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Research article

Ries impact deposits in the North Alpine Foreland Basin of Germany as a terrestrial analog site for impact-produced seismites and sand spikes on planet Mars

  • Received: 29 November 2024 Revised: 11 February 2025 Accepted: 17 February 2025 Published: 24 February 2025
  • The North Alpine foreland basin (NAFB) in Germany is characterized by various types of sedimentologic features that make it an excellent terrestrial analog of regions affected by high-energy asteroid impact-quakes on Mars. Impact events have shaped all planetary bodies in the inner Solar System over the past >4 Gyr. The well-preserved Ries impact crater (Baden-Württemberg and Bavaria), formed around 14.8 Ma, has recently been linked to an earthquake-produced seismite horizon in Mid-Miocene NAFB sediments that exhibits typical dewatering structures and is associated with sand spikes, seismically produced pin-shaped pseudo-concretions. In this terrestrial setting, the sand spike tails systematically point away from the Ries crater. On its path across Gale Crater, the Mars rover Curiosity seems to have observed a similar seismite horizon in early Hesperian lacustrine deposits including clastic dikes, convolute bedding, and, likely, sand spikes. Their orientation suggests that the nearby Slagnos impact crater might be the seismic source for the formation of those seismites. The Ries impact–seismite deposits can be traced over a distance of more than 200 km from the source crater (northern Switzerland), which makes the NAFB an excellent terrestrial analog for similar deposits and their sedimentologic inventory within Gale Crater's lake deposits on Mars.

    Citation: Elmar Buchner, Volker J Sach, Martin Schmieder. Ries impact deposits in the North Alpine Foreland Basin of Germany as a terrestrial analog site for impact-produced seismites and sand spikes on planet Mars[J]. AIMS Geosciences, 2025, 11(1): 68-90. doi: 10.3934/geosci.2025005

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  • The North Alpine foreland basin (NAFB) in Germany is characterized by various types of sedimentologic features that make it an excellent terrestrial analog of regions affected by high-energy asteroid impact-quakes on Mars. Impact events have shaped all planetary bodies in the inner Solar System over the past >4 Gyr. The well-preserved Ries impact crater (Baden-Württemberg and Bavaria), formed around 14.8 Ma, has recently been linked to an earthquake-produced seismite horizon in Mid-Miocene NAFB sediments that exhibits typical dewatering structures and is associated with sand spikes, seismically produced pin-shaped pseudo-concretions. In this terrestrial setting, the sand spike tails systematically point away from the Ries crater. On its path across Gale Crater, the Mars rover Curiosity seems to have observed a similar seismite horizon in early Hesperian lacustrine deposits including clastic dikes, convolute bedding, and, likely, sand spikes. Their orientation suggests that the nearby Slagnos impact crater might be the seismic source for the formation of those seismites. The Ries impact–seismite deposits can be traced over a distance of more than 200 km from the source crater (northern Switzerland), which makes the NAFB an excellent terrestrial analog for similar deposits and their sedimentologic inventory within Gale Crater's lake deposits on Mars.



    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

    (T1(θtω)Gε(t,y),(vεM)p1+)18α1T(θtω)p2O(vεM)2p2+dx+2α1T(θtω)pvεM|Gε(t,y)|2dy. (62)

    All above estimates yield

    ddt(vεM)+pLp(O)(2p312pα1Mp2T(θtω)p2)O(vεM)p+dx+14pα1T(θtω)p2O(vεM)2p2+dx|δ(θrω)|pO|vε|pdx+2T(θtω)pO|ψ1(t,y,εg(y)yn+1)|p2dy+2pα1T(θtω)pO|Gε(t,y)|2dy. (63)

    Multiplying (63) by et0(2p312pα1Mp2T(θrω)p2)dr, then integrating on (τ1,τ) we obtain

    (vε(τ,τt,ω,ψε)M)+pLp(O)
    +14pα1ττ1T(θζω)p2eζτ(2p312pα1Mp2T(θrω)p2)dr
    ×O(vε(ζ,τt,ω,ψε)M)2p2+dxdζ
    eτ1τ(2p312pα1Mp2T(θrω)p2)dr(vε(τ1,τt,ω,ψε)M)+pLp(O)
    +ττ1|δ(θζω)|peζτ(2p312pα1Mp2T(θrω)p2)drvε(ζ,τt,ω,ψε)pLp(O)dζ
    +2|O|ττ1T(θζω)peζτ+ξ(2p312pα1Mp2T(θrω)p2)drψ1(ζ,)p2L(˜O)dζ.
    +2p|O|α1ττ1T(θζω)peζτ+ξ(2p312pα1Mp2T(θrω)p2)drG(ζ,)2L(˜O)dζ, (64)

    where |O| stands for the Lebesgue measure of O. Replacing ω by θτω in (64) we get

    14pα101T(θζω)p2eζ0(2p312pα1Mp2T(θrω)p2)dr
    ×O(vε(ζ+τ,τt,θτω,ψε)M)2p2+dxdζ
    e10(2p312pα1Mp2T(θrω)p2)dr(vε(τ1,τt,θτω,ψε)M)+pLp(O)
    +01|δ(θζ+ξω)|peζ0(2p312pα1Mp2T(θrω)p2)drvε(ζ+τ,τt,θτω,ψε)pLp(O)dζ
    +2|O|01T(θζω)peζ0(2p312pα1Mp2T(θrω)p2)drψ1(ζ+τ,)p2L(˜O)dζ.
    +2p|O|α101T(θζω)peζ0(2p312pα1Mp2T(θrω)p2)drG(ζ+τ,)2L(˜O)dζ. (65)

    Since ω is continuous on [1,0], there exist c1=c1(ω,p,α1)>0 and c2=c2(ω,p,α1)>0 such that

    c112pα1T(θrω)p2c2 for all  r[ρ1,0]. (66)

    By (66) we obtain

    ec2Mp2ζeζ+ξξ12pα1Mp2T(θrω)p2drec1Mp2ζ for all  ζ[1,0]andξ[ρ,0]. (67)

    For the left-hand side of (65), by (67) we find that there exists c3=c3(ω)>0 such that

    14pα101T(θζω)p2eζ0(2p312pα1Mp2T(θrω)p2)dr
    O(vε(ζ+τ,τt,θτω,ψε)M)2p2+dxdζ
    c301ec2Mp2ζO(vε(ζ+τ,τt,θτω,ψε)M)2p2+dxdζ. (68)

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

    e10(2p312pα1Mp2T(θrω)p2)dr(vε(τ1,τt,θτω,ψε)M)+pLp(O)
    e2p3ec1Mp2(vε(τ1,τt,θτω,ψε)M)+pLp(O)
    e2p3ec1Mp2vε(τ1,τt,θτω,ψε)pLp(O). (69)

    Similarly, for the second terms on the right-hand side of (65), we have from (67) there exists c4=c4(ω)>0 such that

    01|δ(θζω)|peζ0(2p312pα1Mp2T(θrω)p2)drvε(ζ+τ,τt,θτω,ψε)pLp(O)dζ
    c401ec1Mp2ζvε(ζ+τ,τt,θτω,ψε)pLp(O)dζ (70)

    Since φ1Lloc(R,L(˜O)) and GL2loc(R,L(˜O)), for the two three terms on the right-hand side of (65), by (67) we obtain there exists c5=c5(τ,ω)>0 such that

    2|O|01T(θζω)peζ0(2p312pα1Mp2T(θrω)p2)drψ1(ζ+τ,)p2L(˜O)dζ.
    +2p|O|α101T(θζω)peζ0(2p312pα1Mp2T(θrω)p2)drG(ζ+τ,)2L(˜O)dζ
    c501ec1Mp2ζdζc11c5M2p. (71)

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

    c301ec2Mp2ζO(vε(ζ+τ,τt,θτω,ψε)M)2p2+dydζ
    e2p3ec1Mp2vε(τ1,τt,θτω,ψε)pLp(O)
    +c401ec1Mp2ζvε(ζ+τ,τt,θτω,ψε)pLp(O)dζ+c11c5M2p,

    which together with Lemma 3.2 and Lemma 3.3 implies that there exist c6=c6(τ,ω)>0 and T=T(τ,ω,D1)2 such that for all tT,

    c301ec2Mp2ζO(vε(ζ+τ,τt,θτω,ψε)M)2p2+dxdζ
    c6ec1Mp2+c601ec1Mp2ζdζ+c11c5M2pc6ec1Mp2+c11(c5+c6)M2p. (72)

    Since p>2, we find that for every η>0, there exists M0=M0(τ,ω,η)>0 such that for all MM0 and tT,

    01ec2Mp2ζO(vε(ζ+τ,τt,θτω,ψε)M)2p2+dydζη. (73)

    Note that |v|2(vM)+ for v2M, which together with (73) yields that for all MM0 and tT,

    01ec2Mp2ζ{yO: vε(ζ+τ,τt,θτω,ψε)2M}|vε(ζ+τ,τt,θτω,ψε)|2p2dydζ22p201ec2Mp2ζO(vε(ζ+τ,τt,θτω,ψε)M)2p2+dxdζ22p2η. (74)

    Similarly, one can verify that there exist M1=M1(τ,ω,η)>0 and T1=(τ,ω,D)2 such that for all MM1 and tT1,

    01ec2Mp2ζ{yO: vε(ζ+τ,τt,θτω,ψε)2M}|vε(ζ+τ,τt,θτω,ψε)|2p2dydζ22p2η. (75)

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

    Note that Aε is a family of linear operators in Yε, for 0εε0, where Yε=Hg(O), for 0<εε0, and Y0=Hg(Q), is self-adjoint and has a compact resolvent. Then, σ(Aε) consists of only eigenvalues {λεn}n=1 with finite multiplicity:

    0λε1λε2λεn+,

    and their associated eigenfunctions {ϖεn}n=1 form an orthonormal basis of Yε.

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

    Next, we introduce the spectral projections. We use Pεm to denote the projection from Yε onto the eigenspace span{ϖεi}mi=1 given by

    Pεn(u)=mi=1(u,ϖεi)YεϖεiforuYε.

    We use Qεm to denote its orthogonal complement projection, i.e., Pεm+Qεm=Iε, where Iε is the identity operators on Yε. It is clear that

    aε(u,u)=(Aεu,u)Hg(O)λεn(u,u)Hg(O),uPεnD(A1/2ε). (76)

    and

    aε(u,u)=(Aεu,u)Hg(O)λεm+1(u,u)Hg(O),uQεmD(A1/2ε). (77)

    Let uε=uε1+uε2 and vε=vε1+vε2, where uε1=Pεmuε, uε2=Qεmuε, vε1=Pεmvε, and vε2=Qεmvε, respectively.

    Lemma 3.5. Assume that (8)-(11), (39) and (42) hold. Then for every τR, ωΩ, η>0 and D1={D1(τ,ω):τR,ωΩ}D1, there exists T=T(τ,ω,D1,η)2, m=m(τ,ω,D,η)N and 0<ε1=ε1(n)<ε0 such that for all tT, 0<ε<ε1 and ϕεD1(τt,θtω), the solution uε of (19)-(20) with ω replaced by θτω satisfies

    uε2(τ,τt,θτω,ϕε)H1(O)η.

    Proof. Taking the inner product (36) with Aεvε2 in Hg(O), we get

    12ddtaε(vε2,vε2)+Aεvε22(δ(θtω)vε2,Aεvε2)+(QεnT1(θtω)hε(t,y,T(θtω)vε),Aεvε2)+(QεnT1(θtω)Gε(t,y),Aεvε2). (78)

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

    (δ(θtω)vε2,Aεvε2)18Aεvε22+2|δ(θtω)|2vε22. (79)

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

    (QεnT1(θtω)hε(t,y,T(θtω)vε),Aεvε2)18Aεvε22+2T1(θtω)2O|hε(t,y,T(θtω)vε)|2dy18Aεvε22+2α2T1(θtω)2O(|T(θtω)vε|p1+ψ2(t,y,εg(y)yn+1))2dy18Aεvε22+4α2T(θtω)2p4v2p22p2+4α2|O|T1(θtω)2ψ2(t,)2L(˜O). (80)

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

    (QεnT1(θtω)Gε(t,y),Aεvε2)18Aεvε22+2|O|T1(θtω)2G(t,)2L(˜O) (81)

    Noting that Aεvε22λεn+1aε(vε2,vε2), we obtain from all above estimates that

    ddtaε(vε2,vε2)+λεn+1aε(vε2,vε2)4δ2(θtω)vε22+8α2T(θtω)2p4vε2p22p2+cT1(θtω)2(ψ2(t,)2L(˜O)+G(t,)2L(˜O)). (82)

    Taking ξ(τ1,τ), multiplying (82) by eλεn+1t, first integrating with respect to t on (ξ,τ), integrating with respect to ξ on (τ1,τ), and then replacing ω by θτω, we get

    aε(vε2(τ,τt,θτω,ψε),vε2(τ,τt,θτω,ψε))ττ1eλεn+1(rτ)aε(vε2(r,τt,θτω,ψε),vε2(r,τt,θτω,ψε))dr+4δ2ττ1eλεn+1(rτ)δ2(θrτω)aε(vε2(r,τt,θτω,ψε),vε2(r,τt,θτω,ψε))dr+8α2ττ1eλεn+1(rτ)T(θrτω)2p4vε(r,τt,θτω,ψε)2p22p2dr+cττ1eλεn+1(rτ)T1(θrτω)2(ψ2(r,)2L(˜O))dr+cττ1eλεn+1(rτ)T1(θrτω)2G(r,)2L(˜O)dr. (83)

    Since φ2,GL2loc(R,L(˜O)), T(θtω) is continuous on [1,0] and λεn+1 approaches λ0n+1 as ε0, we find that there exists c=c(ω)>0 and 0<ε<ε0 such that for 0<ε<ε,

    aε(vε2(τ,τt,θτω,ψε),vε2(τ,τt,θτω,ψε))c01eλεn+1rvε(r+τ,τt,θτω,ψε)2p22p2dr+c01eλεn+1raε(vε(r+τ,τt,θτω,ψε),vε(r+τ,τt,θτω,ψε))dr
    +c01eλεn+1rdrc01e(λ0n+11)rvε(r+τ,τt,θτω,ψε)2p22p2dr+c01e(λ0n+11)raε(vε(r+τ,τt,θτω,ψε),vε(r+τ+s,τt,θτω,ψε))dr+c01e(λ0n+11)rvε(r+τρ0(r+τ+s),τt,θτω,ψε)2dr+c01e(λ0n+11)rdr. (84)

    Given η>0, let T=T(τ,ω,D1)2,γ=γ(ω)>0, M=M(τ,ω,η)1 and 0<ε1<ε be the constants in Lemma 3.4. Choose N1=N1(τ,ω,η)1 large enough such that λ0n+11γMp2 for all nN1. Then, by Lemma 3.4, we obtain, for all tT, nN1 and 0<ε<ε1,

    c01e(λ0n+11)rvε(r+τ,τt,θτω,ψε)2p22p2drc01e(λ0n+11)r{yO:|vε|2M}|vε(r+τ,τt,θτω,ψε)|2p2dydr+c01e(λ0n+11)r{yO:|vε|<2M}|vε(r+τ,τt,θτω,ψε)|2p2dydrc01eγMp2r{yO:|vε|2M}|vε(r+τ,τt,θτω,ψε)|2p2dydr+c01e(λ0n+11)r{yO:|vε|<2M}|vε(r+τ,τt,θτω,ψε)|2p2dydrη+c22p2M2p2|O|01e(λ0n+11)rdrη+c22p2M2p2|O|1λ0n+11. (85)

    For the last three terms on the right-hand side of (84), by Lemma 3.1, we find that there exist c1=c1(τ,ω)>0 and T1=T1(τ,ω,D1)T such that for all tT1,

    c01e(λ0n+11)raε(vε(r+τ,τt,θτω,ψε),vε(r+τ,τt,θτω,ψε))dr+c01e(λ0n+11)rdrc101e(λ0n+11)rdrc11λ0n+11. (86)

    Since λ0n+1 as n, we obtain from (84)-(86) that there exists N2=N2(τ,ω,η)N1 such that for all nN2, tT1 and 0<ε<ε1,

    aε(vε2(τ+s,τt,θτω,ψε),vε2(τ+s,τt,θτω,ψε))2η,

    which together vε(t)=T1(θtω)uε(t) and (77) completes the proof.

    In this subsection, we establish the existence of D1-pullback attractor for the cocycle Φε 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 Φε associated with problem (19)-(20) has a closed measurable D1-pullback absorbing set K={K(τ,ω):τR,ωΩ}D1.

    Proof. We first notice that, by Lemma 3.2, Φε has a closed D1-pullback absorbing set K in H1(O). More precisely, given τR and ωΩ, let

    K(τ,ω)={uH1(O):u2H1(O)L(τ,ω)}, (87)

    where L(τ,ω) is the constant given by the right-hand side of (55). It is evident that, for each τR, L(τ,):ΩR is (F,B(R))-measurable. In addition, for every τR, ωΩ, and DD1, there exists T=T(τ,ω,D)2 such that for all tT,

    Φε(t,τt,θtω,D(τt,θtω))K(τ,ω).

    Thus we find that K={K(τ,ω):τR,ωΩ} is a closed measurable set and pullback-attracts all elements in D1. By the similar argument as in [15] we can obtain easily from (43) that K={K(τ,ω):τR,ωΩ} is tempered. Consequently, K is a closed measurable D1-pullback absorbing set for Φε in D1.

    Lemma 4.2. Assume that (8)-(11), (39) and (43) hold. Then, the cocycle Φε is D1-pullback asymptotically compact in H1(O); that is, for all τR and ωΩ, {Φε(tn,τtn,θtnω,xn)}n=1 has a convergent subsequence in H1(O) whenever tn and xnD1(τtn,θtnω) with {D1(τ,ω):τR, ωΩ}D1.

    Proof. We will show that for every η>0, the sequence {uε(τ,τtn,θτω,ϕε)}n=1 has a finite open cover of balls with radii less than η. By Lemma 3.5, we infer that there exists N1=N1(τ,ω,D1,η)1, m0=m0(τ,ω,D1,η)N and 0<ε1=ε1(m0)<ε0 such that for all nN1 and 0<ε<ε1,

    uε2(τ,τtn,θτω,ϕε)H1(O)=Qm0uε(τ,τtn,θτω,ϕε)H1(O)<η4. (88)

    On the other hand, by Lemma 3.2 we find that the sequence {Pm0uε(τ,τtn,θτω,ϕε)}n=1 is bounded in the finite-dimensional space Pm0H1(O) and hence is precompact, which together with (88) shows that the sequence uε(τ,τtn,θτω,ϕε) has a finite open cover of balls with radii less than η in H1(O), as desired.

    Theorem 4.3. Assume that (8)-(11), (39) and (43) hold. Then, the cocycle Φε has a unique D1-pullback (N,H1(O))-attractor Aε={Aε(τ,ω):τR,ωΩ}.

    Proof. First, we know from Lemma 4.1 that Φε has a a closed measurable D1-pullback absorbing set K(τ,ω). Second, it follows from Lemma 4.2 that Φε is D1-pullback asymptotically compact from N to H1(O). Hence, the existence of a unique D1-pullback (N,H1(O))-attractor for the cocycle Φε 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 Φ0 has a unique D0-pullback (M, H1(Q))-attractor A0={A0(τ,ω):τR,ωΩ}.

    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<εε0, τR, ωΩ, T>0, and λ1>λ0, the solution vε of (36) satisfies, for all t[τ,τ+T],

    tτvε(r,τ,ω,ψε)2H1ε(O)drcψε2N+cτ+Tτ(G(r,)2L(˜O)+ψ1(r,)2L(˜O))dr,

    where c is a positive constant depending on τ, ω, λ0 and T, but independent of ε.

    Similarly, one can prove

    Lemma 5.2. Assume that (8)-(11) and (39) hold. Then for every τR, ωΩ, T>0, and λ1>λ0, the solution v0 of (37) satisfies, for all t[τ,τ+T],

    tτv0(r,τ,ω,ψ0)2H1(Q)drcψ02M+cτ+Tτ(G(r,)2L(˜O)+ψ1(r,)2L(˜O))dr,

    where c is a positive constant depending on τ, ω, λ0 and T, but independent of ε.

    In the sequel, we further assume the functions G and H satisfy that for all t,sR,

    Gε(t,)G0(t,)L2(O)κ1(t)ε (89)

    and

    Hε(t,,s)H0(t,,s)L2(O)κ2(t)ε, (90)

    where κ1(t),κ2(t)L2loc(R).

    By (12) and (90) we have, for all x˜O and t,sR,

    hε(t,,s)h0(t,,s)L2(O)κ2(t)ε. (91)

    Since M can be embedded naturally into N as the subspace of functions independent of yn+1, we can consider the cocycle Φ0 as a mapping from M into N. Therefore we can compare Φ0 with Φε.

    Theorem 5.3. Suppose (8)-(11), (39), and (89)-(90) hold. Given τR, ωΩ, εn0 and a positive number L(τ,ω), if ϕεnH1εn(O)) such that ϕεnH1εn(O))L(τ,ω), then there exists ϕ0M such that, up to a subsequence, for t0,

    limnΦεn(t,τ,ω,ϕεn)Φ0(t,τ,ω,ϕ0)N=0.

    Proof. Since ϕεnH1εn(O), there exists ϕ0M such that ϕεnϕ0 in N. By the similar proof of that of Theorem 5.4 in [14], for any T>0, we have for t[τ,τ+T]

    vεn(t)v0(t)2Ncϕεnϕ02N+cmaxν[τ,t]ξ(θνω)tτvεn(s)v0(s)2Nds
    +cεnmaxν[τ,t]T1(θνω)tτ(vεn(s)2H1εn(O)+v0(s)2H1(Q))ds+cεnmaxν[τ,t]T1(θνω)tτ(κ21(s)+κ22(s))ds+cεntτ(vεn(s)2H1εn(O)+v0(s)2H1(Q))ds, (92)

    where ξ(θtω)=β+|δ(θtω)|. By Lemma 5.1 and Lemma 5.2 we find that there exists a positive constant ϱ=ϱ(τ,ω,T), independent of εn, such that for all t[τ,τ+T],

    vεn(t)v0(t)2Nec(1+maxν[τ,τ+T]ξ(θνω))Tϕεnϕ02N+ϱεnec(1+maxν[τ,τ+T]ξ(θνω))T[ψ02M+ψεn2N+τ+Tτ(κ21(s)+κ22(s))ds+τ+Tτ(G(s,)2L(˜O)+ψ1(s,)2L(˜O))ds]. (93)

    Notice that, for all t[τ,τ+T],

    uεn(t,τ,ω,ϕε)u0(t,τ,ω,ϕ0)2Nmaxν[τ,τ+T]T(θνω)2vεn(t,τ,ω,T1(θτω)ϕε)v0(t,τ,ω,T1(θτω)ϕ02N,

    which together with (93) implies the desired results.

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

    Lemma 5.4. Assume that (8)-(11), (39) and (43) hold. If εn0 and uεnAεn(τ,ω), then there exist a subsequence of (uεn)nN, again denoted by (uεn)nN, and uH1(Q) such that

    limnuεnuH1(O)=0.

    Proof. Take a sequence tn. By the invariance of Aεn there exists ϕεnAεn(τtn,θtnω) such that

    uεn=Φεn(tn,τtn,θtnω,ϕεn). (94)

    By Lemma 4.1, we have ϕεnK(τtn,θtnω)D1. Since εn0 and tn, By Lemma 3.5, for any η>0, there exists a large enough N1N such that for all nN1,

    QεnN1uεn(τ,τtn,θτω,ϕεn)H1(O)η. (95)

    By Lemma 3.2, we have

    PεnN1uεn(τ,τtn,θτω,ϕεn)H1(O)<M. (96)

    It follows from (95) and (96) that (uεn(τ,τtn,θτω,ϕεn))nN is precompact in H1(O). Since the estimate (55) holds, there exists u in H1(Q) and a subsequence of (uεn)nN, again denoted by (uεn)nN, such that

    limnuεnuH1(O)=0. (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 Aε are upper-semicontinuous at ε=0, that is, for every τR and ωΩ,

    limε0distH1(O)(Aε(τ,ω),A0(τ,ω))=0.

    Proof. Given τR and ωΩ, by the invariance of Aε and (55) we find that there exists ε0>0 such that

    u2H1ε(O)L(τ,ω)for all  0<ε<ε0  and uAε(τ,ω), (98)

    where L(τ,ω) is the positive constant given by the right-hand side of (55) which is independent of ε. If the theorem is not true, there exist δ>0, a sequence (εn)nN of positives constants, εn0, and a sequence (zn)nN, znAεn(τ,ω) for all nN, such that

    distH1(O)(zn,A0(τ,ω))δfor allnN. (99)

    By Lemma 5.4 there exists z in H1(Q) and a subsequence of (zn)nN, again denoted by (zn)nN, such that

    limnznzH1(O)=0. (100)

    By the invariance property of the attractor Aεn(τ,ω), for every t>0 there exists ytnAεn(τt,θtω) such that

    zn=Φεn(t,τt,θtω,ytn). (101)

    By Lemma 5.4 again there exists yt in H1(Q) and a subsequence of (ytn)nN, again denoted by (ytn)nN, such that

    limnytnytH1(O)=0. (102)

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

    limnΦεn(t,τt,θtω,ytn)=Φ0(t,τt,θtω,yt)inN. (103)

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

    z=Φ0(t,τt,θtω,yt)inH1(O). (104)

    Notice that Aεn(τt,θtω)K(τt,θtω) and ytnAεn(τt,θtω) for all nN. Thus by (98) we have

    limsupnytnH1(O)K(τt,θtω)H1(O)L(τt,θtω). (105)

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

    ytH1(Q)L(τt,θtω). (106)

    By K0D0 and the attraction property of A0 in D0, we obtain from (104) and (106) that

    distH1(Q)(z,A0(τ,ω))=distH1(Q)(Φ0(t,τt,θtω,yt),A0(τ,ω))distH1(Q)(Φ0(t,τt,θtω,K0(τt,θtω)),A0(τ,ω))0,ast. (107)

    This implies that zA0(τ,ω) since A0(τ,ω) is compact. Therefore, we have

    distH1(O)(zn,A0(τ,ω))distH1(O)(zn,z)0,

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

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



    [1] Sach VJ, Buchner E, Schmieder M (2020) Enigmatic earthquake-generated large-scale clastic dyke in the Biberach area (SW Germany). Sediment Geol 398: 105571. https://doi.org/10.1016/j.sedgeo.2019.105571 doi: 10.1016/j.sedgeo.2019.105571
    [2] Buchner E, Sach VJ, Schmieder M (2020) New discovery of two seismite horizons challenges the Ries–Steinheim double‑impact theory. Sci Rep 10: 22143. https://doi.org/10.1038/s41598-020-79032-4 doi: 10.1038/s41598-020-79032-4
    [3] Stöffler D, Artemieva NA, Wünnemann K, et al. (2013) Ries crater and suevite revisited—Observations and modeling. Part Ⅰ: Observations. Meteorit Planet Sci 48: 515–589. https://doi.org/10.1111/maps.12086 doi: 10.1111/maps.12086
    [4] Artemieva NA, Wünnemann K, Krien F, et al. (2013) Ries crater and suevite revisited—Observations and modeling. Part Ⅱ: Modeling. Meteorit Planet Sci 48: 590–627. https://doi.org/10.1111/maps.12085 doi: 10.1111/maps.12085
    [5] Schmieder M, Kennedy T, Jourdan F, et al. (2018) A high-precision 40Ar/39Ar age for the Nördlinger Ries impact crater, Germany, and implications for the accurate dating of terrestrial impact events. Geochim Cosmochim Acta 220: 146–157. https://doi.org/10.1016/j.gca.2017.09.036 doi: 10.1016/j.gca.2017.09.036
    [6] Schmieder M, Kennedy T, Jourdan F (2018) Response to comment on "A high-precision 40Ar/39Ar age for the Nördlinger Ries impact crater, Germany, and implications for the accurate dating of terrestrial impact events" by Schmieder et al. (Geochim. Cosmochim. Acta 220 (2018) 146–157). Geochim Cosmochim Acta 238: 602–605. https://doi.org/10.1016/j.gca.2018.07.025 doi: 10.1016/j.gca.2018.07.025
    [7] Sach VJ (1999) Litho- und biostratigraphische Untersuchungen in der Oberen Süßwassermolasse des Landkreises Biberach an der Riß (Oberschwaben). Stuttgarter Beitr Naturk B 276: 1–167.
    [8] Sach VJ (2014) Strahlenkalke (Shatter-Cones) aus dem Brockhorizont der Oberen Süßwassermolasse in Oberschwaben (Südwestdeutschland)—Fernauswürflinge des Nördlinger-Ries-Impaktes, Pfeil Verlag, München, 1–17.
    [9] Hofmann B, Hofmann F (1992) An impactite horizon in the upper freshwater molasse in Eastern Switzerland: Distal Ries ejecta. Eclogae Geol Helv 85: 788–789.
    [10] Letsch D (2017) Diamictites and soft sediment deformation related to the Ries (ca. 14.9 Ma) meteorite impact: the "Blockhorizont" of Bernhardzell (Eastern Switzerland). Int J Earth Sci 107: 1379–1380. https://doi.org/10.1007/s00531-017-1542-1 doi: 10.1007/s00531-017-1542-1
    [11] Holm-Alwmark S, Alwmark C, Ferrière L, et al. (2021) Shocked quartz in distal ejecta from the Ries impact event (Germany) found at ~ 180 km distance, near Bernhardzell, eastern Switzerland. Sci Rep 11: 7438. https://doi.org/10.1038/s41598-021-86685-2 doi: 10.1038/s41598-021-86685-2
    [12] Buchner E, Sach VJ, Schmieder M (2022) Event- and biostratigraphic evidence for two independent Ries and Steinheim asteroid impacts in the Middle Miocene. Sci Rep 12: 18603. https://doi.org/10.1038/s41598-022-21409-8 doi: 10.1038/s41598-022-21409-8
    [13] Collins G, Melosh HJ, Marcus R (2005) Earth impact effects program: a web-based computer program for calculating the regional environmental consequences of a meteoroid impact on Earth. Meteorit Planet Sci 40: 817–840. https://doi.org/10.1111/j.1945-5100.2005.tb00157.x doi: 10.1111/j.1945-5100.2005.tb00157.x
    [14] Schmieder M, Sach VJ, Buchner E (2021) The Chöpfi pinnacles near Winterthur, Switzerland: Long-distance effects of the Ries impact-earthquake? Int J Earth Sci 111: 145–147. https://doi.org/10.1007/s00531-021-02082-0 doi: 10.1007/s00531-021-02082-0
    [15] Buchner E, Sach VJ, Schmieder M (2021) Sand spikes pinpoint powerful palaeoseismicity. Nat Commun 12: 6731. https://doi.org/10.1038/s41467-021-27061-6 doi: 10.1038/s41467-021-27061-6
    [16] Maurer H, Buchner E (2007) Rekonstruktion fluvialer Systeme der Oberen Süßwassermolasse im Nordalpinen Vorlandbecken SW-Deutschlands. German J Geosci (ZdGG) 158: 249–270. https://doi.org/10.1127/1860-1804/2007/0158-0249 doi: 10.1127/1860-1804/2007/0158-0249
    [17] Heider J, Wegele A, Amstutz GC (1976) Beobachtungen über Sandrosen und Zapfensande aus der Süßwassermolasse Südwürttembergs. Der Aufschluß 27: 297–307.
    [18] Sanborn WB (1976) Oddities of the Mineral World. Van Nostrand Reinhold Company, NY, USA. 142. Available from: http://allanmccollum.net/amcimages/sanborn.html, (last accessed February 19, 2025).
    [19] Akçiz SO, Grant Ludwig L, Arrowsmith JR, et al. (2010) Century-long average time intervals between earthquake ruptures of the San Andreas fault in the Carrizo Plain, California. Geology 38: 787–790. https://doi.org/10.1130/G30995.1 doi: 10.1130/G30995.1
    [20] McBride EF, Picard MD, Folk RL (1994) Orientated Concretions, Ionian Coast, Italy: Evidence of Groundwater flow direction. J Sediment Res A64: 535–540. https://doi.org/10.1306/D4267DFC-2B26-11D7-8648000102C1865D doi: 10.1306/D4267DFC-2B26-11D7-8648000102C1865D
    [21] McCullough LN, Ritter JB, Zaleha MJ, et al. (2003) Habit, formation, and implications of elongeate, calcite concretions, Victoria, Australia. Department of Geology, Wittenberg University, Ohio, USA. Published Senior Honors Thesis. 25.
    [22] Grant JA, Wilson SA (2018) Possible Geomorphic and Crater Density Evidence for Late Aqueous Activity in Gale Crater. LPI Contrib. 49th Annual Lunar and Planetary Science Conference.
    [23] Metz J, Grotzinger J, Okubo C, et al. (2010) Thin‐skinned deformation of sedimentary rocks in Valles Marineris, Mars. J Geophys Res Planets 115: E11004. https://doi.org/10.1029/2010JE003593 doi: 10.1029/2010JE003593
    [24] NASA Mars Science Laboratory, Curiosity Rover, 2024. Available from: https://mars.nasa.gov/msl/mission/science/.
    [25] Wray JJ (2013) Gale Crater: The Mars Science Laboratory/Curiosity rover landing site. Int J Astrobiol 12: 25–38. https://doi.org/10.1017/S1473550412000328 doi: 10.1017/S1473550412000328
    [26] Grotzinger JP, Sumner DY, Kah LC, et al. (2014) A habitable fluvio-lacustrine environment at Yellowknife Bay, Gale Crater, Mars. Science 343: 6169. https://doi.org/10.1126/science.1242777 doi: 10.1126/science.1242777
    [27] Grotzinger JP, Gupta S, Malin MC, et al. (2015) Deposition, exhumation, and paleoclimate of an ancient lake deposit, Gale Crater, Mars. Science 350: 6257. https://doi.org/10.1126/science.aac7575 doi: 10.1126/science.aac7575
    [28] Buz J, Ehlmann BL, Pan L, et al. (2017) Mineralogy and stratigraphy of the Gale crater rim, wall, and floor units. J Geophys Res Planets 122: 1090–1118. https://doi.org/10.1002/2016JE005163 doi: 10.1002/2016JE005163
    [29] Schwenzer SP, Abramov O, Allen CC, et al. (2012) Gale Crater: Formation and post-impact hydrous environments. Planet Space Sci 70: 84–95. https://doi.org/10.1016/j.pss.2012.05.014 doi: 10.1016/j.pss.2012.05.014
    [30] Montenat C, Barrier P, Ott d'Estevou P, et al. (2007) Seismites: An attempt at critical analysis and classification. Sediment Geol 196: 5–30. https://doi.org/10.1016/j.sedgeo.2006.08.004 doi: 10.1016/j.sedgeo.2006.08.004
    [31] Hargitai H, Levi T (2015) Clastic dikes, Encyclopedia of Planetary Landforms, Hargitai H and Kereszturi A, Eds., Encyclopedia of Planetary Landforms, Springer, NY, USA.
    [32] Sleep NH, Olds EP (2018) Remote faulting triggered by strong seismic waves from the Cretaceous-Paleogene asteroid impact. Seismol Res Lett 89: 570–576. https://doi.org/10.1785/0220170223 doi: 10.1785/0220170223
    [33] DePalma RA, Smit J, Burnham DA, et al. (2019) A seismically induced onshore surge deposit at the K-Pg. boundary, North Dakota. PNAS 116: 8190–8199. https://doi.org/10.1073/pnas.1817407116 doi: 10.1073/pnas.1817407116
    [34] Vaniman DT, Bish DL, Ming DW, et al. (2014) Mineralogy of a mudstone at Yellowknife Bay, Gale Crater, Mars. Science 343: 6169. https://doi.org/10.1126/science.1243480 doi: 10.1126/science.1243480
    [35] Ehlmann BL, Buz J (2015) Mineralogy and fluvial history of the watersheds of Gale, Knobel, and Sharp craters: A regional context for MSL Curiosity's exploration. Geophys Res Lett 42: 264–273. https://doi.org/10.1002/2014GL062553 doi: 10.1002/2014GL062553
    [36] Carter J, Viviano-Beck C, Loizeau D, et al. (2015) Orbital detection and implications of akaganéite on Mars. Icarus 253: 296–310. https://doi.org/10.1016/j.icarus.2015.01.020 doi: 10.1016/j.icarus.2015.01.020
    [37] Tohver E, Schmieder M, Lana C, et al. (2018) End-Permian impactogenic earthquake and tsunami deposits in the intracratonic Paraná Basin of Brazil. GSA Bull 130: 1099–1120. https://doi.org/10.1130/B31626.1 doi: 10.1130/B31626.1
    [38] Weatherley DK, Henley RW (2013) Flash vaporization during earthquakes evidenced by gold deposits. Nature Geosci 6: 294–298. https://doi.org/10.1038/ngeo1759 doi: 10.1038/ngeo1759
    [39] Simms JM (2003) Uniquely extensive seismite from the latest Triassic of the United Kingdom: evidence for bolide impact? Geology 31: 557–560. https://doi.org/10.1130/0091-7613(2003)031<0557:UESFTL>2.0.CO;2 doi: 10.1130/0091-7613(2003)031<0557:UESFTL>2.0.CO;2
    [40] Banham SG, Gupta S, Rubin DM, et al. (2018) Ancient Martian aeolian processes and palaeomorphology reconstructed from the Stimson formation on the lower slope of Aeolis Mons, Gale crater, Mars. Sedimentology 65: 993–1042. https://doi.org/10.1111/sed.12469 doi: 10.1111/sed.12469
    [41] Bohacs KM, Carrol AR, Neal JE (2003) Lessons from large lake systems—Thresholds, nonlinearity, and strange attractors. Special Papers-Geological Society of America, 75–90. https://doi.org/10.1130/0-8137-2370-1.75 doi: 10.1130/0-8137-2370-1.75
    [42] Brož P, Oehler D, Mazzini A, et al. (2023) An overview of sedimentary volcanism on Mars. Earth Surf Dynam 11: 633–661. https://doi.org/10.5194/egusphere-2022-1458 doi: 10.5194/egusphere-2022-1458
    [43] Sturm S, Wulf G, Jung D, et al. (2013) The Ries impact, a double-layer rampart crater on Earth. Geology 41: 531–534. https://doi.org/10.1130/G33934.1 doi: 10.1130/G33934.1
    [44] Wilson SA, Morgan AM, Howard AD, et al. (2021) The global distribution of craters with alluvial fans and deltas on Mars. Geophys Res Lett 48: e2020GL091653. https://doi.org/10.1029/2020GL091653 doi: 10.1029/2020GL091653
    [45] Grotzinger JP, Crisp J, Vasavada AR, et al. (2012) Mars Science Laboratory mission and science investigation. Space Sci Rev 170: 5–56. https://doi.org/10.1007/s11214-012-9892-2 doi: 10.1007/s11214-012-9892-2
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