Review Topical Sections

Urinary VPAC1: A potential biomarker in prostate cancer

  • Prostate cancer is ranked as the fourth most prevalent cancer commonly diagnosed among males over 40 years of age, according to the WHO Cancer Fact Sheet 2020, and it is additionally a leading cause of cancer mortality among males. The incidence of prostate cancer and mortality varied significantly across the globe. Diagnosis of prostate cancer hinders easier management of cases, and prostate-specific antigen (PSA) use for screening of prostate cancer has poor specificity and sensitivity, thereby yielding overdiagnosis and unnecessary biopsies. Radiologically guided (ultrasound/MRI) prostate biopsy, considered the gold standard, is invasive and can miss a significant number of metastatic cancers. Even though mild, other prostate biopsy complications occur on a large scale, and few severe ones are often recorded. Scientists intensify their search for biomarker(s) for non-invasive diagnosis of prostate cancer using proteomics, metabolomics, genomics, and bioinformatics—urinary biomarkers were uniquely on the lookout. Vasoactive intestinal peptide (VIP)/pituitary adenylate cyclase-activating peptide (PACAP) receptor 1 (VPAC1), which is overexpressed (a thousandfold) in prostate cancer at the onset of oncogenesis and is excreted in the urine on tumor cells, is a contender in the prostate cancer biomarker quest. VPAC1 is ubiquitous, expressed by normal and malignant cells, and interwoven in their cell membranes. Therefore, using urine samples limits the possibility of making the wrong diagnosis, since VPAC1 is not normally excreted in the urine. Nevertheless, studying transmembrane receptors is intricate. However, producing monoclonal antibodies against the N-terminal end of VPAC1 can provide a promising target for designing a non-invasive diagnostic assay for early detection of prostate cancer using a urine sample.

    Citation: Mansur Aliyu, Ali Akbar Saboor-Yaraghi, Shima Nejati, Behrouz Robat-Jazi. Urinary VPAC1: A potential biomarker in prostate cancer[J]. AIMS Allergy and Immunology, 2022, 6(2): 42-63. doi: 10.3934/Allergy.2022006

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  • Prostate cancer is ranked as the fourth most prevalent cancer commonly diagnosed among males over 40 years of age, according to the WHO Cancer Fact Sheet 2020, and it is additionally a leading cause of cancer mortality among males. The incidence of prostate cancer and mortality varied significantly across the globe. Diagnosis of prostate cancer hinders easier management of cases, and prostate-specific antigen (PSA) use for screening of prostate cancer has poor specificity and sensitivity, thereby yielding overdiagnosis and unnecessary biopsies. Radiologically guided (ultrasound/MRI) prostate biopsy, considered the gold standard, is invasive and can miss a significant number of metastatic cancers. Even though mild, other prostate biopsy complications occur on a large scale, and few severe ones are often recorded. Scientists intensify their search for biomarker(s) for non-invasive diagnosis of prostate cancer using proteomics, metabolomics, genomics, and bioinformatics—urinary biomarkers were uniquely on the lookout. Vasoactive intestinal peptide (VIP)/pituitary adenylate cyclase-activating peptide (PACAP) receptor 1 (VPAC1), which is overexpressed (a thousandfold) in prostate cancer at the onset of oncogenesis and is excreted in the urine on tumor cells, is a contender in the prostate cancer biomarker quest. VPAC1 is ubiquitous, expressed by normal and malignant cells, and interwoven in their cell membranes. Therefore, using urine samples limits the possibility of making the wrong diagnosis, since VPAC1 is not normally excreted in the urine. Nevertheless, studying transmembrane receptors is intricate. However, producing monoclonal antibodies against the N-terminal end of VPAC1 can provide a promising target for designing a non-invasive diagnostic assay for early detection of prostate cancer using a urine sample.


    Abbreviations

    PSA:

    prostate-specific antigen; 

    VIP:

    vasoactive intestinal peptide; 

    PACAP:

    pituitary adenylate cyclase-activating peptide; 

    VPAC1:

    VIP/PACAP receptor 1; 

    ASR:

    age-standardized rate; 

    BPH:

    benign prostatic hypertrophy; 

    DHT:

    5α-dihydrotestosterone; 

    RTqPCR:

    quantitative reverse transcription polymerase chain reaction; 

    MRI:

    magnetic resonance imaging; 

    mp-MRI:

    multiparametric MRI; 

    DCE-MRI:

    dynamic contrast-enhanced MRI; 

    DWI-MRI:

    diffusion-weighted imaging MRI; 

    TRUS:

    trans-rectal ultrasound scan; 

    ELISA:

    enzyme-linked immunosorbent assay; 

    PCA3:

    prostate cancer antigen 3; 

    TMPRSS2:

    transmembrane protease serine 2; 

    GSTP1:

    glutathione S-transferase P1; 

    DRE:

    digital rectal examination; 

    AUC:

    area under the curve; 

    aHGF:

    hepatocyte growth factor; 

    IGFBP3:

    insulin-like growth factor binding protein 3; 

    OPN:

    Plasma osteopontin; 

    LNCaP:

    lymph node carcinoma of the prostate; 

    GPCRs:

    G protein-coupled receptors; 

    RAMPS:

    receptor activity modifying proteins; 

    7-TMD:

    seven-transmembrane domain; 

    ECD:

    extracellular domain; 

    CHO:

    Chinese hamster ovary; 

    TBP:

    TATA-box binding protein; 

    COX-2:

    cyclooxygenase-2; 

    MMP9:

    metalloproteinase 9; 

    uPA:

    urokinase plasminogen activator; 

    uPAR:

    uPA receptor; 

    DAPI:

    4,6-diamidino-2-phenylindole; 

    AC:

    adenylyl cyclase; 

    CREB:

    cAMP response element-binding protein; 

    PKA:

    phosphokinase A; 

    iNOS:

    inducible nitric oxide synthase; 

    CBP:

    CREB binding protein; 

    NF-κβ:

    nuclear factor-κβ; 

    ERK:

    extracellular signal-regulated kinase; 

    MEKK1:

    MAP/ERK kinase (MEK) kinase 1; 

    IRF-1:

    IFN regulatory factor-1; 

    IKK:

    inhibitory κβ kinase; 

    BB-LP:

    bombesin-like peptides; 

    GRPR:

    gastrin-releasing peptide receptor; 

    PLC:

    phospholipase C; 

    DAG:

    diacylglycerol; 

    IP3:

    1,4,5-triphosphate; 

    PKC:

    protein kinase C; 

    MAPK:

    mitogen-activated protein kinase; 

    MAPKK:

    mitogen-activated protein kinase kinase; 

    ELK:

    ETS like-1 protein; 

    SRE:

    serum response element

    The nonlinear Ginzburg-Landau equation plays an important role in the studies of physics, which describes many interesting phenomena and has been studied extensively (see [1] for a more detailed description). The fractional Ginzburg-Landau equation [2,3,4] is employed to describe processes in media with fractional dispersion or long-range interaction. It becomes very popular because the fractional derivative and fractional integral have broad applications in different fields of science [5,6,7,8,9,10].

    Our work focuses on the existence of invariant measures of the autonomous fractional stochastic delay Ginzburg-Landau equations on Rn:

    du(t)+(1+iν)(Δ)αu(t)dt+(1+iμ)|u(t)|2βu(t)dt+λu(t)dt=G(x,u(tρ))dt,+k=1(σ1,k(x)+κ(x)σ2,k(u(t)))dWk(t), t>0, (1.1)

    with initial condition

    u(s)=φ(s),s[ρ,0], (1.2)

    where u(x,t) is a complex-valued function on Rn×[0,+). In (1.1), i is the imaginary unit, α,β,μ,ν and λ are real constants with β>0,λ>0 and ρ>0. (Δ)α with 0<α<1 is the fractional Laplace operator, σ1,k(x)L2(Rn) and σ2,k(u):CR are nonlinear functions, κ(x)L2(Rn)L(Rn) and {Wk}k=1 is a sequence of independent standard real-valued Wiener process on a complete filtered probability space (Ω, F, {Ft}tR,P), where {Ft}tR is an increasing right continuous family of sub-σ-algebras of F that contains all P-null sets.

    The Ginzburg-Landau equation with fractional derivative was first introduced in [2]. There is a large amount of literature which was used for investigating fractional deterministic Ginzburg-Landau equations such as [1] and stochastic equations such as [11,12,13,14,15,16,17]. These papers had respectively researched the long-time deterministic as well as random dynamical systems of fractional equations with autonomous forms and non-autonomous forms. However, in spite of quite a lot of contribution of the works, no result is provided for the existence of pathwise pullback random attractors and invariant measures for the delay stochastic Ginzburg-Landau equations.

    The delay differential equations [18] was described the dynamical systems that rely on current and past historical states. For the past few years, researchers had made great progress in the study of linear and nonlinear delay differential equations, see [20,21]. Delay differential equations are widely used in many fields, so investigating the solutions of equations has profound significance. Therefore, it's necessary that we establish the dynamics of delay stochastic Ginzburg-Landau equations.

    The goal of this paper is to prove the existence of invariant measures of the stochastic Eqs (1.1) and (1.2) in L2(Ω;C([ρ,0],L2(Rn))) by applying Krylov-Bogolyubov's method. The main difficulty of this paper is that deducing the uniform estimates of solutions (because of the nonlinear term (1+iμ)|u(t)|2βu(t) and complex-valued solutions), proving the weak compactness of a set distribution laws of the segments of solutions in L2(Ω;C([ρ,0],L2(Rn))) (because the standard Sobolev embeddings are not compact on unbounded domains Rn), and establishing the equicontinuity of solutions in L2(Ω;C([ρ,0],L2(Rn))) (because the uniform estimates in L2(Ω;C([ρ,0],L2(Rn))) are not sufficient, and the uniform estimates in L2(Ω;C([ρ,0],H1(Rn))) are needed).

    For the estimates of the nonlinear term (1+iμ)|u(t)|2βu(t), we apply integrating by parts and nonnegative definite quadratic form. There are Several methods to handle the noncompact on unbounded domain, including weighted spaces [22,23,24], weak Feller approach [25,26] and uniform tail-estimates [23,27]. We first obtain the uniform estimates of the tail of the solution as well as the technique of dyadic division, then establish the weak compactness of a set of probability distribution of solutions in C([ρ,0],L2(Rn)) applying the Ascoli-Arzelˊa theorem.

    Let S be the Schwartz space of rapidly decaying C functions on Rn. The fractional Laplace operator (Δ)α for 0<α<1 is defined by, for uS,

    (Δ)αu(x)=12C(n,α)Rnu(x+y)+u(xy)2u(x)|y|n+2αdy,    xRn,

    where C(n,α) is a positive constant given by

    C(n,α)=α4αΓ(n+2α2)πn2Γ(1α).

    By [28], the inner product ((Δ)α2u,(Δ)α2v) in the complex field is defined by

    ((Δ)α2u,(Δ)α2v)=C(n,α)2RnRn(u(x)u(y))(ˉv(x)ˉv(y))|xy|n+2αdxdy,

    for uHα(Rn). The fractional Sobolev space Hα(Rn) is endowed with the norm

    u2Hα(Rn)=u2L2(Rn)+2C(n,α)(Δ)α2u2L2(Rn).

    About the fractional derivative of fractional Ginzburg-Landau equations, there is another statement in [29].

    We organize the article as follows. In Section 2, we establish the well-posedness of (1.1) and (1.2) in L2(Ω;C([ρ,0],H)). In Sections 3 and 4, we derive the uniform estimates of solutions in L2(Ω;C([ρ,0],H)) and L2(Ω;C([ρ,0],V)), respectively. In Section 5, the existence of invariant measures is obtained.

    In this section, we show the nonlinear drift term and the diffusion term in (1.1) which are needed for the well-posedness of the stochastic delay Ginzburg-Landau Eqs (1.1) and (1.2) defined on Rn.

    We assume that G:Rn×CC is continuous and satisfies

    |G(x,u)||h(x)|+a|u|, xRn, uC (2.1)

    and

    |G(x,u)||ˆh(x)|+ˆa|u|, xRn, uC, (2.2)

    where a and ˆa>0 are constants and h(x),ˆh(x)L2(Rn). Moreover, G(x,u) is Lipschitz continuous in uC uniformly with respect to xRn. More precisely, there exists a constant CG>0 such that

    |G(x,u1)G(x,u2)|CG|u1u2|, xRn, u1,u2C. (2.3)

    For the diffusion coefficients of noise, we suppose that for each kN+

    k=1σ1,k2<, (2.4)

    and that σ2,k(u):CR is globally Lipschitz continuous; namely, for every kN+, there exists a positive number αk such that for all s1,s2C,

    |σ2,k(s1)σ2,k(s2)|αk|s1s2|. (2.5)

    We further assume that for each kN+, there exist positive numbers βk, ˆβk, γk and ˆγk such that

    |σ2,k(s)|βk+γk|s|, sC, (2.6)

    and

    |σ2,k(s)|ˆβk+ˆγk|s|, sC, (2.7)

    where k=1(α2k+β2k+γ2k+ˆβ2k+ˆγ2k)<+. In this paper, we deal with the stochastic Eqs (1.1) and (1.2) in the space C([ρ,0],L2(Rn)). In the following discussion, we denote by H=L2(Rn), V=H1(Rn).

    A solution of problems (1.1) and (1.2) will be understood in the following sense.

    Definition 2.1. We suppose that φ(s)L2(Ω,C([ρ,0],H)) is F0-measurable. Then, a continuous H-valued Ft-adapted stochastic process u(x,t) is named a solution of problems (1.1) and (1.2), if

    1) u is pathwise continuous on [0,+), and Ft-adapted for all t0,

    uL2(Ω,C([0,T],H))L2(Ω,L2([0,T],V))

    for all T>0,

    2) u(s)=φ(s) for ρs0,

    3) For all t0 and ξV,

    (u(t),ξ)+(1+iν)t0((Δ)α2u(s),(Δ)α2ξ)ds+t0Rn(1+iμ)|u(s)|2βu(s)ξ(x)dxds+λt0(u(s),ξ)ds=(φ(0),ξ)+t0(G(s,u(sρ)),ξ)ds+k=1t0(σ1,k(x)+κ(x)σ2,k(u(s)),ξ)dWk(s), (2.8)

    for almost all ωΩ.

    By the Galerkin method and the argument of Theorem 3.1 in [30], one can verify that if (2.1)–(2.7) hold true, then, for every F0-measurable function φ(s)L2(Ω,C([ρ,0],H)), the problems (1.1) and (1.2) has a unique solution u(x,t) in the sense of Definition 2.1.

    Now, we establish the Lipschitz continuity of the solutions of the problems (1.1) and (1.2) with respect to the initial data in L2(Ω,C([ρ,0],H)).

    Theorem 2.2. Suppose (2.1)–(2.6) hold, and F0-measurable function φ1,φ2L2(Ω,C([ρ,0],H)). If u1=u(t,φ1) and u2=u(t,φ2) are the solutions of the problems (1.1) and (1.2) with initial data φ1 and φ2, respectively, then, for any t0,

    E[supρstu(s,φ1)u(s,φ2)2]+E[t0u(s,φ1)u(s,φ2)2Vds]
    C1e˜C1tE[supρs0φ1(s)φ2(s)]2],

    where C1 and ˜C1 are positive constants independent of φ1 and φ2.

    Proof. Since both u1 and u2 are the solutions of the problems (1.1) and (1.2), we have, for all t0,

    u1u2+(1+iν)t0(Δ)α(u1u2)ds+(1+iμ)t0(|u1|2βu1|u2|2βu2)ds+λt0(u1u2)ds=φ1(0)φ2(0)+t0(G(x,u1(sρ))G(x,u2(sρ)))ds+k=1t0κ(x)(σ2,k(u1)σ2,k(u2))dWk. (2.9)

    By (2.9), the integration by parts of Ito's formula and taking the real parts, we get, for all t0,

    u1u22+2t0(Δ)α2(u1u2)2ds+2Ret0Rn(ˉu1ˉu2)[|u1|2βu1|u2|2βu2]dxds+2λt0u1u22ds=φ1(0)φ2(0)2+2Ret0(u1u2,G(x,u1(sρ))G(x,u2(sρ)))ds+k=1t0κ(x)(σ2,k(u1)σ2,k(u2))2ds+2Ret0(u1u2,k=1κ(x)(σ2,k(u1)σ2,k(u2)))dWk(s). (2.10)

    For the third term in the first row of (2.10), one has

        2Ret0Rn(ˉu1ˉu2)[|u1|2βu1|u2|2βu2]dxds=t0Rn2|u1|2β+2+2|u2|2β+22Re(u1ˉu2)(|u1|2β+|u2|2β)dxdst0Rn2|u1|2β+2+2|u2|2β+22|u1||u2|(|u1|2β+|u2|2β)dxdst0Rn2|u1|2β+2+2|u2|2β+2(|u1|2+|u2|2)(|u1|2β+|u2|2β)dxds=t0Rn|u1|2β+2+|u2|2β+2|u1|2β|u2|2|u2|2β|u1|2dxds=t0Rn(|u1|2β|u2|2β)(|u1|2|u2|2)dxds0.

    By (2.10), we deduce that for t0,

        E[sup0rtu1(r)u2(r)2]E[supρs0φ1(s)φ2(s)2]+2E[t0u1u2G(x,u1(sρ))G(x,u2(sρ))ds]   +k=1E[t0κ(σ2,k(u1)σ2,k(u2))2ds]   +2E[sup0rt|k=1r0(u1u2,κ(x)(σ2,k(u1)σ2,k(u2))dWk(s))|]. (2.11)

    For the second term on the right-hand side of (2.11), by (2.3), one has

        2E[t0u1u2G(x,u1(sρ))G(x,u2(sρ))ds]E[t0u1u22ds]+E[t0G(x,u1(sρ))G(x,u2(sρ))2ds]E[t0u1u22ds]+C2GE[t0u1(sρ)u2(sρ)2ds]=E[t0u1u22ds]+C2GE[tρρu1u22ds](1+C2G)E[t0u1u22ds]+C2GE[0ρφ1(s)φ2(s)2ds](1+C2G)t0E[sup0rsu1u22]ds+ρC2GE[supρs0φ1(s)φ2(s)2].

    For the third term on the right-hand side of (2.11), by (2.5), we have

      k=1E[t0κ(x)(σ2,k(u1)σ2,k(u2))2ds]κ(x)2Lk=1α2kE[t0u1u22ds]κ(x)2Lk=1α2kt0E[sup0rsu1u22]ds. (2.12)

    For the forth term on the right-hand side of (2.11), by Burkholder-Davis-Gundy's inequality, one has

        2E[sup0rt|k=1r0(u1u2,κ(x)(σ2,k(u1)σ2,k(u2))dWk(s))|]B1E[(t0k=1|(u1u2,κ(x)(σ2,k(u1)σ2,k(u2)))|2ds)12]B1E[(t0k=1u1u22κ2Lσ2,k(u1)σ2,k(u2)2ds)12]B1E[sup0stu1u2κL(k=1α2k)12(t0u1u22ds)12]12E[sup0stu1u22]+12B21κ2Lk=1α2kE[t0sup0rsu1u22ds], (2.13)

    where B1 is a constant produced by Burkholder-Davis-Gundy's inequality.

    It follows from (2.11)–(2.13) that for all t0,

    E[sup0rtu1(r)u2(r)2]2(1+ρC2G)E[supρs0φ1(s)φ2(s)2]+2[1+C2G+(1+12B21)κ2Lk=1α2k]t0E[sup0rsu1(r)u2(r)2]ds. (2.14)

    Applying Gronwall inequality to (2.14), we obtain that for all t0,

    E[sup0rtu1(r)u2(r)2]2(1+ρC2G)ec1tE[supρs0φ1(s)φ2(s)2], (2.15)

    where c1=2[1+C2G+(1+12B21)κ2Lk=1α2k]. By (2.10), there exists c2 such that for all t0,

    E[t0u1u22Vds]˜c2ec2tE[supρs0φ1(s)φ2(s)2].

    We assume that a, αk and γk are small enough in the sense, there exists a constant p2 such that

    2112p(2p1)2p12pa+2p(2p1)κ2Lk=1(α2k+γ2k)<pλ. (3.1)

    By (3.1), one has

    2κ2Lk=1γ2k<λ, (3.2)

    and

    2a+2κ2Lk=1γ2k<λ. (3.3)

    The inequalities (3.1)–(3.3) are used to establish the uniform tail-estimate of the solution of (1.1) and (1.2).

    Lemma 3.1. Suppose (2.1)–(2.6) and (3.2) hold. If φ(s)L2(Ω;C([ρ,0],H)), then, for all t0, there exists a positive constant μ1 such that the solution u of (1.1) and (1.2) satisfies

    E[u(t)2]+t0eμ1(st)E(u(s)2V)ds+t0eμ1(st)E(u(s)2β+2L2β+2)ds
    M1E[supρs0φ(s)2]+~M1, (3.4)

    and

    t+ρ0E[u(s)2V]ds(M1(t+ρ)+1+2aρC(n,α))E[supρs0φ(s)2]+2(t+ρ)aC(n,α)h(x)2
    +2(t+ρ)C(n,α)k=1(σ1,k2+2β2kκ(x)2)+˜M1(t+ρ),

    where ˜M1 is a positive constant independent of φ.

    Proof. By (1.1) and the integration by parts of Ito's formula, we have for all t0,

        u(t)2+2t0(Δ)α2u(s)2ds+2t0u(s)2β+2L2β+2ds+2λt0u(s)2ds=2Ret0(u(s),G(x,u(sρ)))ds+φ(0)2+k=1t0σ1,k(x)+κ(x)σ2,k(u(s))2ds+2Ret0(u(s),k=1σ1,k(x)+κ(x)σ2,k(u(s)))dWk(s). (3.5)

    The system (3.5) can be rewritten as

        d(u(t)2)+2(Δ)α2u(t)2dt+2u(t)2β+2L2β+2dt+2λu(t)2dt=2Re(u(t),G(x,u(tρ)))dt+k=1σ1,k(x)+κ(x)σ2,k(u(t))2dt+2Re(u(t),k=1σ1,k(x)+κ(x)σ2,k(u(t)))dWk(t). (3.6)

    Assume that μ1 is a positive constant, one has

        eμ1tu(t)2+2t0eμ1s(Δ)α2u(s)2ds+2t0eμ1su(s)2β+2L2β+2ds=(μ12λ)t0eμ1su(s)2ds+φ(0)2+2Ret0eμ1s(u(s),G(x,u(sρ)))ds+k=1t0eμ1sσ1,k+κσ2,k(u(s))2ds+2Ret0eμ1s(u(s),k=1σ1,k(x)+κ(x)σ2,k(u(s)))dWk(s).

    Taking the expectation, we have for all t0,

        eμ1tE(u(t)2)+2E[t0eμ1s(Δ)α2u(s)2ds]+2E[t0eμ1su(s)pLpds]=E(φ(0)2)+(μ12λ)E[t0eμ1su(s)2ds]+2E[t0eμ1sRe(u(s),G(x,u(sρ)))ds]+k=1E[t0eμ1sσ1,k(x)+κ(x)σ2,k(u(s))2ds]. (3.7)

    For the third term on the right-hand side (3.7), by (2.1), we have

        2E[t0eμ1sRe(u(s),G(x,u(sρ)))ds]2t0eμ1sE[u(s)G(x,u(sρ))]ds2at0eμ1sE(u(s)2)ds+22at0eμ1sE[G(x,u(sρ))2]ds2at0eμ1sE(u(s)2)ds+2at0eμ1sh(x)2ds+2at0eμ1sE[u(sρ)2]ds2a(1+eμ1ρ)t0eμ1sE[u(s)2]ds+2ah(x)2t0eμ1sds+2aeμ1ρ0ρeμ1sE[φ(s)2]ds2a(1+eμ1ρ)t0eμ1sE[u(s)2]ds+2eμ1taμ1h(x)2+2aρeμ1ρE[supρs0φ(s)2]. (3.8)

    For the forth term on the right-hand side (3.7), by (2.6), we have

        k=1E[t0eμ1sσ1,k+κσ2,k(u(s))2ds]k=1E[t0eμ1s(2σ1,k2+2κσ2,k(u(s))2)ds]2μ1k=1σ1,k2eμ1t+4k=1t0eμ1sE[β2kκ2+γ2kκ2Lu(s)2]ds2μ1k=1(σ1,k2+2β2kκ(x)2)eμ1t+4k=1γ2kκ(x)2Lt0eμ1sE(u(s)2)ds. (3.9)

    By (3.7)–(3.9), we obtain for all t0,

        eμ1tE(u(t)2)+2E[t0eμ1s(Δ)α2u(s)2ds]+2E[t0eμ1su(s)2β+2L2β+2ds](1+2aρeμ1ρ)E[supρs0φ(s)2]+[μ12λ+2a(1+eμ1ρ)+4k=1γ2kκ2L]t0eμ1sE[u2]ds+2aμ1eμ1th(x)2+2μ1k=1(σ1,k2+2β2kκ(x)2)eμ1t. (3.10)

    By (3.2), there exists a positive constant μ1 sufficiently small such that

    2μ1+2a+2aeμ1ρ+4k=1γ2kκ(x)2L2λ.

    Then, we have, for all t0,

        E(u(t)2)+2t0eμ1(st)E((Δ)α2u(s)2)ds     +μ1t0eμ1(st)E(u(s)2)ds+2t0eμ1(st)E(u(s)2β+2L2β+2)ds(1+2aρeμ1ρ)E(supρs0φ(s)2)+1μ1(2ah(x)2+2k=1(σ1,k2+2β2kκ(x)2)),

    which completes the proof of (3.4).

    Integrating (3.6) on [0,t+ρ] and taking the expectation, one has

      E[u(t+ρ)2]+2E[t+ρ0(Δ)α2u(s)2ds]+2E[t+ρ0u(s)2β+2L2β+2ds]+2λE[t+ρ0u(s)2ds]=E[φ(0)2]+2E[t+ρ0Re(u(s),G(x,u(sρ)))ds]+k=1E[t+ρ0σ1,k+κ(x)σ2,k(u(s))ds]. (3.11)

    For the second term on the right-hand side of (3.11), by (2.1), we have

        2E[t+ρ0Re(u,G(x,u(sρ)))ds]22aE[t+ρ0u2ds]+2aρE[supρs0φ(s)2]+2(t+ρ)ah2. (3.12)

    For the third term on the right-hand side of (3.11), one has

        k=1E[t+ρ0σ1,k+κσ2,k(u(s))ds]2(t+ρ)k=1(σ1,k2+2β2kκ2)+4k=1γ2kκ(x)2LE[t+ρ0u2ds]. (3.13)

    Then, by (3.2) and (3.11)–(3.13), for all t0, we obtain,

    2E[t+ρ0(Δ)α2u(s)2ds](1+2aρ)E[supρs0φ(s)2]
    +2(t+ρ)k=1(σ1,k2+2β2kκ(x)2)+2(t+ρ)ah(x)2.

    The result then follows from (3.4).

    The next lemma is used to obtain the uniform estimates of the segments of solutions in C([ρ,0],H).

    Lemma 3.2. Suppose (2.1)–(2.6) and (3.2) hold. Then, for any φ(s)L2(Ω,F0;C([ρ,0],H)), the solution of (1.1) satisfies that, for all tρ,

    E(suptρrtu(r)2)M2E[supρs0φ(s)2]+˜M2,

    where M2 and ˜M2 are positive constants independent of φ.

    Proof. By (1.1) and integration by parts of Ito's formula and taking the real part, we get for all tρ and tρrt,

        u(r)2+2rtρ(Δ)α2u(s)2ds+2rtρu(s)2β+2L2β+2ds+2λrtρu(s)2ds=u(tρ)2+2Rertρ(u(s),G(x,u(sρ)))ds+k=1rtρσ1,k(x)+κ(x)σ2,k(u(s)))2ds+2Rek=1rtρ(u(s),(σ1,k(x)+κ(x)σ2,k(u(s))dWk(s)). (3.14)

    For the second term on the right-hand side of (3.14), by (2.1) we have, for all tρ and tρrt,

        2Rertρ(u(s),G(x,u(sρ)))ds2rtρu(s)G(x,u(sρ))dsrtρu(s)2ds+rtρG(x,u(sρ))2dsrtρu(s)2ds+2rtρh2ds+2a2rtρu(sρ)2dsrtρu(s)2ds+2ρh2+2a2tρt2ρu(s)2ds. (3.15)

    For the third term on the right-hand side of of (3.14), for all tρ and tρrt, by (2.6), we have

        k=1rtρσ1,k(x)+κ(x)σ2,k(u(s))2ds2ρk=1σ1,k2+4ρκ2k=1β2k+4κ2Lk=1γ2krtρu(s)2ds. (3.16)

    By (3.14)–(3.16), we obtain for all tρ and tρrt,

    u(r)2c3+u(tρ)2+c4rt2ρu(s)2ds
    +2Rek=1rtρ(u(s),(σ1,k(x)+κ(x)σ2,k(u(s))dWk(s)), (3.17)

    where c3=2ρh2+2ρk=1σ1,k2+4ρκ2k=1β2k and c4=1+2a2+4κ2Lk=1γ2k. By (3.17), we find that for all tρ,

    E[suptρrtu(r)2]c3+E[u(tρ)2]+c4tt2ρE[u(s)2]ds+2E[suptρrt|k=1rtρ(u(s),(σ1,k(x)+κ(x)σ2,k(u(s))dWk(s))|]. (3.18)

    For the second term and the third term on the right-hand side of (3.18), by Lemma 3.1, we deduce for all tρ,

    E[u(tρ)2]sups0E[u(s)2]M1E[supρs0φ2]+˜M1 (3.19)

    and

    c4tt2ρE[u(s)2]ds2ρc4supsρE[u(s)2]c5E[supρs0φ2]+c5. (3.20)

    For the last term on the right-hand side of (3.18), by Burkholder-Davis-Gundy's inequality and Lemma 3.1, we obtain for all tρ,

        2E[suptρrt|k=1rtρ(u(s),σ1,k(x)+κ(x)σ2,k(u(s))dWk(s))|]2B2E[(k=1ttρ|(u(s),σ1,k+κσ2,k(u(s)))|2ds)12]12E[suptρstu(s)2]+2B22E[k=1ttρσ1,k+κσ2,k(u(s))2ds]12E[suptρstu(s)2]+2B22(2ρk=1σ1,k2+4ρκ2k=1β2k)+8B22ρκ2Lk=1γ2ksups0E[u(s)2]. (3.21)

    By Lemma 3.1 and (3.18)–(3.21), we deduce that for all tρ,

    E[suptρrtu(r)2]M2E[supρs0φ(s)2]+˜M2.

    This completes the proof.

    To establish the tightness of a family of distributions of solutions, we now derive uniform estimates on the tails of solutions to the problems (1.1) and (1.2).

    Lemma 3.3. Suppose (2.1)–(2.6) and (3.2) hold. If φ(s)L2(Ω,C([ρ,0],H)). Then, for all t0, the solution u of (1.1) and (1.2) satisfies

    lim supmsuptρ|x|mE[|u(t,x)|2]dx=0.

    Proof. We suppose that θ(x):RnR is a smooth function with 0θ(x)1, for all xRn defined by

    θ(x)={0if |x|1,1if |x|2.

    For fixed mN, we denote that θm(x)=θ(xm). By (1.1), we have

    d(θmu)+(1+iν)θm(Δ)αudt+(1+iμ)θm|u|2βudt+λθmudt=θmG(x,u(tρ))dt
    +k=1θm(σ1,k+κσ2,k)dWk(t). (3.22)

    By (3.2), We can find μ2 sufficiently small such that

    μ2+22a+4κ2Lk=1γ2k2λ<0. (3.23)

    By (3.22) and integration by parts of Ito's formula and taking the expectation, we obtain

        E[θmu2]+2t0eμ2(st)E[Rnθ2m|u|2β+2dx]ds=eμ2tE[θmφ(0)2]2t0eμ2(st)E[Re(1+iν)((Δ)α2u,(Δ)α2(θ2mu))]ds+(μ22λ)t0eμ2(st)E[θmu2]ds+2t0eμ2(st)E[Re(θmu,θmG(x,u(sρ)))]ds+k=1t0eμ2(st)E[θm(σ1,k+κ(x)σ2,k(u(s)))2]ds. (3.24)

    For the first term in the second row of (3.24), since φ(s)L2(Ω,C([ρ,0],H)), we have for all s[ρ,0], E[φ(0)2]<. It follows that for any ε>0, there exists a positive N1=N1(ε,φ)1, for all mN1, one has |x|mE[φ2(0,x)]dx<ε. Consequently,

        E[θmφ(0)2]=E[Rn|θ(xm)φ(0,x)|2dx]=E[|x|m|θ(xm)φ(0,x)|2dx]|x|mE[|φ(0,x)|2]dx<ε,   mN1. (3.25)

    Now we consider the second term on the right-hand side of (3.24). We first have

        2E[Re(1+iν)((Δ)α2u(s),(Δ)α2(θ2mu(s)))]=C(n,α)E[Re(1+iν)RnRn[u(x)u(y)][θ2m(x)ˉu(x)θ2m(y)ˉu(y)]|xy|n+2α]dxdy=C(n,α)E[Re(1+iν)RnRn[u(x)u(y)][θ2m(x)(ˉu(x)ˉu(y))+ˉu(y)(θ2m(x)θ2m(y))]|xy|n+2α]dxdy=C(n,α)E[Re(1+iν)RnRnθ2m(x)|u(x)u(y)|2|xy|n+2αdxdy]C(n,α)E[Re(1+iν)RnRn(u(x)u(y))(θ2m(x)θ2m(y))ˉu(y)|xy|n+2αdxdy]C(n,α)E[Re(1+iν)RnRn(u(x)u(y))(θ2m(x)θ2m(y))ˉu(y)|xy|n+2αdxdy]C(n,α)1+ν2E[|RnRn(u(x)u(y))(θ2m(x)θ2m(y))ˉu(y)|xy|n+2αdxdy|]2C(n,α)1+ν2E[Rn|ˉu(y)|(Rn|(u(x)u(y))(θm(x)θm(y))||xy|n+2αdx)dy]2C(n,α)1+ν2E[u(s)(Rn(Rn|(u(x)u(y))(θm(x)θm(y))||xy|n+2αdx)2dy)12]2C(n,α)1+ν2E[u(s)(Rn(Rn|u(x)u(y)|2|xy|n+2αdxRn|(θm(x)θm(y))|2|xy|n+2αdx)dy)12]. (3.26)

    We now prove the following inequality:

    Rn|(θm(x)θm(y))|2|xy|n+2αdxc6m2α. (3.27)

    Let xy=h and hm=z, then, we obtain,

        Rn|(θm(x)θm(y))|2|xy|n+2αdx=Rn|θ(y+hm)θ(ym)|2|h|n+2αdh=Rn|θ(ym+z)θ(ym)|2mn+2α|z|n+2αmndz=1m2αRn|θ(ym+z)θ(ym)|2|z|n+2αdz=1m2α|z|1|θ(ym+z)θ(ym)|2|z|n+2αdz+1m2α|z|>1|θ(ym+z)θ(ym)|2|z|n+2αdzc6m2α|z|1|z|2|z|n+2αdz+4m2α|z|>11|z|n+2αdzc6m2α|z|11|z|n+2α2dz+4m2α|z|>11|z|n+2αdzc6ˉc6m2α+4˜c6m2α=c6ˉc6+4˜c6m2α. (3.28)

    This proves (3.27) with c6:=c6ˉc6+4˜c6. By (3.26) and (3.27), we obtain,

        2E[Re(1+iν)((Δ)α2u(s),(Δ)α2θ2mu(s))]2c6(1+ν2)C(n,α)mαE[u(s)RnRn|u(x)u(y)|2|xy|n+2αdxdy]c6(1+ν2)C(n,α)mα(E(u(s)2)+E(RnRn|u(x)u(y)|2|xy|n+2αdxdy))c6(1+ν2)C(n,α)mαE(u(s)2)+2c6(1+ν2)mαE((Δ)α2u(s)2). (3.29)

    By (3.29), for the second term on the right-hand side of (3.24), we get

        2t0eμ2sE[Re(1+iν)((Δ)α2u(s),(Δ)α2θ2mu(s))]dsc6(1+ν2)C(n,α)mαt0eμ2sE[u(s)2]ds+2c6(1+ν2)mαt0eμ2sE[(Δ)α2u(s)2]ds. (3.30)

    By Lemma 3.1, we have

        c6(1+ν2)C(n,α)mαt0eμ2(st)E[u(s)2]dsc6(1+ν2)C(n,α)mα[M1E[supρs0φ(s)2]+˜M1]t0eμ2(st)dsc6(1+ν2)C(n,α)mα1μ2[M1E[supρs0φ(s)2]+˜M1]. (3.31)

    By (3.31), we deduce that there exists N2(ε,φ)N1, for all t0 and mN2,

    c6(1+ν2)C(n,α)mαt0eμ2(st)E[u(s)2]ds<ε.

    By Lemma 3.1, there exists N3(ε,φ)N2 such that for all t0 and mN3,

        2c6(1+ν2)mαt0eμ2(st)E[(Δ)α2u2]ds2c6(1+ν2)mα[M1E[supρs0φ(s)2]+˜M1]<ε.

    For the forth term on the right-hand side of (3.24), we obtain that there exists N4(ε,φ)N3, for all t0 and mN4,

        2t0eμ2(st)E[Re(θmu,θmG(x,u(sρ)))]ds2at0eμ2(st)E[θmu(s)2]ds+12at0eμ2(st)E[θmG(x,u(sρ))2]ds2aμ2|x|mh2(x)dx+2a0ρeμ2(st)E[θmφ(s)2]ds+22at0eμ2(st)E[θmu(s)2]ds2aμ2ε+2a0ρeμ2(st)E[θmφ(s)2]ds+22at0eμ2(st)E[θmu(s)2]ds.

    Since {φ(s)L2(Ω,H)|s[ρ,0]} is compact, it has a open cover of balls with radius ε2 which denoted by {B(φi,ε2)}li=1. Since φi=φ(si)L2(Ω;C([ρ,0],H)) for i=1,2,,l, we obtain that for given ε>0,

    {φ(s)L2(Ω;C([ρ,0],H))}li=1{XL2(Ω,H)|XφiL2(Ω,H)<ε2}.

    Since φiL2(Ω,H), there exists a positive constant N5=N5(ε,φ)N4, for mN5, we have

    supi=1,2,,l|x|mE[|φ(si,x)|2]dx<ε4.

    Then,

    sups[ρ,0]|x|mE[|φ(s,x)|2]dx<ε2,mN5.

    Consequently, one has

        2t0eμ2(st)E[Re(θmu,θmG(x,u(sρ)))]ds2aμ2ε+2aρε2+22at0eμ2(st)E[θmu(s)2]ds. (3.32)

    For the fifth term on the right-hand side of (3.24), by (2.6), we obtain

        k=1t0eμ2(st)E[θm(σ1,k+κ(x)σ2,k(u(s)))2]ds2k=1t0eμ2(st)θmσ1,k2ds+2k=1t0eμ2(st)E[θmκ(x)σ2,k(u(s))2]ds2μ2k=1|x|m|σ1,k(x)|2dx+4μ2k=1β2k|x|mκ2(x)dx+4κ(x)2Lk=1γ2kt0eμ2(st)E[θmu(s)2]ds.

    Since k=1σ1,k2< and κ(x)L2(Rn)L(Rn), there exists N6=N6(ε,φ)N5, for all t0 and mN6, we have

    k=1|x|m|σ1,k(x)|2dx+|x|mκ2(x)dx<ε.

    Consequently, for the fifth term on the right-hand side of (3.24), we get for all t0 and mN6,

    k=1t0eμ2(st)E[θm(σ1,k+κσ2,k)2]ds2μ2(1+2k=1β2k)ε
    +4κ2Lk=1γ2kt0eμ2(st)E[θmu(s)2]ds.

    Therefore, for all t0 and mN6,

    E[θmu(t)2][2+eμ2t+2aμ2+22aρ+2μ2(1+2k=1β2k)]ε
    +(μ22λ+22a+4κ2Lk=1γ2k)t0eμ2(st)E[θmu(s)2]ds.

    Taking the limit in the above equation and by (3.23), we have

    lim supmsuptρ|x|mE[|u(t,x)|2]dx=0,

    which completes the proof.

    Lemma 3.4. Suppose (2.1)–(2.6) and (3.2) hold. If φ(s)L2(Ω,C([ρ,0],H)), then the solution u of (1.1) and (1.2) satisfies

    lim supmsupt0E[supr[tρ,t]|x|m|u(r,x)|2dx]=0.

    Proof. By (3.22) and integration by parts of Ito's formula and taking the real part, for all tρ and r[tρ,t], we have

        eμ2rθmu(r)2+2rtρeμ2sRnθ2m|u|2β+2dxds=eμ2(tρ)θmu(tρ)22rtρeμ2sRe(1+iν)((Δ)α2u(s),(Δ)α2θ2mu(s))ds+(μ22λ)rtρeμ2sθmu(s)2ds+2Rertρeμ2s(θmu(s),θmG(x,u(sρ)))ds+k=1rtρeμ2sθm(σ1,k+κ(x)σ2,k(u(s)))2ds+2Rek=1rtρeμ2s(θmu(s),θm(σ1,k+κσ2,k(u(s))))dWk(s). (3.33)

    By (3.33), we deduce,

        E[suptρrtθmu(r)2]E[θmu(tρ)2]2E[suptρrtrtρeμ2(sr)Re(1+iν)((Δ)α2u,(Δ)α2θ2mu)ds]+|μ22λ|E[suptρrtrtρθmu2eμ2(sr)ds]+2E[suptρrtrtρeμ2(sr)θmuθmG(x,u(sρ))ds]+k=1E[suptρrtrtρeμ2(sr)θm(σ1,k+κσ2,k(u(s)))2ds]+2E[suptρrt|k=1rtρeμ2(sr)(θmu(s),θm(σ1,k+κσ2,k(u(s))))dWk(s)|]. (3.34)

    For the first term on the right-hand side of (3.34), by Lemma 3.3, one has for any ε>0, there exists ˜N1(ε,φ)1 such that for all m˜N1 and tρ,

    E[θmu(tρ)2]|x|mE[|u(tρ,x)|2]dx<ε. (3.35)

    For the second term on the right-hand side of (3.34), by (3.29), we have

        2E[suptρrtrtρeμ2(sr)Re(1+iν)((Δ)α2u(s),(Δ)α2θ2mu(s))ds]2c6(1+ν2)C(n,α)mαE[suptρrt(rtρeμ2(sr)u(s)(Δ)α2u(s)ds)]2c6(1+ν2)C(n,α)mαeμ2ρE[(ttρeμ2(st)u(s)(Δ)α2u(s)ds)]c6(1+ν2)C(n,α)mαeμ2ρ{ttρeμ2(st)E[u2]ds+E[ttρeμ2(st)(Δ)α2u2ds]}c6(1+ν2)C(n,α)mαeμ2ρ{ρsups[tρ,t]E[u(s)2]+E[ttρeμ2(st)(Δ)α2u2ds]}. (3.36)

    By Lemma 3.1 and (3.36), we deduce that there exists ˜N2(ε,φ)˜N1 such that for all m˜N2 and tρ,

    2E[suptρrtrtρeμ2(sr)Re(1+iν)((Δ)α2u(s),(Δ)α2θ2mu(s))ds]<ε. (3.37)

    For the third term on the right-hand side of (3.34), by Lemma 3.3, we obtain that for all m˜N2 and tρ,

    |μ22λ|E[suptρrtrtρθmu(s)2eμ2(sr)ds]|μ22λ|E[ttρθmu(s)2ds]
    |μ22λ|ρsuptρstE[θmu(s)2]<|μ22λ|ρε. (3.38)

    For the forth term on the right-hand side of (3.34), by (2.1), we obtain

    2E[suptρrtrtρeμ2(sr)θmu(s)θmG(x,u(sρ))ds]ttρE[θmu(s)2]ds+2ρθmh2+2a2tρt2ρE[θmu(s)2]dsρsuptρstE[θmu(s)2]+2ρθmh2+2a2ρsupt2ρstρE[θmu(s)2],

    which along with Lemma 3.3, we deduce that there exists ˜N3(ε,φ)˜N2 such that for all m˜N3 and tρ,

    2E[suptρrtrtρeμ2(sr)θmu(s)θmG(x,u(sρ))ds]<(3+2a2)ρε. (3.39)

    For the fifth term on the right-hand side of (3.34), by (2.6), we have

        k=1E[suptρrtrtρeμ2(sr)θm(σ1,k+κσ2,k(u(s)))2ds]2ρk=1θmσ1,k2+2ρk=1suptρstE[θmκ(x)σ2,k(u(s))2]2ρk=1|x|m|σ1,k(x)|2dx+4ρk=1β2k|x|m|κ(x)|2dx+4ρκ(x)2Lk=1γ2ksuptρstE[θmu(s)2].

    By the condition κ(x)L2(Rn)L(Rn), (2.4) and Lemma 3.3, we deduce that there exists ˜N4(ε,φ)˜N3 such that for all m˜N4 and tρ,

    k=1E[suptρrtrtρeμ2(sr)θm(σ1,k+κσ2,k(u(s)))2ds]<2ρ(1+λ+2k=1β2k)ε. (3.40)

    For the sixth term on the right-hand side of (3.34), by (2.6), (3.40) and Burkholder-Davis-Gundy's inequality, we have,

        2E[suptρrt|k=1rtρeμ2(sr)(θmu(s),θm(σ1,k+κσ2,k(u(s))))dWk(s)|]2eμ2(tρ)E[suptρrt|k=1rtρeμ2s(θmu(s),θmσ1,k+θmκ(x)σ2,k(u(s)))dWk(s)|]2˜B2eμ2(tρ)E[(ttρe2μ2sk=1|(θmu(s),θmσ1,k+θmκ(x)σ2,k(u(s)))|2ds)12]2˜B2eμ2(tρ)E[suptρstθmu(s)(ttρe2μ2sk=1θmσ1,k+θmκσ2,k(u(s))2ds)12]12E[suptρstθmu(s)2]+2˜B22E[e2μ2ρttρe2μ2(st)k=1θmσ1,k+θmκ(x)σ2,k(u(s))2ds]12E[suptρstθmu(s)2]+2˜B22e2μ2ρk=1E[suptρrtrtρeμ2(sr)θmσ1,k+θmκ(x)σ2,k(u(s))2ds]12E[suptρstθmu(s)2]+4ρ(1+λ+2k=1β2k)˜B22e2μ2ρε.

    Above all, for all m˜N4 and tρ, we obtain,

    E[suptρrtθmu(r)2][4+2|μ22λ|ρ+(6+4a2)ρ+4ρ(1+2˜B22e2μ2ρ)(1+λ+2k=1β2k)]ε.

    Therefore, we conclude

    lim supmsupt0E[suptρrt|x|m|u(r,x)|2dx]=0.

    Lemma 3.5. Suppose (2.1)–(2.6) and (3.1) hold. If φ(s)L2(Ω,C([ρ,0],H)), then there exists a positive constant μ3 such that the solution u of (1.1) and (1.2) satisfies

     suptρE[u(t)2p]+supt0E[t0eμ3(st)u(s)2p2(Δ)α2u(s)2ds]        (1+aρeμρ2p(4p2)2p12p)E[φ2pCH]+M3, (3.41)

    where M3 is a positive constant independent of φ.

    Proof. By (3.1), there exist positive constants μ and ϵ1 such that

     μ+aeμρ2p2112p(2p1)2p12p+4(p1)(2p1)ϵ2p2p21k=1(σ1,k2+κ2β2k)         +4θ(2p1)κ2Lk=1γ2k2pλ. (3.42)

    Given nN, let τn be a stopping time as defined by

    τn=inf{t0:u(t)>n},

    and as usual, we set τn=+ if {t0:u(t)>n}=. By the continuity of solutions, we have

    limnτn=+.

    Applying Ito's formula, we obtain

        d(u(t)2p)=d((u(t)2)p)=pu(t)2(p1)d(u(t)2)+2p(p1)u(t)2(p2)    ×k=1|(u(t),σ1,k+κσ2,k(u(t)))|2dt. (3.43)

    Substituting (3.6) into (3.43), we infer

    d(u(t)2p)=2pu(t)2(p1)(Δ)α2u(t)2dt2pu(t)2(p1)u(t)2β+2L2β+2dt2pλu(t)2pdt     +2pu(t)2(p1)Re(u(t),G(x,u(tρ)))dt     +pu(t)2(p1)k=1σ1,k(x)+κ(x)σ2,k(u(t))2dt     +2pu(t)2(p1)Re(u(t),k=1σ1,k(x)+κ(x)σ2,k(u(t)))dWk(t)     +2p(p1)u(t)2(p2)k=1|(u(t),σ1,k+κσ2,k(u(t)))|2dt. (3.44)

    We also get the formula

    d(eμtu(t)2p)=μeμtu(t)2pdt+eμtd(u(t)2p). (3.45)

    Substituting (3.44) into (3.45) and integrating on (0,tτn) with t0, we deduce

        eμ(tτn)u(tτn)2p+2ptτn0eμsu(s)2(p1)(Δ)α2u(s)2ds=2ptτn0eμsu(s)2(p1)u(s)2β+2L2β+2ds+φ(0)2p+(μ2pλ)tτn0eμsu(s)2pds      +2ptτn0eμsu(s)2(p1)Re(u(s),G(x,u(sρ)))ds      +pk=1tτn0eμsu(s)2(p1)σ1,k+κσ2,k(u(s))2ds      +2pk=1tτn0eμsu(t)2(p1)Re(u(s),σ1,k+κσ2,k(u(s)))dWk(s)      +2p(p1)k=1tτn0eμsu(s)2(p2)|(u(s),σ1,k+κσ2,k(u(s)))|2ds. (3.46)

    Taking the expectation, we obtain for t0,

        E[eμ(tτn)u(tτn)2p]+2pE[tτn0eμsu(s)2(p1)(Δ)α2u(s)2ds]=2pE[tτn0eμsu(s)2(p1)u(s)2β+2L2β+2ds]+E[φ(0)2p]+(μ2pλ)E[tτn0eμsu(s)2pds]      +2pE[tτn0eμsu(s)2(p1)Re(u(s),G(x,u(sρ)))ds]      +pk=1E[tτn0eμsu(s)2(p1)σ1,k+κσ2,k(u(s))2ds]      +2p(p1)k=1E[tτn0eμsu(s)2(p2)|(u(s),σ1,k+κσ2,k(u(s)))|2ds]E[φ(0)2p]+(μ2pλ)E[tτn0eμsu(s)2pds]      +2pE[tτn0eμsu(s)2(p1)Re(u(s),G(x,u(sρ)))ds]      +pk=1E[tτn0eμsu(s)2(p1)σ1,k+κσ2,k(u(s))2ds]      +2p(p1)k=1E[tτn0eμsu(s)2(p2)|(u(s),σ1,k+κσ2,k(u(s)))|2ds]. (3.47)

    Next, we estimate the terms on the right-hand side of (3.47).

    For the third term on the right-hand side of (3.47), by Young's inequality and (2.1), we infer

        2θE[tτn0eμsu(s)2(p1)Re(u(s),G(x,u(sρ)))ds]2θE[tτn0eμsu(s)2p1G(x,u(sρ))2ds]aeμρ2p2112p(2p1)2p12pE[tτn0eμsu(s)2pds]        +(2p122p1a2peμρ)2p12pE[tτn0eμsG(x,u(sρ))2ds]aeμρ2p2112p(2p1)2p12pE[tτn0eμsu(s)2pds]        +22p1(2p122p1a2peμρ)2p12pE[tτn0eμs(h2p+a2pu(sρ)2p)ds]aeμρ2p2112p(2p1)2p12pE[tτn0eμsu(s)2pds]        +1μ(4p2a2peμρ)2p12ph2peμt+aρeμρ2p(4p2)2p12pE[φ2pCH]. (3.48)

    For the forth term on the right-hand side of (3.47), we infer

        pk=1E[tτn0eμsu(s)2(p1)σ1,k+κσ2,k(u(s))2ds]2pk=1E[tτn0eμsu(s)2(p1)σ1,k2ds]      +2pk=1E[tτn0eμsu(s)2(p1)κσ2,k(u(s))2ds]. (3.49)

    For the first term on the right-hand side of (3.49), we have

        2pk=1E[tτn0eμsu(s)2(p1)σ1,k2ds]2(p1)ϵ2p2p21k=1σ1,k2E[tτn0eμsu(s)2pds]+2μϵp1k=1σ1,k2eμt. (3.50)

    For the second term on the right-hand side of (3.49), we have

        2pk=1E[tτn0eμsu(s)2(p1)κσ2,k(u(s))2ds]4pκ2k=1β2kE[tτn0eμsu(s)2(p1)ds]+4pκ2Lk=1γ2kE[tτn0eμsu(s)2pds]4(p1)ϵ2p2p21κ2k=1β2kE[tτn0eμsu(s)2pds]    +4μϵp1κ2k=1β2keμt+4pκ2Lk=1γ2kE[tτn0eμsu(s)2pds]. (3.51)

    By (3.49)–(3.51), we obtain

        pk=1E[tτn0eμsu(s)2(p1)σ1,k+κσ2,k(u(s))2ds][4(p1)ϵ2p2p21k=1(σ1,k2+κ2β2k)+4pκ2Lk=1γ2k]E[tτn0eμsu(s)2pds]    +2μϵp1k=1(σ1,k2+2κ2β2k))eμt. (3.52)

    For the fifth term on the right-hand side of (3.47), applying (3.52), we have

        2p(p1)k=1E[tτn0eμsu(s)2(p2)|(u(s),σ1,k+κσ2,k(u(s)))|2ds]2p(p1)k=1E[tτn0eμsu(s)2p2σ1,k+κσ2,k(u(s))2ds][8(p1)2ϵ2p2p21k=1(σ1,k2+κ2β2k)+8p(p1)κ2Lk=1γ2k]E[tτn0eμsu(s)2pads]    +4(p1)μϵp1k=1(σ1,k2+2κ2β2k))eμt. (3.53)

    From (3.47), (3.48), (3.52) and (3.53), we obtain that for t0,

        E[eμ(tτn)u(tτn)2p]+2pE[tτn0eμsu(s)2(p1)(Δ)α2u(s)2ds](1+aρeμρ2p(4p2)2p12p)E[φ2pCH]   +(μ2pλ+aeμρ2p2112p(2p1)2p12p+4(p1)(2p1)ϵ2p2p21    ×k=1(σ1,k2+κ2β2k)+4p(2p1)κ2Lk=1γ2k)E[tτn0eμsu(s)2pds]      +1μ(4p2a2peμρ)2p12ph2peμt+4(p1)μϵp1k=1(σ1,k2+2κ2β2k))eμt. (3.54)

    Then by (3.42) and (3.54), we obtain that for t0,

        E[eμ(tτn)u(tτn)2p]+2pE[tτn0eμsu(s)2(p1)(Δ)α2u(s)2ds](1+aρeμρ2p(4p2)2p12p)E[φ2pCH]+1μ(4p2a2paeμρ)2p12ph2peμt   +4(p1)μϵp1k=1(σ1,k2+2κ2β2k))eμt. (3.55)

    Letting n, by Fatou's Lemma, we deduce that for t0,

        E[eμtu(t)2p]+2pE[t0eμsu(s)2(p1)(Δ)α2u(s)2ds](1+aρeμρ2p(4θ2)2θ12p)E[φ2pCH]+1μ(4p2a2peμρ)2p12ph2peμt   +4(p1)μϵp1k=1(σ1,k2+2κ2β2k))eμt.

    Hence, we have for t0,

        E[u(t)2p]+2pE[t0eμ(st)u(s)2(p1)(Δ)α2u(s)2ds](1+aρeμρ2p(4p2)2p12p)E[φ2pCH]+1μ(4p2a2peμρ)2p12ph2p   +4(p1)μϵp1k=1(σ1,k2+2κ2β2k)).

    This implies the desired estimate.

    In this section, we establish the uniform estimates of solutions of problems (1.1) and (1.2) with initial data in C([ρ,0],V). To the end, we assume that for each kN, the function σ1,kV and

    k=1σ1,k2V<. (4.1)

    Furthermore, we assume that the function κV and there exists a constant C>0 such that

    |κ(x)|C. (4.2)

    In the sequel, we further assume that the constant a, ˆγk in (2.7) are sufficiently small in the sense that there exists a constant p2 such that

    ˆa2112p(2p1)2p12p+2p(2p1)κ2Lk=1(β2k+ˆβ2k+γ2k+ˆγ2k)<pλ2. (4.3)

    By (4.3), we can find

    2ˆa+2κ2Lk=1ˆγ2k<λ2. (4.4)

    Lemma 4.1. Suppose (2.1)–(2.7) and (4.4) hold. If φ(s)L2(Ω;C([ρ,0],V)), then, for all t0, there exists a positive constant μ4 such that the solution u of (1.1) and (1.2) satisfies

    supsρE[u(t)2]+sups0E[t0eμ4(st)(Δ)α+12u(s)2ds]M4(E[φ2CV]+1), (4.5)

    where M4 is a positive constant independent of φ.

    Proof. By (4.4), there exists a positive constant μ1 such that

    μ12λ+8κ2Lk=1ˆγ2k<0. (4.6)

    By (1.1) and applying Ito's formula to eμ1tu(t)2, we have for t0,

        eμ1tu(t)2+2t0eμ1s(Δ)α+12u(s)2ds+2t0eμ1sRe((1+iμ)|u(s)|2βu(s),Δu(s))ds=(μ12λ)t0eμ1su(s)2ds+φ(0)2+2Ret0eμ1s(G(x,u(sρ)),Δu(s))ds    +k=1t0eμ1s(σ1,k+κσ2,k(u(s)))2ds     +2k=1Ret0eμ1s(σ1,k(x)+κ(x)σ2,k(u(s)),Δu(s))dWk(s).

    Taking the expectation, we have for all t0,

        eμ1tE[u(t)2]+2E[t0eμ1s(Δ)α+12u(s)2ds]+2E[t0eμ1sRe((1+iμ)|u(s)|2βu(s),Δu(s))ds]=(μ12λ)E[t0eμ1su(s)2ds]+E[φ(0)2]+2E[Ret0eμ1s(G(x,u(sρ)),Δu(s))ds]    +k=1E[t0eμ1s(σ1,k+κσ2,k(u(s)))2ds]. (4.7)

    First, we estimate the third term on the left-hand side of (4.7). Applying integrating by parts, we have

        Re((1+iμ)|u|2βu,Δu)=Re(1+iμ)Rn((β+1)|u|2β|u|2+β|u|2(β1)(u¯u)2)dx=Rn|u|2(β1)((β+1)|u|2|u|2+β(1+iμ)2(u¯u)2+β(1iμ)2(¯uu)2)dx=Rn|u|2(β1)trace(YMYH), (4.8)

    where

    Y=(¯uuu¯u)H,M=(β+12β(1+iμ)2β(1iμ)2β+12),

    and YH is the conjugate transpose of the matrix Y. We observe that the condition β11+μ21 implies that the matrix M is nonpositive definite. Hence, we obtain

    2E[t0eμ1sRe((1+iμ)|u(s)|2βu(s),Δu(s))ds]0. (4.9)

    Next, we estimate the terms on the right-hand side of (4.7). For the third term on the right-hand side of (4.7), applying (2.2) and Gagliardo-Nirenberg inequality, we have

        2E[Ret0eμ1s(G(x,u(sρ)),Δu(s))ds]2E[t0eμ1su(s)G(x,u(sρ))ds]E[t0eμ1su(s)2ds]+E[t0eμ1sG(x,u(sρ))2ds]E[t0eμ1su(s)2ds]+2E[t0eμ1sˆh(x)2ds]+2ˆa2E[t0eμ1su(sρ)2ds]E[t0eμ1s(Δ)α+12u(s)2ds]+2μ1ˆh(x)2eμ1t    +cμ1sups0E[u(s)2]eμ1t+2ˆa2μ1supρs0E[φ(s)2]eμ1t, (4.10)

    where c is a positive constant from Gagliardo-Nirenberg inequality. For the forth term on the right-hand side of (4.7), applying (2.6) and (2.7), we have

        k=1E[t0eμ1s(σ1,k+κσ2,k(u(s)))2ds]2k=1E[t0eμ1s(σ1,k2+(κσ2,k(u(s)))2)ds]2μ1k=1σ1,k2eμ1t+8k=1E[t0eμ1s(β2kκ2+ˆβ2kκ2+γ2kC2u(s)2+ˆγ2kκ2Lu(s)2)ds]2μ1k=1(σ1,k2+4β2kκ2+4ˆβ2kκ2+4C2γ2ksups0E[u(s)2])eμ1t     +8k=1ˆγ2kκ(x)2LE[t0eμ1su(s)2ds]. (4.11)

    By (4.7), (4.10) and (4.11), we obtain

        E[u(t)2]+E[t0eμ1(st)(Δ)α+12u(s)2ds] E[φ(0)2]eμ1t+2μ1ˆh(x)2+(μ12λ+8κ2Lk=1ˆγ2k)E[t0eμ1su(s)2ds]    +2μ1(c2+4(C2k=1γ2k+cκ2Lk=1ˆγ2k))supsρE[u(s)2]    +2μ1k=1(σ1,k2+4(β2k+ˆβ2k)κ2V)+2ˆa2μ1supρs0E[φ(s)2]. (4.12)

    Then by (4.6) and (4.12), we obtain that for all t0,

        E[u(t)2]+E[t0eμ1(st)(Δ)α+12u(s)2ds] E[φ(0)2]eμ1t+2μ1ˆh(x)2+2μ1(c2+4(C2k=1γ2k+cκ2Lk=1ˆγ2k))supsρE[u(s)2]    +2μ1k=1(σ1,k2+4(β2k+ˆβ2k)κ2V)+2ˆa2μ1supρs0E[φ(s)2]. (4.13)

    Then by (4.13) and Lemma 3.1, we obtain the estimates (4.5).

    Lemma 4.2. Suppose (2.1)–(2.7) and (4.4) hold. If φ(s)L2(Ω;C([ρ,0],V)), then the solution u of (1.1) and (1.2) satisfies

    suptρ{E[suptρrtu(r)2]}M5(E[φ2CV]+1), (4.14)

    where M5 is a positive constant independent of φ.

    Proof. By (1.1) and Ito's formula, we get for all tρ and tρrt,

        u(r)2+2rtρ(Δ)α+12u(s)2ds    +2rtρRe((1+iμ)|u(s)|2βu(s),Δu(s))ds+2λrtρu(s)2ds=u(tρ)2+2Rertρ(G(x,u(sρ)),Δu(s))ds    +k=1rtρ(σ1,k+κσ2,k(u(s)))2ds     +2k=1Rertρ(σ1,k(x)+κ(x)σ2,k(u(s)),Δu(s))dWk(s). (4.15)

    For the third term on the left-hand side of (4.15), applying (4.8), we have

    2rtρRe((1+iμ)|u(s)|2βu(s),Δu(s))ds0. (4.16)

    For the second term on the right-hand side of (4.15), applying (2.2) and Gagliardo-Nirenberg inequality, we have

        2Rertρ(G(x,u(sρ)),Δu(s))ds2rtρu(s)G(x,u(sρ))dsrtρu(s)2ds+rtρG(x,u(sρ))2dsrtρu(s)2ds+2rtρˆh(x)2ds+2ˆa2rtρu(sρ)2dsrtρ(Δ)α+12u(s)2ds+2ρˆh(x)2+2ˆa2rρt2ρu(s)2ds+crtρu(s)2ds. (4.17)

    For the third term on the right-hand side of (4.15), applying (2.6) and (2.7), we have

        k=1rtρ(σ1,k+κσ2,k(u(s)))2ds2k=1rtρ(σ1,k2+(κσ2,k(u(s)))2)ds2ρk=1σ1,k2+8ρ(κ2k=1β2k+κ2k=1ˆβ2k)     +8C2k=1γ2krtρu(s)2ds+8κ2Lk=1ˆγ2krtρu(s)2ds. (4.18)

    By (4.4) and (4.15)–(4.18), we infer that for all tρ and tρrt,

        u(r)2c1+u(tρ)2+c2rt2ρu(s)2ds+2ˆa2rt2ρu(s)2ds     +2k=1Rertρ(σ1,k(x)+κ(x)σ2,k(u(s)),Δu(s))dWk(s), (4.19)

    where c1 and c2 are positive constants. By (4.19), we deduce that for all tρ,

        E[suptρrtu(r)2]c1+E[u(tρ)2]+c2rt2ρE[u(s)2+u(s)2]ds     +2E[suptρrt|k=1rtρ(σ1,k(x)+κ(x)σ2,k(u(s)),Δu(s))dWk(s)|]. (4.20)

    For the second term on the right-hand side of (4.20), by Lemma 4.1 we infer that for all tρ,

        E[u(tρ)2]supsρE[u(s)2]c3E[φ2CV]+c3. (4.21)

    For the third term on the right-hand side of (4.20), by Lemmas 3.1 and 4.1 we infer that for all tρ,

        c2rt2ρE[u(s)2+u(s)2]ds2ρc2supsρE[u(s)2+u(s)2]c4E[φ2CV]+c4. (4.22)

    For the last term on the right-hand side of (4.20), by BDG inequality, (4.18), Lemmas 3.1 and 4.1, we deduce that for all tρ,

        2E[suptρrt|k=1rtρ(σ1,k(x)+κ(x)σ2,k(u(s)),Δu(s))dWk(s)|]2c5E[(k=1ttρ|(σ1,k(x)+κ(x)σ2,k(u(s)),Δu(s))|2ds)12]2c5E[(k=1ttρu(s)2(σ1,k(x)+κ(x)σ2,k(u(s)))2ds)12]2c5E[suptρstu(s)(k=1ttρ(σ1,k(x)+κ(x)σ2,k(u(s)))2ds)12]12E[suptρstu(s)2]+2c25E[k=1ttρ(σ1,k(x)+κ(x)σ2,k(u(s)))2ds]12E[suptρstu(s)2]+c6+c6ttρE[u(s)2+u(s)2]ds12E[suptρstu(s)2]+c6+ρc6(sups0Eu(s)2+sups0u(s)2)12E[suptρstu(s)2]+c7E[φ2CV]+c7. (4.23)

    By (4.20)–(4.23), we obtain that for all tρ,

    E[suptρrtu(r)2]c8E[φ2CV]+c9,

    which completes the proof.

    Lemma 4.3. Suppose (2.1)–(2.7) and (3.1) hold. If φ(s)L2p(Ω,C([ρ,0],V)), then there exists a positive constant μ5 such that the solution u of (1.1) and (1.2) satisfies

    (4.24)

    where is a positive constant independent of .

    Proof. By (3.1), there exist positive constants and such that

    (4.25)

    By (1.1) and applying Ito's formula to , we get for ,

    Taking the expectation, we have for ,

    (4.26)

    By (4.8), we get the third term on the left-hand side of (4.26) is nonnegative. Next, we estimate each term on the right-hand side of (4.26). For the third term on the right-hand side of (4.26), applying (2.2), Gagliardo-Nirenberg inequality and Young's inequality, we deduce

    (4.27)

    For the forth term on the right-hand side of (4.26), applying (2.7), we infer

    (4.28)

    Then applying Young's inequality, (4.28) can be estimated by

    (4.29)

    For the fifth term on the right-hand side of (4.26), applying integrating by parts and (4.29), we get

    (4.30)

    By (4.26), (4.27), (4.29) and (4.30), we obtain

    (4.31)

    Then by (4.25) and (4.31), we deduce that for all ,

    (4.32)

    Therefore, by (4.32) and Lemma 3.5, there exists a constant independent of such that

    (4.33)

    For convenience, we write . Then, similar to Theorem 6.5 in [31], the solution of (1.1) and (1.2) can be expressed as

    (4.34)

    The next lemma is concerned with the Hlder continuity ofsolutions in time which is needed to prove the tightness of distributions of solutions.

    Lemma 4.4. Suppose (2.1)–(2.7) and (3.1) hold. If , then the solution of (1.1) and (1.2) satisfies, for any ,

    (4.35)

    where is a positive constant depending on , but independent of and .

    Proof. By (4.34), we get for ,

    (4.36)

    Then we infer

    (4.37)

    Taking the expectation of (4.36), we have for all ,

    (4.38)

    For the first term on the right-hand side of (4.38), by Theorem 1.4.3 in [32], we find that there exists a positive number depending on such that for all ,

    Applying Lemmas 3.5 and 4.3, we obtain for all ,

    (4.39)

    For the second term on the right-hand side of (4.38), by the contraction property of , we infer that for all ,

    We deduce the estimate similarly to Lemma 3.5 together with Lemma 3.3 in [1]. Hence, the second term on the right-hand side of (4.38) can be estimated by

    (4.40)

    For the third term on the right-hand side of (4.38), by the contraction property of and (2.1) and Lemma 3.5, we deduce that for all ,

    (4.41)

    For the forth term the right-hand side of (4.38), from the BDG inequality, the contraction property of , (2.6) Hlder's inequality and Lemma 3.5, we deduce

    (4.42)

    Therefore, from (4.38)–(4.42), we obtain there exists independent of and , such that for all ,

    The proof is complete.

    In this section, we first recall the definition of invariant measure and transition operator. Then we construct a compact subset of in order to prove the tightness of the sequence of invariant measure on .

    Recall that for any initial time and every -measurable function , problems (1.1) and (1.2) has a unique solution for . For convenience, given and -measurable function , the segment of on is written as

    Then for all . We introduce the transition operator for (1.1). If is a bounded Borel function, then for initial time with and , we write

    Particularly, for , and , we have

    where is the characteristic function of . Then is the distribution of in . In the following context, we will write as .

    Recall that a probability measure on is called an invariant measure, if for all and every bounded and continuous function

    According to [33], we infer that the transition operator has the following properties.

    Lemma 5.1. Suppose (2.1)–(2.7) and (4.1)–(4.3) hold. One has

    (a) The family is Feller; that is, if is bounded and continuous, then for any , the function is also bounded and continuous.

    (b) The family is homogeneous (in time); that is, for any ,

    (c) Given and , the process is a -valued Markov process. Consequently, if is a bounded Borel function, then for any , -almost surely,

    and the Chapman-Kolmogorov equation is valid:

    for any and

    Now, we establish the existence of invariant measures of problems (1.1) and (1.2).

    Theorem 5.2. Suppose (2.1)–(2.7) and (4.1)–(4.3) hold. Then (1.1) and (1.2) processes an invariant measure on .

    Proof. We employ Krylov-Bogolyubov's method to the solution of problems (1.1) and (1.2), where the initial condition at the initial time 0. Because of this particular , we know that all results obtained in the previous Sections 3 and 4 are valid. For simplicity, the solution is written as and the segment as . For , we set

    (5.1)

    Step 1. We prove the tightness of in . Applying Lemmas 3.2 and 4.2, we get that there exists such that for all ,

    (5.2)

    By (5.2) and Chebyshev's inequality, we have that for all ,

    and hence for every , there exists such that for all ,

    (5.3)

    By Lemma 4.4, we get that there exists such that for all and ,

    and hence for all and ,

    (5.4)

    Since , applying (5.4) and the usual technique of dyadic division, we obtain that there exists such that for all ,

    (5.5)

    By Lemma 3.4, we get that for given and , there exists an integer such that for all ,

    which implies that for all and ,

    (5.6)

    By (5.6), we infer that for all ,

    and hence for all ,

    (5.7)

    Let

    (5.8)
    (5.9)
    (5.10)

    and

    (5.11)

    From (5.3), (5.5) and (5.7)–(5.11), we obtain that for all ,

    (5.12)

    By (5.1) and (5.12), we deduce that for all ,

    (5.13)

    Next, we prove the set is precompact in . First, we prove for every the set is a precompact subset of . By (5.8) and (5.11), we obtain that for every , the set is bounded in . Let . Then we get that the set is bounded in and hence precompact in due to compactness of the embedding . This implies that the set has a finite open cover of balls with radius in . Note that for every , there exists such that for all ,

    (5.14)

    Hence, by (5.14), the set has a finite open cover of balls with radius in . Since is arbitrary, we obtain that the set is percompact in . Then from (5.9) and (5.11), we obtain that is equicontinuous in . Therefore, by the Ascoli-Arzel theorem we deduce that is precompact in , which along with (5.13) shows that is tight on .

    Step 2. We prove the existence of invariant measures of problems (1.1) and (1.2). Since the sequence is tight on , there exists a probability measure on , we take a subsequence of (not rebel) such that In the following, we prove is an invariant measure of (1.1) and (1.2). Applying (5.1) and the Chapman-Kolmogorov equation, we obtain that for every and every ,

    which completes the proof.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors are grateful to the anonymous referees whose suggestions have in our opinion, greatly improved the paper. This work is partially supported by the NSF of Shandong Province (No. ZR 2021MA055) and USA Simons Foundation (No. 628308).

    The authors declare there is no conflict of interest.



    Conflicts of interest



    The authors declare no conflict of interest.

    Author contributions



    MA conceived the idea and drafted the manuscript. MA, AASY, SN, and BRJ searched for the literature, while MA and SN wrote the manuscript and drew the diagrams. MA, AASY, SN, and BRJ revised the manuscript critically and approved the final draft. AASY supervised the whole process.

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