \left( {\tau ,\omega } \right)} \right) \\ & = & \text{dist}_{H^1(\mathcal Q)} \left( {\Phi _0 \left( {t,\tau - t,\theta _{ - t} \omega ,y_ * ^t } \right),\mathcal A_0 \left( {\tau ,\omega } \right)} \right) \\ &\le& \text{dist}_{H^1(\mathcal Q)} \left( {\Phi _0 \left( {t,\tau - t, \theta _{ - t} \omega ,K_0 \left( {\tau - t,\theta _{ - t} \omega } \right)} \right),\mathcal A_0 \left( {\tau ,\omega } \right)} \right)\\ && \to 0,\quad \text{as}\,\,t \to \infty . \end{eqnarray} (107)

    This implies that z_{\ast}\in \mathcal{A}_0(\tau,\omega) since \mathcal{A}_0(\tau,\omega) is compact. Therefore, we have

    \text{dist}_{H^1(\mathcal O)} \left( {z_n ,\mathcal{A}_0 \left( \tau,\omega \right)} \right) \le \text{dist}_{H^1(\mathcal O)} \left( {z_n ,z_ * } \right) \to 0,

    a contradiction with (99). This completes the proof.

    The authors would like to thank the anonymous referee for the useful suggestions and comments.



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