International Journal of Analysis and Applications Volume 19, Number 6 (2021), 858-889 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-19-2021-858 STOCHASTIC CHEMOTAXIS MODEL WITH FRACTIONAL DERIVATIVE DRIVEN BY MULTIPLICATIVE NOISE ALI SLIMANI∗, AMIRA RAHAI, AMAR GUESMIA, LAMINE BOUZETTOUTA Laboratory of Applied Mathematics and History and Didactics of Mathematics (LAMAHIS) University of 20 August 1955, Skikda, Algeria ∗Corresponding author: alislimani21math@gmail.com Abstract. We introduce stochastic model of chemotaxis by fractional Derivative generalizing the deter- ministic Keller Segel model. These models include fluctuations which are important in systems with small particle numbers or close to a critical point. In this work, we study of nonlinear stochastic chemotaxis model with Dirichlet boundary conditions, fractional Derivative and disturbed by multiplicative noise. The required results prove the existence and uniqueness of mild solution to time and space-fractional, for this we use analysis techniques and fractional calculus and semigroup theory, also studying the regularity properties of mild solution for this model. 1. Introduction In this study, we consider on the following generalized SKSM with time-space fractional derivative on a bounded domain D ⊂ Rd(1 ≤ d ≤ 3) : (1.1)   cD β t u + (−∆) α 2 u−∇(u∇c) = g(u)Ẇ(t), (t,x) ∈ [0,T] ×D, cD β t c + (−∆) α 2 c− c∇c = f(c)Ẇ(t), (t,x) ∈ [0,T] ×D, with subject to the initial conditions: (1.2)   u(0,x) = u0(x), x ∈ D,c(0,x) = c0(x), x ∈ D, Received April 15th, 2021; accepted May 7th, 2021; published October 28th, 2021. 2010 Mathematics Subject Classification. 92C17, 35K58, 82C22. Key words and phrases. stochastic Keller-Segel model; chemotaxis; fractional derivative; mild solution; regularity properties. ©2021 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 858 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-858 Int. J. Anal. Appl. 19 (6) (2021) 859 and the Dirichlet boundary conditions: (1.3)   u(t,x) |∂D = 0, t ∈ [0,T],c(t,x) |∂D = 0, t ∈ [0,T], where u = u(t,x) denots the population density of biological individuals, c = c(t,x) denots the concentration of chemical substance, and ∇(u∇c) is called a chemotactic term that is used to model the fact that cells are attracted by chemical stimulus. In which the terms g(u)Ẇ(t) = g(u) dW(t) dt , and f(c)Ẇ(t) = f(c) dW(t) dt They describe the case-dependent random noise, where W(t)t∈[0T] is Ft− adapted Wiener process defined on a completed probability space (Ω,F,P) with the expectation E and associate with the normal filtration Ft = σ{W(s) : 0 ≤ s ≤ t}, The operator (−∆) α 2 , α ∈ (1, 2) stands for the fractional power of the Laplacian (see [1]). We denote by cD β t the Caputo derivative of order β, which is defined by (see [17]) (1.4) cD β t u(t,x) =   1 Γ(1−β) t∫ 0 ∂u(s,x) ∂s ds (t−s)β , 0 < β < 1, ∂u(s,x) ∂s , β = 1, cD β t c(t,x) =   1 Γ(1−β) t∫ 0 ∂c(s,x) ∂s ds (t−s)β , 0 < β < 1, ∂c(s,x) ∂s , β = 1, where Γ(.) stands for the gemma function Γ(β) = ∫ ∞ 0 tβ−1e−tdt. The rest of the paper is organized as follows. In Section 2, we will introduce some notations and preliminaries, which play a crucial role in our theorem analysis. In Section 3, the existence and uniqueness of mild solution to the problem of time-space fractional (2.1) and in Section 4, the spatial and temporal regularity properties of mild solution to this time-space fractional (2.1) are proved. In Section 5, the existence and uniqueness of mild solution to the problem of time-space fractional (2.6). Finally, the spatial and temporal regularity properties of mild solution to this time-space fractional (2.6) are proved. We use stochastic analysis techniques, fractional calculus and semigroup theory. Next, we mention some Notations and preliminaries the task at work. 2. Notations and preliminaries Denote the basic functional space Lp(D), 1 ≤ p < ∞ and Hs(D) by the usual Lebesgue and Sobolev spaces, respectively. We assume that A is the negative Laplacian −∆ in a bounded domain D with zero Dirichlet boundary conditions in a Hilbert space H = L2(D), which are given by A = −∆, D(A) = H10 (D) ∩H 2(D). Int. J. Anal. Appl. 19 (6) (2021) 860 Since the operator A is self-adjoint on H with discrete spectral, i.e., there exists the eigenvectors en with corresponding eigenvalues λn such that Aen = λnen,en = √ 2 sin(nπ),λn = π 2n2,n ∈ N+. For any s > 0, let Ḣs be the domain of the fractional power A s 2 = (−∆) s 2 , which can be defined by A s 2 en = λ s 2 n, n = 1, 2, ..., and Ḣs = D(A s 2 ) = {v ∈ L2(D),s.t. ‖ v ‖2 Ḣs = ∞∑ n=1 λ s 2 nv 2 n < ∞}, where vn := 〈v,en〉 with the inner product 〈., .〉 in L2(D). We denote that ‖ v ‖Ḣs=‖ A s 2 v ‖, and the corresponding dual space Ḣ−s with the inverse operator A− s 2 .We also denote As for A s 2 and the bilinear operators B(u,c) = O(uOc), and D(B) = H10 and L(c,v) = cOv, and D(L) = H 1 0 with a slight abuse of notation L(c,c) = L(c). Then the eqs (1.1) and (1.3) can be rewritten as the following abstract formulation: (2.1)   cDβu(t) = −Aαu(t) + B(u(t),c(t)) + g(u(t)) dW(t) dt , t > 0, u(0) = u0, and (2.2)   cDβc(t) = −Aαc(t) + L(c(t)) + f(c(t)) dW(t) dt , t > 0, c(0) = c0, where {W(t)}t≥0 is a Q− Wiener process with linear bounded covariance operator Q such that Tr(Q) < ∞. Further, there exists the eigenvalues λn and corresponding eigenfunctions en satisfy Qn = λnen, n = 1, 2, ..., then the Wiener process is given by W(t) = ∞∑ n=1 λ 1 2 nβn(t)en, in which {βn}n≥1 is a sequence of real-valued standard Brownian motions. Let L20 = L2(Q 1 2 (H),H) denote the space of Hillbert-Schmidt operators from Q 1 2 (H) to H with the norm ‖ Φ ‖L20 =‖ ΦQ 1 2 ‖Hs= ( ∞∑ n=1 ΦQ 1 2 en) 1 2 , i.e., L20 = {Φ ∈ L(H) : ∞∑ n=1 ‖ ΦQ 1 2 ‖2< ∞}, where L(H) is the space of bounded linear operators from H to H. For an arbitrary Banach space B, we denote ‖ . ‖Lp(Ω;B) by the norm in Lp(Ω,F,P; B) , which defined as ‖ v ‖Lp(Ω;B)= (E[‖ v ‖ p B]) 1 p , ∀v ∈ Lp(Ω,F,P; B), for any p ≥ 2. We shall also need the following result with respect to the fractional operator Aα (see Ref. [18]). Int. J. Anal. Appl. 19 (6) (2021) 861 Lemma 2.1. For any α > 0, an analytic semigroup Sα(t) = e −tAα, t ≥ 0 is generated by the operator −Aα on Lp, and for any ν ≥ 0, there exists a constant Cαν dependent on α and ν such that (2.3) ‖ AνSα(t) ‖£(Lp)≤ Cανt− ν α , t > 0, in which £(B) denotes the Banach space of all linear bounded operators from B to itself. Next, we will introduce the following lemma to estimate the stochastic integrals, which contains the Burkholder-Davis-Gundy’s inequality. Lemma 2.2. ( [8]) For any 0 ≤ t1 < t2 ≤ T and p ≥ 2, and for any predictable stochastic process v : [0,T] × Ω → L20, which satisfies E[( T∫ 0 ‖ v(s) ‖2L20 ds) p 2 ] < ∞, then we have (2.4) E[‖ t2∫ t1 ‖ v(s)dW(s) ‖p ds] ≤ C(p)E[( t2∫ t1 ‖ v(s) ‖2L20 ds) p 2 ], where C(p) = [ p(p−1) 2 ] p 2 ( p p−1 ) p( p 2 −1) is a constant. Now, we give the following definition of mild solution for our time-space fractional stochastic Keller-Segel model. Definition 2.1. A Ft adapted process (u(t),c(t))t∈[0,T] is called a mild solution (1.1), if (u(t),c(t))t∈[0,T] ∈ C ( [0,T]; Ḣν ) P - a. e, and it holds, (2.5) u(t) = Eβ(t)u0 + ∫ t 0 (t−s)β−1Eββ(t−s)B (u(s),c(s)) ds + ∫ t 0 (t−s)β−1Eββ(t−s)g(u(s))dW(s), and (2.6) c(t) = Eβ(t)c0 + ∫ t 0 (t−s)β−1Eββ(t−s)L(c(s))ds + ∫ t 0 (t−s)β−1Eββ(t−s)f(c(s))dW(s), respectefily for a. s. ω ∈ Ω, where the generalized Mittag-Leffler operators Eβ(t) and Eββ(t) are defined as Eβ(t) = ∫ ∞ 0 Mβ(θ)Sα(t βθ)dθ, and Eββ(t) = ∫ ∞ 0 βθMβ(θ)Sα(t βθ)dθ, Int. J. Anal. Appl. 19 (6) (2021) 862 which contain the Mainardi’s Wright-type function with β ∈ (0, 1) given by Mβ(θ) = ∞∑ n=0 (−1)nθn n!Γ(1 −β(1 + n)) , in which the Mainardi function Mβ(θ) act as a bridge between the classical integral-order and fractional derivatives of differential equations, for more details see [19, 20]. Here, the derivation of mild solution (2.5) and (2.6) can be found in Appendix (7) and Appendix (8) (respectively). Lemma 2.3. [2] For any β ∈ (0, 1) and −1 < ε < ∞, it is not difficult to verity that (2.7) Mβ(θ) ≥ 0, and ∫ ∞ 0 θεMβ(θ)dθ = Γ(1 + ε) Γ(1 + βε) , for all θ ≥ 0. Theorem 2.1. For any t > 0, Eβ(t) and Eββ(t) are linear and bounded operators. Moreover, for 0 ≤ ν < α < 2, there exist constants Cα = C(α,β,ν) > 0 and Cβ = C(α,β,ν) > 0 such that (2.8) ‖ Eβ(t)v ‖Ḣν≤ Cαt −βν α ‖ v ‖, ‖ Eββ(t)v ‖Ḣν≤ Cβt −βν α ‖ v ‖ . Proof. For t > 0 and 0 ≤ ν < α < 2, by means of the Lemma (2.1) and Lemma (2.3), we have ‖ Eβ(t)v ‖Ḣν ≤ ∫∞ 0 Mβ(θ) ‖ AνSα(tβθ)v ‖ dθ ≤ ∫∞ 0 Cανt −βν α θ −ν α Mβ(θ) ‖ v ‖ dθ = CανΓ(1−να ) Γ(1−βν α ) t −βν α ‖ v ‖ v ∈ L2(D), and ‖ Eββ(t)v ‖Ḣν ≤ ∞∫ 0 βθMβ(θ) ‖ AνSα(tβθ)v ‖ dθ ≤ ∞∫ 0 Cανβt −βν α θ1− ν α Mβ(θ) ‖ v ‖ do = CανβΓ(2−να ) Γ(1+β(1−ν α )) t −βν α ‖ v ‖, v ∈ L2(D), which imply that the estimates (2.8) hold, so it is easy to know that Eβ(t) and Eββ(t) are linear and bounded operators. � Theorem 2.2. For any t > 0, the operators Eβ(t) and Eββ(t) are strongly continuous. Moreover, for any 0 ≤ t1 < t2 ≤ T and for 0 < ν < α < 2, there exist constants Cαν = C(α,β,ν) > 0 and Cβν = C(α,β,ν) > 0 such that (2.9) ‖ (Eβ(t2) −Eβ(t1))v ‖Ḣν≤ Cαν(t2 − t1) βν α ‖ v ‖, Int. J. Anal. Appl. 19 (6) (2021) 863 and (2.10) ‖ (Eββ(t2) −Eββ(t1))v ‖Ḣν≤ Cβν(t2 − t1) βν α ‖ v ‖ . Proof. for any 0 ≤ t1 < t2 ≤ T, it is easy to deduce that (2.11) t2∫ t1 dSα(t βθ) dt dt = Sα(t β 2 θ) −Sα(t β 1 θ) = − t2∫ t1 βtβ−1θAαSα(t βθ)dt, for 0 < ν < α < 2, making use of the above expression, the Lemma (2.1) and Lemma (2.3), we can arrive at ‖ (Eβ(t2) −Eβ(t1))v ‖Ḣν = ‖ Aν(Eβ(t2) −Eβ(t1))v ‖ = ‖ ∞∫ 0 Mβ(θ)Aν((Sα(t β 2 θ) −Sα(t β 1 θ))vdθ)vdθ ‖ ≤ ∞∫ 0 βθMβ(θ) t2∫ t1 tβ−1 ‖ Aα+νSα(tβθ)v ‖L2 dtdθ ≤ ∞∫ 0 Cανβθ −ν α Mβ(θ)( t2∫ t1 t −βν α −1dt) ‖ v ‖ dθ = αCανΓ(1−να ) νΓ(1−βν α ) (t −βν α 2 − t −βν α 1 ) ‖ v ‖ ≤ αCανΓ(1− ν α ) νT 2βν α Γ(1 βν α ) (t2 − t1) βν α ‖ v ‖, v ∈ L2(D), and ‖ (Eββ(t2) −Eββ(t1))v ‖Ḣν = ‖ Aν(Eββ(t2) −Eββ(t1))v ‖ = ‖ ∞∫ 0 βθMβ(θ)Aν(Sα(t β 2 θ) −Sα(t β 1 θ))vdθ ‖ ≤ ∞∫ 0 β2θ2Mβ(θ) t2∫ t1 tβ−1 ‖ Aα+νSα(tβθ)v ‖L2 dtdθ ≤ ∞∫ 0 Cανβ 2θ1− ν α Mβ(θ)( t2∫ t1 t −βν α −1dt) ‖ v ‖ dθ = αCανΓ(2−να ) νΓ(1+β(1−ν α )) (t −βν α 2 − t −βν α 1 ) ‖ v ‖ ≤ αCανΓ(1− ν α ) νT 2βν α 0 Γ(1+β(1− ν α )) (t2 − t1) βν α ‖ v ‖, v ∈ L2(D), Int. J. Anal. Appl. 19 (6) (2021) 864 It is obviously to see that the term ‖ (Eβ(t2) −Eβ(t1))v ‖Ḣν→ 0 and ‖ (Eββ(t2) −Eββ(t1))v ‖Ḣν→ 0 as t1 → t2. Which mean that the operators Eβ(t) and Eββ(t) are strongly continuous. � Remark 2.1. Assume ν = 0 in theorem (2.2), then there exist constants Cα = C(α,β) > 0 and Cβ = C(α,β) > 0 such that (2.12) ‖ (Eβ(t2) −Eβ(t1))v ‖Ḣν≤ Cα(t2 − t1) ‖ v ‖, and (2.13) ‖ (Eββ(t2) −Eββ(t1))v ‖Ḣν≤ Cβ(t2 − t1) ‖ v ‖ . Proof. for any 0 < T0 ≤ t1 < t2 ≤ T, the same as the proof of Theorem (2.2), we get ‖ (Eβ(t2) −Eβ(t1))v ‖Ḣν = ‖ ∞∫ 0 Mβ(θ)((Sα(t β 2 θ) −Sα(t β 1 θ))vdθ)vdθ ‖L2 ≤ ∞∫ 0 βθMβ(θ) t2∫ t1 tβ−1 ‖ AαSα(tβθ)v ‖L2 dtdθ ≤ ∞∫ 0 CααβθMβ(θ)( t2∫ t1 t−1dt) ‖ v ‖L2 dθ ≤ Cααβ(ln t2 − ln t1) ‖ v ‖ = Cααβ T0 (t2 − t1) ‖ v ‖, v ∈ L2(D), and ‖ (Eββ(t2) −Eββ(t1))v ‖Ḣν = ‖ ∞∫ 0 βθMββ(θ)((Sα(t β 2 θ) −Sα(t β 1 θ))vdθ)vdθ ‖L2 ≤ ∞∫ 0 β2θ2Mβ(θ) t2∫ t1 tβ−1 ‖ AαSα(tβθ)v ‖L2 dtdθ ≤ ∞∫ 0 Cααβ 2θMβ(θ)( t2∫ t1 t−1dt) ‖ v ‖ dθ ≤ Cααβ 2Γ(2) Γ(1+β) (ln t2 − ln t1) ‖ v ‖ ≤ Cααβ 2Γ(2) T0Γ(1+β) (t2 − t1) ‖ v ‖, v ∈ L2(D). This completes the proof. � Int. J. Anal. Appl. 19 (6) (2021) 865 3. Existence and uniqueness of mild solution Our main purpose of this section is to prove the existence and uniqueness of mild solution to the problem (2.1). To do this, the following assumptions are imposed. 3.1. Assumption . The measurable function g : Ω × H → L20 satisfies the following globalLipschitz and growth conditions: (3.1) ‖ g(v) ‖L20≤ C ‖ v ‖, ‖ g(u) −g(v) ‖L20≤ C ‖ u−v ‖, for all u,v ∈ H. 3.2. Assumption . Let C, C1 are a positive real number, then the bounded bilinear operator B : L20(D) → H−1(D) satisfies the following properties: ‖ B(u,c) ‖Ḣ−1 ≤ C ‖ u ‖‖ c ‖ ≤ C1 ‖ u ‖2, and (3.2) ‖ B(u,c) −B(v,c) ‖Ḣ−1≤ CC1(‖ u ‖ + ‖ v ‖) ‖ u−v ‖, where C1 depend a norm the c in L 2 0(D), and for all u,v,c ∈ L20(D). 3.3. Assumption . Assume that the initial value u0 : Ω → Ḣν is a F0-measurable random variable, it holds that (3.3) ‖ u0 ‖Lp(Ω,Ḣν)< ∞, for any 0 ≤ ν < α < 2. Theorem 3.1. Let Assumptiom (3.1) to (3.3) be satisfied for some p ≥ 2, then there exists a unique mild solution (u(t))t∈[0,T] in the space L p(Ω,Ḣν) with 0 ≤ ν < α < 2. Proof. We fix an ω ∈ Ω and use the standard Picard’s iteration argument to prove the existence of mild solution. To begin with, the sequence of stochastic process {u(t)}n≥0 is constructed as (3.4)   un+1(t) = Eβ(t)u0 + N1(un(t)) + N2(un(t)),u0(t) = u0, Int. J. Anal. Appl. 19 (6) (2021) 866 where (3.5)   N1(un(t)) = t∫ 0 (t−s)β−1Eβ,β(t−s)B(un(s),c(s))ds, N2(un(t)) = t∫ 0 (t−s)β−1Eβ,β(t−s)g(un(s))dW(s). The proof will be split into three steps. Step1 For each n ≥ 0, we show that sup E[‖ un(t) ‖ p Ḣν ] < ∞, Note that (3.6) E[‖ un+1(t) ‖ p Ḣν ] ≤ 3p−1E[‖ Eβ(t)u0 ‖ p Ḣν ] + 3p−1E[‖ N1(un(t)) ‖ p Ḣν ] + 3p−1E[‖ N2(um(t)) ‖ p Ḣν ]. The application of the Lemma (2.1) gives (3.7) E[‖ Eβ(t)u0 ‖ p Ḣν ] ≤ E[ ∞∫ 0 Mβ(θ)(‖ AνSα(tβθ)u0 ‖2) 1 2 dθ] = E[ ∞∫ 0 Mβ(θ)( ∑∞ n=1〈Aαe −tβθAαu0,en〉2) 1 2 dθ] = E[ ∞∫ 0 Mβ(θ)( ∑∞ n=1〈Aαu0,e −tβθλ α 2 en〉2) 1 2 dθ] ≤ E[ ∞∫ 0 Mβ(θ) ‖ u0 ‖Ḣν dθ] = E[‖ u0 ‖Ḣν ] Applying the following Hölder inequality to the second term of the right-hand side of (3.6) (3.8) E[‖ N1(un(t)) ‖ p Ḣν ] ≤ E[( t∫ 0 ‖ (t−s)β−1A1Eβ,β(t−s)Aν−1B(un(s),c(s)) ‖ ds)p] ≤ Cpβ( t∫ 0 (t−s) p(β−1− β α ) p−1 ds)p−1 t∫ 0 E[‖ Aν−1B(un(s),c(s)) ‖p]ds ≤ K1 t∫ 0 E[‖ un(s) ‖ p Ḣν ]ds, where K1 = C p βC pC p 1 [ p−1 pβ(1− 1 alpha )−1 ] p−1Tpβ(1 1 α )−1( max t∈[0,T] E[‖ un(t) ‖ p Ḣν . Making use of the Hölder inequality and Lemma (2.2) to the third term of the right-hand side of (3.6), we get (3.9) E[‖ N2(un(t)) ‖ p Ḣν ] ≤ C(p)E[( t∫ 0 ‖ (t−s)β−1Eβ,β(t−s) ‖2 Aνg(un(s)) ‖2L20 ds) p 2 ] ≤ K2 t∫ 0 E[‖ un(s) ‖2Ḣν ]ds, where K2 = C(p)C p βC pC p 1 [ p−2 p(2β−1)−2 ] p−2 2 T p(2β−1)−2 2 . Int. J. Anal. Appl. 19 (6) (2021) 867 Using the above estimates (3.6) and (3.9), we have E[‖ un+1(t) ‖ p Ḣν ] ≤ 3p−1E[‖ u0 ‖ p Ḣν ] + 3p−1(K1 + K2) t∫ 0 E[‖ un(s) ‖ p Ḣν ]ds By means of the extension of Gronwall’s lemma, it holds that sup t∈[0,T] E[‖ un+1(t) ‖ p Ḣν ] < ∞, for each n ≥ 0. Step2 Show that the sequence {un(t)}n≥0 is a Cauchy sequence in the space Lp(Ω; Ḣν). For any n ≥ m ≥ 1, applying the similar arguments employed to obtain (3.8) and (3.9), we get (3.10) E[‖ un(t) −um(t) ‖ p Ḣν ] ≤ 2p−1E[‖ N1(un−1(t)) −N1(um−1(t)) ‖ p Ḣν ] + 2p−1E[‖ N2(un−1(t)) −N2(um−1(t)) ‖ p Ḣν ] ≤ K t∫ 0 E[‖ un−1(s) −um−1(s) ‖ p Ḣν ]ds, in wich K = 2p−1{CpβC pC p 1}[ p−1 p(β−β ν )−1 ]p−1Tp(β− β α )( max t∈[0,T] E[‖ un−1(t) ‖ p Ḣν ]) + max t∈[0,T] E[‖ um−1(t) ‖ p Ḣν ]) + C(p)C p βC pC p 1 [ p−2 p(2β−1)−2 ] p−2 2 T p(2β−1)−2 2 }. A direct application of Gronwall’s lemma yields sup t∈[0,T] E[‖ un(t) −um(t) ‖ p Ḣν ] = 0, for all T > 0. Taking limits to the stochastic sequence {un(t)}n≥0 in (3.4) as n →∞, we finish the proof of the existence of mild solution to (2.1). Step3 We show the uniqueness of mild solution. Assume u and v are two mild solutions of the problem (2.1), using the similar calculations as in Step 2, we can obtain (3.11) sup t∈[0,T] E[‖ u(t) −v(t) ‖p Ḣν ] = 0, for all T > 0, which implies that u = v, it follows that the uniqueness of mild solution. Obviously, when ν = 0, the above three steps still work. Thus the proof of Theorem 3.1 is completed. � 4. Regularity of mild solution In this section, we will prove the spatial and temporal regularity properties of mild solution to time-space fractional SKSM based on the analytic semigroup. Int. J. Anal. Appl. 19 (6) (2021) 868 Theorem 4.1. Let Assumptions (3.1) to (3.3) hold with 1 ≤ ν < α < 2 and p ≥ 2 let u(t) be a unique mild solution of the problem (2.1) with P(u(t) ∈ Ḣν) = 1 for any t ∈ [0,T], then there exists a constant C such that (4.1) sup t∈[0,T] ‖ u(t) ‖Lp(Ω;Ḣν)≤ C(‖ u0 ‖Lp(Ω;H) + sup t∈[0,T] ‖ u(t) ‖Lp(Ω;Ḣ1)). Proof. For any 0 ≤ t ≤ T and 1 ≤ ν < α < 2, we have (4.2) ‖ u(t) ‖Lp(Ω;Ḣν) = (E[‖ u(t) ‖ p Ḣν ]) 1 p =‖ Aνu(t) ‖Lp(Ω;H) ≤ ‖ AνEβ(t)u0 ‖Lp(Ω;H) + ‖ Aν t∫ 0 (t−s)β−1Eβ,β(t−s)B(u(s),c(s))ds ‖Lp(Ω;H) + ‖ Aν t∫ 0 (t−s)β−1Eβ,β(t−s)g(u(s))dW(s) ‖ Lp(Ω;H) = I + II + III. Using Theorem (2.1), the first term can be estimated by (4.3) I =‖ AνEβ(t)u0 ‖Lp(Ω;H)≤ Cαt− βν α ‖ u0 ‖Lp(Ω;H)< ∞. It is easy to know that (4.4) T∫ 0 Cαt −βν α ‖ u0 ‖Lp(Ω;H) dt = αCα α−βν T α−βν α ‖ u0 ‖Lp(Ω;H) The application of Theorem (2.1) and Assumptions (3.2), we get (4.5) (II)p ≤ E[(‖ Aν t∫ 0 (t−s)β−1AνEβ,β(t−s)B(u(s),c(s)) ‖ ds)p] ≤ Cpβ( t∫ 0 (t−s) p[β−1− β(ν+1) α ] p−1 ds)p−1 t∫ 0 E[‖ A−1B(u(s),c(s)) ‖ p Ḣ1 ]ds ≤ C2 sup t∈[0,T] E[‖ u(s) ‖p Ḣ1 ], Int. J. Anal. Appl. 19 (6) (2021) 869 where C2 = C p βC pC p 1{ p−1 p[β−β(ν+1) α ]−1 }p−1Tp[β− β(ν+1) α ]−1 + ( max t∈[0,T] E[‖ u(t) ‖Ḣ1 ]). By means of Theorem (2.1), Assumptions (3.1) and Lemma (2.2), we can deduce (4.6) (III)p ≤ C(p)E[(‖ Aν t∫ 0 ‖ (t−s)β−1Aν−1Eβ,β(t−s) ‖2‖ A1g(u(s)) ‖2L02 ds) p 2 ] ≤ C(p)Cpβ( t∫ 0 (t−s) 2p[β−1− β(ν−1) α ] p−2 ds) p−2 2 t∫ 0 E ‖ A1g(u(s)) ‖ p L02 ds ≤ C3 sup t∈[0,T] E[‖ u(s) ‖p Ḣ1 ], where C3 = C(p)C p βC pC p 1 [ p−2 p[2β−1−β(ν−1) α ]−2 ] p−2 2 T p[2β−1− β(ν−1) α ]−2 2 . Thus, we conclude the proof of Theorem (4.2) by combining with the estimates (4.2)- (4.6). � Next, we will devote to the temporal regularity of the mild solution. Theorem 4.2. Let Assumptions (3.1) to (3.3) hold with 0 < ν < α < 2 and p ≥ 2 for any 0 ≤ t1 < t2 ≤ T , the unique mild solution u(t)to the problem (2.1) is Hölder continuous with respect to the norm ‖ . ‖Lp(Ω;Ḣν) and satisfies (4.7) ‖ u(t2) −u(t1) ‖Lp(Ω;Ḣν)≤ C(t2 − t1) γ. Proof. Foor any 0 ≤ t1 < t2 ≤ T, for the mild solution (2.5), we have (4.8) u(t2) −u(t1) = Eβ(t2)u0 −Eβ(t1)u0 + t2∫ 0 (t2 −s)β−1Eβ,β(t2 −s)B(u(s),c(s))ds − t1∫ 0 (t1 −s)β−1Eβ,β(t1 −s)B(u(s),c(s))ds + t2∫ 0 (t2 −s)β−1β−1Eβ,β(t2 −s)g(u(s))dW(s) − t1∫ 0 (t2 −s)β−1β−1Eβ,β(t1 −s)G(u(s))dW(s) = I1 + I2 + I3, where I1 = Eβ(t2)u0 −Eβ(t1)u0, Int. J. Anal. Appl. 19 (6) (2021) 870 and (4.9) I2 = t2∫ 0 (t2 −s)β−1Eβ,β(t2 −s)B(u(s),c(s))ds − t1∫ 0 (t1 −s)β−1Eβ,β(t1 −s)B(u(s),c(s))ds = t1∫ 0 (t1 −s)β−1[Eβ,β(t2 −s) −Eβ,β(t1 −s)]B(u(s),c(s))ds + t1∫ 0 [(t1 −s)β−1 − (t2 −s)β−1]Eβ,β(t2 −s)B(u(s),c(s))ds + t2∫ t1 (t2 −s)β−1Eβ,β(t2 −s)B(u(s),c(s))ds = I21 + I22 + I23, and (4.10) I3 = t2∫ 0 (t2 −s)β−1Eβ,β(t2 −s)g(u(s))dW(s) − t1∫ 0 (t1 −s)β−1Eβ,β(t1 −s)g(u(s))dW(s) = t1∫ 0 (t1 −s)β−1[Eβ,β(t2 −s) −Eβ,β(t1 −s)]g(u(s))dW(s) + t1∫ 0 [(t2 −s)β−1 − (t1 −s)β−1]Eβ,β(t2 −s)g(u(s))dW(s) + t2∫ t1 (t2 −s)β−1Eβ,β(t2 −s)g(u(s))dW(s) = I31 + I32 + I33. For any 0 < ν < α < 2 and p ≥ 2, by by virtue of Theorem (2.2), it follows that (4.11) E[‖ I1 ‖ p Ḣν ] = E[‖ Aν[Eβ(t2) −Eβ(t1)]u0 ‖p] ≤ Cpα,ν(t2 − t1) pβν α E[‖ u0 ‖p]. Int. J. Anal. Appl. 19 (6) (2021) 871 For the first term I21 in (4.9), applying the Assumption (3.2) and Theorem (2.2) and Hölder’s inequality, we have (4.12) E[‖ I21 ‖ p Ḣν ] = E[‖ t1∫ 0 (t1 −s)β−1Aν[Eβ,β(t2 −s) −Eβ,β(t1 −s)]B(u(s),c(s))ds ‖p] ≤ Cpβν(t2 − t1) pβ(ν+1) α ( t1∫ 0 (t1s) p(β−1) p−1 ds)p−1 t1∫ 0 E[‖ A−1B(u(s),c(s)) ‖ p Ḣ1 ]ds ≤ CpCp1 C p βνT p( p−1 pβ−1 ) p−1(supt∈[0,T] E[‖ u(s) ‖ 2p Ḣ1 ])(t2 − t1) pβ(ν+1) α . Using the Assumptions (3.2), Theorem 2.1 and Hölder’s inequality, we get (4.13) E[‖ I22 ‖ p Ḣν = E[‖ t1∫ 0 [(t2 −s)β−1 − (t1 −s)β−1]AνEβ,β(t2 −s)B(u(s),c(s))ds ‖p] ≤ Cpβ( t1∫ 0 {[(t2 −s)β−1 − (t1 −s)β−1] × (t2 −s)− β(ν+1) α } p p−1 ds)p−1 × t1∫ 0 E[‖ A1B(u(s),c(s)) ‖ p Ḣ1 ]ds ≤ CpCp1 C p βT{ p−1 p[β−β(ν+1) α ]−1 }p−1(supt∈[0,T] E[‖ u(s) ‖ 2p Ḣ1 ])(t2 − t1) pβ(α−v−1)−α α , and (4.14) E[‖ I23 ‖ p Ḣν ] = E[‖ t2∫ t1 (t2s) β−1AνEββ(t2 −s)B(u(s),c(s))ds ‖p] ≤ Cpβ( t2∫ t1 [(t2 −s)β−1− β(ν+1) α ] p p−1 ds)p−1 × t2∫ t1 E[‖ A1B(u(s),c(s)) ‖ p Ḣ1 ]ds ≤ CpCp1 C p β{ p−1 p[β−β(ν+1) α ]−1 }p−1( sup t∈[0,T] E[‖ u(s) ‖2p Ḣ1 ])(t2 − t1) pβ(α−v−1) α . Int. J. Anal. Appl. 19 (6) (2021) 872 Next, by following the similar arguments as in the proof of (4.12)- (4.14) and using the Lemma (2.2), there holds E[‖ I31 ‖ p Ḣν ] = E[‖ t1∫ 0 (t1 −s)β−1Aν[Eβ,β(t2 −s) −Eβ,β(t1 −s)]g(u(s))dW(s) ‖p] ≤ C(p)E[( t1∫ 0 ‖ (t1 −s)β−1Aν[Eβ,β(t2 −s) −Eβ,β(t1 −s)] ‖2‖ g(u(s)) ‖2L20 ds) p 2 ] ≤ C(p)Cpβν(t2 − t1) pβν α ( t1∫ 0 (t1 −s) 2p(β−1) p−2 ds) p−2 2 t1∫ 0 E ‖ g(u(s)) ‖p L20 ds ≤ C(p)CpβνT 2pβ−p−1 2 ( p−1 2pβ−p−2 ) p−1( sup t∈[0,T] E[‖ u(t) ‖p])(t1 − t2) pβν α , and (4.15) E[‖ I32 ‖] = E[‖ t1∫ 0 [(t2 −s)β−1 − (t1 −s)β−1]AνEββ(t2 −s)g(u(s))dW(s) ‖p] ≤ C(p)E[( t1∫ 0 ‖ ((t2 −s)β−1 − (t1 −s)β−1)AνEββ(t2 −s) ‖2‖ g(u(s)) ‖2L20 ) p 2 ]ds ≤ C(p)Cpβ( t1∫ 0 {[(t2 −s)β−1 − (t1 −s)β−1] × (t2 −s)− βν 2 } 2p p−2 ds) p−2 2 × t1∫ 0 E ‖ g(u(s)) ‖p L20 ds ≤ C(p)CpβC pT[ α(p−2) 2pβ(α−ν)−(p+2)α] p−2 2 ( sup t∈[0,T] E[‖ u(t) ‖p])(t2 − t1) 2pβ(α−ν)−(p+2)α 2α , and (4.16) E[‖ I33 ‖] = E[‖ t2∫ t1 (t2 −s)β−1AνEββ(t2 −s)g(u(s))dW(s) ‖p] ≤ C(p)E[ t2∫ t1 (t2 −s)β−1AνEββ(t2 −s) ‖2‖ g(u(s)) ‖2L20 ds) p 2 ] ≤ C(p)Cpβ( t2∫ t1 [(t2 −s)β−1− βν α ] 2p p−2 ds) p−2 2 t2∫ t1 E ‖ g(u) ‖p L20 ds ≤ C(p)CpβC ρ[ α(p−2) 2pβ(α−ν)−(p+2)α] p−2 2 ( sup t∈[0,T] E[‖ u(t) ‖p])(t2 − t1) 2pβ(α−ν)−pα 2α . � Int. J. Anal. Appl. 19 (6) (2021) 873 Taking expectation on the both side of (4.8), and in view of the estimates (4.11)- (4.16), we conclude that (4.17) ‖ u(t2) −u(t1) ‖Lp(Ω;Ḣv)≤ C(t2 − t1) γ, in which we tak γ = min{βν α , pβ(α−ν−1)−α pα , 2pβ(α−ν)−(p+2)α 2pα }, whene 0 < t2 − t1 < 1. Otherwise, if t2 − t1 ≥ 1 then we set γ = max{ β(ν+1) α , β(α−ν−1) α , 2pβ(α−ν)−pα 2pα }. This completes the proof of Theorem (4.2) 5. Existence and uniqueness of mild solution Our main purpose of this section is to prove the existence and uniqueness of mild solution to the problem (2.6). To do this, the following assumptions are imposed. 5.1. Assumption. The measurable function f : Ω × H → L20 satisfies the following global Lipschitz and growth conditions: (5.1) ‖ f(v) ‖L20≤ C ‖ v ‖, ‖ f(u) −f(v) ‖L20≤ C ‖ u−v ‖, for all u,v ∈ H. 5.2. Assumption. Let C, is a positive real number, then the bounded bilinear operator L : L20(D) → H−1(D) satisfies the following properties: (5.2) ‖ L(c) ‖Ḣ−1≤ C ‖ c ‖ 2, and (5.3) ‖ L(c) −L(v) ‖Ḣ−1≤ C(‖ c ‖ + ‖ v ‖) ‖ c−v ‖, and for all v,c ∈ L20(D). 5.3. Assumption. Assume that the initial value c0 : Ω → Ḣν is a F0-measurable random variable, it holds that (5.4) ‖ c0 ‖Lp(Ω,Ḣν)< ∞, for any 0 ≤ ν < α < 2. Theorem 5.1. Let Assumptiom (5.1) to (5.3) be satisfied for some p ≥ 2, then there exists a unique mild solution (c(t))t∈[0,T] in the space L p(Ω,Ḣν) with 0 ≤ ν < α < 2. Int. J. Anal. Appl. 19 (6) (2021) 874 Proof. We fix an ω ∈ Ω and use the standard Picard’s iteration argument to prove the existence of mild solution. To begin with, the sequence of stochastic process {cn(t)}n≥0 is constructed as (5.5)   cn+1(t) = Eβ(t)c0 + N1(cn(t)) + N2(cn(t)),c0(t) = c0, where (5.6)   N1(cn(t)) = t∫ 0 (t−s)β−1Eβ,β(t−s)L(cn(s))ds, N2(cn(t)) = t∫ 0 (t−s)β−1Eβ,β(t−s)f(cn(s))dW(s). The proof will be split into three steps. Step1 For each n ≥ 0, we show that sup E[‖ cn(t) ‖ p Ḣν ] < ∞, note that (5.7) E[‖ cn+1(t) ‖ p Ḣν ] ≤ 3p−1E[‖ Eβ(t)c0 ‖ p Ḣν ] + 3p−1E[‖ N1(cn(t)) ‖ p Ḣν ] + 3p−1E[‖ N2(cn(t)) ‖ p Ḣν ]. The application of the Lemma (2.1) gives (5.8) E[‖ Eβ(t)c0 ‖ p Ḣν ] ≤ E[ ∞∫ 0 Mβ(θ)(‖ AνSα(tβθ)c0 ‖2) 1 2 dθ] = E[ ∞∫ 0 Mβ(θ)( ∑∞ n=1〈Aαe −tβθAαc0,en〉2) 1 2 dθ] = E[ ∞∫ 0 Mβ(θ)( ∑∞ n=1〈Aαu0,e −tβθλ α 2 en〉2) 1 2 dθ] ≤ E[ ∞∫ 0 Mβ(θ) ‖ c0 ‖Ḣν dθ] = E[‖ c0 ‖Ḣν ]. Applying the following Hölder inequality to the second term of the right-hand side of (5.7) (5.9) E[‖ N1(cn(t)) ‖ p Ḣν ] ≤ E[( t∫ 0 ‖ (t−s)β−1A1Eβ,β(t−s)Aν−1L(cn(s)) ‖ ds)p] ≤ Cpβ( t∫ 0 (t−s) p(β−1− β α ) p−1 ds)p−1 t∫ 0 E[‖ Aν−1L(cn(s)) ‖p]ds ≤ K1 t∫ 0 E[‖ cn(s) ‖ p Ḣν ]]ds, Int. J. Anal. Appl. 19 (6) (2021) 875 where K1 = C p βC p[ p−1 pβ(1− 1 α )−1 ] p−1Tpβ(1− 1 α )−1( max t∈[0,T] E[‖ cn(t) ‖ p Ḣν ). Making use of the Hölder inequality and Lemma (2.2) to the third term of the right-hand side of (5.7), we get (5.10) E[‖ N2(cn(t)) ‖ p Ḣν ] ≤ C(p)E[( t∫ 0 ‖ (t−s)β−1Eβ,β(t−s) ‖2 Aνf(cn(s)) ‖2L20 ds) p 2 ] ≤ K2 t∫ 0 E[‖ cn(s) ‖2Ḣν ]ds, where K2 = C(p)C p βC p[ p−2 p(2β−1)−2 ] p−2 2 T p(2β−1)−2 2 . Using the above estimates (5.7)- (5.10), we have E[‖ cn+1(t) ‖ p Ḣν ] ≤ 3p−1E[‖ c0 ‖ p Ḣν ] + 3p−1(K1 + K2) t∫ 0 E[‖ cn(s) ‖ p Ḣν ]ds . By means of the extension of Gronwall’s lemma, it holds that sup t∈[0,T] E[‖ cn+1(t) ‖ p Ḣν ] < ∞, for each n ≥ 0. Step1: Show that the sequence {cn(t)}n≥0 is a Cauchy sequence in the space Lp(Ω; Ḣν). For any n ≥ m ≥ 1, applying the similar arguments employed to obtain (5.9) and (5.10), we get (5.11) E[‖ cn(t) − cm(t) ‖ p Ḣν ] ≤ 2p−1E[‖ N1(cn−1(t)) −N1(cm−1(t)) ‖ p Ḣν ] + 2p−1E[‖ N2(cn−1(t)) −N2(cm−1(t)) ‖ p Ḣν ] ≤ K t∫ 0 E[‖ cn−1(s) − cm−1(s) ‖ p Ḣν ]ds, in wich (5.12) K = 2p−1{CpβC p}[ p−1 p(β−β ν )−1 ]p−1Tp(β− β α )( max t∈[0,T] E[‖ cn−1(t) ‖ p Ḣν ]) + max t∈[0,T] E[‖ cm−1(t) ‖ p Ḣν ]) + C(p)C p βC p[ p−2 p(2β−1)−2 ] p−2 2 T p(2β−1)−2 2 }. A direct application of Gronwall’s lemma yields sup t∈[0,T] E[‖ cn(t) − cm(t) ‖ p Ḣν ] = 0, for all T > 0. Taking limits to the stochastic sequence {cn(t)}n≥0 in (5.5) as n → ∞, we finish the proof of the existence of mild solution to (2.6). Step 3: We show the uniqueness of mild solution. Assume c and v are two mild solutions of the problem Int. J. Anal. Appl. 19 (6) (2021) 876 (2.6), using the similar calculations as in Step 2, we can obtain (5.13) sup t∈[0,T] E[‖ c(t) −v(t) ‖p Ḣν ] = 0, for all T > 0, which implies that c = v, it follows that the uniqueness of mild solution. Obviously, when ν = 0, the above three steps still work. Thus the proof of Theorem (6.1) is completed. � 6. Regularity of mild solution In this section, we will prove the spatial and temporal regularity properties of mild solution to time-space fractional SKSM based on the analytic semigroup. Theorem 6.1. Let Assumptions (5.1) to (5.3) hold with 1 ≤ ν < α < 2 and p ≥ 2, let c(t) be a unique mild solution of the problem (2.6) with P(c(t) ∈ Ḣν) = 1 for any t ∈ [0,T], then there exists a constant C such that (6.1) sup t∈[0,T] ‖ c(t) ‖Lp(Ω;Ḣν)≤ C(‖ c0 ‖Lp(Ω;H) + sup t∈[0,T] ‖ c(t) ‖Lp(Ω;Ḣ1)). Proof. For any 0 ≤ t ≤ T and 1 ≤ ν < α < 2, we have (6.2) ‖ c(t) ‖Lp(Ω;Ḣν) = (E[‖ c(t) ‖ p Ḣν ]) 1 p =‖ Aνc(t) ‖Lp(Ω;H) ≤ ‖ AνEβ(t)c0 ‖Lp(Ω;H) + ‖ Aν t∫ 0 (t−s)β−1Eβ,β(t−s)L(c(s))ds ‖Lp(Ω;H) + ‖ Aν t∫ 0 (t−s)β−1Eβ,β(t−s)f(c(s))dW(s) ‖ Lp(Ω;H) = I + II + III. Using Theorem (2.1), the first term can be estimated by (6.3) I =‖ AνEβ(t)c0 ‖Lp(Ω;H)≤ Cαt− βν α ‖ c0 ‖Lp(Ω;H)< ∞. It is easy to know that (6.4) T∫ 0 Cαt −βν α ‖ c0 ‖Lp(Ω;H) dt = αCα α−βν T α−βν α ‖ c0 ‖Lp(Ω;H) . Int. J. Anal. Appl. 19 (6) (2021) 877 The application of Theorem (2.1) and Assumptions (5.2), we get (6.5) (II)p ≤ E[(‖ Aν t∫ 0 (t−s)β−1AνEβ,β(t−s)L(c(s)) ‖ ds)p] ≤ Cpβ( t∫ 0 (t−s) p[β−1− β(ν+1) α ] p−1 ds)p−1 t∫ 0 E[‖ A−1L(c(s)) ‖ p Ḣ1 ]ds ≤ C2 sup t∈[0,T] E[‖ c(s) ‖p Ḣ1 ], where C2 = C p βC p{ p−1 p[β−β(ν+1) α ]−1 }p−1Tp[β− β(ν+1) α ]−1 + ( max t∈[0,T] E[‖ c(t) ‖Ḣ1 ]). By means of Theorem (2.1), Assumptions (5.1) and Lemma (2.2), we can deduce (6.6) (III)p ≤ C(p)E[(‖ Aν t∫ 0 ‖ (t−s)β−1Aν−1Eβ,β(t−s) ‖2‖ A1f(c(s)) ‖2L02 ds) p 2 ] ≤ C(p)Cpβ( t∫ 0 (t−s) 2p[β−1− β(ν−1) α ] p−2 ds) p−2 2 t∫ 0 E ‖ A1f(c(s)) ‖ p L02 ds ≤ C3 sup t∈[0,T] E[‖ c(s) ‖p Ḣ1 ], where C3 = C(p)C p βC p[ p−2 p[2β−1−β(ν−1) α ]−2 ] p−2 2 T p[2β−1− β(ν−1) α ]−2 2 . Thus, we conclude the proof of Theorem (6.1) by combining with the estimates (6.2)- (6.6). � Next, we will devote to the temporal regularity of the mild solution. Theorem 6.2. Let Assumptions (5.1) to (5.3) hold with 0 < ν < α < 2 and p ≥ 2, for any 0 ≤ t1 < t2 ≤ T , the unique mild solution c(t)to the problem (2.6) is Hölder continuous with respect to the norm ‖ . ‖Lp(Ω;Ḣν) and satisfies (6.7) ‖ c(t2) − c(t1) ‖Lp(Ω;Ḣν)≤ C(t2 − t1) γ. Int. J. Anal. Appl. 19 (6) (2021) 878 Proof. Foor any 0 ≤ t1 < t2 ≤ T, for the mild solution (2.6), we have (6.8) c(t2) − c(t1) = Eβ(t2)c0 −Eβ(t1)c0 + t2∫ 0 (t2 −s)β−1Eβ,β(t2 −s)L(c(s))ds − t1∫ 0 (t1 −s)β−1Eβ,β(t1 −s)L(c(s))ds + t2∫ 0 (t2 −s)β−1β−1Eβ,β(t2 −s)f(c(s))dW(s) − t1∫ 0 (t2 −s)β−1β−1Eβ,β(t1 −s)G(u(s))dW(s) = I1 + I2 + I3, where I1 = Eβ(t2)c0 −Eβ(t1)c0, and (6.9) I2 = t2∫ 0 (t2 −s)β−1Eβ,β(t2 −s)L(c(s))ds − t1∫ 0 (t1 −s)β−1Eβ,β(t1 −s)L(c(s))ds = t1∫ 0 (t1 −s)β−1[Eβ,β(t2 −s) −Eβ,β(t1 −s)]L(c(s))ds + t1∫ 0 [(t1 −s)β−1 − (t2 −s)β−1]Eβ,β(t2 −s)L(c(s))ds + t2∫ t1 (t2 −s)β−1Eβ,β(t2 −s)L(c(s))ds = I21 + I22 + I23, Int. J. Anal. Appl. 19 (6) (2021) 879 and (6.10) I3 = t2∫ 0 (t2 −s)β−1Eβ,β(t2 −s)f(c(s))dW(s) − t1∫ 0 (t1 −s)β−1Eβ,β(t1 −s)f(c(s))dW(s) = t1∫ 0 (t1 −s)β−1[Eβ,β(t2 −s) −Eβ,β(t1 −s)]f(c(s))dW(s) + t1∫ 0 [(t2 −s)β−1 − (t1 −s)β−1]Eβ,β(t2 −s)g(u(s))dW(s) + t2∫ t1 (t2 −s)β−1Eβ,β(t2 −s)f(c(s))dW(s) = I31 + I32 + I33. For any 0 < ν < α < 2 and p ≥ 2, by virtue of Theorem (2.2), it follows that (6.11) E[‖ I1 ‖ p Ḣν ] = E[‖ Aν[Eβ(t2) −Eβ(t1)]c0 ‖p] ≤ Cpαν(t2 − t1) pβν α E[‖ c0 ‖p]. For the first term I21 in (6.9), applying the Assumption (5.2) and Theorem (6.2) and Hölder’s inequality, we have (6.12) E[‖ I21 ‖ p Ḣν ] = E[‖ t1∫ 0 (t1 −s)β−1Aν[Eβ,β(t2 −s) −Eβ,β(t1 −s)]L(c(s))ds ‖p] ≤ Cpβν(t2 − t1) pβ(ν+1) α ( t1∫ 0 (t1 −s) p(β−1) p−1 ds)p−1 t1∫ 0 E[‖ A−1L(c(s)) ‖ p Ḣ1 ]ds ≤ CpCpβνT p( p−1 pβ−1 ) p−1( sup t∈[0,T] E[‖ c(s) ‖2p Ḣ1 ])(t2 − t1) pβ(ν+1) α . Int. J. Anal. Appl. 19 (6) (2021) 880 Using the Assumptions (5.2), Theorem (6.2) and Hölder’s inequality, we get (6.13) E[‖ I22 ‖ p Ḣν = E[‖ t1∫ 0 [(t2 −s)β−1 − (t1 −s)β−1]AνEβ,β(t2 −s)L(c(s))ds ‖p] ≤ Cpβ( t1∫ 0 {[(t2 −s)β−1 − (t1 −s)β−1] × (t2 −s)− β(ν+1) α } p p−1 ds)p−1 × t1∫ 0 E[‖ A1L(c(s)) ‖ p Ḣ1 ]ds ≤ CpCpβT{ p−1 p[β−β(ν+1) α ]−1 }p−1( sup t∈[0,T] E[‖ c(s) ‖2p Ḣ1 ])(t2 − t1) pβ(α−v−1)−α α , and (6.14) E[‖ I23 ‖ p Ḣν ] = E[‖ t2∫ t1 (t2 −s)β−1AνEββ(t2 −s)L(c(s))ds ‖p] ≤ Cpβ( t2∫ t1 [(t2 −s)β−1− β(ν+1) α ] p p−1 ds)p−1 × t2∫ t1 E[‖ A1L(c(s)) ‖ p Ḣ1 ]ds ≤ CpCpβ{ p−1 p[β−β(ν+1) α ]−1 }p−1( sup t∈[0,T] E[‖ c(s) ‖2p Ḣ1 ])(t2 − t1) pβ(α−v−1) α . Next, by following the similar arguments as in the proof of (6.12)- (6.14) and using the Lemma (2.2), there holds E[‖ I31 ‖ p Ḣν ] = E[‖ t1∫ 0 (t1 −s)β−1Aν[Eβ,β(t2 −s) −Eβ,β(t1 −s)]f(u(s))dW(s) ‖p] ≤ C(p)E[( t1∫ 0 ‖ (t1 −s)β−1Aν[Eβ,β(t2 −s) −Eβ,β(t1 −s)] ‖2‖ f(c(s)) ‖2L20 ds) p 2 ] ≤ C(p)Cpβν(t2 − t1) pβν α ( t1∫ 0 (t1 −s) 2p(β−1) p−2 ds) p−2 2 t1∫ 0 E ‖ f(c(s)) ‖p L20 ds ≤ C(p)CpβνT 2pβ−p−1 2 ( p−1 2pβ−p−2 ) p−1( sup t∈[0,T] E[‖ c(t) ‖p])(t1 − t2) pβν α , Int. J. Anal. Appl. 19 (6) (2021) 881 and (6.15) E[‖ I32 ‖] = E[‖ t1∫ 0 [(t2 −s)β−1 − (t1 −s)β−1]AνEββ(t2 −s)f(c(s))dW(s) ‖p] ≤ C(p)E[( t1∫ 0 ‖ ((t2 −s)β−1 − (t1 −s)β−1)AνEββ(t2 −s) ‖2‖ f(c(s)) ‖2L20 ) p 2 ]ds ≤ C(p)Cpβ( t1∫ 0 {[(t2 −s)β−1 − (t1 −s)β−1] × (t2 −s)− βν 2 } 2p p−2 ds) p−2 2 × t1∫ 0 E ‖ f(c(s)) ‖p L20 ds ≤ C(p)CpβC pT [ α(p−2) 2pβ(α−ν)−(p+2)α] p−2 2 ( sup t∈[0,T] E[‖ c(t) ‖p])(t2 − t1) 2pβ(α−ν)−(p+2)α 2α , and (6.16) E[‖ I33 ‖] = E[‖ t2∫ t1 (t2 −s)β−1AνEββ(t2 −s)f(c(s))dW(s) ‖p] ≤ C(p)E[ t2∫ t1 (t2 −s)β−1AνEββ(t2 −s) ‖2‖ f(c(s)) ‖2L20 ds) p 2 ] ≤ C(p)Cpβ( t2∫ t1 [(t2 −s)β−1− βν α ] 2p p−2 ds) p−2 2 t2∫ t1 E ‖ f(u) ‖p L20 ds ≤ C(p)CpβC ρ[ α(p−2) 2pβ(α−ν)−(p+2)α] p−2 2 ( sup t∈[0,T] E[‖ c(t) ‖p])(t2 − t1) 2pβ(α−ν)−pα 2α Taking expectation on the both side of (6.8), and in view of the estimates (6.11)- (6.16), we conclude that (6.17) ‖ c(t2) − c(t1) ‖Lp(Ω;Ḣv)≤ C(t2 − t1) γ, in which we tak γ = min{βν α , pβ(α−ν−1)−α pα , 2pβ(α−ν)−(p+2)α 2pα }, whene 0 < t2 − t1 < 1. Otherwise, if t2 − t1 ≥ 1 then we set γ = max{ β(ν+1) α , β(α−ν−1) α , 2pβ(α−ν)−pα 2pα }. � This completes the proof of Theorem (6.2) Int. J. Anal. Appl. 19 (6) (2021) 882 7. Appendix A Considering the following abstract formulation of time-space fractional stochastic of equation (2.1) (7.1)   cD β t u(t) = −Aαu(t) + B(u(t),c(t)) + g(u(t)) dW(t) dt , t > 0, u(0) = u0. We derive the mild solution to (7.1) by means of Laplace transform, which denoted by a. Let λ > 0, and we define that û(λ) = ∞∫ 0 e−λsu(s)ds, B̂(λ) = ∞∫ 0 e−λsB(u(s),c(s))ds, and Ĝ(λ) = ∞∫ 0 e−λs[g(u(s)) dW(s) ds ]ds = ∞∫ 0 e−λsg(u(s))dW(s). Upon Laplace transform, using the formula cD̂ β t u(λ) = λ βû−λβ−1u0. Then applying the Laplace transform to (7.1), we obtain (7.2) û(λ) = 1 λ u0 + 1 λβ (−Aα)û(λ) + 1λβ [B̂(λ) + Ĝ(λ)] = λβ−1(λβI + Aα) −1u0 + (λ βI + Aα) −1[B̂(λ) + Ĝ(λ)] = λβ−1 ∞∫ 0 e−λ βsSα(s)u0ds + ∞∫ 0 e−λ βsSα(s)[B̂(λ) + Ĝ(λ)]ds, in which I is the identity operator, and Sα(t) = e −tAα is an analytic semigroup generated by the operator −Aα. We introduce the following one-sided stable probability density function: (7.3) Wβ = 1 π ∞∑ n=1 (−1)n−1θβn−1 Γ(βn + 1) n! sin(nπβ), θ ∈ (0,∞), whose Laplace transform is given by (7.4) ∞∫ 0 e−λθWβ(θ)dθ = e −λβ, 0 < β < 1. Int. J. Anal. Appl. 19 (6) (2021) 883 Making use of above expression (7.4), then the terms on the right-hand side of (7.2) can be written as (7.5) λβ−1 ∞∫ 0 e−λ βsSα(s)u0ds = ∞∫ 0 λβ−1e−λ βtβSα(t β)u0dt = ∞∫ 0 β(λt)β−1e−(λt) β Sα(t β)u0dt = ∞∫ 0 1 λ d dt [e−(λt) β ]Sα(t β)u0dt = ∞∫ 0 ∞∫ 0 θWβ(θ)e −λtθSα(t β)u0dθdt = ∞∫ 0 e−λt[ ∞∫ 0 Wβ(θ)Sα( tβ θβ )u0dθ]dt, and (7.6) ∞∫ 0 e−λ βsSα(s)B̂(λ)ds = ∞∫ 0 βtβ−1e−(λt) β Sα(t β)B̂(λ)dt = ∞∫ 0 ∞∫ 0 βtβ−1e−(λt) β Sα(t β)e−λstβ−1B(u(s),c(s))dsdt = ∞∫ 0 ∞∫ 0 ∞∫ 0 βWβ(θ)e −λtθSα(t β)e−λstβ−1B(u(s),c(s))dθdsdt = ∞∫ 0 ∞∫ 0 ∞∫ 0 βWβ(θ)e −λ(t+s)Sα( tβ θβ )t β−1 θβ B(u(s),c(s))dθdsdt = ∞∫ 0 e−λt[β t∫ 0 ∞∫ 0 Wβ(θ)Sα( (t−s)β θβ ) (t−s)β−1 θβ B(u(s),c(s))dθds]dt, and ∞∫ 0 e−λ βsSα(s)Ĝ(λ)ds = ∞∫ 0 βtβ−1e−(λt) β Sα(t β)Ĝ(λ)dt = ∞∫ 0 ∞∫ 0 βtβ−1e−(λt) β Sα(t β)e−λsg(u(s))dW(s)dt Int. J. Anal. Appl. 19 (6) (2021) 884 (7.7) = ∞∫ 0 ∞∫ 0 ∞∫ 0 βWβ(θ)e −λtθSα(t β)e−λstβ−1g(u(s))dθdW(s)dt = ∞∫ 0 ∞∫ 0 ∞∫ 0 βWβ(θ)e −λ(t+s)Sα( tβ θβ )t β−1 θβ g(u(s))dθdW(s)dt = ∞∫ 0 e−λt[β t∫ 0 ∞∫ 0 Wβ(θ)Sα( (t−s)β θβ ) (t−s)β−1 θβ g(u(s))dθdW(s)]dt. Together with (7.2) and (7.5)- (7.7) helps us to get (7.8) û(λ) = ∞∫ 0 e−λt[ ∞∫ 0 Wβ(θ)Sα( tβ θβ )u0dθ]dt + ∞∫ 0 e−λt[β t∫ 0 ∞∫ 0 Wβ(θ)Sα( (t−s)β θβ ) (t−s)β−1 θβ B(u(s),c(s))dθds]dt + ∞∫ 0 e−λt[β t∫ 0 ∞∫ 0 Wβ(θ)Sα( (t−s)β θβ ) (t−s)β−1 θβ g(u(s))dθdW(s)]dt, Now, by means of inverse Laplace transform to (7.8), we have achieved that (7.9) u(t) = ∞∫ 0 Wβ(θ)Sα( tβ θβ )u0dθ + β t∫ 0 ∞∫ 0 Wβ(θ)Sα( (t−s)β θβ ) (t−s)β−1 θβ B(u(s),c(s))dθds + β t∫ 0 ∞∫ 0 Wβ(θ)Sα( (t−s)β θβ ) (t−s)β−1 θβ g(u(s))dθdW(s) = ∞∫ 0 1 β θ− 1 β −1Wβ(θ − 1 β )Sα(t βθ)u0dθ + t∫ 0 ∞∫ 0 θ− 1 β Wβ(θ − 1 β )Sα((t−s)βθ)(t−s)β−1B(u(s),c(s))dθds t∫ 0 ∞∫ 0 θ− 1 β Wβ(θ − 1 β )Sα((t−s)βθ)(t−s)β−1g(u(s))dθdW(s). Here, we also introduce the Mainardi’s Wright-type function Mβ(θ) = ∞∑ n=0 (−1)nθn nθ!Γ(1 −β(1 + n)) = 1 π ∞∑ n=1 (−1)n−1θn−1 (n− 1)! Γ(nβ) sin(nπβ), Int. J. Anal. Appl. 19 (6) (2021) 885 where 0 < β < 1 and θ ∈ (0,∞). Further, the relationships between the probability density function Wβ(θ) and Mainardi’s Wright-type function Mβ(θ) are shown that Mβ(θ) = 1 β θ− 1 β −1Wβ(θ − 1 β ). We denote the generalized Mittag-Leffler operators Eα(t) and Eββ(t) as Eα(t) = ∞∫ 0 Mβ(θ)Sα(t βθ)dθ, and Eββ(t) = ∞∫ 0 βθMβ(θ)Sα(t βθ)dθ. Therefore, the equation (7.9) can be written as (7.10) u(t) = Eβ(t)u0 + t∫ 0 (t−s)β−1Eββ(t−s)B (u(s),c(s)) ds + t∫ 0 (t−s)β−1Eββ(t−s)g(u(s))dW(s), Up to now, we have deduced the mild solution (7.10) to the time-space fractional stochastic equation (2.1). 8. Appendix B Considering the following abstract formulation of time-space fractional stochastic of equation (2.6) (8.1)   cD β t c(t) = −Aαc(t) + L(c(t)) + f(c(t)) dW(t) dt , t > 0, c(0) = c0, We derive the mild solution to (8.1) by means of Laplace transform, which denoted by ˆ. λ > 0, and we define that ĉ(λ) = ∞∫ 0 e−λsc(s)ds, L̂(λ) = ∞∫ 0 e−λsL(c(s))ds, and Ĥ(λ) = ∞∫ 0 e−λs[f(c(s)) dW(s) ds ]ds = ∞∫ 0 e−λsf(c(s))dW(s). Upon Laplace transform, using the formula cD̂ β t c(λ) = λ βĉ−λβ−1c0. Then applying the Laplace transform to (8.1), we obtain (8.2) ĉ(λ) = 1 λ c0 + 1 λβ (−Aα)ĉ(λ) + 1λβ [L̂(λ) + Ĥ(λ)] = λβ−1(λβI + Aα) −1c0 + (λ βI + Aα) −1[L̂(λ) + Ĥ(λ)] = λβ−1 ∞∫ 0 e−λ βsSα(s)c0ds + ∞∫ 0 e−λ βsSα(s)[L̂(λ) + Ĥ(λ)]ds Int. J. Anal. Appl. 19 (6) (2021) 886 in which I is the identity operator, and Sα(t) = e −tAα is an analytic semigroup generated by the operator −Aα. We introduce the following one-sided stable probability density function: (8.3) Wβ = 1 π ∞∑ n=1 (−1)n−1θβn−1 Γ(βn + 1) n! sin(nπβ), θ ∈ (0,∞), whose Laplace transform is given by (8.4) ∞∫ 0 e−λθWβ(θ)dθ = e −λβ, 0 < β < 1. Making use of above expression (8.4), then the terms on the right-hand side of (8.2) can be written as (8.5) λβ−1 ∞∫ 0 e−λ βsSα(s)c0ds = ∞∫ 0 λβ−1e−λ βtβSα(t β)c0dt = ∞∫ 0 β(λt)β−1e−(λt) β Sα(t β)c0dt = ∞∫ 0 1 λ d dt [e−(λt) β ]Sα(t β)c0dt = ∞∫ 0 ∞∫ 0 θWβ(θ)e −λtθSα(t β)c0dθdt = ∞∫ 0 e−λt[ ∞∫ 0 Wβ(θ)Sα( tβ θβ )c0dθ]dt, and (8.6) ∞∫ 0 e−λ βsSα(s)L̂(λ)ds = ∞∫ 0 βtβ−1e−(λt) β Sα(t β)L̂(λ)dt = ∞∫ 0 ∞∫ 0 βtβ−1e−(λt) β Sα(t β)e−λstβ−1L(c(s))dsdt = ∞∫ 0 ∞∫ 0 ∞∫ 0 βWβ(θ)e −λtθSα(t β)e−λstβ−1L(c(s))dθdsdt = ∞∫ 0 ∞∫ 0 ∞∫ 0 βWβ(θ)e −λ(t+s)Sα( tβ θβ )t β−1 θβ L(c(s))dθdsdt = ∞∫ 0 e−λt[β t∫ 0 ∞∫ 0 Wβ(θ)Sα( (t−s)β θβ ) (t−s)β−1 θβ L(c(s))dθds]dt, Int. J. Anal. Appl. 19 (6) (2021) 887 and (8.7) ∞∫ 0 e−λ βsSα(s)Ĥ(λ)ds = ∞∫ 0 βtβ−1e−(λt) β Sα(t β)Ĥ(λ)dt = ∞∫ 0 ∞∫ 0 βtβ−1e−(λt) β Sα(t β)e−λsf(c(s))dW(s)dt = ∞∫ 0 ∞∫ 0 ∞∫ 0 βWβ(θ)e −λtθSα(t β)e−λstβ−1f(c(s))dθdW(s)dt = ∞∫ 0 ∞∫ 0 ∞∫ 0 βWβ(θ)e −λ(t+s)Sα( tβ θβ )t β−1 θβ f(c(s))dθdW(s)dt = ∞∫ 0 e−λt[β t∫ 0 ∞∫ 0 Wβ(θ)Sα( (t−s)β θβ ) (t−s)β−1 θβ f(c(s))dθdW(s)]dt. Together with (8.2) and (8.5)- (8.7) helps us to get (8.8) ĉ(λ) = ∞∫ 0 e−λt[ ∞∫ 0 Wβ(θ)Sα( tβ θβ )c0dθ]dt + ∞∫ 0 e−λt[β t∫ 0 ∞∫ 0 Wβ(θ)Sα( (t−s)β θβ ) (t−s)β−1 θβ L(c(s))dθds]dt + ∞∫ 0 e−λt[β t∫ 0 ∞∫ 0 Wβ(θ)Sα( (t−s)β θβ ) (t−s)β−1 θβ f(c(s))dθdW(s)]dt. Now, by means of inverse Laplace transform to (8.8), we have achieved that c(t) = ∞∫ 0 Wβ(θ)Sα( tβ θβ )c0dθ + β t∫ 0 ∞∫ 0 Wβ(θ)Sα( (t−s)β θβ ) (t−s)β−1 θβ L(c(s))dθds + β t∫ 0 ∞∫ 0 Wβ(θ)Sα( (t−s)β θβ ) (t−s)β−1 θβ f(c(s))dθdW(s) Int. J. Anal. Appl. 19 (6) (2021) 888 (8.9) = ∞∫ 0 1 β θ− 1 β −1Wβ(θ − 1 β )Sα(t βθ)c0dθ + t∫ 0 ∞∫ 0 θ− 1 β Wβ(θ − 1 β )Sα((t−s)βθ)(t−s)β−1L(c(s))dθds t∫ 0 ∞∫ 0 θ− 1 β Wβ(θ − 1 β )Sα((t−s)βθ)(t−s)β−1f(c(s))dθdW(s). Here, we also introduce the Mainardi’s Wright-type function Mβ(θ) = ∞∑ n=0 (−1)nθn nθ!Γ(1 −β(1 + n)) = 1 π ∞∑ n=1 (−1)n−1θn−1 (n− 1)! Γ(nβ) sin(nπβ), where 0 < β < 1 and θ ∈ (0,∞). Further, the relationships between the probability density function Wβ(θ) and Mainardi’s Wright-type function Mβ(θ) are shown that Mβ(θ) = 1 β θ− 1 β −1Wβ(θ − 1 β ). We denote the generalized Mittag-Leffler operators Eα(t) and Eββ(t) as Eα(t) = ∞∫ 0 Mβ(θ)Sα(t βθ)dθ, and Eββ(t) = ∞∫ 0 βθMβ(θ)Sα(t βθ)dθ. Therefore, the equation (7.9) can be written as (8.10) c(t) = Eβ(t)c0 + t∫ 0 (t−s)β−1Eββ(t−s)L(c(s))ds + t∫ 0 (t−s)β−1Eββ(t−s)f(c(s))dW(s). Up to now, we have deduced the mild solution (8.10) to the time-space fractional stochastic equation (2.6). 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