Well-posedness of stochastic time fractional 2D-stokes models with finite and infinite delay

Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 86, pp. 1–29. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu WELL-POSEDNESS OF STOCHASTIC TIME FRACTIONAL 2D-STOKES MODELS WITH FINITE AND INFINITE DELAY JIAOHUI XU, TOM´ AS CARABALLO Abstract. We analyze the well-posedness of two versions of a stochastic time delay fractional 2D-Stokes model with nonlinear multiplicative noise. The main tool to prove the existence and uniqueness of mild solutions is a fixed point argument. The results for the first model can only be proved for α∈ (1/2,1), and the global existence in time is shown only when the noise is additive. As for the second model, all results are true for α∈(0,1), and the global solutions in time is shown for general nonlinear multiplicative noise. The analyzes for the finite and infinite delay cases, follow the same lines, but they require different phase spaces and estimates. This article can be considered as a first approximation to the challenging model of stochastic time fractional Navier-Stokes (with or without delay) which so far remains as an open problem. 1. Introduction Xu et al. [16] studied the well-posedness of the stochastic time delay fractional 2D-Stokes equations of order α∈(0,1) with nonlinear multiplicative noise, ∂α tu−κ∆u+∇p=f(t, ut) + g(t, ut)dW(t) dt in R2, t > 0, ∇ · u= 0 in R2, t > 0, u(t, x) = ϕ(t, x) in R2, t ∈[−h, 0] . (1.1) Here fand gare external forcing terms containing some hereditary or delay characteristics, ϕis the initial datum on the time interval [−h, 0] and his a fixed positive number (in the case of bounded delay) or h=∞(for unbounded delay), W(t) is a standard Brownian motion/Wiener process on an underlying complete filtered probability space (Ω,F,{Ft}t≥0,P). This model can be considered as a first linear approximation of the most challenging problem of the stochastic time fractional 2D-Navier-Stokes equations, ∂α tu−κ∆u+u· ∇u+∇p=f(t, ut) + g(t, ut)dW(t) dt in R2, t > 0, ∇ · u= 0 in R2, t > 0, u(t, x) = ϕ(t, x) in R2, t ∈[−h, 0], (1.2) 1991 Mathematics Subject Classification. 35R11, 35Q30, 65F08, 60H15, 65F10. Key words and phrases. Well-posedness; stochastic time fractional 2D-Stokes equations; mild solution; finite delay; infinite delay; multiplicative noise. c 2022. This work is licensed under a CC BY 4.0 license. Submitted April 23, 2022. Published December 21, 2022. 1 2 J. XU, T. CARABALLO EJDE-2022/86 which is our future final goal because of the importance of Navier-Stokes problems in the research of turbulent fluids. The deterministic model (i.e. f=g= 0) has been analyzed by Carvalho-Neto and Planas [4], and the existence (and eventual uniqueness) of mild solutions were proved by them. However, because of the reasons mentioned in [16], new techniques are necessary to handle the stochastic problem (1.2) in an appropriate way, mainly because of the difficulties generated by the nonlinear term u· ∇u. In [16], a detailed introduction and background with many references related to problem (1.2) were described, thus we recommend the reader to read it first for a better understanding of the analysis in this article. Let us now recall that in [16], the authors proved the existence and uniqueness of mild solutions to problem (1.1), for α∈(0,1), based on [16, Definition 2]. However, this definition given in [16] does not match the standard one of mild solution for an evolution equation with time fractional derivative like (1.1). Despite of the fact that all results in [16] were correctly proved according to such definition, it is desirable and necessary to state a more accurate definition that matches the usual ones for time fractional derivative which requires α∈(1/2,1). Hence, our first objective in this paper is to re-establish the results in [16] according to the new definition that we will state in this paper (see Definition 3.2 in Section 3). In addition, we will analyze another problem which contains more time regularity in the coefficients, ensuring better results for the asymptotic behavior of the system. In particular, the well-posedness can be proved for α∈(0,1). More precisely, we will consider the problem ∂α tu−κ∆u+∇p=J1−α t[f(t, ut)] + J1−α t[g(t, ut)dW(t) dt ] in R2, t > 0, ∇ · u= 0 in R2, t > 0, u(t, x) = ϕ(t, x) in R2, t ∈[−h, 0], (1.3) where J1−α tis the Riemann-Liouville fractional integral of order 1 −αdefined in Section 2. The main reasons supporting this kind of models with more regular coefficient can be found in the previous literature (see, e.g. Li and Wang [7] and the references therein), but to mention some of them we can say that, in addition that the well-posedness is proved for α∈(0,1), the expression for the mild solution (see Definition 4.1 in Section 4) only involves one of the Mittag-Leffler operators and does not contain singular kernel, which allows us to obtain better estimates in the computations, also to prove global in time existence of solutions for general multiplicative noise (not only for additive one as in the first case). Furthermore, one can have a much better asymptotic behavior of solutions because of the compactness properties of the Mittag-Leffler operator involved in the formulation. Although there are several possibilities to handle the problems with time fractional derivative, we have chosen the Caputo fractional time derivative, whose advantage, amongst others, is that the derivative of a constant function is zero. Thus, time-independent solutions are also solutions of the time-dependent problem [1]. Also, compared with Riemann-Liouville derivative [12], Caputo derivative removes singularities at the origin and shares many similarities with the classical derivative, so that they are suitable for initial value problems. We have structured our paper as follows. Section 2 is devoted to briefly recall some relevant preliminaries for our analysis. In Section 3, we first investigate the well-posedness of (1.1) by considering the case of bounded/finite delay and proving EJDE-2022/86 STOCHASTIC TIME FRACTIONAL 2D-STOKES MODELS 3 the existence, uniqueness and continuous dependence of mild solutions with respect to initial data. We can only prove that the mild solution is globally defined in time when the noise is additive. The results are shown under two sets of assumptions. Next we consider the case of unbounded/infinite delay and prove similar results but with substantial differences concerning the phase space and estimates. In contrast, these results are only valid for α∈(1/2,1). In Section 4, a parallel analysis is carried out but for (1.3). In this case, since the external forcing terms possess more regularity, we can perform the complete analysis of well-posedness and global existence in time for general nonlinear multiplicative noise term, as well as for any α∈(0,1). Eventually, some conclusions and comments about future directions are included in Section 5. 2. Preliminaries Let us recall some basic background, notation and properties of Mittag-Leffler functions; see [2, 3, 4, 11, 12, 13, 14] and the references therein for more information. 2.1. Stochastic theory and notation. First we fix a stochastic basis, S:= (Ω,F,{Ft}t≥0,P, W), where Pis a probability measure on Ω, Fis a σ-algebra, {Ft}t≥0is a rightcontinuous filtration on (Ω,F) such that F0contains all the P-negligible subsets and W(t) = W(ω, t), ω∈Ω is a standard Brownian motion defined on (Ω,F,{Ft}t≥0,P). To set problems (1.1) and (1.3) in an abstract framework, we consider the standard notation L2 σto describe the subspace of the divergence-free vector fields in L2: L2 σ={u∈L2:∇ · u= 0 in R2}, with norm k·k, where L2denotes the vector-valued Lebesgue space and for u∈L2, kuk2= 2 X j=1 ZR2 |uj(x)|2dx. In a similar way, we define Lr σfor r > 1. Besides, let S⊂Rand Xbe a Banach space. We denote the space of continuous functions from Sto Xby C(S;X) (equipped with its usual supremum norm). L2(S;X) denotes the Banach space of L2integrable functions u:S→X.H1(S;X) = W1,2(S;X) is the subspace of L2(S;X) consisting of functions such that the weak derivative ∂u ∂t belongs to L2(S;X). Both spaces L2(S;X) and W1,2(S;X) are endowed with their standard norms. Moreover, we denote a∧b= min{a, b}. Consider a fixed T > 0, given u: [−h, T]→L2 σ, for each t∈[0, T], we denote by utthe function defined on [−h, 0] as ut(s) = u(t+s), s ∈[−h, 0], where h > 0 denotes the delay. When h=∞, it denotes an infinite or unbounded delay. Furthermore, let L2(Ω; X) be the Banach space of X-valued random variables with norm ku(·)k2 L2(Ω;X)=Eku(·)k2 X, where the expectation Eis defined by Eu= RΩu(·)dP. 4 J. XU, T. CARABALLO EJDE-2022/86 2.2. Fractional setting and Mittag-Leffler operators. Let us now introduce some basic and useful results concerning the fractional calculus theory. For α > 0, define the function gα:R→Rby gα(t) := (1 Γ(α)tα−1, t > 0, 0, t ≤0, where Γ(α) denotes the Euler Gamma function. Assume that T > 0, for a function u∈L1([0, T]; X), the Riemann-Liouville fractional integral of order αof uis given by Jα tu(t) := gα∗u(t) = 1 Γ(α)Zt 0 (t−s)α−1u(s)ds, t ∈[0, T]. Thus, based on the definition of Riemann-Liouville fractional integral operator, we present the Caputo fractional differential operator (see [4, 6, 12] and the references therein for more details). Definition 2.1 ([4, Def. 1]).Let α∈(0,1) and T > 0. Consider u∈C([0, T]; X) such that the convolution g1−α∗u∈W1,1([0, T]; X). The expression Dα tu(t) := d dtJ1−α t[u(t)−u(0)]=d dt1 Γ(1 −α)Zt 0 (t−s)−α[u(s)−u(0)]ds is called the Caputo fractional derivative of order αof the function u. We recall now some properties of the Mainardi function [3] denoted by Mα. This function is a particular case of the Wright type function introduced by Mainardi in [11]. More precisely, for α∈(0,1), the entire function Mα:C→Cis Mα(z) := ∞ X n=0 (−z)n n!Γ(1 −α(1 + n)). Some basic properties of the Mainardi function are necessary in this paper to deal with most estimates in the proofs. Proposition 2.2 ([4, Pro. 2]).For α∈(0,1) and −1< r < ∞, when we restrict Mαto the positive real line, it holds that Mα(t)≥0for all t≥0,and Z∞ 0 trMα(t)dt =Γ(r+ 1) Γ(αr + 1). The next results are classical computations in the literature dealing with Mittag- Leffler operators, as can be seen, for instance, in [4]. Indeed, let Xbe a Banach space and −A :D(A)⊂X→Xbe the infinitesimal generator of an analytic semigroup {T(t) : t≥0}. Then, for each α∈(0,1), we define the Mittag-Leffler families {Eα(−tαA) : t≥0}and {Eα,α(−tαA) : t≥0}by Eα(−tαA) = Z∞ 0 Mα(s)T(stα)ds, Eα,α(−tαA) = Z∞ 0 αsMα(s)T(stα)ds. Note that the Mainardi function acts as a bridge between the fractional and the classical abstract theories. This relation is based on the inversion of certain Laplace transforms in order to obtain the fundamental solutions of the fractional diffusionwave equations (see, e.g., [3, 4, 8] and the references therein). The following lemma contains the main assertions on Mittag-Leffler operators. EJDE-2022/86 STOCHASTIC TIME FRACTIONAL 2D-STOKES MODELS 5 Lemma 2.3 ([4, Theorem 3]).Operators Eα(−tαA)and Eα,α(−tαA)are well defined from Xto X. Moreover, for x∈X, it holds: (i) Eα(−tαA)x|t=0 =x; (ii) The functions t∈[0,+∞)7→ Eα(−tαA)xand t∈[0,+∞)7→ Eα,α(−tαA)x are analytic from [0,+∞)to X. 3. Analysis of problem (1.1) We can rewrite the time fractional stochastic 2D-Stokes delay differential equations (1.1) in the abstract form Dα tu=−Au +F(t, ut) + G(t, ut)dW(t) dt , t > 0, u(t) = ϕ(t), t ∈(−h, 0], (3.1) where A=−P∆ = −∆P,F(t, ut) = Pf(t, ut), and G(t, ut) = Pg(t, ut). Here, P:L2→L2 σis the Helmholtz-Leray projector and A:D(A)⊂L2 σ→L2 σis the Stokes operator. Before stating the correct definition of mild solutions to problem (3.1), let us recall the properties of both families of Mittag-Leffler operators, which furnish the essential tools used throughout the whole article, see [4, 5] for more details. Lemma 3.1. Consider α∈(0,1), and r1,r2real numbers satisfying 1< r1≤r2<∞and r2N/(2r2+N)< r1. Then, for any v∈Lr1 σ, there exists a constant C1=C1(r1, r2, N, α)>0such that (i) kEα(−tαAr1)vkLr2≤C1t−α(N/r1−N/r2)/2kvkLr1, t > 0, (ii) kEα,α(−tαAr1)vkLr2≤C1t−α(N/r1−N/r2)/2kvkLr1, t > 0, where Ardenotes the Stokes operator from D(Ar)⊂Lr σ→Lr σ. Definition 3.2. Let S= (Ω,F,{F}t≥0,P) be a fixed stochastic basis generated by a standard Brownian motion Wand T > 0. Consider α∈(1/2,1). Let ϕbe an initial function such that ϕ(t) is F0-measurable for t∈(−h, 0]. A mild solution to problem (3.1) on (−h, T] is a stochastic process such that u(t) = ϕ(t) for t∈(−h, 0], and fulfills, for t∈[0, T], u(t) = Eα(−tαA)ϕ(0) + Zt 0 (t−s)α−1Eα,α(−(t−s)αA)F(s, us)ds +Zt 0 (t−s)α−1Eα,α(−(t−s)αA)G(s, us)dW(s),P-a.s. (3.2) Remark 3.3. The Stokes operator −Ais the infinitesimal generator of an analytic semigroup {e−tA :t≥0}. Hence, the Mittag-Leffler families Eα(−tαA) and Eα,α(−tαA) are well defined. It is worth mentioning that the analysis in this paper can be easily extended to the case in which system (3.1) is driven by a Hilbert valued Brownian motion/Wiener process in infinite dimensions. However we prefer to consider this simpler formulation for the sake of clarity to the reader. 6 J. XU, T. CARABALLO EJDE-2022/86 The definition of mild solutions established in [16] omitted the kernel (t−s)α−1 in both integrals in (3.2). Consequently, the concept of mild solution used in [16] does not provide the usual one of mild solution in the deterministic case, i.e., G≡0. 3.1. Well-posedness of problem (3.1) with bounded delay. In this subsection, we justify the well-posedness of the fractional stochastic 2D-Stokes equations with bounded delay Dα tu=−Au +F(t, ut) + G(t, ut)dW(t) dt , t > 0, u(t) = ϕ(t), t ∈[−h, 0], (3.3) where his a positive fixed constant (finite delay). First, we need to introduce suitable Banach spaces, which aim to capture the essence of the problem. For any fixed T > 0, consider the Banach space XT 2which is the set of continuous functions u: [−h, T ]→L2(Ω; L2 σ) equipped with its natural norm kukXT 2=sup t∈[−h,T ] Eku(t)k21/2. When no confusion is possible we will omit Tin XT 2. Next, we state the hypotheses imposed on both external forcing terms in our problem. Let F,G: [0,+∞)×C([−h, 0]; L2(Ω; L2 σ)) →L2(Ω; L2 σ). (A1) For each ξ∈C([−h, 0]; L2(Ω; L2 σ)), the mappings t∈[0,+∞)→F(t, ξ) and t∈[0,+∞)→G(t, ξ) are measurable. (A2) F(·,0) = G(·,0) = 0 (for simplicity). (A3) There exist positive constants LFand LG, such that, for all t∈[0,∞) and ξ, η ∈C([−h, 0]; L2(Ω; L2 σ)), kF(t, ξ)−F(t, η)k2 L2(Ω;L2 σ)≤LFkξ−ηk2 C([−h,0];L2(Ω;L2 σ)), kG(t, ξ)−G(t, η)k2 L2(Ω;L2 σ)≤LGkξ−ηk2 C([−h,0];L2(Ω;L2 σ)). (A4) There exists a constant Lf>0, such that the function F: [0,∞)× C([−h, 0]; L2(Ω; L2 σ)) →L2(Ω; L2 σ) satisfies Zt 0 EkF(s, us)−F(s, vs)k2ds ≤LfZt −h Eku(s)−v(s)k2ds, for all u,v∈C([−h, T]; L2(Ω; L2 σ)). (A5) There exists a constant Lg>0, such that the function G: [0,∞)× C([−h, 0]; L2(Ω; L2 σ)) →L2(Ω; L2 σ) satisfies Zt 0 EkG(s, us)−G(s, vs)k2ds ≤LgZt −h Eku(s)−v(s)k2ds, for all u, v ∈C([−h, T ]; L2(Ω; L2 σ)). We can now establish a first result on local existence and uniqueness of mild solutions to problem (3.3) by a fixed point argument. Theorem 3.4. Let α∈(1/2,1),(A1)–(A3) hold, and ϕ∈C([−h, 0]; L2(Ω; L2 σ)) such that ϕ(t)is F0-measurable for all t∈[−h, 0]. Then, there exists T > 0 (small enough) such that problem (3.3) admits a unique mild solution in the sense of Definition 3.2 on [−h, T]. EJDE-2022/86 STOCHASTIC TIME FRACTIONAL 2D-STOKES MODELS 7 Proof. Let ϕ∈C([−h, 0]; L2(Ω; L2 σ)) be the initial function, and choose R > 0 such that (3C2 1+ 1)kϕk2 C([−h,0];L2(Ω;L2 σ)) ≤R2 2, where C1is the constant in Lemma 3.1 for r1=r2= 2. For some T > 0 which will be fixed later on, we define the space Cϕ Rfor the previous R: Cϕ R=u∈C([−h, T]; L2(Ω; L2 σ)) : u(t) = ϕ(t) for all t∈[−h, 0],kukX2≤R. Let us moreover define the operator Non Cϕ Rwith α∈(1/2,1) as follows, (Nu)(t) =        ϕ(t), t ∈[−h, 0], Eα(−tαA)ϕ(0) + Rt 0(t−s)α−1Eα,α(−(t−s)αA)F(s, us)ds +Rt 0(t−s)α−1Eα,α(−(t−s)αA)G(s, us)dW(s), t ∈(0, T],P-a.s. Assertion 1: Nu∈C([−h, T]; L2(Ω; L2 σ)), for every u∈C([−h, T]; L2(Ω; L2 σ)). Observe that, if t∈[−h, 0], then (Nu)(t) = ϕ(t) and ϕ∈C([−h, 0]; L2(Ω; L2 σ)). It only remains to prove the continuity of Nuon [0, T]. But this follows immediately from the proof of [4, Lemma 11], with the help of the analytical property of the Mittag-Leffler operators in time (see Lemma 2.3(ii)). We omit the details here. Assertion 2: There exists T > 0 (sufficiently small) such that kN ukX2≤R, for all u∈ Cϕ R. Indeed, for any u∈ Cϕ R, it holds kNukX2=sup t∈[−h,T ] Ek(Nu)(t)k21/2.(3.4) On the one hand, for t∈[−h, 0], we have Ek(Nu)(t)k2=Ekϕ(t)k2≤sup t∈[−h,0] Ekϕ(t)k2=kϕk2 C([−h,0];L2(Ω;L2 σ)).(3.5) On the other hand, for t∈(0, T], we have Ek(Nu)(t)k2≤3EkEα(−tαA)ϕ(0)k2 + 3E Zt 0 (t−s)α−1Eα,α(−(t−s)αA)F(s, us)ds  2 + 3E Zt 0 (t−s)α−1Eα,α(−(t−s)αA)G(s, us)dW(s)  2 := I1+I2+I3. (3.6) Now we estimate each term on the right-hand side of (3.6). For I1, by Lemma 3.1(i), it is obvious that I1= 3EkEα(−tαA)ϕ(0)k2≤3C2 1Ekϕ(0)k2≤3C2 1sup t∈[−h,0] Ekϕ(t)k2.(3.7) 8 J. XU, T. CARABALLO EJDE-2022/86 For I2, by Lemma 3.1(i), (A1)–(A3), the Cauchy-Schwarz inequality and Fubini’s theorem, we obtain I2= 3E Zt 0 (t−s)α−1Eα,α(−(t−s)αA)F(s, us)ds  2 ≤3EZt 0 (t−s)α−1kEα,α(−(t−s)αA)F(s, us)kds2 ≤3C2 1t2α−1 2α−1Zt 0 EkF(s, us)k2ds ≤3C2 1LFt2α−1 2α−1Zt 0 kusk2 C([−h,0];L2(Ω;L2 σ))ds ≤3C2 1LFt2α−1 2α−1Zt 0 sup θ∈[−h,0] Eku(s+θ)k2ds ≤3C2 1LFt2α−1 2α−1Zt 0 sup θ∈[−h,T ] Eku(θ)k2ds ≤3C2 1LFt2α 2α−1R2. (3.8) For I3, by Lemma 3.1(i), Itˆo’s isometry and (A1)–(A3), we have I3= 3E Zt 0 (t−s)α−1Eα,α(−(t−s)αA)G(s, us)dW(s)  2 ≤3EZt 0 (t−s)2α−2kEα,α(−(t−s)αA)G(s, us)k2ds ≤3C2 1LGZt 0 (t−s)2α−2kusk2 C([−h,0];L2(Ω;L2 σ))ds ≤3C2 1LGt2α−1 2α−1R2. (3.9) Replacing (3.7)–(3.9) into (3.6), combining with (3.5), we obtain Ek(Nu)(t)k2≤3C2 1kϕk2 C([−h,0];L2(Ω;L2 σ)) +3C2 1LFt2αR2 2α−1+3C2 1LGt2α−1R2 2α−1. Consequently, thanks to the choice of R, we can choose Tsmall enough such that kNukX2=sup t∈[−h,T ] Ek(Nu)(t)k21/2 ≤(3C2 1+ 1)kϕk2 C([−h,0];L2(Ω;L2 σ)) +3C2 1LFT2αR2 2α−1 +3C2 1LGT2α−1R2 2α−11/2≤R. (3.10) Assertion 3: Operator N:Cϕ R→ Cϕ Ris a contraction. To this end, for any u, v∈ Cϕ R, it follows that kNu− N vkX2:= sup t∈[−h,T ] Ek(Nu)(t)−(Nv)(t)k21/2.(3.11) EJDE-2022/86 STOCHASTIC TIME FRACTIONAL 2D-STOKES MODELS 9 Noticing that, for t∈[−h, 0], one has (Nu)(t) = (Nv)(t) = ϕ(t), it is sufficient to consider t∈(0, T]. Observe that Ek(Nu)(t)−(Nv)(t)k2 ≤2E Zt 0 (t−s)α−1Eα,α(−(t−s)αA)(F(s, us)−F(s, vs))ds  2 + 2E Zt 0 (t−s)α−1Eα,α(−(t−s)αA)(G(s, us)−G(s, vs))dW(s)  2 := J1+J2. (3.12) For J1, by Lemma 3.1(i), (A3), the Cauchy-Schwarz inequality and Fubini’s theorem, we obtain J1= 2E Zt 0 (t−s)α−1Eα,α(−(t−s)αA)(F(s, us)−F(s, vs))ds  2 ≤2EZt 0 (t−s)α−1kEα,α(−(t−s)αA)(F(s, us)−F(s, vs))kds2 ≤2C2 1LFt2α−1 2α−1Zt 0 kus−vsk2 C([−h,0];L2(Ω;L2 σ))ds ≤2C2 1LFt2α 2α−1sup s∈[0,t] Eku(s)−v(s)k2. (3.13) As for J2, by Lemma 3.1(i), (A3) and Itˆo’s isometry, we deduce J2= 2E Zt 0 (t−s)α−1Eα,α(−(t−s)αA)(G(s, us)−G(s, vs))dW(s)  2 ≤2C2 1LGZt 0 (t−s)2α−2kus−vsk2 C([−h,0];L2(Ω;L2 σ))ds ≤2C2 1LGt2α−1 2α−1sup s∈[0,t] Eku(s)−v(s)k2. (3.14) Hence, substituting (3.13)-(3.14) into (3.12), it follows that kNu− N vkX2≤2C2 1LFT2α 2α−1+2C2 1LGT2α−1 2α−1sup t∈[0,T ] Eku(t)−v(t)k21/2, := Mku−vkX2, where M2=2C2 1LFT2α 2α−1+2C2 1LGT2α−1 2α−1. Therefore, we can choose Tsmall enough such that 0 <M<1. In other words, operator Nmaps Cϕ Rinto itself and it is a contraction. The Banach fixed-point Theorem ensures that operator Npossesses a fixed point in Cϕ R. Namely, problem (3.3) has a unique local mild solution on [−h, T].  With similar arguments as in the proof of the previous theorem, we can prove the local existence and uniqueness of mild solutions to problem (3.3) under assumptions (A4) and (A5) instead of (A3). We establish the result in the next theorem but we omit the proof. 16 J. XU, T. CARABALLO EJDE-2022/86 Consequently, thanks to the choice of R, we can choose Tsmall enough such that kLukX2=sup t∈[−h,T ] Ek(Lu)(t)k21/2 ≤3C2 1+1+C2 1hLfT+C2 1hLgsup t∈[−h,0] Ekϕ(t)k2 + 3C2 1T2LfR2+ 3C2 1TLgR21/2≤R. (4.11) Assertion 3: Operator L:Bϕ R→ Bϕ Ris a contraction. To this end, for any u, v∈ Bϕ R, it follows that kLu− LvkX2:= sup t∈[−h,T ] Ek(Lu)(t)−(Lv)(t)k21/2.(4.12) For t∈[−h, 0], one has (Lu)(t) = (Lv)(t) = ϕ(t). Thus, it is sufficient to consider the case t∈(0, T]. Observe that Ek(Lu)(t)−(Lv)(t)k2 ≤2E Zt 0 Eα(−(t−s)αA)(F(s, us)−F(s, vs))ds  2 + 2E Zt 0 Eα(−(t−s)αA)(G(s, us)−G(s, vs))dW(s)  2 := J1+J2. (4.13) For J1, by Lemma 3.1(i), (A4), the Cauchy-Schwarz inequality and Fubini’s theorem, we obtain J1= 2E Zt 0 Eα(−(t−s)αA)(F(s, us)−F(s, vs))ds  2 ≤2EZt 0 kEα(−(t−s)αA)(F(s, us)−F(s, vs))kds2 ≤2C2 1LftZt −h Eku(s)−v(s)k2ds = 2C2 1LftZt 0 Eku(s)−v(s)k2ds ≤2C2 1Lft2sup s∈[0,t] Eku(s)−v(s)k2. (4.14) For J2, by Lemma 3.1(i), (A5) and Itˆo’s isometry, J2= 2E Zt 0 Eα(−(t−s)αA)(G(s, us)−G(s, vs))dW(s)  2 ≤2C2 1LgZt −h Eku(s)−v(s)k2ds = 2C2 1LgZt 0 Eku(s)−v(s)k2ds ≤2C2 1Lgtsup s∈[0,t] Eku(s)−v(s)k2. (4.15) EJDE-2022/86 STOCHASTIC TIME FRACTIONAL 2D-STOKES MODELS 17 Hence, substituting (4.14)-(4.15) into (4.13), it follows that kLu− LvkX2≤2C2 1T(LfT+Lg) sup t∈[0,T ] Eku(t)−v(t)k21/2 := Mku−vkX2, (4.16) where M2= 2C2 1T(LfT+Lg). Therefore, we can choose Tsmall enough such that 0<M<1, and the Banach fixed-point theorem implies that operator Lpossesses a fixed point in Bϕ R. Unlike the result proved in Section 3, where we could only prove global existence for additive noise, in this situation we can prove this global in time result for a general multiplicative noise term. Theorem 4.3. Assume hypotheses of Theorem 4.2 hold. Then, for every initial value ϕ∈C([−h, 0]; L2(Ω; L2 σ)) such that ϕ(t)is F0-measurable for all t∈[−h, 0], the initial value problem (4.3) has a unique mild solution defined globally in the sense of Definition 4.1. Proof. We proceed with the proof under assumptions (A1), (A2), (A4),(A5). The proof under assumptions (A1)–(A3) is similar and we omit it. Assume that there exist two solutions to problem (4.3), uand von [0, T1] and [0, T2], respectively. Let us prove that u=von [−h, T1∧T2]. It is clear that u(t) = v(t) = ϕ(t) on [−h, 0], so we only need to show that u(t) = v(t) for any t∈[0, T1∧T2]. Notice that ku−vk2 X2:= sup t∈[−h,T1∧T2] Eku(t)−v(t)k2,(4.17) and Eku(t)−v(t)k2≤2E Zt 0 Eα(−(t−s)αA)(F(s, us)−F(s, vs))ds  2 + 2E Zt 0 Eα(−(t−s)αA)(G(s, us)−G(s, vs))dW(s)  2 := I1+I2. (4.18) For I1, by Lemma 3.1(i), (A4) and the Cauchy-Schwarz inequality, it follows that I1≤2EZt 0 kEα(−(t−s)αA)(F(s, us)−F(s, vs))kds2 ≤2C2 1EZt 0 kF(s, us)−F(s, vs)kds2 ≤2C2 1LftZt 0 Eku(s)−v(s)k2ds ≤2C2 1LftZt 0 sup σ∈[0,s] Eku(σ)−v(σ)k2ds. (4.19) 18 J. XU, T. CARABALLO EJDE-2022/86 For I2, by Lemma 3.1(i), (A5) and Itˆo’s isometry, we have I2≤2Zt 0 EkEα(−(t−s)αA)(G(s, us)−G(s, vs))k2ds ≤2C2 1LgZt 0 Eku(s)−v(s)k2ds ≤2C2 1LgZt 0 sup σ∈[0,s] Eku(σ)−v(σ)k2ds. (4.20) Substituting (4.19)-(4.20) to (4.18), we deduce that Eku(t)−v(t)k2≤2C2 1(Lft+Lg)Zt 0 sup σ∈[0,s] Eku(σ)−v(σ)k2ds. Denoting M1= 2C2 1(Lf(T1∧T2) + Lg), for all t∈[0, T1∧T2], we have sup s∈[−h,t] Eku(s)−v(s)k2≤ M1Zt 0sup σ∈[−h,s] Eku(σ)−v(σ)k2ds. Then Gronwall’s Lemma implies that sup s∈[−h,t] Eku(s)−v(s)k2= 0,for all t∈[0, T1∧T2]. Therefore, u=von [−h, T1∧T2]. Now we prove that for each given T > 0, the mild solution to problem (4.3) is bounded with X2norm. Taking into account Lemma 3.1(i), (A2), (A4), (A5), Itˆo’s isometry, the Cauchy-Schwarz inequality, and Fubini’s theorem, we have Eku(t)k2≤3EkEα(−tαA)ϕ(0)k2+ 3E Zt 0 Eα(−(t−s)αA)F(s, us)ds  2 + 3E Zt 0 Eα(−(t−s)αA)G(s, us)dW(s)  2 ≤3C2 1sup t∈[−h,0] Ekϕ(t)k2+ 3C2 1tELfZ0 −h kϕ(s)k2ds +LfZt 0 ku(s)k2ds + 3C2 1ELgZ0 −h kϕ(s)k2ds +LgZt 0 ku(s)k2ds ≤3C2 1(1 + Lfth +Lgh) sup t∈[−h,0] Ekϕ(t)k2 + 3C2 1(Lft+Lg)Zt 0 sup σ∈[0,s] Eku(σ)k2ds. Therefore, for all t∈[0, T], sup s∈[−h,t] Eku(s)k2≤3(C2 1+1+C2 1LfTh +C2 1Lgh) sup s∈[−h,0] Ekϕ(s)k2 + 3C2 1(LfT+Lg)Zt 0 sup σ∈[0,s] Eku(σ)k2ds := A(ϕ, T, F, G) + M2Zt 0 sup σ∈[−h,s] Eku(σ)k2ds, EJDE-2022/86 STOCHASTIC TIME FRACTIONAL 2D-STOKES MODELS 19 where A(ϕ, T, F, G) := 3(C2 1+1+C2 1LfTh +C2 1Lgh) sup t∈[−h,0] Ekϕ(t)k2, M2:= 3C2 1(LfT+Lg). Applying the Gronwall lemma, for any fixed T > 0, we obtain kuk2 X2≤A(ϕ, T, F, G) exp(M2T). Because of the arbitrariness of T, together with the conclusion of uniqueness of u on [−h, T], it is straightforward that the mild solution to problem (4.3) is defined globally.  As a consequence, we can now state the continuous dependence on initial values. Proposition 4.4. Under the assumptions of Theorem 4.2, the mild solution to (4.3) is continuous with respect to the initial data ϕ∈C([−h, 0]; L2(Ω; L2 σ)). In particular, if u(·),w(·)are the corresponding mild solutions on [−h, T]to the initial data ζand ψ, then the following estimate holds, ku−wkX2≤3kζ−ψkC([−h,0];L2(Ω;L2 σ)) exp(2C2 1(LfT+Lg)T). This proposition is proved by using similar arguments as those concerning the uniqueness of previous theorem. 4.2. Well-posedness of problem (4.1) with unbounded delay. In this section, we analyze the well-posedness of the following stochastic time fractional 2D-Stokes equations with unbounded delay: Dα tu=−Au +J1−α t[F(t, ut)] + J1−α t[G(t, ut)dW(t) dt ], t > 0, u(t) = ϕ(t), t ∈(−∞,0]. (4.21) Theorem 4.5. Let α∈(0,1) and (A1’)–(A3’) hold. Then for each initial function ϕ∈ CX(L2(Ω; L2 σ)) such that ϕ(t)is F0-measurable for all t∈[−h, 0], problem (4.21) admits a unique mild solution on (−∞, T]in the sense of Definition 4.1, for T > 0small enough. Proof. Let ϕ∈ CX(L2(Ω; L2 σ)) be such an initial value and choose R > 0 such that 3(C2 1+ 1)kϕk2 CX(L2(Ω;L2 σ)) ≤R2 3. Now consider the space Vϕ R=u∈C((−∞, T]; L2(Ω; L2 σ)) : u(t) = ϕ(t) for all t∈(−∞,0], and |uTkCX(L2(Ω;L2 σ)) ≤R, and define the operator Kon Vϕ Ras (Ku)(t) =      ϕ(t), t ∈(−∞,0], Eα(−tαA)ϕ(0) + Rt 0Eα(−(t−s)αA)F(s, us)ds +Rt 0Eα(−(t−s)αA)G(s, us)dW(s), t ∈(0, T],P-a.s. (4.22) Note that if u∈ Vϕ R, then kutkCX(L2(Ω;L2 σ)) ≤Rfor all t∈[0, T]. 20 J. XU, T. CARABALLO EJDE-2022/86 Assertion 1: Ku∈C((−∞, T ]; L2(Ω; L2 σ)) for all u∈C((−∞, T]; L2(Ω; L2 σ)). Observe that if t∈(−∞,0], then (Ku)(t) = ϕ(t). Therefore, we only need to check the continuity of Kuon [0, T]. By slightly modifying the proof in [4, Lemma 11], with the help of the analyticity in time of Mittag-Leffler operators (see Lemma 2.3(ii)), the result holds. Assertion 2: There exists T > 0 such that k(Ku)tkCX(L2(Ω;L2 σ)) ≤Rfor all t∈ [0, T] and u∈ Vϕ R. For every u∈ Vϕ R, we have to show that k(Ku)tkCX(L2(Ω;L2 σ)) := sup θ∈(−∞,0] Ek(Ku)(t+θ)k21/2≤R. For t+θ∈(−∞,0], namely, t∈(−∞,−θ), we have Ek(Ku)(t+θ)k2=Ekϕ(t+θ)k2≤sup t∈(−∞,0] Ekϕ(t)k2.(4.23) If t+θ∈(0, T], namely t∈(−θ, T] (for convenience, here we denote by t:= t+θ∈ [0, T]), then it follows that Ek(Ku)(t)k2≤3EkEα(−tαA)ϕ(0)k2 + 3E Zt 0 Eα(−(t−s)αA)F(s, us)ds  2 + 3E Zt 0 Eα(−(t−s)αA)G(s, us)dW(s)  2 := I1+I2+I3. (4.24) We estimate now each term. For I1, by Lemma 3.1(i), it is obvious that I1= 3EkEα(−tαA)ϕ(0)k2≤3C2 1Ekϕ(0)k2≤3C2 1sup t∈(−∞,0] Ekϕ(t)k2.(4.25) For I2, by Lemma 3.1(i), (A2’)-(A3’), the Cauchy-Schwarz inequality and Fubini’s theorem, we obtain I2= 3E Zt 0 Eα(−(t−s)αA)F(s, us)ds  2 ≤3EZt 0 kEα(−(t−s)αA)F(s, us)kds2 ≤3C2 1tZt 0 EkF(s, us)k2ds ≤3C2 1L0 FtZt 0 kusk2 CX(L2(Ω;L2 σ))ds ≤3C2 1L0 Ft2kutk2 CX(L2(Ω;L2 σ)) ≤3C2 1L0 Ft2R2. (4.26) EJDE-2022/86 STOCHASTIC TIME FRACTIONAL 2D-STOKES MODELS 21 For I3, by Lemma 3.1(i), Itˆo’s isometry, (A2’), and (A3’), we have I3= 3E Zt 0 Eα(−(t−s)αA)G(s, us)dW(s)  2 ≤3C2 1Zt 0 EkG(s, us)k2ds ≤3C2 1L0 GZt 0 kusk2 CX(L2(Ω;L2 σ))ds ≤3C2 1L0 Gtkutk2 CX(L2(Ω;L2 σ)) ≤3C2 1L0 GtR2. (4.27) Replacing (4.25)-(4.27) into (4.24), combining with (4.23), it is obvious that Ek(Ku)tk2≤3(C2 1+ 1) sup t∈(−∞,0] Ekϕ(t)k2+C2 1t2L0 FR2+C2 1tL0 GR2. Consequently, by the choice of R, we can choose Tsuch that k(Ku)tkCX(L2(Ω;L2 σ)) =sup θ∈(−∞,0] Ek(Ku)(t+θ)k21/2 ≤3(C2 1+ 1)kϕk2 CX(L2(Ω;L2 σ)) + 3C2 1T2L0 FR2+ 3C2 1TL0 GR21/2≤R, for all t∈(0, T]. Assertion 3: Operator K:Vϕ R→ Vϕ Ris a contraction. To this end, for each u, v∈ Vϕ Rand t∈[0, T], it follows that k(Ku)t−(Kv)tk2 CX(L2(Ω;L2 σ)) := sup θ∈(−∞,0] Ek(Ku)(t+θ)−(Kv)(t+θ)k2 = sup t∈(−∞,T ] Ek(Ku)(t)−(Kv)(t)k2.(4.28) For t∈(−∞,0], one has (Ku)(t)=(Kv)(t) = ϕ(t). Thus, we only need to consider the case t∈(0, T]. Observe that Ek(Ku)(t)−(Kv)(t)k2 ≤2E Zt 0 Eα(−(t−s)αA)(F(s, us)−F(s, vs))ds  2 + 2E Zt 0 Eα(−(t−s)αA)(G(s, us)−G(s, vs))dW(s)  2 := J1+J2. (4.29) 22 J. XU, T. CARABALLO EJDE-2022/86 For J1, by Lemma 3.1(i), (A3’), the Cauchy-Schwarz inequality, and Fubini’s theorem, we obtain J1= 2E Zt 0 Eα(−(t−s)αA)(F(s, us)−F(s, vs))ds  2 ≤2EZt 0 kEα(−(t−s)αA)(F(s, us)−F(s, vs))kds2 ≤2C2 1L0 FtZt 0 kus−vsk2 CX(L2(Ω;L2 σ))ds ≤2C2 1L0 Ft2kut−vtk2 CX(L2(Ω;L2 σ)). (4.30) For J2, by Lemma 3.1(i), (A3’), and Itˆo’s isometry, one has J2= 2E Zt 0 Eα(−(t−s)αA)(G(s, us)−G(s, vs))dW(s)  2 ≤2C2 1L0 GZt 0 kus−vsk2 CX(L2(Ω;L2 σ))ds ≤2C2 1L0 Gtkut−vtk2 CX(L2(Ω;L2 σ)). (4.31) Hence, substituting (4.29)-(4.31) into (4.28), it follows that k(Ku)t−(Kv)tkCX(L2(Ω;L2 σ)) ≤2C2 1(L0 FT2+L0 GT)kut−vtk2 CX(L2(Ω;L2 σ))1/2 := Wkut−vtkCX(L2(Ω;L2 σ)), where W2= 2C2 1(L0 FT2+L0 GT). Therefore, we can choose Tsmall enough such that 0 <W<1, which means that, the operator Kmaps Vϕ Rinto itself. This is a contraction. The Banach fixed-point theorem yields that operator Khas a fixed point in Vϕ R. Namely, problem (4.21) has a unique local mild solution on (−∞, T].  Theorem 4.6. Assume the hypotheses of Theorem 4.5 hold. Then, for every initial value ϕ∈ CX(L2(Ω; L2 σ)) such that ϕ(t)is F0-measurable for all t∈[−h, 0], the initial value problem (4.21) has a unique mild solution defined globally in the sense of Definition 4.1. Proof. The proof follows the same lines as in Section 3, but with differences in the estimates. Assume that there exist two solutions to problem (4.21), uand v on [0, T1] and [0, T2], respectively. Let us prove that u=von (−∞, T1∧T2]. It is remarkable that u(t) = v(t) = ϕ(t) on (−∞,0], so we only need to prove that u(t) = v(t) for any t∈(0, T1∧T2]. Observe that kut−vtk2 CX(L2(Ω;L2 σ)) := sup s∈(−∞,t] Eku(s)−v(s)k2,(4.32) and Eku(t)−v(t)k2≤2E Zt 0 Eα(−(t−s)αA)(F(s, us)−F(s, vs))ds  2 + 2E Zt 0 Eα(−(t−s)αA)(G(s, us)−G(s, vs))dW(s)  2 := I1+I2. (4.33) EJDE-2022/86 STOCHASTIC TIME FRACTIONAL 2D-STOKES MODELS 23 For I1, by Lemma 3.1(i), (A3’) and the Cauchy-Schwarz inequality, it follows that I1≤2EZt 0 kEα(−(t−s)αA)(F(s, us)−F(s, vs))k2 ≤2C2 1EZt 0 kF(s, us)−F(s, vs)kds2 ≤2C2 1L0 FtZt 0 kus−vsk2 CX(L2(Ω;L2 σ))ds. (4.34) For I2, by Lemma 3.1(i), (A3’) and Itˆo’s isometry, we have I2≤2Zt 0 EkEα(−(t−s)αA)(G(s, us)−G(s, vs))k2ds ≤2C2 1L0 GZt 0 kus−vsk2 CX(L2(Ω;L2 σ))ds. (4.35) Substituting (4.34)-(4.35) to (4.33), we have Eku(t)−v(t)k2≤2C2 1(L0 Ft+L0 G)Zt 0 kus−vsk2 CX(L2(Ω;L2 σ))ds. Denoting by W1= 2C2 1(L0 F(T1∧T2) + L0 G), we have for all t∈[0, T1∧T2], kut−vtk2 CX(L2(Ω;L2 σ)) ≤ W1Zt 0 kus−vsk2 CX(L2(Ω;L2 σ))ds. The Gronwall Lemma implies that kut−vtkCX(L2(Ω;L2 σ)) = 0,for all t∈[0, T1∧T2], and therefore u=von (−∞, T1∧T2]. Now we prove that for each given T > 0, the mild solution to (4.21) is bounded with CX(L2(Ω; L2 σ)) norm. Taking into account Lemma 3.1(i), (A1’)–(A3’), Itˆo’s isometry, the Cauchy-Schwarz inequality, and Fubini’s theorem, we have Eku(t)k2≤3EkEα(−tαA)ϕ(0)k2 + 3E Zt 0 Eα(−(t−s)αA)F(s, us)ds  2 + 3E Zt 0 Eα(−(t−s)αA)G(s, us)dW(s)  2 ≤3C2 1kϕk2 CX(L2(Ω;L2 σ)) + 3C2 1L0 FtZt 0 kusk2 CX(L2(Ω;L2 σ))ds + 3C2 1L0 GZt 0 kusk2 CX(L2(Ω;L2 σ))ds ≤3C2 1kϕk2 CX(L2(Ω;L2 σ)) + 3C2 1(L0 Ft+L0 G)Zt 0 kusk2 CX(L2(Ω;L2 σ))ds. Applying the Gronwall lemma to the above inequality, for any fixed T > 0 and t∈[0, T], we obtain kutk2 CX(L2(Ω;L2 σ)) ≤(3C2 1+ 1)kϕk2 CX(L2(Ω;L2 σ)) exp(3C2 1(L0 FT+L0 G)T). 24 J. XU, T. CARABALLO EJDE-2022/86 Thank to the arbitrariness of T, together with the conclusion of uniqueness of uon (−∞, T], it is straightforward that the mild solution to (4.21) is defined globally.  As a consequence, we can state the continuous dependence of mild solutions with respect to the initial data. Proposition 4.7. Under the assumptions of Theorem 4.5, the mild solution to (4.21) is continuous with respect to the initial data ϕ∈ CX(L2(Ω; L2 σ)). In particular, if u(·),w(·)are the mild solutions corresponding to the initial data ζand ψon (−∞, T], then kuT−wTkCX(L2(Ω;L2 σ)) ≤3C2 1kζ−ψkCX(L2(Ω;L2 σ)) exp(3C2 1(L0 FT+L0 G)T). This proposition is proved by similar arguments to those for the uniqueness of previous theorem. 5. Conclusion and final remarks In this article, we considered quite general time-fractional stochastic Stokes models driven by multiplicative Brownian motion with finite and infinite delay. As we said, this is only a first approach to our goal concerning the case of stochastic time fractional delay Navier-Stokes equations with multiplicative noise. However, to that end, a new technique has to be designed because the fixed point theorem used in our proofs is not appropriate to handle the nonlinear term: the appearance of expectation in the norm does not allow us to bound that term in an appropriate way, as it is done in the deterministic case, especially for the contraction property. Therefore, this is a challenging problem to be analyzed shortly. But, it is not surprising that the problem cannot be analyzed with this technique since, to the best of our knowledge, even the non-fractional stochastic time derivative system has not been solved for the multiplicative noise case. We plan to work on this first case and combine the ideas of both techniques to achieve our goal for the timefractional stochastic Navier-Stokes equations with delays. On a different note, concerning the models analyzed in the current paper, we are interested in analyzing the long time behavior of both problems (1.1) and (1.3) with general multiplicative noise. To this end, it is necessary to ensure first the existence of solutions globally defined in time. Motivated by this fact, we will investigate, in a forthcoming paper, the idea of approximating our problems with nonlinear multiplicative noise by the so-called colored noise. This means to consider Wong- Zakai approximations to provide information about the stochastic problem (see Lu et al. [10] for more details). 6. Appendix Proof of Theorem 3.8. Consider an initial function ϕ∈ CX(L2(Ω; L2 σ)) and choose R > 0 such that (3C2 1+ 1)kϕk2 CX(L2(Ω;L2 σ)) ≤R2 3. Let α∈(1/2,1) and define the following space Uϕ Rfor R > 0, Uϕ R=u∈C((−∞, T] : L2(Ω; L2 σ)) : u(t) = ϕ(t),∀t∈(−∞,0] and kuTkCX≤R. EJDE-2022/86 STOCHASTIC TIME FRACTIONAL 2D-STOKES MODELS 25 Notice that if u∈ Uϕ R, then ut∈ CX(L2(Ω; L2 σ)) for all t∈[0, T]. Define now the operator Qon Uϕ Ras (Qu)(t) =      ϕ(t), t ∈(−∞,0], Eα(−tαA)ϕ(0) + Rt 0(t−s)α−1Eα,α(−(t−s)αA)F(s, us)ds +Rt 0(t−s)α−1Eα,α(−(t−s)αA)G(s, us)dW(s), t ∈[0, T],P-a.s. Assertion 1: Qu∈C((−∞, T]; L2(Ω; L2 σ)) for all u∈C((−∞,0]; L2(Ω; L2 σ)). Observe that if t∈(−∞,0], then (Qu)(t) = ϕ(t). Therefore, we only need to check the continuity of Quon [0, T]. By slightly modifying the proof in [4, Lemma 11], with the help of the analyticity of Mittag-Leffler operator in time (see Lemma 2.3 (ii)), the result follows. Assertion 2: There exists T > 0 such that k(Qu)tkCX(L2(Ω;L2 σ)) ≤R, for all t∈[0, T] and u∈ Uϕ R. For every u∈ Uϕ Rand t∈[0, T], we have to show that k(Qu)tk2 CX(L2(Ω;L2 σ)) := sup θ∈(−∞,0] Ek(Qu)(t+θ)k2= sup t∈(−∞,T ] Ek(Qu)(t)k2≤R2. For t∈(−∞,0], we have Ek(Qu)(t)k2=Ekϕ(t)k2≤sup t∈(−∞,0] Ekϕ(t)k2.(6.1) If t∈(0, T], then Ek(Qu)(t)k2≤3EkEα(−tαA)ϕ(0)k2 + 3E Zt 0 (t−s)α−1Eα,α(−(t−s)αA)F(s, us)ds  2 + 3E Zt 0 (t−s)α−1Eα,α(−(t−s)αA)G(s, us)dW(s)  2 := I1+I2+I3. (6.2) We estimate now each term. For I1, by Lemma 3.1(i), it is obvious that I1= 3EkEα(−tαA)ϕ(0)k2≤3C2 1Ekϕ(0)k2≤3C2 1sup t∈(−∞,0] Ekϕ(t)k2.(6.3) For I2, by Lemma 3.1(i), (A1’)–(A3’), the Cauchy-Schwarz inequality, and Fubini’s theorem, we obtain I2= 3E Zt 0 (t−s)α−1Eα,α(−(t−s)αA)F(s, us)ds  2 ≤3EZt 0 (t−s)α−1kEα,α(−(t−s)αA)F(s, us)kds2 ≤3C2 1t2α−1 2α−1Zt 0 EkF(s, us)k2ds ≤3C2 1L0 Ft2α−1 2α−1Zt 0 kusk2 CX(L2(Ω;L2 σ))ds ≤3C2 1L0 Ft2α 2α−1kutk2 CX(L2(Ω;L2 σ)) ≤3C2 1L0 Ft2α 2α−1R2. (6.4)