Existence and Stability of Square-Mean S-Asymptotically Periodic Solutions to a Fractional Stochastic Diffusion Equation with Fractional Brownian Motion

Key Laboratory of Streaming Data Computing Technologies and Application, Northwest Minzu University, Lanzhou 730000, China School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730000, China School of Mathematics and Computer Science, Xiangtan University, Xiangtan, Hunan 411105, China Nonlinear Analysis and Applied Mathematics Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia


Introduction
Originated in 1695, fractional calculus has been widely applied in physics, chemistry, economics, biology, and other fields. Recent decades have witnessed the rapid development of fractional calculus, with the emergence of many related researches [1][2][3][4][5][6][7][8][9]. e dynamic behavior of some complex processes in reality can be explained by fractional differential equations. For example, anomalous diffusion phenomena can be described with fractional diffusion equations. Compared with the traditional diffusion equations (first order), with fractional diffusion equations, subdiffusion or supdiffusion phenomenon can be described when its order is between 0 and 1 or between 1 and 2, respectively.
In addition, the diffusion phenomenon in real life is often affected by random factors, promoting the generation of fractional stochastic diffusion equations. However, this research has not aroused much concern until recent years. In [10], a class of nonautonomous fractional stochastic reactiondiffusion equations was studied, obtaining the regularity of random attractors.
e Galerkin method was applied by Wang [11] to investigate the existence of tempered pullback random attractors for nonautonomous fractional reactiondiffusion equations with multiplicative noise. Chen [12] studied the stochastic time-fractional diffusion equations with multiplicative white noise, obtaining the Hölder continuity of the solution. Peng and Huang [13] established the existence of mild solutions for a nonlocal backward problem for fractional stochastic diffusion equations.
We also notice that some researches focus on the stability of solutions to fractional stochastic differential equations of order α ∈ (0, 1). Li and Wang [14] studied the existence and asymptotic behavior of solutions to fractional stochastic delay evolution equations with integral term and Wiener process by using fractional resolvent operator theory and the Schauder fixed-point theorem. Mathiyalagan and Balachandran [15] studied the finite-time stochastic stability of fractional-order singular systems with time delay and white noise utilizing the Gronwall approach and stochastic analysis technique. In [16], applying the Laplace transform method, the authors obtained the existence, uniqueness, and Hyers-Ulam stability of solutions to a class of linear fractional differential equations involving Mittag-Leffler kernel. e resolvent operator technique and contraction mapping principle were used in [17] to study the existence and uniqueness of mild solution to fractional neutral stochastic integrodifferential equations involving impulses driven by fractional Brownian motion (FBM), and a new impulsiveintegral inequality was used to obtain the exponential stability for these equations. Moreover, the existence and asymptotic stability in the p-th moment of mild solutions to a class of fractional stochastic partial differential equations with Wiener process was investigated by Zhang et al. [18]. Because the form of the equations in this paper is different from those in the above studies, the methods to prove the stability in these studies cannot be directly applied to this paper.
Inspired by the above researches, in order to study the stability and periodicity of anomalous diffusion phenomena affected by random factors, we consider the fractional stochastic diffusion equation involving Dirichlet boundary conditions: where z α 0,t denotes the Caputo fractional derivative, α ∈ ((1/2), (3/2) − H), and Ω ⊂ R n is a bounded open domain, whose boundary zΩ is sufficiently smooth. Functions f and D satisfy some appropriate conditions, with the initial data u 0 for u, (zB H Q /zt) is the fractional Brownian motion (FBM), and H ∈ ((1/2), 1). In addition, (2) and the functions a ij satisfy where C 0 > 0 represents a constant, and b( Firstly, a new generalized Gronwall inequality is given. en, we obtain the existence and uniqueness of the S-asymptotically periodic solutions for problem (1) based on the characteristics of Mittag-Leffler functions, Hölder inequality, and the inequality for fractional stochastic integral with FBM and Banach's fixed-point theorem. In addition, with the help of the generalized Gronwall inequality, some conditions are given to ensure the asymptotic stability of the S-asymptotically periodic solutions for problem (1). We notice that the generalized Gronwall inequality in [19] cannot be applied to eorem 2 because the estimation obtained by that method is not stable.
Compared with previous research results, the innovations of this paper include the following: (1) the equations studied contain both fractional differential operators and FBM. It is worth mentioning that the standard Brownian motion, without long memory, cannot represent all types of noise. A good long-term memory noise could be described by FBM of Hurst parameter H ∈ ((1/2), 1) [20]. For instance, the continuous disturbance and long-term dependence in the financial market model can be considered as a kind of FBM [21], with the impact of nuclear waste on the environment being seen as FBM in ecological models. Other research studies on FBM can be referred to [22][23][24][25][26][27][28][29][30]. (2) Stability of the S-asymptotically periodic solutions is studied by means of a new generalized Gronwall inequality. e paper is organized as follows: the readers are allowed to review Section 2 for the necessary basic knowledge, followed by some results of the existence and uniqueness of S-asymptotically periodic solutions in Section 3. Subsequently, the asymptotic stability of S-asymptotically periodic solutions is studied in Section 4, with a numerical simulation example in Section 5.

Preliminaries
For the sake of convenience in writing, throughout this paper, by ∞, we mean +∞. (Ω, F, F t t≥0 , P) denotes a complete filtered probability space, and U and H are two separable Hilbert spaces. e space of bounded linear operators from U into H is written as L(U, H). For convenience, the same notation ‖·‖ is applied to denote norms in U, H and L(U, H); (·, ·) is applied to denote the inner produce of U and H. Moreover, L 2 (Ω; H) is the space of all strongly measurable and square-integrable H-valued random variables under the Banach norm (E‖·‖ 2 ) 1/2 .
In the following, we introduce the definition and properties of FBM. We denote by β H (t) t∈R (H ∈ (0, 1)) a two-sided one-dimensional FBM [23]. en, β H is a continuous-centered Gaussian process, whose variance function is In addition, if W is a Wiener process, then Let Q ∈ L(H, H) be an operator with T r (Q) � ∞ n�1 λ n < ∞ and Qe n � λ n e n for constants λ n ≥ 0(n � 1, 2, . . .) and a complete orthonormal basis e n ∞ n�1 in H. e infinite dimensional FBM on H can be expressed by where Q is the covariance operator and β H n (t) ∞ n�1 are twosided one-dimensional FBMs, which are mutually independent on (Ω, F, F t t≥0 , P).
Let L 0 2 (U, H) be the collection of all Q-Hilbert-Schmidt operators ξ: U ⟶ H, where ξQ 1/2 is a Hilbert-Schmidt operator, and the norm is For convenience, set L 0 2 (H): � L 0 2 (H, H). e space L 0 2 (U, H) is a separable Hilbert space whose inner product 〈φ, ϕ〉 L 0 2 � ∞ n�1 〈φe n , ϕe n 〉. en, we define the stochastic integral of ϕ with regard to B H Q by where ϕ(t) t∈[0,T] is the deterministic function with values in L 0 2 (U, H). Now, we recall the Mittag-Leffler function and the probability density functions which play important roles in fractional differential equations [31].

Lemma 1.
e Mittag-Leffler functions where Γ is the Gamma function, have the following properties: We notice that the Mittag-Leffler function E α (t) is a generalization of exponential function e t , which is E 1 (t). Set We suppose that − A generates an exponentially stable where M > 0 and δ > 0 are constants. us, (1) can be transformed into where D α 0 is the Caputo-fractional derivative. Later, in the paper, u 0 ∈ L 2 (Ω; H).

Remark 1.
We see that (13) is much easy to be verified. For example, let en, A has eigenvalues n 2 (n ∈ N), whose normalized eigenvectors w n (t) � ��� 2/π √ sin(nt)(n ∈ N), − A generates an analytic, compact, and exponentially stable semigroup (14) if where and the probability density function [36,37] Later, in this paper, we need the following results.
Proof. e proof of (3) is as follows. In fact, for In order to get the stability of the square-mean S-asymptotically ω-periodic solution, we need the following generalized Gronwall inequality for fractional differential equations. □ Lemma 3. Let u 0 , λ 1 , λ 2 ∈ R be two constants. If a continuous function u: [0, +∞) ⟶ R satisfies then Proof. We find that the solution of the equation [31] is given by In view of the uniqueness of solution to (23), we get (22).

□ 4 Complexity
Remark 3. Compared with the generalized Gronwall inequality in [19], E α (λ 1 t α )u 0 does not have to be a nondecreasing function, and λ 2 is not necessarily nonnegative. is is very important to prove the stability of the solution. Next, we give a result which is very useful for the estimations of fractional stochastic integral with FBM.

Proof
Step 1: for ∀u(t) ∈ SAP ω ([0, ∞), L 2 (Ω; H)), we prove that Firstly, we have en, Complexity 5 In view of Lemma 2, it is obvious that By combining Hölder inequality with (H 1 ) and using Lemma 2, we have e last formula and Lemma 1 yield that Since Due to Lemma 1, it is obvious that (H 1 ) implies that
On the one hand, for a given t 0 ≥ 0, we have Lemma 2 (1) implies that lim t⟶t 0 K 1 (t) � 0. Arguing similarly as in (33), we see that which means that lim t⟶t 0 K 2 (t) � 0.
For an arbitrary sequence of real numbers t n with t n ⟶ t 0 as n ⟶ ∞, for ∀u(t) ∈ SAP ω ([0, ∞), L 2 (Ω; H)), we have which is due to (H 1 ). Hence, for every n sufficiently large. In view of Additionally, according to the arbitrariness of t n , we have that which gives lim t⟶t 0 K 3 (t) � 0. en, we know that (Γ 1 u)(t) is stochastically continuous.
For a given number t 0 ≥ 0 and t > t 0 , we get It follows from (H 3 ) and Lemmas 2 and 4 that en, by a similar argument to that used in (46), we deduce lim t⟶t 0 N 2 (t) � 0.
Moreover, we apply (H 3 ) − (H 4 ) and Lemma 2 and use Lemma 4 to conclude that (61) Applying this and the above arguments, we conclude that (Γ 2 u)(t) is stochastically bounded and continuous.
Proof. From the proof process of eorem 1, we get the existence and uniqueness of the S-asymptotically ω-periodic solution u * (t) similarly. In addition, for ∀u 1 ∈ L 2 (Ω; H), equation (14) has a unique mild solution u(t) with the new initial value u(0) � u 1 . And then from (17)  14 Complexity