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Research article Special Issues

Multiple datasets to explore the tumor microenvironment of cutaneous squamous cell carcinoma

  • Received: 12 February 2022 Revised: 17 March 2022 Accepted: 18 March 2022 Published: 08 April 2022
  • Background 

    Cutaneous squamous cell carcinoma (cSCC) is one of the most frequent types of cutaneous cancer. The composition and heterogeneity of the tumor microenvironment significantly impact patient prognosis and the ability to practice precision therapy. However, no research has been conducted to examine the design of the tumor microenvironment and its interactions with cSCC.

    Material and Methods 

    We retrieved the datasets GSE42677 and GSE45164 from the GEO public database, integrated them, and analyzed them using the SVA method. We then screened the core genes using the WGCNA network and LASSO regression and checked the model's stability using the ROC curve. Finally, we performed enrichment and correlation analyses on the core genes.

    Results 

    We identified four genes as core cSCC genes: DTYMK, CDCA8, PTTG1 and MAD2L1, and discovered that RORA, RORB and RORC were the primary regulators in the gene set. The GO semantic similarity analysis results indicated that CDCA8 and PTTG1 were the two most essential genes among the four core genes. The results of correlation analysis demonstrated that PTTG1 and HLA-DMA, CDCA8 and HLA-DQB2 were significantly correlated.

    Conclusions 

    Examining the expression levels of four primary genes in cSCC aids in our understanding of the disease's pathophysiology. Additionally, the core genes were found to be highly related with immune regulatory genes, suggesting novel avenues for cSCC prevention and treatment.

    Citation: Jiahua Xing, Muzi Chen, Yan Han. Multiple datasets to explore the tumor microenvironment of cutaneous squamous cell carcinoma[J]. Mathematical Biosciences and Engineering, 2022, 19(6): 5905-5924. doi: 10.3934/mbe.2022276

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  • Background 

    Cutaneous squamous cell carcinoma (cSCC) is one of the most frequent types of cutaneous cancer. The composition and heterogeneity of the tumor microenvironment significantly impact patient prognosis and the ability to practice precision therapy. However, no research has been conducted to examine the design of the tumor microenvironment and its interactions with cSCC.

    Material and Methods 

    We retrieved the datasets GSE42677 and GSE45164 from the GEO public database, integrated them, and analyzed them using the SVA method. We then screened the core genes using the WGCNA network and LASSO regression and checked the model's stability using the ROC curve. Finally, we performed enrichment and correlation analyses on the core genes.

    Results 

    We identified four genes as core cSCC genes: DTYMK, CDCA8, PTTG1 and MAD2L1, and discovered that RORA, RORB and RORC were the primary regulators in the gene set. The GO semantic similarity analysis results indicated that CDCA8 and PTTG1 were the two most essential genes among the four core genes. The results of correlation analysis demonstrated that PTTG1 and HLA-DMA, CDCA8 and HLA-DQB2 were significantly correlated.

    Conclusions 

    Examining the expression levels of four primary genes in cSCC aids in our understanding of the disease's pathophysiology. Additionally, the core genes were found to be highly related with immune regulatory genes, suggesting novel avenues for cSCC prevention and treatment.



    The delay effect originates from the boundary controllers in engineering. The dynamics of a system with boundary delay could be described mathematically by a differential equation with delay term subject to boundary value condition such as [20]. There are many results available in literatures on the well-posedness and pullback dynamics of fluid flow models with delays especially the 2D Navier-Stokes equations, which can be seen in [1], [2], [8] and references therein. Inspired by these works, in this paper, we study the stability of pullback attractors for 3D Brinkman-Forchheimer (BF) equation with delay, which is also a continuation of our previous work in [6]. The existence and structure of attractors are significant to understand the large time behavior of solutions for non-autonomous evolutionary equations. Furthermore, the asymptotic stability of trajectories inside invariant sets determines many important properties of trajectories. The 3D Brinkman-Forchheimer equation with delay is given below:

    {utνΔu+αu+β|u|u+γ|u|2u+p=f(t,ut)+g(x,t),u=0, u(t,x)|Ω=0,u|t=τ=uτ(x), xΩ,uτ(θ,x)=u(τ+θ,x)=ϕ(θ), θ(h,0), h>0. (1)

    Here, (x,t)Ω×R+ with ΩR3 be a bounded domain with sufficiently smooth boundary Ω. u=(u1,u2,u3) is the velocity vector field, p is the pressure, ν>0 and α>0 denotes the Brinkman kinematic viscosity and the Darcy coefficients respectively, β>0 and γ>0 are the Forchheimer coefficients, the external force g(x,t)L2loc(R;H) is a locally square integrable function and the delay term is considered either

    (1). a general delay f(t,ut) with ut:[h,0]H defined as ut=u(t+s) which denotes constant, variable and distributed delays, see Caraballo and Real [1], [2].

    or

    (2). the special application of f(t,ut) as a sub-linear operator

    f(t,ut)=F(u(tρ(t))) (2)

    for a smooth function ρ() defined in Section 4, which satisfies subadditive and positive homogeneous property with second variable component, see Marín-Rubio and Real [8].

    The BF equation describes the conservation law of fluid flow in a porous medium that obeys the Darcy's law. The physical background of 3D BF model can be seen in [14], [9], [18], [19]. For the dynamic systems of problem (1) without delay, i.e., f(t,ut)=0, we can refer to [6], [10], [15], [16], [21] and literature therein for the existence of global weak solution and attractors. For the problem (1) with delay f(t,ut), the global existence of mild solution, continuous dependence on initial data and the minimal family of pullback attractors have been obtained in [6]. In addition, the upper semi-continuous property of pullback attractors as delay vanishes has been proved by virtue of some regular estimates. Furthermore, as a special application of the delay f(t,ut), the pullback dynamics of problem (1) with sub-linear operator (2) has also been shown. However, the asymptotic stability of trajectories inside pullback attractors is still open. Motivated by [3] and [17], applying regularity for weak solution and iteration technique with variable indices, we present some sufficient conditions with the generalized Grashof number to achieve the stability of pullback attractors in this paper. The main features and results can be summarized as follows.

    (a) For problem (1) with delay f(t,ut), we use the regular estimate to achieve an upper bound of Grashof number, which implies the exponential stability of trajectories inside pullback attractors. The proof does not depend on initial data with more regularity, see Section 3. Here we use, the delay f(t,ut) in problem (1) can be the constant, variable and distributed delays F(u(th)), F(u(tρ(t))) and 0hk(t,s)u(t+s)ds respectively, here F() is an appropriate function, see [1], [2].

    (b) For problem (1) with special application of f(t,ut) as the sub-linear operator, the variable indices have been introduced to deal with nonlinear term β|u|u+γ|u|2u and sub-linear operator by iterative argument, see Section 4.

    (c) The asymptotic stability of trajectories inside pullback attractors is further research of the results established in [6]. However, the stability of pullback attractors for (1) with infinite delay is still unknown.

    In this section, we give some notations and the equivalent abstract form of (1) in this section.

    Denoting E:={u|u(C0(Ω))3,divu=0}, H is the closure of E in (L2(Ω))3 topology, H and (,) denote the norm and inner product in H respectively. V is the closure of set E in (H1(Ω))3 topology, V and ((,)) denote the norm and inner product in V respectively. H and V are dual spaces of H and V respectively. Clearly, VHHV, H and V are dual spaces of H and V respectively, where the injection is dense, continuous. The norm denotes the norm in V, , be the dual product in V and V. Let P be the Helmholz-Leray orthogonal projection from (L2(Ω))3 onto the space H, we define A:=PΔ as the Stokes operator with domain D(A)=(H2(Ω))3V and λ is the first eigenvalue of A, the sequence {ωj}j=1 is an orthonormal system of eigenfunctions of A, and {λj}j=1 (0<λ1λ2) are eigenvalues of A corresponding to the eigenfunctions {ωj}j=1, see more details in [13].

    By the Helmholz-Leray projection defined above, (1) can be transformed to the abstract equivalent form

    {ut+νAu+P(αu+β|u|u+γ|u|2u)=Pf(t,ut)+Pg(t,x),u|Ω=0,u|t=τ=uτ(x),uτ(θ,x)=ϕ(θ,x) for θ(h,0), (3)

    then we show our results for (3) with f(t,ut) as either general case or its special case F(u(tρ(t))) in Sections 3 and 4, respectively.

    We also define some Banach spaces on delayed interval as CH=C([h,0];H), CV=C([h,0];V) with the norms

    ϕCH=supθ[h,0]ϕ(θ)H,  ϕCV=supθ[h,0]ϕ(θ)V,

    respectively. The Lebesgue integrable spaces on delayed interval can be denoted as LpH=Lp(h,0;H), LpV=Lp(h,0;V). The product space is defined as XH=C([τh,T];H)×C([τ,T];H) and MH=H×(CHL2V) for purpose of phase space in next sections.

    Some assumptions on the external forces and parameters which will be imposed in our main results are the following:

    (Hf) The function f:R×CHH satisfies:

    (a) For any ξCH, the function f(,ξ) is measurable and f(,0)0.

    (b) There exists a Lf>0 such that

    f(t,ξ)f(t,η)HLfξηCH, for ξ,ηCH.

    (c) There exists Cf>0, such that for all u,vC([τh,t];H)

    tτf(r,ur)f(r,vr)2HdrC2ftτhu(r)v(r)2Hdr, for τt. (4)

    (Hg) The function g(,)L2loc(R,V) satisfies that there exists η>0 such that

    teηsg(s,)2Vds<. (5)

    holds for any tτ.

    (H0) The coefficients satisfy α+3β+2γη2C2fαeηh0.

    Lemma 3.1. (The Gronwall inequality with differential form) Let m()C1[R+,R+], v(), h()C[R+,R+] and

    ddtm(t)v(t)m(t)+h(t), m(t=τ)=mτ, tτ. (6)

    Then

    m(t)mτetτv(s)ds+tτh(s)etsv(σ)dσds, tτ. (7)

    In this part, we shall present some retarded integral inequalities from Li, Liu and Ju [5]. Consider the following retarded integral inequalities:

    y(t)XE(t,τ)yτX+tτK1(t,s)ysXds+tK2(t,s)ysXds+ρ,  tτ, (8)

    where E, K1 and K2 are non-negative measurable functions on R2, ρ0 denotes a constant. Let X be a Banach space with spatial variable, based on the retarded Banach space above, then we use CX denotes the norm of space C([h,0];X) for some h0, y(t)0 is a continuous function defined on C([h,T];X), yt(s)=y(t+s) for s[h,0].

    Let L(E,K1,K2,ρ)={yC([h,T];X)|y0 and satisfies the inequality (8)}, and

    κ(K1,K2)=suptτ(tτK1(t,s)ds+tK2(t,s)ds).

    We assume that

    limt+E(t+s,s)=0 (9)

    uniformly with respect to sR+. Moreover, we suppose that κ(K1,K2)<+.

    Lemma 3.2. (The retarded Gronwall inequality) Denoting ϑ=suptsτE(t,s) and κ=κ(K1,K2), then we have the following estimates:

    (1) If κ<1, then for any R, ε>0, there exists ˜T>0 such that

    ytX<μρ+ε, (10)

    for t>˜T and all bounded functions yL(E,K1,K2,ρ) with y0XR, where μ=11κ.

    (2) If κ<11+ϑ, then there exist M, λ>0 which are independent on ρ such that

    ytXMy0Xeλt+γρ,  tτ (11)

    for all bounded functions yL(E,K1,K2,ρ), where γ=μ+11κc and c=max{ϑ1κ,1}.

    (3) If κ<11+ϑ, then the solution reduces to trivial for the occasion κc<1.

    Proof. See Li, Liu and Ju [5].

    Remark 1. (The special case: K2=0) Denote (K1,K2)=(K1,0) and let ϑ, κ, μ, γ be the constants defined in Lemma 3.2. Then we have the similar estimates as in Lemma 3.2.

    The minimal family of pullback attractors will be stated here in preparation for our main result.

    Some inequalities

    Lemma 3.3. (1) (See [7], [11]) Assume that β2, then for any a,bRn, we have

    (|a|β2a|b|β2b)(ab)γ0|ab|β,

    where γ0>0 is a constant which is determined by the volume of domain and its dimension, such as minγ0=126β2 in a 3-dimensional smooth domain.

    (2) The following Cq-inequality holds

    |xqyq|Cq(|x|q1+|y|q1)|xy|

    for the integer q2.

    Well-posedness

    Theorem 3.4. Assume that the external forces g(t,x) and f(t,ut) satisfy the hypothesis (Hg) and (Hf), the initial data (uτ,ϕ)MH=H×(CHL2V) and (H0) are also true. Then there exists a unique global weak solution u=u(,τ,uτ,ϕ)C([τh,T];H)L2(τ,T;V)L4(τ,T;L4(Ω)) of equation (1) on [τh,T].

    Proof. Step 1. Existence of local approximate solution.

    By the property of the Stokes operator A, the sequence of eigenfunctions {wi, i=1,2,} of the Stokes operator is an orthonormal complete basis of H formed by elements of V(H2(Ω))3 such that

    Awi=λiwi, i=1,2,. (12)

    Let Hm=span{w1,w2,,wm}, Pm:HHm be a projection, then the approximate solutions can be written as um(t)=mj=1hjm(t)wj (where hjm(t)=(vm(t),wj) is to be determined) which solve the problem

    {(tum,wj)+ν(um,wj)+(αum+β|um|um+γ|um|2um,wj)=(f(t,umt),wj)+g,wj,um(τ)=Pmuτ=uτm,umτ(θ,x)=Pmϕ(θ)=ϕm(θ) for θ[h,0], (13)

    Then it is easy to check that (13) is equivalent to an ordinary differential equations with unknown variable function hjm(t). By the Cauchy-Peano Theorem of ordinary differential equation, the problem (13) possesses a local solution over the time interval [0,tm].

    Step 2. Uniform estimates of approximate solutions.

    Multiplying (13) by hjm(t), and then summing from j=1 to m, it yields

    12ddtum2H+νum2V+αum2H+βum3L3(Ω)+γum4L4(Ω)|(g(t)+f(s,umt(s)),um)|αum2H+ν2um2V+12νg(t)2V+14αf(t,umt)2H. (14)

    Integrating in time, using the hypotheses on f(,) and g(t,x), by the Young inequality, we get

    um2H+νtτum2Vds+2βtτum3L3(Ω)ds+2γtτum4L4(Ω)dsuτ2H+C2f4α0hϕ(s)2Hds+12νtτg(s)2Vds+C2f4αtτum2Hds. (15)

    Using the Gronwall Lemma of integrable form, we conclude that

    {um}  is bounded in the spaceL(τ,T;H)L2(τh,T;V)L3(τ,T;L3(Ω))L4(τ,T;L4(Ω)).

    Step 3. Compact argument and passing to limit for deriving the global weak solutions.

    In this step, we shall prove {um} has a strong convergence subsequence by the Aubin-Lions Lemma along with the uniformly bounded estimate of dumdt in L2(0,T;V). By the estimates of um above step and continuous embedding VLp(Ω) with p[1,6] for three dimension, we can obtained that |um|umL2(τ,T;H) and |um|2umL2(τ,T;H). From the equation

    dumdt=νAumαumβ|um|umγ|um|2um+P(g(t)+f(t,umt) (16)

    and assumptions (Hf) and (Hg), we can see that {dum/dt} is bounded in L2(τ,T;V).

    By virtue of the Aubin-Lions Lemma, we obtain that {um} has a strong convergent subsequence (also denoted as {um} without confusion) with uL2(τh,T;V) and du/dtL2(τ,T;V) such that

    {um(t)u(t) weakly * in L(τ,T;H),um(t)u(t) stongly in L2(τ,T;H),um(t)u(t) weakly in L2(τ,T;V),dum/dtdu/dt weakly in L2(τ,T;V),f(,um)f(,u) weakly in L2(τ,T;H),umu(t) weakly in L3(τ,T;L3(Ω)),umu(t) weakly in L4(τ,T;L4(Ω)) (17)

    which coincides with the initial data um(τ)=Pmuτu(τ)=uτ and ϕm(s)ϕ(s).

    For the purpose of passing to limit in (13), denoting v=umu, we point out that we can deal with the nonlinear terms as the following novelty. Since wj is an eigenfunction of Stokes operator, we claim that

    Tτ(β|um|umβ|u|u,wj)dsCλ1βum4L4(τ,T;L4(Ω))umu4L4(τ,T;L4(Ω))+CβumuL(τ,T;H)u2L2(τh,T;H)

    and

    Tτ(γ|um|2umγ|u|2u,wj)dsCγum2L2(τ,T;V)umu4L4(τ,T;L4(Ω))+Cγumu4L4(τ,T;L4(Ω))(u2L2(τh,T;V)+um4L4(τ,T;L4(Ω))) (18)

    and the convergence of delayed external force f(t,umt) can be verified by the hypotheses.

    Thus, passing to the limit of (13), we conclude that u is at least one of global weak solutions for problem (1).

    The regularity

    Proposition 1. Assume that the external forces g(t) and f(t,ut) satisfy the hypothesis (Hg) and (Hf), the initial data (uτ,ϕ)MH=H×(CHL2V) and (H0) are also true. Then the global weak solution u in Theorem 3.4 has the regular boundedness in L(τ,T;V).

    Proof. Taking inner product of (3) with Au, it yields

    12ddtA1/2u2H+νAu2H+αA1/2u2H+βΩ|u|uAudx+γΩ|u|2uAudx=(f(t,ut),Au)+(g(t),Au). (19)

    According to Lemma 3.3, the nonlinear terms have the following estimates

    |β(|u|u,Au)|ν2Au2H+β4νu4L4 (20)

    and

    γΩ|u|2uAudx=γ2Ω|(|u|2)|2dx+γΩ|u|2|u|2dx (21)

    and

    (f(t,ut),Au)+(g(t),Au)12νf(t,ut)2H+12νg(t)2H+ν2Au2H, (22)

    hence, we conclude that

    ddtA1/2u2H+2αA1/2u2H+γΩ|(|u|2)|2dx+2γΩ|u|2|u|2dxβ2νu4L4+1νf(t,ut)2H+1νg(t)2H. (23)

    Letting t1st, neglecting the third and fourth terms on the left hand side of (23), integrating (23) with time variable from s to t, it yields

    A1/2u(t)2H+2αtsA1/2u(r)2HdrA1/2u(s)2H+β2νtsu(r)4L4dr+2νtsf(r,ur)2Hdr+2νtsg(r)2Hdr (24)

    and

    tsf(r,ur)2HdrL2fϕ(θ)2L2H+L2ftsu(r)2Hdr. (25)

    Then integrating with s from t1 to t, using the uniform boundedness of u in Theorem 3.4, we deduce that

    A1/2u(t)2Htt1A1/2u(s)2Hds+β2νtt1u(r)4L4dr+2L2fνϕ(θ)2L2H+2L2fνtτu(r)2Hdr+2νtt1g(r)2HdrC[ϕ2L2H+uτ2H]+Ctτg2Hds+2L2fνλ1tτu(r)2Vdr, (26)

    which means the uniform boundedness of the global weak solution u in L(τ,T;V). The proof has been finished.

    Uniqueness

    Proposition 2. Assume the hypotheses in Theorem 3.4 hold. Then the global weak solution u is unique.

    Proof. Using the same energy estimates as above, we can deduce the uniqueness easily, here we skip the details.

    To description of pullback attractors, the functional space MH=H×(CHL2V) is used as our phase space equipped with the norm (ξ,ζ)MH=ξH+ζL2V+ζCH for (ξ,ζ)MH. Based on the well-posedness, we shall verify the pullback dissipation and asymptotic compactness for the process to achieve the existence of pullback attractors, which also needs the following assumption:

    (H1) For every uL2(τh,T;V), there exists a η(0,νλ1) which is independent on u such that

    tτeηsf(s,us)2Hds<C2ftτheηsu(s)2Hds. (27)

    for any tT.

    The continuous process

    Proposition 3. For given f:R×CHH and gL2loc(R;V) satisfying (Hf),(Hg), (H1) and (H0). Then, the solution of problem (1) generates a biparametric family of mappings U(t,τ):MHMH by U(t,τ)(uτ,ϕ)=(u(t),ut), which is a continuous process.

    Pullback dissipation

    Lemma 3.5. Assume that f:R×CHH and gL2loc(R;V) satisfying (Hf),(Hg), (H1) and (H0). Then, for any (uτ,ϕ)MH, the solution u of (1) satisfies the estimates

    u(t)2He8ηCfα(tτ)(uτ2H+Cfϕ(r)2L2H)+e8ηCfαtνηλ1tτeηrg(r)2Vdr (28)

    and

    νtsu(r)2Vdru(s)2H+8Cfαus2L2H+1νtsg(r)2Vdr+8Cfαtsu(r)2Hdr, (29)
    βtsu(r)3L3(Ω)dru(s)2H+8Cfαus2L2H+1νtsg(r)2Vdr+8Cfαtsu(r)2Hdr, (30)
    γtsu(r)4L4(Ω)dru(s)2H+8Cfαus2L2H+1νtsg(r)2Vdr+8Cfαtsu(r)2Hdr. (31)

    Proof. By the energy estimate of (1) and using Young's inequality, we arrive at

    ddtu2H+2νu2V+2αu2H+2βu3L3(Ω)+2γu2L4(Ω)1νηλ1g2V+(νηλ1)u2V+2αu2H+8αf(t,ut)2H, (32)

    where η(0,νλ1).

    Multiplying the above inequality by eηt, we obtain

    ddt(eηtu2H)+eηtνλ1u2H+2βeηtu3L3(Ω)+2γeηtu2L4(Ω)1νηλ1eηtg2V+8Cfαeηtf(t,ut)2H.

    Thus integrating with respect to time variable, it yields

    eηtu2H+νλ1tτeηru(r)2Hdreητ(uτ2H+Cf0hϕ(r)2Hdr)+1νηλ1tτeηrg(r)2Vdr+8Cfαtτeηru(r)2Hdr (33)

    and by the Gronwall Lemma, we can derive the estimate in our theorem.

    Using the energy estimate of (1) again, we can check that

    ddtu2H+2νu2V+2αu2H+2βu3L3(Ω)+2γu2L4(Ω)1νg2V+νu2V+2αu2H+8αf(t,ut)2H, (34)

    Integrating from s to t, using the estimate of u in H, we can derive the desired result.

    Based on Lemma 3.5, we can present the pullback dissipation based on the following universes for the tempered dynamics.

    Definition 3.6. (Universe). (1) We will denote by DMHη the class of all families of nonempty subsets ˆD={D(t):tR}P(MH) such that

    limτ(eητsup(ξ,ζ)D(τ)(ξ,ζ)2MH)=0. (35)

    (2) DMHF denotes the class of families ˆD={D(t)=D:tR} with D a fixed nonempty bounded subset in MH.

    Remark 2. The universes DMHη and DMHη satisfy include closed property.

    Proposition 4. (The DMHη and DMHF pullback absorbing sets in MH) For given f:R×CHH and gL2loc(R;V) satisfying (Hf),(Hg), (H1) and (H0) holds. Then, the family ˆD0={D0(t):tR}MH is defined by

    D0(t)=¯BH(0,ρH(t))×(¯BL2V(0,ρL2H(t))¯BCH(0,ρCH(t)))

    is the pullback DMHη-absorbing set for the process U(t,τ) on MH and ˆD0DMHη, where the balls is defined as centered in the point zero and measured by the radius

    ρ2H(t)=1+e8ηCfα(th)νηλ1teηrg(r)2Vdr,ρ2L2V(t)=1ν[1+uτ2H+8Cfαϕ2L2H+g(r)2L2(th,t;V)ν+8Cfhαρ2H(t)].

    Moreover, the pullback DMHF-absorbing set can be defined as the same technique.

    Proof. Using the estimates in Lemma 3.5, choosing any ˆDDMHη(t), there exists a pullback time τ(ˆD,t)th such that

    u(t,τ;uτ,ϕ)2Hρ2H(t)=1+e8ηCfα(th)νηλ1teηrg(r)2Vdr (36)

    holds for any ττ(ˆD,t) and (uτ,ϕ)D(τ). Moreover, in particular, it yields that ut2CHρ2H(t). By the similar technique and estimate in Lemma 3.5, we derive that ut2L2Vρ2L2V(t). Combining the above estimate and the definition of universe, we conclude that ˆD0DMHη. The proof has been finished.

    Pullback asymptotic compactness

    Theorem 3.7. Assume that f:R×CHH and gL2loc(R;H) satisfying (Hf),(Hg), (H1) and (H0) holds. Then, the processes U(t,τ):MHMH generated by the solution of problem (1) is DMHη-pullback asymptotically compact.

    Proof. Step 1. Weak convergence of the sequence {un(t,x)} in the interval [th,t] for arbitrary tτ and weak convergence of {un(t)} in H.

    For arbitrary fixed tτ, consider a family ˆDDMHη, let {τn}(,t] with τn and {(uτn,ϕn)} with (uτn,ϕn)D(τn) be two sequences for all n, then we denote {(un,unt)}ˆD as a sequence with un()=u(;τn,uτn,ϕn).

    By using the similar energy estimate in Theorem 3.4 and technique in Proposition 4, there exists a pullback time τ(ˆD,t)t3h1, such that the sequence {un} with ττ(ˆD,t) is bounded in L(t3h1,t;H)L2(t2h1,t;V)L3(t2h1,t;L3(Ω))L4(t2h1,t;L4(Ω)). From the equation, we can check that

    (un)L2(th1,t;V)νunL2(th1,t;V)+αλ11unL2(th1,t;V)+βunL4(th1,t;L4(Ω))+Cλ1,|Ω|γunL2(th1,t;V)+Cαf(t,unt)L2(th1,t;H)+CνgL2(th1,t;V). (37)

    From the hypotheses (Hf), f(t,unt) is bounded in L2(th1,t;H), which implies {(un)} is bounded in L2(th1,t;V). Hence, by the Aubin-Lions Lemma and the diagonal procedure, there exists a subsequence (relabeled also as {un}) such that un(t)u(t) strongly in L2(th1,t;H). Combining the uniform boundedness of sequence above, it yields that

    {unu weakly * in L(t3h1,t;H),unu weakly in L2(t2h1,t;V),(un)u weakly in L2(th1,t;V),umu(t) weakly in L3(t2h1,t;L3(Ω)),umu(t) weakly in L4(t2h1,t;L4(Ω)),unu stongly in L2(th1,t;H),un(s)u(s) stongly in H, a.e. s(th1,t). (38)

    By Theorem 3.4, from the hypothesis on f, it follows that

    f(,un)f(,u) weakly in L2(th1,t;H). (39)

    Thus, from (38) and (39), we can conclude that uC([th1,t];H) is a weak solution for problem (1) with the initial data of u(,x) at the initial time th1 denoted as u_{t-h-1} .

    From the uniform bounded estimate of u^n by Proposition 1 in L^{\infty}(t-h-1,t;V) and (u^n)' is uniform bounded in L^2(t-h-1,t;V') , using the Aubin-Lions-Simon Lemma (see [12]), we can derive that

    \begin{eqnarray} u^n\rightarrow u\ \mbox{strongly in}\ C([t-h-1,t];H). \end{eqnarray} (40)

    Therefore, we can conclude that

    \begin{eqnarray} u^n(s_n) \rightharpoonup u(s)\ \ \mbox{weakly in } H \end{eqnarray} (41)

    for any \{s_n\}\subset [t-h-1,t] , s_n\rightarrow s\in [t-h-1,t] , which implies

    \begin{eqnarray} \liminf\limits_{n\rightarrow\infty}\|u^{n}(s_n)\|_{H}\geq \|u(s)\|_{H}. \end{eqnarray} (42)

    Step 2. The strong convergence of corresponding sequences via energy equation method: u^n(s_n)\rightarrow u(s) strongly in C([t-h,t];H) .

    The asymptotic compactness of sequence u^n in H will be presented in sequel, i.e.,

    \begin{eqnarray} \|u^{n}(s_n)-u(s)\|_{H}\rightarrow 0\ \mbox{as}\ n\rightarrow+\infty, \end{eqnarray} (43)

    which is equivalent to prove (42) combining with

    \begin{eqnarray} \limsup\limits_{n\rightarrow\infty}\|u^{n}(s_n)\|_H\leq \|u(s)\|_H \end{eqnarray} (44)

    for a sequence \{s_n\}\subset [t-h,t] and s_n\rightarrow s as n\rightarrow +\infty , which will be proved next.

    Using the energy estimate to all u^n and u , we obtain that for all t-h-1\leq s_1\leq s_2\leq t ,

    \begin{eqnarray} &&\|u^n(s_2)\|^2_H+\nu \int^{s_2}_{s_1}\|u^n(r)\|^2_Vdr+2\beta \int^{s_2}_{s_1}\|u^n(r)\|^3_{\mathbb{L}^4(\Omega)}dr+2\gamma \int^{s_2}_{s_1}\|u^n(r)\|^4_{\mathbb{L}^4(\Omega)}\\ &\leq& \frac{2C_f^2}{\alpha}\int^{s_2}_{s_2}\|u^n_r\|^2_Hdr+\frac{8}{\nu}\int^{s_2}_{s_1}\|g(r)\|^2_{V'}dr \end{eqnarray} (45)

    and

    \begin{eqnarray} &&\|u(s_2)\|^2_H+\nu \int^{s_2}_{s_1}\|u(r)\|^2_Vdr+2\beta \int^{s_2}_{s_1}\|u(r)\|^3_{\mathbb{L}^4(\Omega)}dr+2\gamma \int^{s_2}_{s_1}\|u(r)\|^4_{\mathbb{L}^4(\Omega)}\\ &\leq& \frac{2C_f^2}{\alpha}\int^{s_2}_{s_2}\|u_r\|^2_Hdr+\frac{8}{\nu}\int^{s_2}_{s_1}\|g(r)\|^2_{V'}dr. \end{eqnarray} (46)

    Then, we define the functionals J_n(s) and J(s) defined for s\in [t-h-1,t] as following

    \begin{eqnarray} J_n(s)& = &\frac{1}{2}\|u^n\|^2_H-\int^{s}_{t-h-1}\langle g(r),u^n(r)\rangle dr-\int^{s}_{t-h-1}(f(r,u^n_r),u^n(r))dr \end{eqnarray} (47)

    and

    \begin{eqnarray} J(t)& = &\frac{1}{2}\|u(s)\|^2_H-\int^{s}_{t-h-1}\langle g(r),u(r)\rangle dr-\int^{s}_{t-h-1}(f(r,u_r),u(r))dr. \end{eqnarray} (48)

    Combining the convergence in (38), observing that J_n(s) and J(s) are continuous and non-increasing in [t-h-1,t] , we derive that

    \begin{eqnarray} &&\int^t_{t-h-1}\langle g(r),u^n(r)\rangle dr\rightarrow 2\int^t_{t-h-1}\langle g(r),u(r)\rangle dr \end{eqnarray} (49)

    and

    \begin{eqnarray} &&\int^t_{t-h-1}(f(r,u^n_r),u^n(r))dr\rightarrow 2\int^t_{t-h-1}(f(r,u_r),u(r))dr \end{eqnarray} (50)

    as n\rightarrow +\infty , which implies that

    \begin{eqnarray} J_n(s)\rightarrow J(s)\ \ \mbox{a.e.} s\in (t-h-1,t), \end{eqnarray} (51)

    i.e., for \forall\ \varepsilon>0 , there exists a n_k\in\mathbb{N} , for all n\geq n_k and s_k\in [t-h-1,t] , such that

    \begin{eqnarray} |J_n(s_k)-J(s_k)|\leq \frac{\varepsilon}{2}. \end{eqnarray} (52)

    Since J(s) is continuous and J_n(s) is uniformly continuous with respect to time s , then for any \varepsilon>0 , there exists \tilde{n}_k\in\mathbb{N} such that for the sequence \{s_k\}\subset [t-h-1,t] with s_k\rightarrow s for all n\geq \tilde{n}_k ,

    \begin{eqnarray} |J(s_{k})-J(s)|\leq \frac{\varepsilon}{2}, \end{eqnarray} (53)

    Choosing \bar{n}_k = \max\{n_k,\tilde{n}_k\} , then for all n>\bar{n}_k , it yields that

    \begin{eqnarray} |J_n(s_n)-J(s)|\leq |J_n(s_n)-J(s_n)|+|J(s_n)-J(s)| < \varepsilon. \end{eqnarray} (54)

    Therefore, for any \{s_n\}\subset [t-h-1,t] , we have

    \begin{eqnarray} \limsup\limits_{n\rightarrow\infty}J_n(s_n)\leq J(s), \end{eqnarray} (55)

    which implies

    \begin{eqnarray} \limsup\limits_{n\rightarrow \infty}\|u^n(s_n)\|_H \leq \|u(s)\|_H. \end{eqnarray} (56)

    we conclude the strong convergence u^n(s_n)\rightarrow u(s) in C([t-h,t];H) .

    Step 3. The strong convergence: u^n(s_n)\rightarrow u(s) strongly in L^2(t-h,t;V) .

    Combining the energy estimates in (45) and (46), noting the energy functionals J_n(\cdot) and J(\cdot) , using the convergence in (38), we can deduce the norm convergence

    \begin{eqnarray} \|u^n(s)\|_{L^2(t-h,t;V)}\rightarrow \|u(s)\|_{L^2(t-h,t;V)}. \end{eqnarray} (57)

    Hence jointing with the weak convergence in (38), we can derive that u^n(s_n)\rightarrow u(s) strongly in L^2(t-h,t;V) .

    Step 4. The \mathcal{D}^{M_H}_{\eta} -pullback asymptotic compactness.

    By using the results from Steps 2 to 4 and noting the definition of universe, we can conclude that the processes is \mathcal{D}^{M_H}_{\eta} -pullback asymptotic compact in M_H , which means the proof has been finished.

    Remark 3. Using the similar technique, we can derive the processes U(t,\tau): M_H\rightarrow M_H generated by the solution of problem (1) is \mathcal{D}^{M_H}_{F} -pullback asymptotic compact.

    Theorem 3.8. Assume that f:\mathbb{R}\times C_H\rightarrow H and g\in L^2_{loc}(\mathbb{R};H) satisfying (H_{f}),(H_{g}) , (H_1) and (H_0) holds. Then, the process U(t,\tau): M_H\rightarrow M_H generated by the solution of problem (1) possess the minimal pullback attractors \mathcal{A}_{\mathcal{D}^{M_H}_{\eta}}(t) and \mathcal{A}_{\mathcal{D}^{M_H}_{F}}(t) in M_H , which satisfy the following relation

    \begin{eqnarray} \mathcal{A}_{\mathcal{D}^{M_H}_{F}}(t) \subset \mathcal{A}_{\mathcal{D}^{M_H}_{\eta}}(t). \end{eqnarray} (58)

    Proof. From Proposition 3, we observe that the process U(t,\tau) is continuous in M_H . The \mathcal{D}^{M_H}_{F} and \mathcal{D}^{M_H}_{\eta} pullback absorbing sets are established by Proposition 4. By Theorems 3.7 and Remark 3 give the \mathcal{D}^{M_H}_{\eta} and \mathcal{D}^{M_H}_{F} pullback asymptotic compactness of the processes. Using the existence theory of pullback attractors in [3] or [4], we can conclude our desired results.

    Based on the universes defined in Definition 3.6, the relation between \mathcal{A}_{\mathcal{D}^{M_H}_{F}}(t) \mathcal{A}_{\mathcal{D}^{M_H}_{\eta}}(t) holds easily.

    Definition 3.9. The pullback attractors is asymptotically stable if the trajectories inside attractor reduces to a single orbit as \tau\rightarrow -\infty .

    Theorem 3.10. Assume that 2\nu\lambda_1-\frac{L^2_f}{\alpha}>0 , the external forces g\in L^2_{loc}(\mathbb{R};H) and f(t,u_t) satisfy the hypothesis (H_{f}) , (H_{g}) and (H_1) , the initial data (\phi,u_{\tau})\in M_H and (H_0) holds. Then the trajectories inside pullback attractors \mathcal{A}_{\mathcal{D}^{M_H}_{\eta}}(t) is asymptotically stable if

    \mathit{\mbox{G}}(t)\leq K_0,

    where \mathit{\mbox{G}}^2(t) = \frac{\langle\|g\|^2_{H}\rangle|_{\leq t}}{\nu^2\lambda_1} is a generalized Grashof number for the fluid flow, and

    \begin{equation} K_0 = \Big\{[\nu^2\lambda_1(2\nu\lambda_1+\alpha)]\Big/\Big[4C_{|\Omega|}\beta \Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\Big]\Big\}^{1/2},\nonumber \end{equation}

    here C_{|\Omega|}>0 is a constant which depends on the volume of \Omega .

    Proof. Let u(t) and v(t) be two weak solutions of problem (3) with delay f(t,u_t) which subject to initial data

    \begin{eqnarray} u(\tau+\theta)|_{\theta\in [-h,0]} = \phi(\theta),\ \ \ u|_{t = \tau} = u_{\tau} \end{eqnarray} (59)

    and

    \begin{eqnarray} v(\tau+\theta)|_{\theta\in [-h,0]} = \tilde{\phi}(\theta), \ \ v|_{t = \tau} = \tilde{u}_{\tau} \end{eqnarray} (60)

    respectively. Denoting

    \begin{eqnarray} (u,u_t) = U(t,\tau)(u_{\tau},\varphi)\ \ \mbox{and}\ \ (v,v_t) = U(t,\tau)(\tilde{u}_{\tau},\tilde{\varphi}) \end{eqnarray} (61)

    as two trajectories inside the pullback attractors, letting w = u(t)-v(t) and w_t = u_t-v_t , then it is easy to check that w satisfies the following problem

    \begin{equation} \begin{cases} \frac{\partial w}{\partial t}+\nu A w +P\Big(\alpha w+\beta(|u|u-|v|v)+\gamma (|u|^2u-|v|^2v)\Big)&\\ \quad = P(f(t, u_t)-f(t,v_t)),& \\ w|_{\partial\Omega} = 0,& \\ w(t = \tau) = u_{\tau}-\tilde{u}_{\tau},&\\ w(\tau+\theta) = \phi(\theta)-\tilde{\phi}(\theta),\ \theta\in [-h,0].& \end{cases} \end{equation} (62)

    Taking inner product of (62) with w in H , using Poincaré's inequality and Lemma 3.3, it follows

    \begin{eqnarray} \gamma(|u|^2u-|v|^2v, u-v)\geq \gamma \gamma_0 \|u-v\|^4_{\bf{L}^4} \end{eqnarray} (63)

    and

    \begin{eqnarray} &&\frac{1}{2}\frac{d}{d t}\|w\|^2_H+\nu\|w\|^2_V+\alpha \|w\|_H^2+\gamma \gamma_0 \|w\|^4_{\bf{L}^4}\\ &\leq& \Big|\beta(|u|u-|v|v,w)\Big|+\Big|(f(t,u_t)-f(t,v_t),w)\Big|\\ &\leq&\beta\Big(\int_{\Omega}|u|^2|w|dx+\int_{\Omega}|w||v|^2dx\Big)+\frac{\alpha}{2}\|w\|^2_H+\frac{L_f^2}{2\alpha}\|w_t\|^2_H \end{eqnarray}
    \begin{eqnarray} &\leq& \beta(\|u\|^2_{\bf{L}^4}+\|v\|^2_{\bf{L}^4})\|w\|^2_{H}+\frac{\alpha}{2}\|w\|^2_H+\frac{L_f^2}{2\alpha}\|w_t\|^2_H\\ &\leq& C_{|\Omega|}\beta(\|u\|^2_{V}+\|v\|^2_{V})\|w\|^2_{H}+\frac{\alpha}{2}\|w\|^2_H+\frac{L_f^2}{2\alpha}\|w_t\|^2_H. \end{eqnarray} (64)

    Using the Poincaré inequality and Lemma 3.1, noting that if

    \begin{eqnarray} 2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V) > 0, \end{eqnarray} (65)

    then we can obtain

    \begin{eqnarray} \|w\|^2_H&\leq& e^{\int^t_{\tau}[2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)-(2\nu\lambda_1+\alpha)]ds}\Big[\|u_{\tau}-\tilde{u}_{\tau}\|^2_H+\\ &&+\frac{L_f^2}{\alpha}\int^t_{\tau}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma}\|w_t\|^2_Hds\Big]. \end{eqnarray} (66)

    Denoting

    \begin{eqnarray} E(t,\tau) = e^{-\int^t_{\tau}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]ds} \end{eqnarray} (67)

    and

    \begin{eqnarray} K_1(t,s) = \frac{L_f^2}{\alpha}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma} \end{eqnarray} (68)

    and

    \begin{eqnarray} \Theta = \sup\limits_{t\geq s\geq \tau}E(t,s),\ \ \ \ \kappa(K_1,0) = \sup\limits_{t\geq\tau}\int^t_{\tau}K_1(t,s)ds, \end{eqnarray} (69)

    by virtue of Lemma 3.2, choosing \kappa(K_1,0)<\frac{1}{1+\Theta} , then there exists M>0 and \lambda>0 , such that we can obtain the estimate

    \begin{eqnarray} \|w_t\|^2_H&\leq& M\|u_{\tau}-\tilde{u}_{\tau}\|^2_H e^{-\lambda (t-\tau)}. \end{eqnarray} (70)

    Substituting (70) into (64), using Lemma 3.1 again, we can conclude the following estimate

    \begin{eqnarray} \|w\|^2_H &\leq& \|u_{\tau}-\tilde{u}_{\tau}\|^2_He^{-\int^t_{\tau}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]ds}\\ &&+\frac{L_f^2}{\alpha}M\|u_{\tau}-\tilde{u}_{\tau}\|^2_H e^{-\lambda (t-\tau)}\int^t_{\tau}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma}ds.\\ \end{eqnarray} (71)

    From (70) and (71), if we fixed u_{\tau} and \tilde{u}_{\tau} and let \tau\rightarrow -\infty , then we can conclude that the trajectories inside pullback attractors reduce to a point, which implies the pullback attractors is asymptotically stable provided that

    \begin{equation} 2\nu\lambda_1+\alpha > 2C_{|\Omega|}\beta \langle \|u\|^2_V+\|v\|^2_V\rangle_{\leq t}, \end{equation} (72)

    where

    \begin{equation} \langle h \rangle_{\leq t} = \limsup\limits_{\tau\rightarrow -\infty}\frac{1}{t-\tau}\int^t_{\tau}h(r)dr. \end{equation} (73)

    Since u and v are the global weak solutions for problem (3), we will use some delicate estimates to make (72) more explicit next. Multiplying (3) with u and integrating by parts over \Omega , we have

    \begin{eqnarray} &&\frac{1}{2}\frac{d}{dt}\|u\|^2_H+\nu\|A^{1/2}u\|^2_H+\alpha\|u\|^2_H+\beta \|u\|^3_{\bf{L}^3}+\gamma \|u\|^4_{\bf{L}^4}\\ &\leq&\alpha \|u\|^2_H+\frac{1}{2\alpha}\Big[\|f(t,u_t)\|^2_H+\|g\|^2_{H}\Big]\\ &\leq&\alpha \|u\|^2_H+\frac{L_f^2}{2\alpha}\|u_t\|^2_H+\frac{1}{2\alpha}\|g\|^2_{H}. \end{eqnarray} (74)

    Using the Poincaré inequality and Lemma 3.1, then we can obtain

    \begin{eqnarray} \|u\|^2_H&\leq& e^{-2\nu\lambda_1(t-\tau)}\|u_{\tau}\|^2_H+\\ &&+\frac{L_f^2}{\alpha}\int^t_{\tau}e^{-2\nu\lambda_1(t-s)}\|u_s\|^2_Hds+\frac{1}{\alpha}\int^t_{\tau}e^{-2\nu\lambda_1(t-s)}\|g\|^2_Hds. \end{eqnarray} (75)

    Denoting

    \begin{eqnarray} E(t,\tau) = e^{-2\nu\lambda_1(t-\tau)} \end{eqnarray} (76)

    and

    \begin{eqnarray} K_1(t,s) = \frac{L_f^2}{\alpha}e^{-2\nu\lambda_1(t-s)} \end{eqnarray} (77)

    and

    \begin{equation} \rho = \frac{1}{\alpha}\int^t_{\tau}e^{-2\nu\lambda_1(t-s)}\|g\|^2_Hds, \end{equation} (78)

    letting

    \begin{eqnarray} \Theta = \sup\limits_{t\geq s\geq \tau}E(t,s),\ \ \ \ \kappa(K_1,0) = \sup\limits_{t\geq\tau}\int^t_{\tau}K_1(t,s)ds, \end{eqnarray} (79)

    by virtue of Lemma 3.2, choosing \kappa(K_1,0)<\frac{1}{1+\Theta} , then there exists \hat{M}>0 and \hat{\lambda}>0 , such that we can obtain the estimate

    \begin{eqnarray} \|u_t\|^2_H&\leq& \hat{M}\|u_{\tau}\|^2_H e^{-\lambda (t-\tau)}+\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}\int^t_{\tau}e^{-2\nu\lambda_1(t-s)}\|g\|^2_Hds\\ &\leq& \hat{M}\|u_{\tau}\|^2_H e^{-\lambda (t-\tau)}+\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}\int^t_{\tau}\|g\|^2_Hds. \end{eqnarray} (80)

    Substituting (80) into (75), using Lemma 3.1 again, we can conclude the following estimate

    \begin{eqnarray} \|u\|^2_H &\leq& C\|u_{\tau}\|^2_He^{-\lambda (t-\tau)}+\Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\int^t_{\tau}\|g\|^2_Hds. \end{eqnarray} (81)

    Integrating (74) from \tau to t , it follows

    \begin{eqnarray} &&\|u\|^2_H+2\nu\int^t_{\tau}\|u\|^2_Vds+2\beta\int^t_{\tau} \|u\|^3_{\bf{L}^3}ds+2\gamma \int^t_{\tau}\|u\|^4_{\bf{L}^4}ds\\ &\leq& \Big[\frac{1}{\alpha}\|\phi\|^2_{L^2_H}+\|u_{\tau}\|^2_H\Big]+\frac{L_f^2}{\alpha}\int^{t}_{\tau}\|u_t(s)\|^2_Hds+\frac{1}{\alpha}\int^t_{\tau}\|g\|^2_{H}ds. \end{eqnarray} (82)

    By the estimate of (80) and (81), we derive

    \begin{eqnarray} \int^t_{\tau}\|u(r)\|^4_{\bf{L}^4}dr\leq C\Big[\frac{1}{\alpha}\|\phi\|^2_{L^2_H}+\|u_{\tau}\|^2_H\Big]+\Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\int^t_{\tau}\|g\|^2_{H}ds \end{eqnarray} (83)

    and

    \begin{eqnarray} \int^t_{\tau}\|u(r)\|^2_{V}dr\leq C\Big[\frac{1}{\alpha}\|\phi\|^2_{L^2_H}+\|u_{\tau}\|^2_H\Big]+\Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\int^t_{\tau}\|g\|^2_{H}ds. \end{eqnarray} (84)

    Combining (72), (73) with (84), we conclude that

    \begin{eqnarray} && \langle \|u\|^2_V+\|v\|^2_V\rangle|_{\leq t}\leq 2 \Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\langle \|g\|^2_H\rangle|_{\leq t} \end{eqnarray} (85)

    and hence the asymptotic stability holds provided that

    \begin{eqnarray} 4C_{|\Omega|}\beta \Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\langle \|g\|^2_H\rangle|_{\leq t} \leq 2\nu\lambda_1+\alpha. \end{eqnarray} (86)

    If we define the generalized Grashof number as G(t) = \Big(\frac{\langle\|g\|^2_{H}\rangle|_{\leq t}}{\nu^2\lambda_1}\Big)^{1/2} , then a sufficient condition for the asymptotic stability of trajectories inside pullback attractors can be conclude as

    \begin{eqnarray} G(t)\leq \Big\{(2\nu\lambda_1+\alpha)\Big/\Big[4C_{|\Omega|}\nu^2\beta\lambda_1 \Big(\frac{L_f^2}{\alpha}\frac{2-\frac{L^2_f}{\alpha}}{1-\frac{2L^2_f}{\alpha}}+\frac{1}{\alpha}\Big)\Big]\Big\}^{1/2} = K_0, \end{eqnarray} (87)

    which completes the proof for our first result.

    Remark 4. Theorem 3.10 is a further research for the existence of pullback attractor in [6].

    We first state some hypothesis on the external forces and sub-linear operator.

    \bf{(H_{f})} The function f(t,u_t) = F(u(t-\rho(t))) satisfies the following assumptions.

    \bf{(H_{f})}-\bf{(a)} The function \rho\in C^{1}([0,+\infty);[0,h]) , and there exists a constant \rho^{\ast} satisfying

    \Big|\frac{d\rho}{dt}\Big|\leq\rho^{\ast} < 1, \ \ \forall t\geq 0.

    \bf{(H_{f})}-\bf{(b)} The external force f(\cdot,y):[\tau,+\infty)\times H\rightarrow H is measurable for all y\in H and f(\cdot,0) = 0 . In addition, there exist functions a,b: [\tau,+\infty)\rightarrow[0,+\infty) , with a\in L^{q}_{loc}(\mathbb{R}) and b\in L^{1}_{loc}(\mathbb{R}) for all T\geq\tau and 1\leq q\leq +\infty with {\limsup_{\tau\rightarrow -\infty}}\int^t_{\tau}b(s)ds = \tilde{b}_0\in (0,+\infty) , such that

    \begin{eqnarray} \|F(y)\|^2_H\leq a(t)\|y\|^2_H+b(t), \ \ \forall t\geq\tau, y\in H. \end{eqnarray} (88)

    \bf{(H_{f})}-\bf{(c)} There exist \kappa\in L^{\infty}(\tau,T) , L(R)>0 and R>0 , such that

    \begin{eqnarray} \|F(u)-F(v)\|_H\leq L(R)\kappa^\frac{1}{2}(t)\|u-v\|_H, \ u,v\in H. \end{eqnarray} (89)

    holds for \|u\|_H\leq R, \|v\|_H\leq R and t\in [\tau,T] .

    \bf{(H_{g})} The function g(\cdot,\cdot)\in L^{2}_{loc}(\mathbb{R},H) , which satisfies that there exists a m>0 , such that

    \begin{eqnarray} \int^{t}_{-\infty}e^{ms}\|g(s,\cdot)\|^{2}_Hds < \infty, \ \ \forall t\in\mathbb{R}. \end{eqnarray} (90)

    \bf{(\widetilde{H}_{0})} When a\in L^{q}_{loc}(\mathbb{R}) , it holds

    \begin{eqnarray} \frac{\nu}{2}-\frac{\|a\|_{L^q_{loc}(\mathbb{R})}}{1-\rho^\ast} > 0. \end{eqnarray} (91)

    In this part, the well-posedness and pullback attractors for problem (1) with sub-linear operator will be stated for our discussion in sequel.

    \bullet Well-posedness

    Assume that the initial date u_{\tau}\in H and \phi\in C_H\cap L^{2q'}(-h,0;H) with \frac{1}{q}+\frac{1}{q'} = 1 and recall that (1) with sub-linear operator has the following abstract form:

    \begin{equation} \begin{cases} u(t)+\int^t_\tau P(\nu Au+\alpha u+\beta|u|u+\gamma |u|^2u)ds &\\ \quad = u(\tau) +\int^t_\tau P\Big(F\big(u(s-\rho(s))\big)+g(s,x)\Big)ds,& \\ w|_{\partial\Omega} = 0,& \\ u(t = \tau) = u_{\tau},&\\ u(\tau+t) = \phi(t),\ t\in [-h,0],& \end{cases} \end{equation} (92)

    which possesses a global mild solution as the following theorem.

    Theorem 4.1. Assume that the external forces g(t) and f(t,u_t) = F(u(t-\rho(t))) satisfy the hypothesis \bf{(H_{f})} and \bf{(H_{g})} , the initial data u_{\tau}\in H and \phi\in C_H\cap L^{2q'}_{H} and \bf{(\widetilde{H}_{0})} are also true. Then there exists a unique global mild solution u = u(\cdot,\tau,u_{\tau},\phi) \in L^{\infty}(\tau-h,T;H)\cap L^{2}(\tau,T;V)\cap L^{4}(\tau,T;\bf{L}^{4}(\Omega)) of problem (1) for special case of f(t,u_t) = F(t-\rho(t)) , such that it satisfies (92) in distributed sense and the following energy equality

    \begin{eqnarray} &&\|u(t)\|^{2}_H+2\nu\int^t_{\tau}\|u(s)\|^{2}_Vds+2\alpha \int^t_{\tau}\|u(s)\|^{2}_Hds\\ &&+2\beta\int^t_{\tau}\|u(s)\|^{3}_{\bf{L}^3}ds+2\gamma\int^t_{\tau}\|u(s)\|^{4}_{\bf{L}^4}ds\\ & = &\|u_{\tau}\|^{2}_H+2\int^t_{\tau}\Big[\big(F(u(s-\rho(s))),u(s)\big)+2(g(s,x),u(s))\Big]ds. \end{eqnarray} (93)

    Moreover, we can define a continuous process \{U(t,\tau)|t\geq\tau,\tau\in\mathbb{R}\} as </italic> <italic> U(t,\tau): (C_H\cap L^{2q'}_H)\times H\rightarrow C_H\times H , i.e., U(t,\tau)(\phi,u_{\tau}) = (u_t,u) .

    Proof. Using the Galerkin method and compact argument as in Section 3.3, we can easily derive the result.

    \bullet The pullback dynamics

    After obtaining the existence of the global well-posedness, we establish the existence of the pullback attractors to (1) with sub-linear operator.

    Theorem 4.2. (The pullback attractors in H ) Assume that \bf{(H_{f})} and \bf{(H_{g})} hold, the initial data u_{\tau}\in H and \phi\in C_H\cap L^{2q'}_{H} and \bf{(\widetilde{H}_{0})} are also true. then the process U(t,\tau) associated to problem (1) with f(t,u_t) = F(u(t-\rho(t))) has two families of pullback attractors \mathcal{A}_{C_H\times H}(t) similar as in Theorem 3.7.

    Proof. Using the similar technique as in Section3.3, we can obtain the existence of pullback attractors, here we skip the details.

    Theorem 4.3. We assume that the external forces g(t) and f(t,u_t) = F(u(t-\rho(t))) satisfy the hypothesis \bf{(H_{f})} and \bf{(H_{g})} , the initial data (\phi,u_{\tau})\in (C_H\cap L^{2q'}_H)\times H and \bf{(\widetilde{H}_{0})} holds true.

    Then the trajectories inside pullback attractors \mathcal{A}_{C_H} is asymptotically stable if

    \begin{equation} \mathit{\mbox{G}}(t)\leq \tilde{K}_0, \end{equation} (94)

    where \mathit{\mbox{G}}^2(t) = \frac{\langle\|g\|^2_{H}\rangle|_{\leq t}}{\nu^2\lambda_1} is defined as the generalized Grashof number for the fluid flow, \langle \|g\|^2_H\rangle|_{\leq t} = {\lim_{\tau\rightarrow -\infty}}\frac{1}{t-\tau}\int^t_{\tau}\|g(s)\|^2_Hds and

    \begin{equation} \tilde{K}_0 = \Big\{(2\nu\lambda_1+\alpha)\Big/\Big[2C_{|\Omega|}\beta\nu\lambda_1 (\frac{1}{\alpha^2(1-\rho^*)}+\|\tilde{a}(t)\|_{L^1})\Big]\Big\}^{1/2} > 0,\nonumber \end{equation}

    here C_{|\Omega|}>0 is a constant dependent on the volume of \Omega .

    Proof. Step 1. The inequality for asymptotic stability of trajectories.

    Let u(t) and v(t) be two solutions of problem (92) with initial data

    \begin{eqnarray} u(\theta+\tau)|_{\theta\in [-h,0]} = \phi(\theta)|_{\theta\in[-h,0]},\ \ u|_{t = \tau} = u_{\tau} \end{eqnarray} (95)

    and

    \begin{eqnarray} v(\theta+\tau)|_{\theta\in [-h,0]} = \tilde{\phi}(\theta)|_{\theta\in[-h,0]},\ \ v|_{t = \tau} = \tilde{u}_{\tau} \end{eqnarray} (96)

    respectively, then u, v can be represented by

    \begin{eqnarray} (u,u_t) = (U(t,\tau)u_{\tau},U(t,\tau)\phi),\ \ (v,v_t) = (U(t,\tau)\tilde{u}_{\tau},U(t,\tau)\tilde{\phi}). \end{eqnarray} (97)

    If we denote w = u(t)-v(t) and w(t-\rho(t)) = u(t-\rho(t))-v(t-\rho(t)) , then it is easy to check that w satisfies the following problem

    \begin{equation} \begin{cases} \frac{\partial w}{\partial t}+\nu A w +P\Big(\alpha w+\beta(|u|u-|v|v)+\gamma (|u|^2u-|v|^2v)\Big)&\\ \quad = P\Big(F\big(u(t-\rho(t))\big)-F\big(v(t-\rho(t))\big)\Big),& \\ w|_{\partial\Omega} = 0,& \\ w(t = \tau) = u_{\tau}-\tilde{u}_{\tau},&\\ w(\tau+\theta) = \phi(\theta)-\tilde{\phi}(\theta),\ \theta\in [-h,0].& \end{cases} \end{equation} (98)

    Multiplying (98) with w and integrating by parts in \Omega , using the Poincaré and Young inequalities, from (63) we obtain that

    \begin{eqnarray} &&\frac{1}{2}\frac{d}{d t}\|w\|^2_H+\nu\|w\|^2_V+\alpha \|w\|_H^2+\gamma \gamma_0 \|w\|^4_{\bf{L}^4}\\ &\leq& |\beta(|u|u-|v|v,w)|+\Big|\Big(F\big(u(t-\rho(t))\big)-F\big(v(t-\rho(t))\big),w\Big)\Big|\\ &\leq& C_{|\Omega|}\beta(\|u\|^2_{V}+\|u\|^2_{V})\|w\|^2_{H}+\frac{\alpha}{2}\|w\|^2_H\\ &&+\frac{1}{\alpha}\|F\big(u(t-\rho(t))\big)-F\big(v(t-\rho(t))\big)\|^2_H\\ &\leq& C_{|\Omega|}\beta(\|u\|^2_{V}+\|u\|^2_{V})\|w\|^2_{H}+\frac{\alpha}{2}\|w\|^2_H\\ &&+\frac{L^2(R)\kappa(t)}{\alpha}\|w(t-\rho(t))\|^2_H. \end{eqnarray} (99)

    Using the Poincaré inequality and Lemma 3.1, noting that if

    \begin{eqnarray} 2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V) > 0, \end{eqnarray} (100)

    then we can obtain

    \begin{eqnarray} \|w\|^2_H&\leq& e^{\int^t_{\tau}[2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)-(2\nu\lambda_1+\alpha)]ds}\Big[\|u_{\tau}-\tilde{u}_{\tau}\|^2_H+\\ &&+\frac{L^2(R)\|\kappa(t)\|_{L^{\infty}}}{\alpha}\int^t_{\tau}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma}\\ && \quad \times\|w(t-\rho(t))\|^2_Hds\Big]. \end{eqnarray} (101)

    Denoting

    \begin{eqnarray} E(t,\tau) = e^{-\int^t_{\tau}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]ds} \end{eqnarray} (102)

    and

    \begin{eqnarray} K_1(t,s) = \frac{L^2(R)\|\kappa(t)\|_{L^{\infty}}}{\alpha}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma} \end{eqnarray} (103)

    and

    \begin{eqnarray} \Theta = \sup\limits_{t\geq s\geq \tau}E(t,s),\ \ \ \ \kappa(K_1,0) = \sup\limits_{t\geq\tau}\int^t_{\tau}K_1(t,s)ds, \end{eqnarray} (104)

    by virtue of Lemma 3.2, choosing \kappa(K_1,0)<\frac{1}{1+\Theta} , then there exists \tilde{M}>0 and \tilde{\lambda}>0 , such that we can obtain the estimate

    \begin{eqnarray} \|w(t-\rho(t))\|^2_H&\leq& \tilde{M}\|u_{\tau}-\tilde{u}_{\tau}\|^2_H e^{-\tilde{\lambda} (t-\tau)}. \end{eqnarray} (105)

    Substituting (105) into (99), using Lemma 3.1 again, we can conclude the following estimate

    \begin{eqnarray} \|w\|^2_H &\leq& \|u_{\tau}-\tilde{u}_{\tau}\|^2_He^{-\int^t_{\tau}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]ds}\\ &&+\frac{L^2(R)\|\kappa(t)\|_{L^{\infty}}}{\alpha}\tilde{M}\|u_{\tau}-\tilde{u}_{\tau}\|^2_H e^{-\tilde{\lambda} (t-\tau)}\\ && \quad \times\int^t_{\tau}e^{-\int^t_{s}[2\nu\lambda_1+\alpha-2C_{|\Omega|}\beta(\|u\|^2_V+\|v\|^2_V)]d\sigma}ds. \end{eqnarray} (106)

    From the result in last section, we can find that the pullback attractors is asymptotically stable as \tau\rightarrow -\infty provided that

    \begin{equation} 2\nu\lambda_1+\alpha > 2C_{|\Omega|}\beta \langle \|u\|^2_V+\|v\|^2_V\rangle_{\leq t}. \end{equation} (107)

    Step 2.Some energy estimate for (1) with sub-linear operator.

    Multiplying (3) with u in H , and then integrating by parts over \Omega , we have

    \begin{eqnarray} &&\frac{1}{2}\frac{d}{dt}\|u\|^2_H+\nu\|A^{1/2}u\|^2_H+\alpha\|u\|^2_H+\beta \|u\|^3_{\bf{L}^3}+\gamma \|u\|^4_{\bf{L}^4}\\ &\leq&\alpha \|u\|^2_H+\frac{1}{2\alpha}\Big[\|f\big(t,u(t-\rho(t))\big)\|^2_H+\|g\|^2_{H}\Big]. \end{eqnarray} (108)

    Moreover, let \theta = s-\rho(s) , then it yields

    \begin{equation} d\theta = (1-\rho'(s))ds,\ a(t)\rightarrow \tilde{a}(\bar{t})\in L^p(\tau,T), \end{equation} (109)

    which means \tilde{a}\in L^q(\tau-h,\tau) and

    \begin{align} &\int^t_\tau\|f(s,u(s-\rho(s)))\|^2_Hds\\ \leq&\int^t_\tau a(s)\|u(s-\rho(s))\|^2_Hds+\int^T_\tau b(s)ds\\ \leq& \ \dfrac{1}{1-\rho^*}\int_{\tau-\rho(\tau)}^{t-\rho(t)} \tilde{a}(s)\|u(s)\|^2_Hds+\int^t_\tau b(s)ds\\ \leq& \ \dfrac{1}{1-\rho^*}\left(\int_{-\rho(\tau)}^{0}\tilde{a}(t+\tau)\|\phi(t)\|^2_Hdt +\int^t_\tau\tilde{a}(s)\|u(s)\|^2_Hds\right)+\int^t_\tau b(s)ds\\ \leq& \dfrac{1}{1-\rho^*}\left(\|\phi(t)\|^2_{L^{2q}_{H}}\|\tilde{a}\|_{L^q(\tau-h,\tau)} +\int^t_\tau\tilde{a}(s)\|u(s)\|^2_Hds\right)+\int^t_\tau b(s)ds, \end{align} (110)

    Integrating (108) with time variable from \tau to t , we conclude that

    \begin{eqnarray} &&\|u\|^2_H+2\nu\int^t_{\tau}\|u\|^2_Vds+2\beta\int^t_{\tau} \|u\|^3_{\bf{L}^3}ds+2\gamma \int^t_{\tau}\|u\|^4_{\bf{L}^4}ds\\ &\leq& \dfrac{\|\tilde{a}\|_{L^q(\tau-h,\tau)}}{\alpha(1-\rho^*)}\|\phi(t)\|^2_{L^{2q}_{H}}+\|u_{\tau}\|^2_H+\frac{1}{\alpha(1-\rho^*)}\int^{t}_{\tau}\tilde{a}(s)\|u(s)\|^2_Hds\\ &&+\frac{1}{\alpha}\int^t_{\tau}\|g\|^2_{H}ds+\frac{1}{\alpha}\int^t_\tau b(s)ds, \end{eqnarray} (111)

    then we can achieve that

    \begin{eqnarray} \|u(t)\|^2_H&\leq& \Big[\dfrac{\|\tilde{a}\|_{L^q(\tau-h,\tau)}}{\alpha(1-\rho^*)}\|\phi(t)\|^2_{L^{2q}_{H}}+\|u_{\tau}\|^2_H\Big]e^{-\chi_{\sigma}(t,\tau)}\\ &&+\frac{1}{\alpha}\int^t_{\tau}\|g\|^2_{H}e^{-\chi_{\sigma}(t,s)}ds+\frac{1}{\alpha}\int^t_{\tau}b(s)e^{-\chi_{\sigma}(t,s)}ds, \end{eqnarray} (112)

    where the new variable index \chi_{\sigma}(\cdot,\cdot) is introduced as

    \begin{eqnarray} \chi_{\sigma}(t,s) = (2\nu\lambda_1-\sigma)(t-s)-\frac{1}{\alpha(1-\rho^*)}\int^t_{s}\tilde{a}(r)dr, \end{eqnarray} (113)

    which satisfies the relations

    \begin{eqnarray} \chi_{\sigma}(0,t)-\chi_{\sigma}(0,s) = -\chi_{\sigma}(t,s) \end{eqnarray} (114)

    and

    \begin{eqnarray} \chi_{\sigma}(0,r)\leq \chi_{\sigma}(0,t)+\Big(2\nu\lambda_1-\delta\Big)h,\ \ \mbox{if}\ 2\nu\lambda_1+\alpha-\delta > 0 \end{eqnarray} (115)

    for r\in [t-h,t] .

    Moreover, using the variable index introduced above, we can conclude that

    \begin{eqnarray} &&2\nu\int^t_{\tau}\|u(r)\|^2_{V}dr\\ &\leq& \dfrac{\|\tilde{a}\|_{L^q(\tau-h,\tau)}}{\alpha(1-\rho^*)}\|\phi(t)\|^2_{L^{2q}_{H}}+\|u_{\tau}\|^2_H\\ &&+\frac{1}{\alpha}\int^t_{\tau}\|g\|^2_{H}ds+\frac{1}{\alpha}\int^t_\tau b(s)ds\\ &&+\frac{1}{\alpha(1-\rho^*)}\Big[\dfrac{\|\tilde{a}\|_{L^q(\tau-h,\tau)}}{\alpha(1-\rho^*)}\|\phi(t)\|^2_{L^{2q}_{H}}+\|u_{\tau}\|^2_H\Big]\int^t_{\tau}\tilde{a}(s)e^{-\chi_{\sigma}(s,\tau)}ds\\ &&+\frac{1}{\alpha^2(1-\rho^*)}\int^t_{\tau}\|g(s)\|^2_{H}ds\int^t_{\tau}\tilde{a}(s)ds+\frac{\|b\|_{L^1(\tau,T)}}{\alpha^2(1-\rho^*)}\int^t_{\tau}\tilde{a}(s)ds. \end{eqnarray} (116)

    Step 3. The sufficient condition for asymptotic stability of trajectories inside pullback attractors.

    Combining (107) with (116), we conclude that

    \begin{eqnarray} &&2C_{|\Omega|}\beta\langle \|u\|^2_V+\|v\|^2_V\rangle|_{\leq t}\\ &\leq& \frac{2C_{|\Omega|}\beta}{\nu}\Big[\Big(\frac{1}{\alpha^2(1-\rho^*)}+\int^t_{\tau}\tilde{a}(s)ds\Big)\langle \|g(t)\|_H^2\rangle|_{\leq t} \\ &&+\frac{1}{\alpha}\langle \|b(t)\|_{L^1}\rangle|_{\leq t}+\frac{\|b\|_{L^1(\tau,T)}}{\alpha^2(1-\rho^*)}\langle \|\tilde{a}(t)\|_{L^1}\rangle|_{\leq t}\Big]. \end{eqnarray} (117)

    and hence the asymptotic stability holds provided that

    \begin{eqnarray} &&\Big(\frac{1}{\alpha^2(1-\rho^*)}+\|\tilde{a}(t)\|_{L^1}\Big)\langle \|g(t)\|_H^2\rangle|_{\leq t}+\frac{1}{\alpha}\langle \|b(t)\|_{L^1}\rangle|_{\leq t}+\frac{\|b\|_{L^1(\tau,T)}}{\alpha^2(1-\rho^*)}\langle \|\tilde{a}(t)\|_{L^1}\rangle|_{\leq t}\\ &&\leq \frac{\nu(2\nu\lambda_1+\alpha)}{2C_{|\Omega|}\beta}. \end{eqnarray} (118)

    If we define the generalized Grashof number as G(t) = \Big(\frac{\langle\|g\|^2_{H}\rangle|_{\leq t}}{\nu^2\lambda_1}\Big)^{1/2} , neglecting the positive terms on the left-hand side of (118), then a sufficient condition for the asymptotic stability of trajectories inside pullback attractors can be conclude as

    \begin{eqnarray} G(t)\leq \Big\{(2\nu\lambda_1+\alpha)\Big/\Big[2C_{|\Omega|}\beta\nu\lambda_1 (\frac{1}{\alpha^2(1-\rho^*)}+\|\tilde{a}(t)\|_{L^1})\Big]\Big\}^{1/2} = \tilde{K}_0, \end{eqnarray} (119)

    which completes the proof for our first result.

    Remark 5. If we denote

    \begin{eqnarray} \limsup\limits_{\tau\rightarrow -\infty}\frac{1}{t-\tau}\int^t_{\tau}b(r)dr = b_0\in [0,+\infty) \end{eqnarray} (120)

    and

    \begin{eqnarray} \limsup\limits_{\tau\rightarrow -\infty}\frac{1}{t-\tau}\int^t_{\tau}\tilde{a}(r)dr = \tilde{a}_0\in [0,+\infty), \end{eqnarray} (121)

    such that there exists some \sigma>0 , the following assumption

    \begin{eqnarray} \frac{\nu(2\nu\lambda_1+\alpha)}{2C_{|\Omega|}\beta} > \frac{b_0}{\alpha}+\frac{\|b\|_{L^1(\tau,T)}\tilde{\alpha}_0}{\alpha^2(1-\rho^*)}+\delta \end{eqnarray} (122)

    holds. Then more precise sufficient condition for the asymptotic stability of pullback attractors is

    \begin{eqnarray} G(t)\leq \Big[\frac{\frac{\nu(2\nu\lambda_1+\alpha)}{2C_{|\Omega|}\beta}-\frac{b_0}{\alpha}-\frac{\|b\|_{L^1(\tau,T)}\tilde{\alpha}_0}{\alpha^2(1-\rho^*)}}{\nu^2\lambda_1(\frac{1}{\alpha^2(1-\rho^*)}+\|\tilde{a}(t)\|_{L^1})}\Big]^{1/2} \end{eqnarray} (123)

    which has smaller upper boundedness than (119).

    The structure and stability of 3D BF equations with delay are investigated in this paper. A future research in the pullback dynamics of (1) is to study the geometric property of pullback attractors, such as the fractal dimension.

    Xin-Guang Yang was partially supported by the Fund of Young Backbone Teacher in Henan Province (No. 2018GGJS039) and Henan Overseas Expertise Introduction Center for Discipline Innovation (No. CXJD2020003). Xinjie Yan was partly supported by Excellent Innovation Team Project of "Analysis Theory of Partial Differential Equations" in China University of Mining and Technology (No. 2020QN003). Ling Ding was partly supported by NSFC of China (Grant No. 1196302).

    The authors want to express their most sincere thanks to refrees for the improvement of this manuscript. The authors also want to thank Professors Tomás Caraballo (Universidad de Sevilla), Desheng Li (Tianjin University) and Shubin Wang (Zhengzhou University) for fruitful discussion on this subject.



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