Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 49, pp. 1–20. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: https://doi.org/10.58997/ejde.2023.49 STOCHASTIC BURGERS EQUATIONS WITH FRACTIONAL DERIVATIVE DRIVEN BY FRACTIONAL NOISE YUBO DUAN, YIMING JIANG, YANG TIAN, YAWEI WEI Abstract. In this article, we study fractional stochastic Burgers equations perturbed by fractional noise. Existence and uniqueness of a mild solution is given by a fixed point argument. Then, we explore Hölder regularity of the mild solution in C([0, T∗];Lp(Ω; Ḣγ)) for some stopping time T∗. 1. Introduction In this article, we study the fractional stochastic Burgers equation with fractional noise Dβ t u(t, x)− u(t, x) ∂u(t, x) ∂x + (−∆)α/2u(t, x) = ẆH(t), u(0, x) = u0(x), u(t, x) ∣∣ ∂D = 0, (1.1) where (t, x) ∈ [0, T ]×D, D is a bounded interval in R, and u0 ∈ L2(D). The operator (−∆)α/2 is the fractional power of −∆, with α ∈ (1, 2), defined as (−∆)α/2en := λα/2n en, n = 1, 2, . . . , (1.2) where λn is the eigenvalue of −∆, with the corresponding eigenvector en. The fractional derivative Dβ t is the Caputo derivative of order β ∈ (0, 1] in the time variable, which is defined as follows (see [15]) Dβ t u(t, x) = { 1 Γ(1−β) ∫ t 0 ∂u(s,x) ∂s ds (t−s)β , 0 < β < 1, ∂u(t,x) ∂t , β = 1, (1.3) in which the gamma function is defined as Γ(β) = ∫∞ 0 tβ−1e−tdt. The process {WH(t), t ∈ [0, T ]} is a cylindrical fractional Brownian motion on a real and sepa- rable Hilbert space, with Hurst parameter H ∈ (1/2, 1). The classical stochastic Burgers equation ∂u ∂t = ∂2u ∂x2 − 1 2 u ∂u ∂x + σ(u) ∂2W ∂t∂x . 2020 Mathematics Subject Classification. 60H15, 35R60, 35K05. Key words and phrases. Stochastic Burgers equation; Caputo derivative; fractional noise; Mittag-Leffler operator; mild solution. ©2023. This work is licensed under a CC BY 4.0 license. Submitted May 25, 2023. Published July 17, 2023. 1 2 Y. DUAN, Y. JIANG, Y. TIAN, Y. WEI EJDE-2023/49 models a turbulent flow and is solved by the Hopf-Cole transformation [5, 6, 7]. In the past few years, stochastic Burgers equations perturbed by different random noises have been studied intensively. This equation plays an important role in nonlinear acoustics, cosmology, and statistical physics [1, 2, 3, 21]. The author in [16] considers one dimensional stochastic Burgers equation driven by white noise term, and obtains existence of a weak solution by proving tightness for a sequence of polygonal approximations and solving a martingale problem for the weak limit. In [9] the authors explore the existence and uniqueness of the global solution of a stochastic Burgers equation perturbed by white noise, and the existence of an invariant measure the corresponding transition semigroup. Stochastic Burgers equations with fractional Laplacian in spatial variable have been also explored. For instance, the researchers in [25] study a model involving the Lipschitz continuity of the inhomogeneous term, and a diffusion coefficient with space-time white noise in local subspace. In [4] the authors explore existence and uniqueness of invariant measures for the stochastic Burgers equation driven by fractional Laplacian and space-time white noise. They show that the transition measures of the solution converge to the invariant measure in the norm of total variation. Many researchers have developed interests in the time-fractional diffusion equa- tions [24, 27, 30] which are also applied for describing the memory effect of the wall friction through the boundary layer [13]. The authors in [8] studied the non- linear stochastic equation of fractional derivative both in space and time variables with space-time white noise. They obtained the existence and uniqueness of solu- tion with the moment bounds of solutions under Dalang’s condition. In [31], it is proved that there is a unique mild solution of the stochastic Burgers equation with time- and space-fractional derivative driven by white noise, by a Picard iteration method. Different from the white noise in the model in [31], here we consider the fractional noise in time variable. The fractional Brownian motion was first introduced with a Hilbert space frame- work by Kolmogorov in [17]. In recent years, fractional Brownian motion has been attracted attention because of their useful feature of preserving long term memory, and a large number of interesting results from scaling invariance to the description of their laws as random fields have been established by various authors. The study of these Gaussian processes has its historical motivation from their applications in hy- drology and telecommunication, and has been applied to the mathematical finance, biotechnology and biophysics, see for example [11, 18, 23] and their references. In [14], the researchers explore that the existence, uniqueness, and moment estimate for the solution of the stochastic Burgers equation driven by multi-parameter frac- tional noise. The authors in [26] show the local and global existence and uniqueness results for the stochastic Burgers equation driven by fractional Brownian motion with H > 1 4 . In our work, we consider a U-valued Q-cylindrical fractional Brownian motion with Hurst parameter H ∈ (1/2, 1). The above research work motivates us to obtain the existence and uniqueness of the mild solution to the problem (1.1) with boundary and initial conditions, and explore Hölder regularity of the mild solution. EJDE-2023/49 FRACTIONAL STOCHASTIC EVOLUTION EQUATION 3 Definition 1.1. The domain of the fractional Laplace operator (−∆)α/2 in (1.2) is defined as Ḣα := { v ∈ L2(D) : ∞∑ n=1 λαn〈v, en〉2 <∞ } , with the inner product 〈·, ·〉 in L2(D). Thus, we define the norm as ‖v‖2 Ḣα := ‖Aαv‖2L2(D) = ∞∑ n=1 λαn〈v, en〉2. (1.4) In this article we use the following notation. • Aα := (−∆)α/2. • The eigenvalues of −∆ are λ1 ≤ λ2 ≤ · · · ≤ Λ, where Λ denote the maxi- mum of the eigenvalues in D. • ‖ · ‖ := ‖ · ‖L2(D). • B(u, v) := u ∂v∂x , B(u) := B(u, u). • The domain of the operator B is D(B) := H1 0 (D)×H1 0 (D). Thus, problem (1.1) can be rewritten as Dβ t u(t) = −Aαu(t) +B(u(t)) + ẆH(t), u(0) = u0. (1.5) Now, we introduce the Bochner spaces Lp(Ω; G) = Lp((Ω,F ,P); G) as Lp(Ω; G) = { f : E‖f‖pG = ∫ Ω ‖f(ω)‖pGdP(ω) <∞, ω ∈ Ω } , with the norm ‖f‖Lp(Ω;G) = (E‖f‖pG)1/p, where G is a Banach space. Next, we define mild solutions of problem (1.5), which is inspired by the def- inition of mild solution to the fractional stochastic Burgers equations driven by multiplicative white noise [31]. Definition 1.2. Let {u(t), t ∈ [0, T ]} be a random field that is continuous with respect to t. A function u ∈ C([0, T ];Lp(Ω; Ḣγ)) is a mild solution of (1.5) if u(t) = Lαβ(t)u0 + ∫ t 0 (t− s)β−1Lαβ,β(t− s)B(u(s))ds + ∫ t 0 (t− s)β−1Lαβ,β(t− s) dWH(s), (1.6) where Lαβ(t) and Lαβ,β(t) are the generalized Mittag-Leffler operators defined by (2.5) and (2.6). A derivation of the mild solution is shown in the Appendix, which applies the Laplace transform method and the properties of the semigroup generated from the fractional Laplace operator. In the following, we give some assumptions about the operator B and the initial condition u0. Assumption 1.3. The bounded bilinear operator B satisfies ‖B(u)‖ 6M‖u‖2, ‖B(u)−B(v)‖ 6M(‖u‖+ ‖v‖)‖u− v‖, for all u, v ∈ L2(D), where M is a positive constant. 4 Y. DUAN, Y. JIANG, Y. TIAN, Y. WEI EJDE-2023/49 Assumption 1.4. Let the initial value u0 : Ω → Ḣγ be F0-measurable random variable, satisfying ‖u0‖Lp(Ω;Ḣγ) <∞ for all 0 < γ < α < 2. We set a subspace of C([0, T ];Lp(Ω; Ḣγ)) for a stopping time T ′ as follows ST ′ := {u ∈ C([0, T ];Lp(Ω; Ḣγ)) : sup t∈[0,T ′] E‖u(t)‖p Ḣγ 6 K}, (1.7) where γ > 0, 0 < T ′ 6 T . The main results in this article reads as follows. Theorem 1.5. Let Assumption 1.3 and 1.4 be fulfilled with p > 2, 1/2 < β < 1, 1 < α < 2, and 0 < γ < α(2β−1) 2β . Then there exists a unique mild solution of (1.1) in the space ST∗ , with some stopping time T∗ ∈ [0, T ]. Theorem 1.6. Let Assumption 1.3 and 1.4 be fulfilled with p > 2, 1/2 < β < 1, 1 < α < 2, and 0 < γ < α(2β−1) 2β . Let u be a solution of (1.1) in ST∗ , with T∗ satisfying the conditions in Theorem 1.5. Then for any 0 6 t1 < t2 6 T∗, the solution u(t) is Hölder continuous with respect to the norm ‖·‖Lp(Ω;Ḣγ) and satisfies ‖u(t2)− u(t1)‖p Lp(Ω;Ḣγ) 6 C(t2 − t1)τ , with τ = { min { p ( β − βγ α − 1 2 ) , pβγα } , 0 < t2 − t1 < 1, p(β − βγ α ), t2 − t1 > 1. Now we highlight the contribution of this article in the field of the fractional stochastic Burgers equations. Firstly, our model with time- and space-fractional stochastic Burgers equation driven by fractional noise is new, compared to the problems studied in [4, 9, 14, 31]. Secondly, the U-valued Q-cylindrical fractional Brownian motion makes some difficulties in the analysis, we apply the embedding theorem to solve these difficulties. Finally, we compose the fractional Laplacian and the generalized Mittag-Leffler operators to estimate the norm of the mild solution of problem (1.1). This article is organized as follows. In Section 2, we present some notation and in- troduce fractional Brownian motion, the generalized Mittag-Leffler operators. Then we give properties of fractional Laplacian and the generalized Mittag-Leffler opera- tors. In Section 3, we prove Theorem 1.5 to obtain the existence and uniqueness of mild solution by the Banach Fixed Point Theorem for some stopping time. In Sec- tion 4, we prove Theorem 1.6, to obtain the Hölder continuity of the mild solution finally. 2. Preliminaries 2.1. Fractional Brownian motion. We provide an overview and systematization of stochastic calculus with respect to fractional Brownian motion. First, we intro- duce the one-dimensional fractional Brownian motion briefly; see [22] for details. A one-dimensional fractional Brownian motion with Hurst parameter H ∈ (0, 1) is a centered Gaussian process BH := {BH(t), t ≥ 0} with the covariance function RH(t, s) = E[BH(t)BH(s)] = 1 2 (s2H + t2H − |t− s|2H). EJDE-2023/49 FRACTIONAL STOCHASTIC EVOLUTION EQUATION 5 Note that B1/2(t) is standard Brownian motion. We denote by E the set of step functions on [0, T ]. Let H be the Hilbert space defined as the closure of E with respect to the scalar product 〈1[0,t], 1[0,s]〉H = RH(t, s). The mapping 1[0,t] → BH(t) can be extended to an isometry between H and the Gaussian space associated with BH . When H > 1 2 , it has been proved the covari- ance of fractional Brownian motion can be written as RH(t, s) = H(2H − 1) ∫ t 0 ∫ s 0 |r − u|2H−2 du dr. Consider the square integrable kernel KH(t, s) := cHs H− 1 2 ∫ t s (u− s)H− 3 2uH− 1 2 du, cH = ( H(2H − 1) B(2− 2H,H − 1 2 ) )1/2 (2.1) where t > s > 0, B(·, ·) is the beta function. We deduce that this kernel satisfies RH(t, s) = ∫ t∧s 0 KH(t, u)KH(s, u)du. From the definition of KH , we obtain that ∂KH ∂t (t, s) = cH ( t s )H− 1 2 (t− s)H− 3 2 . We consider the linear operator K∗H from E to L2([0, T ]) defined by (K∗Hϕ)(s) = ∫ T s ϕ(t) ∂KH ∂t (t, s)dt. Notice that ( K∗H1[0,t] ) (s) = KH(t, s)1[0,t](s). The operator K∗H is an isometry between E and L2([0, T ]) that can be extended to the Hilbert space H. In fact, for any s, t ∈ [0, T ] we have 〈K∗H1[0,t],K ∗ H1[0,s]〉L2([0,T ]) = 〈KH(t, ·)1[0,t],KH(s, ·)1[0,s]〉L2([0,T ]) = ∫ t∧s 0 KH(t, u)KH(s, u)du = RH(t, s) = 〈1[0,t], 1[0,s]〉H. Since the operator K∗H provides an isometry between the Hilbert space H and L2([0, T ]), it follows that for any t ∈ [0, T ] there exists a Brownian motion Bt = BH((K∗H)−1(1[0,t])) such that BHt = ∫ t 0 KH(t, s)dBs. Moreover, for any ϕ ∈ H, we have∫ T 0 ϕ(t)dBHt = ∫ T 0 (K∗Hϕ)(t)dBt. 6 Y. DUAN, Y. JIANG, Y. TIAN, Y. WEI EJDE-2023/49 Next we introduce the fractional Brownian motion with values in a Hilbert space and give the definition of the corresponding stochastic integral. Let U , V be sepa- rable Hilbert spaces, and L(U, V ) denote the space of all bounded linear operators from U to V . Let Q ∈ L(U,U) be a nonnegative self-adjoint operator, and let {σn}n∈N be a bounded sequence of nonnegative real numbers such that Qςn = σnςn with ∑∞ n=1 σn <∞, where {ςn}n∈N is a complete orthonormal basis in U . We denote by LQ(U, V ) the space of all ϕ ∈ L(U, V ) such that ϕQ1/2 is a Hilbert-Schmidt operator with the norm ‖ϕ‖2LQ(U,V ) = ∞∑ n=1 ‖ √ σnϕςn‖2V . (2.2) Then ϕ is called a Q-Hilbert-Schmidt operator from U to V . Let {BHn (t)}n∈N be a sequence of two-sided one-dimensional standard fractional Brownian motions mutually independent on (Ω,F ,P). When one considers the series ∞∑ i=1 BHn (t)ςn, t > 0, which not necessarily converges in the space U . Then we consider the U -valued stochastic process WH(t) = ∞∑ i=1 BHn (t)Q1/2ςn, t > 0. Since Q is a nonnegative self-adjoint operator, the above series converges in the space U , that is, it holds that WH(t) ∈ L2(Ω, U). Then, we say that WH(t) is a well defined U -valued Q-cylindrical fractional Brownian motion with covariance operator Q such that WH(t) = ∞∑ n=1 BHn (t)Q1/2ςn = ∞∑ n=1 √ σnB H n (t)ςn, t > 0. Definition 2.1. Let ϕ : [0, T ]→ LQ(U, V ) satisfy ∞∑ n=1 ‖K∗H(ϕQ1/2ςn)‖L2([0,T ];V ) <∞. (2.3) Its stochastic integral with respect to the U -valued Q-cylindrical fractional Brow- nian motion WH is defined, for t ≥ 0, as∫ t 0 ϕ(s) dWH(s) := ∞∑ n=1 ∫ t 0 ϕ(s)Q1/2ςndB H n (s) = ∞∑ n=1 ∫ t 0 K∗H(ϕ(s)Q1/2ςn)dBs. The following lemma estimates the stochastic integrals, see [29] for details. Lemma 2.2. For each ϕ : [0, T ]→ LQ(U, V ) satisfying ∫ T 0 ‖ϕ(s)‖2 L2 Q(U,V ) ds <∞, the integral ∫ t 0 ϕ(s) dWH(s) is well defined as an V -valued random variable, and for any t1, t2 ∈ [0, T ] with t1 < t2 we have E ∥∥∫ t2 t1 ϕ(s) dWH(s) ∥∥p V 6 C(H, p)(t2 − t1) p(2H−1) 2 (∫ t2 t1 ‖ϕ(s)‖2LQ(U,V )ds )p/2 , where the constant C(H, p) is positive. EJDE-2023/49 FRACTIONAL STOCHASTIC EVOLUTION EQUATION 7 2.2. Fractional Laplace Operator. For the operator Aα introduced in the pre- vious section, we have the following property; see [28]. Lemma 2.3. For any α > 0, the operator −Aα generates an analytic semigroup Sα(t) = e−tAα , t > 0 on L2(D). And for each γ > 0, there exists a constant C(α, γ) such that ‖AγSα(t)‖L(L2) ≤ C(α, γ)t−γ/α, t > 0. Here L(L2) denotes the Banach space of linear bounded operators from L2(D) to itself. In this article, the constant C is different from line to line. 2.3. Mittag-Leffler Operator. In this subsection, we introduce the one-sided stable probability density function. For each β ∈ (0, 1), θ ∈ (0,+∞), there exists wβ(θ) = 1 π ∞∑ n=1 (−1)n−1θ−βn−1 Γ(βn+ 1) n! sin(nπβ), and the Mainardi’s Wright-type function (see [19, 20]) given by Mβ(θ) = ∞∑ n=0 (−1)nθn n!Γ(1− β(1 + n)) = 1 π ∞∑ n=1 (−1)n−1θn−1 (n− 1)! Γ(nβ)sin(nπβ). Thus, we can obtain the following properties∫ ∞ 0 Mβ(θ) dθ = 1, Mβ(θ) = 1 β θ− 1 β−1wβ(θ−1/β). (2.4) The above Mainardi function Mβ(θ) acts as a bridge between the following gener- alized Mittag-Leffler operators and the fractional differential equation (1.5). The generalized Mittag-Leffler operators are defined as Lαβ(t) := ∫ ∞ 0 Mβ(θ)Sα(tβθ) dθ, (2.5) Lαβ,β(t) := ∫ ∞ 0 βθMβ(θ)Sα(tβθ) dθ. (2.6) Now we give some properties of these two operators; see [31]. Lemma 2.4. For each β ∈ (0, 1) and −1 < ε <∞, it holds Mβ(θ) > 0, ∫ ∞ 0 θεMβ(θ) dθ = Γ(1 + ε) Γ(1 + βε) , for all θ > 0. Lemma 2.5. For each t > 0, both Lαβ(t) and Lαβ,β(t) are linear and bounded op- erators. Moreover, for any η such that 0 ≤ η < α < 2 and any f ∈ L2(D), it holds ‖Lαβ(t)f‖Ḣη ≤ C(α, β, η)t− βη α ‖f‖, ‖Lαβ,β(t)f‖Ḣη ≤ C(α, β, η)t− βη α ‖f‖, where the constant C(α, β, η) is positive. 8 Y. DUAN, Y. JIANG, Y. TIAN, Y. WEI EJDE-2023/49 Proof. For t > 0, 0 6 η < α < 2, because of Lemmas 2.3 and 2.4, we have ‖Lαβ(t)f‖Ḣη = ‖AηLαβ(t)f‖ ≤ ∫ ∞ 0 Mβ(θ)‖AηSα(tβθ)f‖ dθ ≤ ∫ ∞ 0 C(α, η)t− βη α θ− η αMβ(θ)‖f‖ dθ = C(α, η) Γ(1− η α ) Γ(1− βη α ) t− βη α ‖f‖ = C(α, β, η)t− βη α ‖f‖. Then ‖Lαβ,β(t)f‖Ḣη = ‖AηLαβ,β(t)f‖ ≤ ∫ ∞ 0 βθMβ(θ)‖AηSα(tβθ)f‖ dθ ≤ ∫ ∞ 0 C(α, η)βt− βη α θ1− ηαMβ(θ)‖f‖ dθ = C(α, η)β Γ(2− η α ) Γ(1 + β(1− η α )) t− βη α ‖f‖ = C(α, β, η)t− βη α ‖f‖, Obviously, the linearity of Lαβ(t) and Lαβ,β(t) is the same as in the semigroup Sα(t). Thus, Lαβ and Lαβ,β are linear and bounded operators. � Lemma 2.6. For each t > 0, the operators Lαβ(t) and Lαβ,β(t) are strongly contin- uous with respect to t. Moreover, for t0 > 0, η such that 0 < η < α < 2, it holds that for any f ∈ L2(D) and t ∈ (t0, T ], ‖(Lαβ(t)− Lαβ(t0))f‖Ḣη ≤ C(α, β, η)(t− t0) βη α ‖f‖, ‖(Lαβ,β(t)− Lαβ,β(t0))f‖Ḣη ≤ C(α, β, η)(t− t0) βη α ‖f‖, where C(α, β, η) > 0. Proof. We know from the properties of the semigroup Sα(t) and Aα that d dt Sα(t)f = AαSα(t)f, AηAsf = Aη+sf. Since 0 < t0 < t ≤ T , we can deduce that for each f ∈ L2(D), ‖(Lαβ(t)− Lαβ(t0))f‖Ḣη ≤ ∫ ∞ 0 Mβ(θ)‖Aη(Sα(tβθ)− Sα(tβ1θ))f‖ dθ = ∫ ∞ 0 Mβ(θ) ∥∥Aη ∫ t t0 dSα(tβθ) dt f ∥∥ dθ = ∫ ∞ 0 Mβ(θ) ∥∥∫ t t0 βsβ−1θAηAαSα(sβθ)fds ∥∥ dθ ≤ ∫ ∞ 0 βθMβ(θ) ∫ t t0 ∥∥sβ−1Aη+αSα(sβθ)f ∥∥ ds dθ. EJDE-2023/49 FRACTIONAL STOCHASTIC EVOLUTION EQUATION 9 From Lemmas 2.3 and 2.4, we have∫ ∞ 0 βθMβ(θ) ∫ t t0 ‖sβ−1Aη+αSα(sβθ)f‖ds dθ ≤ C(α, η) ∫ ∞ 0 βθ− η αMβ(θ)‖f‖ dθ ∫ t t0 s−1− βηα ds = C(α, η) Γ(1− η α ) Γ(1− βη α )t 2βη α 0 (t − βηα 0 − t− βη α )‖f‖ ≤ C(α, β, η)(t− t0) βη α ‖f‖. Also, we use a similar method to obtain that ‖(Lαβ,β(t)− Lαβ,β(t0))f‖Ḣη ≤ ∫ ∞ 0 βθMβ(θ)‖Aη(Sα(tβ2θ)− Sα(tβ0θ))f‖L2(D) dθ ≤ ∫ ∞ 0 β2θ2Mβ(θ) ∫ t t0 ‖sβ−1Aη+αSα(sβθ)f‖ds dθ ≤ C(α, η) ∫ ∞ 0 β2θ1− ηαMβ(θ)‖f‖ dθ ∫ t t0 s−1− βηα ds = C(α, β, η) Γ(2− η α ) Γ(1 + β(1− η α ))t 2βη α 0 (t − βηα 0 − t− βη α )‖f‖ ≤ C(α, β, η)(t− t0) βη α ‖f‖. Thus, ‖(Lαβ(t)− Lαβ(t0))f‖Ḣη , ‖(Lαβ,β(t)− Lαβ,β(t0))f‖Ḣη → 0 as t → t0, and the operators Lαβ(t) and Lαβ,β(t) are strongly continuous. � Corollary 2.7. If we assume η = 0 in Lemma 2.6, then for each f ∈ L2(D), t ∈ (t0, T ], we have ‖(Lαβ(t)− Lαβ(t0))f‖ ≤ C(α, β)(t− t0)‖f‖, ‖(Lαβ,β(t)− Lαβ,β(t0))f‖ ≤ C(α, β)(t− t0)‖f‖. Proof. Following as similar method as in Lemma 2.6, we have that ‖(Lαβ(t)− Lαβ(t0))f‖ = ∥∥∫ ∞ 0 Mβ(θ)(Sα(tβθ)− Sα(tβ0θ))f dθ ∥∥ 6 ∫ ∞ 0 βθMβ(θ) ∫ t t0 sβ−1‖AαSα(sβθ)f‖ds dθ 6 ∫ ∞ 0 C(α)βMβ(θ) (∫ t t0 s−1ds ) ‖f‖ dθ = C(α)β(ln t− ln t0)‖f‖ 6 C(α, β)(t− t0)‖f‖, and ‖(Lαβ,β(t)− Lαβ,β(t0))f‖ = ∥∥∫ ∞ 0 βθMβ(θ)(Sα(tβθ)− Sα(tβ0θ))f dθ ∥∥ 6 ∫ ∞ 0 β2θ2Mβ(θ) ∫ t t0 sβ−1‖AαSα(sβθ)f‖ds dθ 10 Y. DUAN, Y. JIANG, Y. TIAN, Y. WEI EJDE-2023/49 = C(α)β2Γ(2) Γ(1 + β) (ln t− ln t0)‖f‖ 6 C(α, β)(t− t0)‖f‖. � 3. Mild solution In this section, we prove, Theorem 1.5, the existence and uniqueness of a mild solution of (1.1), by the Banach Fixed Point Theorem for some stopping time T∗ in the space ST∗ := {u ∈ C([0, T ];Lp(Ω; Ḣγ)) : sup t∈[0,T∗] E‖u(t)‖p Ḣγ 6 K}. Proof of Theorem 1.5. We define a map F : ST → C([0, T ];Lp(Ω; Ḣγ)) for u ∈ ST as follows (Fu)(t) = Lαβ(t)u0 + ∫ t 0 (t− s)β−1Lαβ,β(t− s)B(u(s))ds + ∫ t 0 (t− s)β−1Lαβ,β(t− s) dWH(s). (3.1) Firstly, we show that the map F is well defined. Indeed, for any u ∈ ST , from ‖f‖Ḣγ = ‖Aγf‖ in (1.4) and the definition of the operators Lαβ and Lαβ,β in (2.5) and (2.6), we have that E‖Fu(t)‖p Ḣγ = E ∥∥∥Lαβ(t)u0 + ∫ t 0 (t− s)β−1Lαβ,β(t− s)B(u(s))ds + ∫ t 0 (t− s)β−1Lαβ,β(t− s) dWH(s) ∥∥∥p Ḣγ ≤ C ( E‖ ∫ ∞ 0 Mβ(θ)AγSα(tβθ)u0 dθ‖p + E‖ ∫ t 0 (t− s)β−1Lαβ,β(t− s)AγB(u(s))ds‖p + E‖ ∫ t 0 (t− s)β−1AγL α β,β(t− s) dWH(s)‖p ) =: C(I1 + I2 + I3). (3.2) From the properties of Aα in Lemma 2.3, and Assumption 1.4, we deduce that I1 = E‖ ∫ ∞ 0 Mβ(θ)AγSα(tβθ)u0 dθ‖p 6 E‖ ∫ ∞ 0 Mβ(θ)(‖AγSα(tβθ)u0‖2)1/2 dθ‖p = E‖ ∫ ∞ 0 Mβ(θ) ( ∞∑ n=1 〈Aγe−t βθAαu0, en〉2 )1/2 dθ‖p = E‖ ∫ ∞ 0 Mβ(θ) ( ∞∑ n=1 〈Aγu0, e −tβθλα/2n en〉2 )1/2 dθ‖p 6 E‖ ∫ ∞ 0 Mβ(θ) dθ‖u0‖Ḣγ‖ p EJDE-2023/49 FRACTIONAL STOCHASTIC EVOLUTION EQUATION 11 = E‖u0‖pḢγ <∞. (3.3) From the properties of Lαβ,β(t) in Lemma 2.5, we have I2 = E ∥∥∫ t 0 (t− s)β−1Lαβ,β(t− s)AγB(u(s))ds ∥∥p 6 E (∫ t 0 ‖(t− s)β−1AγL α β,β(t− s)B(u(s))‖ds )p 6 C(α, β)E (∫ t 0 ‖(t− s)β−1− βγα B(u(s))‖ds )p . (3.4) Since u ∈ ST , by Assumption 1.3, ‖B(u)‖ 6 M‖u‖2, and the Hölder inequality, (3.4) implies that E (∫ t 0 ‖(t− s)β−1− βγα B(u(s))‖ds )p 6 (∫ t 0 (t− s) p(β−1− βγ α ) p−1 ds )p−1 ∫ t 0 E‖B(u(s))‖pds 6 C(α, β, γ,M, λ1,Λ)tpβ− pβγ α −1 ∫ t 0 ( E‖u(s)‖p Ḣγ )2 ds 6 C(α, β, γ,M, λ1,Λ)K2tpβ− pβγ α . (3.5) It follows that I2 ≤ C(α, β, γ,M, λ1,Λ)K2T pβ− pβγ α <∞, (3.6) with 0 < γ < α(pβ−1) pβ , where λ1 and Λ are the minimum and maximum of the eigenvalues of the operator −∆ relatively in the notation in (1). For I3, since Q is a bounded operator, set ∑∞ n=1 σn < R0, for some constant R0 > 0. From the norm (2.2) of LQ(U,L2(D)), Lemma 2.2 and 1/2 < β < 1, 1 < α < 2, 0 < γ < α(2β−1) 2β , we have that I3 = E‖ ∫ t 0 (t− s)β−1AγL α β,β(t− s) dWH(s)‖p 6 C(H, p)t p(2H−1) 2 (∫ t 0 ‖(t− s)β−1AγL α β,β(t− s)‖2LQ(U,L2(D))ds )p/2 = C(H, p)t p(2H−1) 2 (∫ t 0 ∞∑ n=1 ‖ √ σn(t− s)β−1AγL α β,β(t− s)ςn‖2ds )p/2 6 C(H, p, α, β, γ)t p(2H−1) 2 (∫ t 0 (t− s)2(β−1− βγα )ds ( ∞∑ n=1 ‖ √ σnςn‖2U ))p/2 6 C(H, p, α, β, γ,R0)T p(H+β− βγα −1). (3.7) where C(H, p, α, β, γ,R0) is a positive constant depending on H, p, α, β, γ,R0, and the second last inequality holds because of Lemma 2.5. From estimates (3.2)-(3.7), we have that sup t∈[0,T ] E‖Fu(t)‖p Ḣγ ≤ C ( T pβ− pβγ α + T p(H+β− βγα −1) ) <∞. where the constant C is positive and depends on α, β, γ,H,R0, T,M,K, λ1,Λ. Thus, the map F is well defined. 12 Y. DUAN, Y. JIANG, Y. TIAN, Y. WEI EJDE-2023/49 Secondly, we want to find T0 ∈ (0, T ] such that F : ST0 → ST0 . By the same arguments as the above analysis (3.3)-(3.7), we obtain that sup t∈[0,T ] E‖(Fu)(t)‖p Ḣγ ≤ C ( E‖u0‖pḢγ +K2T pβ− pβγ α + T p(H+β− βγα −1) ) , where the constant C is positive and depends on α, β, γ,H,R0, T,M,K, λ1,Λ. Then, we choose T0 such that E‖Fu‖p Ḣv 6 K, for any t ∈ [0, T0], C ( E‖u0‖pḢγ +K2T pβ− pβγα 0 + T p(H+β− βγα −1) 0 ) ≤ K, (3.8) where γ such that 0 < γ < α(2β−1) 2β < α(pβ−1) pβ . Finally, we show F is a contraction mapping on ST∗ with suitable selected T∗ such that 0 < T∗ < T0. For any u, h ∈ ST0 , taking similar method as in the estimate (3.4)-(3.6), and by Assumption 1.3 and Hölder inequality, we have that E‖(Fu)(t)− (Fh)(t)‖p Ḣγ = E‖ ∫ t 0 (t− s)β−1Lαβ,β(t− s)(B(u(s))−B(h(s)))ds‖p Ḣγ 6 E (∫ t 0 ‖(t− s)β−1AγL α β,β(t− s)(B(u(s))−B(h(s)))‖ds )p 6 C(α, β, γ) (∫ t 0 (t− s) p(β−1− βγ α ) p−1 ds )p−1 ∫ t 0 E‖B(u(s))−B(h(s))‖pds 6 C(M,p, α, β, γ)tp(β− βγ α )−1 ∫ t 0 E ((‖u(s)‖+ ‖h(s)‖)‖u(s)− h(s)‖)p ds 6 C(M,p, α, β, γ,Λ)tp(β− βγ α )−1 ∫ t 0 E ( (‖u(s)‖Ḣγ + ‖h(s)‖Ḣγ )‖u(s)− h(s)‖Ḣγ )p ds 6 C(M,K, p, α, β, γ,Λ, λ1)tp(β− βγ α )−1 ∫ t 0 E‖u(s)− h(s)‖p Ḣγ ds, where Λ, λ1 are the maximum and minimum of the eigenvalues of (−∆) relatively. Then, it further implies that with 0 < γ < α(2β−1) 2β < α(pβ−1) pβ sup t∈[0,T0] E‖(Fu)(t)− (Fh)(t)‖p Ḣγ 6 C(M,K, p, α, β, γ,Λ, λ1)T pβ− pβγα 0 sup t∈[0,T0] E‖u(t)− h(t)‖p Ḣγ . We take T∗ ∈ (0, T0) such that C(M,K, p, α, β, γ,Λ, λ1)T pβ− pβγα ∗ < 1, By the Banach Fixed Point Theorem, there exist a unique point u ∈ ST∗ , which is a unique mild solution to the problem (1.5). Then by the equivalency of the problem (1.5) and (1.1), the Theorem 1.5 is proved. � 4. Hölder continuity In this section, we prove Theorem 1.6, and obtain the Hölder continuity of the mild solution in (1.1). EJDE-2023/49 FRACTIONAL STOCHASTIC EVOLUTION EQUATION 13 Proof of Theorem 1.6. For any 0 6 t1 < t2 6 T∗, since u is a mild solution of (1.5), we have E‖u(t2)− u(t1)‖p Ḣγ = E ∥∥∥Lαβ(t2)u0 − Lαβ(t1)u0 + ∫ t2 0 (t2 − s)β−1Lαβ,β(t2 − s)B(u(s))ds − ∫ t1 0 (t1 − s)β−1Lαβ,β(t1 − s)B(u(s))ds + ∫ t2 0 (t2 − s)β−1Lαβ,β(t2 − s) dWH(s) − ∫ t1 0 (t1 − s)β−1Lαβ,β(t1 − s) dWH(s) ∥∥∥p Ḣγ 6 CE‖Lαβ(t2)u0 − Lαβ(t1)u0‖pḢγ + C ( E ∥∥∥ ∫ t2 0 (t2 − s)β−1Lαβ,β(t2 − s)B(u(s))ds − ∫ t1 0 (t1 − s)β−1Lαβ,β(t1 − s)B(u(s))ds ∥∥∥p Ḣγ ) + C ( E ∥∥∥ ∫ t2 0 (t2 − s)β−1Lαβ,β(t2 − s) dWH(s) − ∫ t1 0 (t1 − s)β−1Lαβ,β(t1 − s) dWH(s) ∥∥∥p Ḣγ ) =: C(I1 + I2 + I3). (4.1) Firstly, we consider the term I1. From Lemma 2.6 and Assumption 1.4, we deduce that I1 = E‖Aγ(Lαβ(t2)− Lαβ(t1))u0‖pḢγ 6 C(α, β, γ, p)(t2 − t1) pβγ α E‖u0‖pḢγ 6 C(α, β, γ, p)(t2 − t1) pβγ α . (4.2) Secondly, for the term I2, we divide it into three parts as follows I2 = E ∥∥∥∫ t2 0 (t2 − s)β−1Lαβ,β(t2 − s)B(u(s))ds − ∫ t1 0 (t1 − s)β−1Lαβ,β(t1 − s)B(u(s))ds ∥∥∥p Ḣγ 6 CE ∥∥∫ t1 0 (t1 − s)β−1 ( Lαβ,β(t2 − s)− Lαβ,β(t1 − s) ) B(u(s))ds ∥∥p Ḣγ + CE ∥∥∫ t1 0 ( (t2 − s)β−1 − (t1 − s)β−1 ) Lαβ,β(t2 − s)B(u(s))ds ∥∥p Ḣγ + CE ∥∥∫ t2 t1 (t2 − s)β−1Lαβ,β(t2 − s)B(u(s))ds ∥∥p Ḣγ =: C(I21 + I22 + I23). (4.3) 14 Y. DUAN, Y. JIANG, Y. TIAN, Y. WEI EJDE-2023/49 For I21, by Assumption 1.3 and Lemma 2.6, we have I21 = E‖ ∫ t1 0 (t1 − s)β−1 ( Lαβ,β(t2 − s)− Lαβ,β(t1 − s) ) B(u(s))ds‖p Ḣγ = E‖ ∫ t1 0 (t1 − s)β−1Aγ ( Lαβ,β(t2 − s)− Lαβ,β(t1 − s) ) B(u(s))ds‖p 6 E (∫ t1 0 (t1 − s)β−1‖Aγ ( Lαβ,β(t2 − s)− Lαβ,β(t1 − s) ) B(u(s))‖ds )p 6 C(α, β, γ)(t2 − t1) pβγ α E (∫ t1 0 (t1 − s)β−1‖B(u(s))‖ds )p . (4.4) Then, by using the Hölder inequality, we obtain (t2 − t1) pβγ α E (∫ t1 0 (t1 − s)β−1‖B(u(s))‖ds )p 6 C(λ1,Λ,M)(t2 − t1) pβγ α (∫ t1 0 (t1 − s) p(β−1) p−1 ds )p−1 ∫ t1 0 E‖u(s)‖p Ḣγ ds 6 C(α, β, γ, λ1,Λ,M,K, T∗, p)(t2 − t1) pβγ α , (4.5) where the last inequality in (4.5) holds because u ∈ ST∗ and 1/2 < β < 1, p > 2. Thus, we have I21 6 C(α, β, γ, λ1,Λ,M,K, T∗, p)(t2 − t1) pβγ α . (4.6) Next we estimate I22 and I23 similarly as for (4.4) and (4.5). By applying Lemma 2.5, we can deduce that with 0 < γ < α(pβ−1) pβ and p > 2, I22 = E‖ ∫ t1 0 ( (t2 − s)β−1 − (t1 − s)β−1 ) Lαβ,β(t2 − s)B(u(s))ds‖p Ḣγ = E‖ ∫ t1 0 ( (t2 − s)β−1 − (t1 − s)β−1 ) AγL α β,β(t2 − s)B(u(s))ds‖p 6 C(α, β, γ)E (∫ t1 0 ‖((t2 − s)β−1 − (t1 − s)β−1)(t2 − s)− βγ α ‖‖B(u(s))‖ds )p 6 C(α, β, γ,M, p) (∫ t1 0 ( (t1 − s)β−1 − (t2 − s)β−1 ) p p−1 (t2 − s)− pβγ α(p−1) ds )p−1 × ∫ t1 0 ( E‖u‖p Ḣγ )2 ds 6 C(α, β, γ,M,K, p, T∗)(t2 − t1) pβ(α−γ) α −1, and I23 = E‖ ∫ t2 t1 Aγ(t2 − s)β−1Lαβ,β(t2 − s)B(u(s))ds‖p 6 C(α, β, γ)E (∫ t2 t1 (t2 − s)β−1− βγα ‖B(u(s))‖ )p 6 C(α, β, γ,M) (∫ t2 t1 (t2 − s) p(β−1− βγ α ) p−1 ds )p−1 ∫ t2 t1 ( E‖u‖p Ḣγ )2 ds 6 C(α, β, γ,M,K, p)(t2 − t1) pβ(α−γ) α . (4.7) EJDE-2023/49 FRACTIONAL STOCHASTIC EVOLUTION EQUATION 15 From (4.3)-(4.7), we obtain I2 6 C(α, β, γ, λ1,Λ,M,K, T∗, p) ( (t2 − t1) pβγ α + (t2 − t1) pβ(α−γ) α −1 + (t2 − t1) pβ(α−γ) α ) , (4.8) where C(α, β, γ, λ1,Λ,M,K, T∗, p) is a positive constant. Finally, we estimate the term I3 in (4.1). Here as in (4.3), the term I3 is divided into three parts, I3 = E ∥∥∫ t2 0 (t2 − s)β−1Lαβ,β(t2 − s) dWH(s) − ∫ t1 0 (t1 − s)β−1Lαβ,β(t1 − s) dWH(s) ∥∥p Ḣγ 6 C ( E‖ ∫ t1 0 (t1 − s)β−1 ( Lαβ,β(t2 − s)− Lαβ,β(t1 − s) ) dWH(s)‖p Ḣγ + E‖ ∫ t1 0 ( (t2 − s)β−1 − (t1 − s)β−1 ) Lαβ,β(t2 − s) dWH(s)‖p Ḣγ + E‖ ∫ t2 t1 (t2 − s)β−1Lαβ,β(t2 − s) dWH(s)‖p Ḣγ ) =: C(I31 + I32 + I33). For I31, since ‖ϕ‖2LQ(U,L2(D)) = ∑∞ n=1 ‖ √ σnϕςn‖2 in(2.2), by applying Lemmas 2.2 and 2.6, we obtain I31 = E ∥∥∫ t1 0 (t1 − s)β−1Aγ ( Lαβ,β(t2 − s)− Lαβ,β(t1 − s) ) dWH(s) ∥∥p 6 C(H)t p(2H−1) 2 1 (∫ t1 0 ‖(t1 − s)β−1Aγ(Lαβ,β(t2 − s) − Lαβ,β(t1 − s))‖2LQ(U,L2(D))ds )p/2 = C(H)t p(2H−1) 2 1 (∫ t1 0 (t1 − s)2(β−1) ∞∑ n=1 ‖ √ σnAγ ( Lαβ,β(t2 − s) − Lαβ,β(t1 − s) ) ςn ∥∥2 ds )p/2 . (4.9) Since Q is a bounded operator and ∑∞ n=1 σn 6 R0, 1/2 < β < 1 and 1 < α < 2, the above inequality implies∫ t1 0 (t1 − s)2(β−1) ∞∑ n=1 ∥∥√σnAγ(Lαβ,β(t2 − s)− Lαβ,β(t1 − s) ) ςn ∥∥2 ds 6 C(α, β, γ)(t2 − t1) 2βγ α ∫ t1 0 (t1 − s)2(β−1)ds ( ∞∑ n=1 ‖ √ σnςn‖2U ) 6 C(α, β, γ,H,R0, T∗)(t2 − t1) 2βγ α , (4.10) Thus, from (4.9) it follows that I31 6 C(α, β, γ,H,R0, T∗)(t2 − t1) pβγ α . (4.11) 16 Y. DUAN, Y. JIANG, Y. TIAN, Y. WEI EJDE-2023/49 Similarly, we consider I32 and I33. Set 0 < γ < α(2β−1) 2β , by Lemmas 2.2 and 2.5, it follows that I32 = E‖ ∫ t1 0 ((t2 − s)β−1 − (t1 − s)β−1)AγL α β,β(t2 − s) dWH(s)‖p 6 C(H)t p(2H−1) 2 1 (∫ t1 0 ‖ ( (t2 − s)β−1 − (t1 − s)β−1 ) ×AγLαβ,β(t2 − s)‖2LQ(U,L2(D))ds )p/2 = C(H)t p(2H−1) 2 1 (∫ t1 0 ∞∑ n=1 ‖ √ σn ( (t2 − s)β−1 − (t1 − s)β−1 ) ×AγLαβ,β(t2 − s)ςn‖2 )p/2 6 C(α, β, γ,H, p, T∗) (∫ t1 0 ( (t2 − s)β−1 − (t1 − s)β−1 )2 × (t2 − s)− 2βγ α ds )p/2( ∞∑ n=1 ‖ √ σnςn‖2U )p/2 6 C(α, β, γ,H,R0, T∗, p)(t2 − t1) pβα−pβγ α − p2 , (4.12) and I33 = E‖ ∫ t2 t1 (t2 − s)β−1AγL α β,β(t2 − s) dWH(s)‖p 6 C(H)(t2 − t1) p(2H−1) 2 (∫ t2 t1 (t2 − s)2(β−1)‖AγLαβ,β(t2 − s)‖2LQ(U,L2(D))ds )p/2 = C(H)(t2 − t1) p(2H−1) 2 (∫ t2 t1 (t2 − s)2(β−1)‖ √ σnAγL α β,β(t2 − s)ςn‖2ds )p/2 6 C(α, β, γ,H)(t2 − t1) p(2H−1) 2 (∫ t2 t1 (t2 − s)2(β−1− βγα )ds )p/2( ‖ √ σnςn‖2U )p/2 6 C(α, β, γ,H,R0, p)(t2 − t1)p(H+β− βγα −1). From (4.11), (4.12), and the above inequality, for 0 < γ < α(2β−1) 2β < α(pβ−1) pβ we have I3 6 C(α, β, γ,H,R0, T∗, p) × ( (t2 − t1) pβγ α + (t2 − t1)p(β− βγ α − 1 2 ) + (t2 − t1)p(H+β− βγα −1) ) . (4.13) Thus, by (4.1), (4.2), (4.8), and (4.13), we conclude that E‖u(t2)− u(t1)‖2 Ḣγ 6 C(t2 − t1)τ , where C depends on α, β, γ, λ1,Λ, H, T∗,M,K,R0, p, and τ = { min{p(β − βγ α − 1 2 ), pβγα }, 0 < t2 − t1 < 1, p(β − βγ α ), t2 − t1 > 1. Therefore, we have Hölder continuity of the mild solutions of problems (1.5) and (1.1). This completes the proof. � EJDE-2023/49 FRACTIONAL STOCHASTIC EVOLUTION EQUATION 17 5. Appendix Here, we give the derivation of the mild solution of the abstract problem (1.6); for more details see [31]. Proof. Laplace transform of a function is denoted by f̂ = L (f): û(λ) = ∫ ∞ 0 e−λsu(s) ds, B̂(λ) = ∫ ∞ 0 e−λsB(u(s)) ds, Ĝ(λ) = ∫ ∞ 0 e−λs dWH(s). Applying the fractional integral operator Iβh(t) := 1 Γ(β) ∫ t 0 (t− s)β−1h(s)ds to the equation in (1.5), we obtain u(t) = u0 + 1 Γ(β) ∫ t 0 (t− s)β−1(Aαu(s) +B(u(s)))ds+ 1 Γ(β) ∫ t 0 (t− s)β−1 dWH(s). Then, by the Laplace transform, λβ û− λβ−1u0 = −Aαû+ B̂ + Ĝ; that is, û = λβ−1(λβI +Aα)−1u0 + (λβI +Aα)−1(B̂(λ) + Ĝ(λ)) = λβ−1 ∫ ∞ 0 e−λ βsSα(s)u0ds+ ∫ ∞ 0 e−λ βsSα(s)(B̂(λ) + Ĝ(λ))ds =: I1 + I2 + I3. (5.1) Considering I1, we have I1 = λβ−1 ∫ ∞ 0 e−λ βsSα(s)u0ds = ∫ ∞ 0 λβ−1e−λ βtβSα(tβ)u0d(tβ) = ∫ ∞ 0 λβ−1βtβ−1e−(λt)βSα(tβ)u0dt = ∫ ∞ 0 − 1 λ d dt e−(λt)βSα(tβ)u0dt = ∫ ∞ 0 ∫ ∞ 0 θwβ(θ)e−λθtSα(tβ)u0 dθdt = ∫ ∞ 0 e−λt (∫ ∞ 0 wβ(θ)Sα( tβ θβ )u0 dθ ) dt. (5.2) Next, we estimate the terms I2 and I3 as follows I2 = ∫ ∞ 0 e−λ βsSα(s)B̂(λ)ds = ∫ ∞ 0 e−λ βtβSα(tβ)B̂(λ)d(tβ) = ∫ ∞ 0 ∫ ∞ 0 βtβ−1e−(λt)βSα(tβ)e−λsB(u(s)) ds dt = ∫ ∞ 0 ∫ ∞ 0 ∫ ∞ 0 βe−λtθwβ(θ)Sα(tβ)e−λstβ−1B(u(s)) dθ ds dt = ∫ ∞ 0 ∫ ∞ 0 ∫ ∞ 0 βwβ(θ)e−λ(s+t)Sα( tβ θβ ) tβ−1 θβ B(u(s)) dθ ds dt = ∫ ∞ 0 e−λt ( β ∫ t 0 ∫ ∞ 0 wβ(θ)Sα( (t− s)β θβ ) (t− s)β−1 θβ B(u(s)) dθ ds ) dt. (5.3) 18 Y. DUAN, Y. JIANG, Y. TIAN, Y. WEI EJDE-2023/49 and I3 = ∫ ∞ 0 e−λ βsSα(s)Ĝ(λ)ds = ∫ ∞ 0 βtβ−1e−(λt)βSα(tβ)Ĝ(λ)dt = ∫ ∞ 0 ∫ ∞ 0 βtβ−1e−(λt)βSα(tβ)e−λs dWH(s)dt = ∫ ∞ 0 ∫ ∞ 0 ∫ ∞ 0 βwβ(θ)e−λtθSα(tβ)e−λstβ−1 dθ dWH(s)dt = ∫ ∞ 0 ∫ ∞ 0 ∫ ∞ 0 βwβ(θ)e−λ(t+s)Sα( tβ θβ ) tβ−1 θβ dθ dWH(s)dt = ∫ ∞ 0 e−λt ( β ∫ t 0 ∫ ∞ 0 wβ(θ)Sα( (t− s)β θβ ) (t− s)β−1 θβ dθ dWH(s) ) dt. (5.4) Based on estimates (5.2)-(5.4) and using the inverse Laplace transform, we obtain that the mild solution satisfies u(t) = ∫ ∞ 0 wβ(θ)Sα( tβ θβ )u0 dθ + β ∫ t 0 ∫ ∞ 0 wβ(θ)Sα ( (t− s)β θβ ) (t− s)β−1 θβ B(u(s)) dθ ds + β ∫ t 0 ∫ ∞ 0 wβ(θ)Sα ( (t− s)β θβ ) (t− s)β−1 θβ dθ dWH(s) = ∫ ∞ 0 1 β θ− 1 β−1wβ(θ−1/β)Sα(tβθ)u0 dθ + ∫ t 0 ∫ ∞ 0 θ−1/βwβ(θ−1/β)Sα((t− s)βθ)(t− s)β−1B(u(s)) dθ ds + ∫ t 0 ∫ ∞ 0 θ−1/βwβ(θ−1/β)Sα((t− s)βθ)(t− s)β−1 dθ dWH(s). According to (2.4), (2.5), and (2.6), the mild solution can be written as (1.6), i.e., u(t) = Lαβ(t)u0 + ∫ t 0 (t− s)β−1Lαβ,β(t− s)B(u(s))ds + ∫ t 0 (t− s)β−1Lαβ,β(t− s) dWH(s). � Acknowledgments. This work was supported by the NSFC grant 12271269, and by the Fundamental Research Funds for the Central Universities. References [1] S. Albeverio, F. Flandoli, Y. G. Sinai; SPDE in hydrodynamic: recent progress and prospects, Lecture Notes in Mathematics. 1942. Springer-Verlag C.I.M.E. florence, 2008. [2] L. Bertini, N. Cancrini, G. 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Wang; Stochastic Burgers equation with fractional derivative driven by white noise, Computers and Mathematics with Applications, 74 (2017), 3195-3208. 20 Y. DUAN, Y. JIANG, Y. TIAN, Y. WEI EJDE-2023/49 Yubo Duan School of Mathematical Sciences, Nankai University, Tianjin 300071, China Email address: 1120190017@mail.nankai.edu.cn Yiming Jiang School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China Email address: ymjiangnk@nankai.edu.cn Yang Tian School of Mathematical Sciences, Nankai University, Tianjin 300071, China Email address: 2120180052@mail.nankai.edu.cn Yawei Wei School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China Email address: weiyawei@nankai.edu.cn 1. Introduction 2. Preliminaries 2.1. Fractional Brownian motion 2.2. Fractional Laplace Operator 2.3. Mittag-Leffler Operator 3. Mild solution 4. Hölder continuity 5. Appendix Acknowledgments References