The Delayed Effect of Multiplicative Noise on the Blow-Up for a Class of Fractional Stochastic Differential Equations

Abstract

We investigated the blow-up of the weak solution to a class of fractional nonlinear stochastic differential equations driven by multiplicative noise in this paper. The a priori estimates and Galerkin method were applied to demonstrate the existence and uniqueness of the weak solution. Underlying the hypotheses of the nonlinear function and the initial data, for finite time, we prove that the solution does not blow up. Additionally, under further assumptions, we verified that the presence of multiplicative noise can delay the blow-up of the solution to infinity.

1. Introduction

Stochastic differential equations (SDEs) have been a valuable tool in modern science due to their accuracy in describing and explaining stochastic processes and dynamic behaviors in complex real-world scenarios. Many researchers have extensively studied the applications of such models in various fields [1,2,3,4,5]. However, in complex systems, there are often significant amounts of noise disturbances, and the intensity and distribution characteristics of the noise have a crucial effect on the behavior and characteristics of the system. Therefore, investigating the noise in stochastic differential equations is one of the most-decisive steps in understanding properties such as the existence, stability, and convergence of the solution. There have been several significant research achievements in addressing these problems. For instance, in [6], Da Prato et al. showed that the path uniqueness of a class of non-uniqueness evolutionary equations in Hilbert space can be restored by appropriate non-degenerate additive noise. Galeati studied a class of random transport linear equations with multiplicative spatio-temporal noise and demonstrated that, under certain assumptions about the noise, the solution of the stochastic transport linear equation converges to the solution of the deterministic parabolic equation [7]. The researchers in [8] proved the well-posedness of systems under Brownian-type multiplicative random perturbations. Furthermore, noise can give rise to random drifts that affect the system’s behavior. Detailed in [9], Gess and Maurelli investigated the influence of linear multiplicative noise on nonlinear scalar conservation laws and showed that such random drifts still exhibit certain averaging and regularization effects, which can mitigate the impact of spatial inhomogeneity on the system behavior.

In the study of nonlinear systems, a prominent challenge frequently encountered is the occurrence of the blow-up of the solution. The notion of blow-up can be summarized as the solution ceasing to exist in time as the variable describing the evolution process grows infinitely [10]. Such unstable situations make it impossible for us to make accurate predictions and analyze the behavior of the system. To address this problem, researchers have begun exploring whether introducing noise can delay the occurrence of blow-up. For example, the blow-up problem of nonlinear SDEs driven by Brownian motion was discussed in [11], and the existence of noise was verified to delay the time of solution blow-up. Later, the authors in [12] considered a non-local and nonlinear transport equation under stochastic perturbations and provided a class of noise that can prevent blow-up, ensuring the existence and uniqueness of the global solution. Furthermore, under specific linear noise, it has been proven that the probability of singularities occurring within finite time is positive. Flandoli et al. pointed out that the blow-up of the solutions can be effectively delayed by transport noise [13].

Meanwhile, with the increasing demand for modeling and analyzing complex nonlinear systems in science and engineering, fractional-order operators, by virtue of their hereditary and memory properties, have been developing rapidly in both theoretical analysis and applications. Therefore, compared with [11,13], this paper considers the nonlinear SDE driven by multiplicative noise in the framework of fractional-order operators, shaped as

$$\left\{\begin{array}{c}{\partial }_t^{\beta }u=\left[-{\left(-\Delta \right)}^{s}u+\zeta \left(u\right)\right]+\mu \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}{\sum }_{j=1}^{d-1}{\eta }_m{\omega }_{m,j}\left(x\right)\circ {\partial }_t{W}^{m,j},x\in {\mathbb{T}}^d,t\in \left[0,T\right],\hfill \\ u\left(x,0\right)=u_0,x\in {\mathbb{T}}^d,\hfill \end{array}\right.$$

(1)

where $u_0\in L^2\left({\mathbb{T}}^d\right)$, $s\ge 1$, $0<\beta <1$, and ${\mathbb{T}}^d={\mathbb{R}}^d/{\mathbb{Z}}^d$, while T is finite and d represents the dimension. The symbol ${\partial }_t^{\beta }$ represents the Caputo fractional derivative of order $\beta$; $\Delta$ is the periodic Laplace operator; the rest of the parameters we will describe in Section 2.

The a priori estimates and Galerkin method are tools commonly used to derive the existence and uniqueness of solutions. The former is the process of analyzing the characteristics and properties of an equation to derive estimates about some properties and restrictions of the solution [14]. The key to the Galerkin method is to project the original stochastic differential equation onto a finite-dimensional subspace and find the existence of the solution by means of dimensionality reduction [15,16]. The novelty of this work is that we applied the above methods to truncate and orthogonalize the stochastic process and derive the existence and uniqueness of the weak solution under the Caputo fractional derivative; then, we show the effect of multiplicative noise in delaying the blow-up time of the solution.

The format of this paper is as follows. We introduce some background knowledge and assumptions in Section 2. The proof of the existence and uniqueness of the weak solution for the problem (1) is given in Section 3. Section 4 focuses on proving the delayed blow-up of the solution.

2. Preliminaries and Main Results

We begin by introducing several important definitions and lemmas, along with elucidating the implications of the parameters involved in the model (1) in this section. These definitions and theorems are of great significance for the subsequent analysis and derivation, while the explanation of the meanings of the parameters in the model helps us to better understand the physical meaning and mathematical expression of the model. Subsequently, we present the critical hypotheses and outline the main results.

Definition 1

([17]). Let ${\partial }_t^{\beta }f\left(t\right)$ represent the left-sided Caputo fractional derivative of function $f:[0,T]\to X$ with respect to time, which is defined as

$${\partial }_t^{\beta }f\left(t\right)=\frac{1}{\Gamma \left(1-\beta \right)}{\int }_0^t{\left(t-s\right)}^{-\beta }\frac{\partial }{\partial s}f\left(s,x\right)ds,$$

where $\Gamma \left(·\right)$ is the Gamma function and defined as $\Gamma \left(\beta \right):={\int }_0^{\infty }{t}^{\beta -1}{e}^{-t}dt.$

Lemma 1

([18]).  For an absolutely continuous function $\phi \left(t\right)$ on the interval $\left[0,T\right]$, it holds that

$$k{\phi }^{k-1}\left(t\right){\partial }_t^{\beta }\phi \left(t\right)\ge {\partial }_t^{\beta }{\phi }^k\left(t\right),0<\beta <1,k>1.$$

Next, we will introduce the specific meanings of the parameters in the following equation:

$${\partial }_t^{\beta }u=\left[-{\left(-\Delta \right)}^{s}u+\zeta \left(u\right)\right]+\mu \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}{\sum }_{j=1}^{d-1}{\eta }_m{\omega }_{m,j}\left(x\right)\circ {\partial }_t{W}^{m,j}.$$

The function $\zeta (·)$ is nonlinear, for which the relevant hypotheses are given later. The symbol ${\mathbb{Z}}^d\setminus \left\{0\right\}$ represents the set of integers with zero elements removed. $\mu >0$ is associated with the noise intensity q, and for the sake of simplifying the proof, we would take it to be $\sqrt{dq/\phantom{dq\left(\left(d-1\right){∥\eta ∥}_{{\ell }^2}^2\right)}\left(\left(d-1\right){∥\eta ∥}_{{\ell }^2}^2\right)}$. The sequence ${\left\{{\eta }_m\right\}}_{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}$ is squared summable, with the conditions ${∥\eta ∥}_{{\ell }^2}<\infty$ and ${\eta }_m={\eta }_{-m}$. Similar to the definition of the vector field in [19], we define the vector field ${\omega }_{m,j}$ fulfilling

$${\omega }_{m,j}=q_{m,j}{e}^{2\pi im·x},x\in {\mathbb{T}}^d,m\in {\mathbb{Z}}^d\setminus \left\{0\right\},j=1,2,\dots d-1,$$

where $\left\{q_{m,1},q_{m,2},\dots ,q_{m,d-1}\right\}$ denotes a set of orthonormal bases of $m^{\perp }=\left\{z\in {\mathbb{R}}^d\left|z·m=0\right.\right\}$ and i is the imaginary unit. In particular, for any $m<0$, it holds that $q_{m,j}=q_{-m,j}$. The complex-valued Brownian motion $W^{m,j}$ is defined on a given probability space $\left(\Omega ,\mathcal{F},P\right)$ and takes the form:

$$W^{m,j}=\left\{\begin{array}{c}{B}^{m,j}+i{B}^{-m,j},m\in {\mathbb{Z}}^d\setminus \left\{0\right\},m>0,\hfill \\ B^{-m,j}-i{B}^{m,j},m\in {\mathbb{Z}}^d\setminus \left\{0\right\},m<0,\hfill \end{array}\right.$$

the family $\left\{B^{m,j}:m\in {\mathbb{Z}}^d\setminus \left\{0\right\},j=1,2,\dots ,d-1\right\}$ represents the independent standard real Brownian motions, whenever $m_1\ne \pm m_2$, $j_1\ne j_2$, $W^{m_1,j_1}$ and $W^{m_2,j_2}$ are independent. Moreover, its quadratic covariation process satisfies

$${\left[W^{m_1,j_1},W^{m_2,j_2}\right]}_t=2t{\delta }_{m_1,-m_2}{\delta }_{j_1,j_2}.$$

The symbol ∘ denotes the Stratonovich-type integral.

In summary, for any $x\in {\mathbb{T}}^d$, it satisfies

$$\begin{array}{cc}\hfill {\displaystyle \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}}{\displaystyle \sum _{j=1}^{d-1}}{\eta }_m^2\left({\omega }_{m,j}\left(x\right)\otimes {\omega }_{-m,j}\left(x\right)\right)& ={\displaystyle \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}}{\eta }_m^2{\displaystyle \sum _{j=1}^{d-1}}\left(q_{m,j}{e}^{2\pi im·x}\otimes q_{-m,j}{e}^{-2\pi im·x}\right)\hfill \\ & =\frac{d-1}{d}{∥\eta ∥}_{{\ell }^2}^2{I}_d,\hfill \end{array}$$

where $I_d$ represents the identity matrix of order d and ⊗ denotes the Kronecker product. On the basis of the correlation between the Stratonovich integral and the Itô integral, we obtain the equivalent form of the system (1) as follows:

$$\begin{array}{cc}{\partial }_t^{\beta }u\hfill & =\left[-{\left(-\Delta \right)}^{s}u+\zeta \left(u\right)\right]+\mu {\displaystyle \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}}{\displaystyle \sum _{j=1}^{d-1}}{\eta }_m{\omega }_{m,j}\nabla u\circ {\partial }_t{W}^{m,j}\hfill \\ \hfill & =\left[-{\left(-\Delta \right)}^{s}u+\zeta \left(u\right)\right]+\mu {\displaystyle \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}}{\displaystyle \sum _{j=1}^{d-1}}{\eta }_m{\omega }_{m,j}\nabla u·{\partial }_t{W}^{m,j}\hfill \\ \hfill & +\frac{\mu }{2}{\displaystyle \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}}{\displaystyle \sum _{j=1}^{d-1}}{\eta }_m{\partial }_t\left[{\omega }_{m,j}\nabla u,W^{m,j}\right]\hfill \\ \hfill & =\left[-{\left(-\Delta \right)}^{s}u+q\Delta u+\zeta \left(u\right)\right]+\mu {\displaystyle \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}}{\displaystyle \sum _{j=1}^{d-1}}{\eta }_m{\omega }_{m,j}\nabla u·{\partial }_t{W}^{m,j}.\hfill \end{array}$$

(2)

Inspired by [13,20] and aiming to establish more-general conclusions, it is always assumed in this paper that the nonlinear function $\zeta \left(·\right)$ and the initial condition $u_0$ in system (1) satisfy the following hypotheses:

Hypothesis 1

(H1). The mapping $\zeta :H^{s-{\lambda }_1}\to H^{-s}$ is continuous, and its norm satisfies

$${∥\zeta \left(u\right)∥}_{H^{-s}}\le \mathcal{C}\left(1+{∥u∥}_{L^2}^{a_1}\right)\left(1+{∥u∥}_{H^s}\right),$$

where $a_1\ge 0$, $0<{\lambda }_1<s$.

Hypothesis 2

(H2).  The mapping $\zeta \left(·\right)$ satisfies the growth, namely

$$\left|〈\zeta \left(u\right),u〉\right|\le \mathcal{C}\left(1+{∥u∥}_{L^2}^{a_2}\right)\left(1+{∥u∥}_{H^s}^{{\lambda }_2}\right),$$

where $a_2\ge 0,0<{\lambda }_2<2$. By further applying interpolation inequalities, we can obtain the following bounds:

$$\begin{array}{c}\left|〈\zeta \left(u\right),u〉\right|\le \mathcal{C}\left(1+{∥u∥}_{H^s}^{{\lambda }_2}\right)\left(1+{∥u∥}_{H^s}^{\frac{a_2\kappa }{s+\kappa }}\right)\left(1+{∥u∥}_{H^{-\kappa }}^{\frac{a_{2}s}{s+\kappa }}\right)\hfill \\ \le \mathcal{C}\left(1+{∥u∥}_{H^s}^{{\lambda }_2+\frac{a_2\kappa }{s+\kappa }}\right)\left(1+{∥u∥}_{H^{-\kappa }}^{\frac{a_{2}s}{s+\kappa }}\right),\hfill \end{array}$$

where κ can be chosen sufficiently small.

Hypothesis 3

(H3).  The function $\zeta \left(·\right)$ is local monotonicity, satisfying

$$\left|〈u-v,\zeta \left(u\right)-\zeta \left(v\right)〉\right|\le \mathcal{C}{∥u-v∥}_{L^2}^{a_3}{∥u-v∥}_{H^s}^{{\lambda }_3}\left(1+{∥u∥}_{H^s}^{\eta }+{∥v∥}_{H^s}^{\eta }\right),$$

where $a_3,{\lambda }_3,\eta$ are both positive, and

$$\left\{\begin{array}{c}{a}_3+{\lambda }_3\ge 2,{\lambda }_3<2,\hfill \\ \eta +{\lambda }_3\le 2.\hfill \end{array}\right.$$

Hypothesis 4

(H4).  The initial function $u_0\in L^2\left({\mathbb{T}}^d\right)$ is bounded, closed, and convex, which satisfies the following condition: for any $T>0$, there exists a sufficiently large noise q, which allows the time-fractional deterministic equation:

$${\partial }_t^{\beta }u=-{\left(-\Delta \right)}^{s}u+q\Delta u+\zeta \left(u\right),$$

(3)

to have a global solution with initial condition $u_0$. For any $t\in \left[0,T\right]$, its $L^2$-norm is bounded.

Remark 1.

Combining with Lemma 1, we claim that the solution of the system (3) satisfies the inequality

$${\partial }_t^{\beta }{∥u∥}_{L^2}^2={\int }_{{\mathbb{T}}^d}{\partial }_t^{\beta }{u}^{2}dx\le {\int }_{{\mathbb{T}}^d}2u{\partial }_t^{\beta }udx=2〈\zeta \left(u\right),u〉-2{∥{\left(-\Delta \right)}^{\frac{s}{2}}u∥}_{L^2}^2-2q{∥\nabla u∥}_{L^2}^2,$$

and for the Sobolev space, its norm satisfies

$${∥u∥}_{L^2}^2+{∥{\left(-\Delta \right)}^{s/2}u∥}_{L^2}^2\ge 2^{1-s}{∥u∥}_{H^s}^2;$$

(4)

combining the condition (H2) with the Poincaré inequality and choosing the constant as $\frac{1}{C_2}$, we can simplify the above inequality as

$$\begin{array}{c}{\partial }_t^{\beta }{∥u∥}_{L^2}^2\le C_1\left(1+{∥u∥}_{L^2}^{a_4}\right)-2\left(2C_{2}q-1\right){∥u∥}_{L^2}^2,\hfill \end{array}$$

where $C_1,C_2,a_4=\frac{2a_2}{2-{\lambda }_2}$ are both positive constants.

Throughout the following statements, we mention the solution of the problem (1) by $u\left(u_0,\eta \right)$ and the solution of the problem (3) by $u\left(u_0\right)$ to avoid confusion. The primary outcome of this paper is as follows:

Theorem 1.

Under the hypotheses (H1)–(H4) for the function $\zeta \left(·\right)$ and initial function $u_0$, there exists a locally unique weak solution for the problem (1) for any $0<T<\infty$ and noise intensity q. Furthermore, there exists a square summable real sequence ${\left\{{\eta }_m\right\}}_{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}$, which makes the solution of the system (1) satisfy

$$P\left(\underset{0\le t\le T}{sup}{∥u\left(u_0,\eta \right)∥}_{H^{-\kappa }}<\infty \right)\ge 1-\theta ,\forall \theta >0,$$

namely, the solution does not blow-up within the finite time.

Theorem 1 demonstrates the blow-up of solutions in finite time. Consequently, it is natural to investigate the behavior of the solution as time tends to infinity. To address this question, we further propose the following assumptions:

Hypothesis 5

(H5).  The norm of the solution to the problem (3) satisfies the decay inequality ${∥u\left(u_0\right)∥}_{L^2}\le C{∥u_0∥}_{L^2}{e}^{-\lambda t}$, where λ and C are positive constants.

Hypothesis 6

(H6).  For sufficiently small initial function $u_0$, the time-fractional stochastic system (3) has a unique and global solution.

Based on the assumptions mentioned above, we can derive the following:

Theorem 2.

Assume that the function $\zeta \left(·\right)$ and initial function $u_0$ satisfy the assumptions (H1)–(H4) and that the assumptions (H5) and (H6) hold, there exists a sequence ${\left\{{\eta }_m\right\}}_{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}\in {\ell }^2$, such that the solutions of the system (1) satisfy

$$P\left(\underset{0\le t\le \infty }{sup}{∥u\left(u_0,\eta \right)∥}_{H^{-\kappa }}<\infty \right)\ge 1-\theta ,\forall \theta >0;$$

this means that, as time approaches infinity, the solution does not blow up.

3. Existence and Uniqueness

First, due to the nonlinearfunction $\zeta \left(·\right)$, the solution of the problem (1) may not exist globally. Hence, we introduce a smooth and decreasing truncation function, which takes the form:

$$L_S\left({∥u∥}_{H^{-\kappa }}\right)=\left\{\begin{array}{c}1,0\le {∥u∥}_{H^{-\kappa }}\le S,\hfill \\ 0,\mathrm{others},\hfill \end{array}\right.$$

(5)

where $S>0$ is a constant. Next, we discuss the system:

$${\partial }_t^{\beta }u=\left[-{\left(-\Delta \right)}^{s}u+q\Delta u+L_S\left({∥u∥}_{H^{-\kappa }}\right)\zeta \left(u\right)\right]+\mu \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}\sum _{j=1}^{d-1}{\eta }_m{\omega }_{m,j}\nabla u\left(t\right)·{\partial }_t{W}^{m,j},$$

(6)

where $\beta ,s,\mu$, and $\kappa$ take the same value as above.

Definition 2.

Given a probability space $\left(\Omega ,\mathcal{F},P\right)$, ${\left\{W^{m,j}\right\}}_{m\in {\mathbb{Z}}^d\setminus \left\{0\right\},j=1,\dots ,d-1}$ is a set of complex Brownian motions defined on this probability space, $u\in L^{\infty }\left(0,T;L^2\right)\cap L^2\left(0,T;H^s\right)$ is called a weak solution to Equation (6), and if for any $t\ge 0$, it holds almost surely that

$$\begin{array}{c}〈u,\psi 〉=〈u_0,\psi 〉-\frac{\mu }{\Gamma \left(\beta \right)}{\displaystyle \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}}{\displaystyle \sum _{j=1}^{d-1}}{\eta }_m{\int }_0^t{\left(t-\tau \right)}^{\beta -1}〈u,{\omega }_{m,j}·\nabla \psi 〉d{W}_{\tau }^{m,j}\hfill \\ -\frac{1}{\Gamma \left(\beta \right)}{\int }_0^t{\left(t-\tau \right)}^{\beta -1}\left[〈{\Lambda }^{s}u,{\Lambda }^s\psi 〉+q〈\nabla u,\nabla \psi 〉\right]d\tau \hfill \\ +\frac{1}{\Gamma \left(\beta \right)}{\int }_0^t{\left(t-\tau \right)}^{\beta -1}{L}_S\left({∥u∥}_{H^{-\kappa }}\right)〈\zeta \left(u\right),\psi 〉d\tau ,\hfill \end{array}$$

where $\psi \in H^s\left({\mathbb{T}}^d\right)$, $\Lambda ={\left(-\Delta \right)}^{\frac{1}{2}}$, and $u_0\in L^2\left({\mathbb{T}}^d\right)$.

Remark 2.

For the above definition of weak solutions, we refer to the relevant definitions in the existing literature, such as Definition 2.18 in [17] and Theorem 1.3 in [19].

Subsequently, we utilize the Galerkin method to construct a finite-dimensional subspace ${\mathcal{H}}_M=\mathrm{span}\left\{e^{2\pi mi·x}:\left|m\right|\le M,M\in \mathbb{R}\right\}$ from the space $L^2\left({\mathbb{T}}^d\right)$ and combine it with a priori estimates to derive the existence and uniqueness of the solution. Additionally, we introduce an orthogonal projection operator $\Pi :L^2\left({\mathbb{T}}^d\right)\to {\mathcal{H}}_M$. On the subspace ${\mathcal{H}}_M$, we obtain the following conclusions:

Lemma 2.

For the sequence ${\left\{{\eta }_m\right\}}_{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}\in {\ell }^2$, the time-fractional stochastic differential equation:

$$\left\{\begin{array}{c}{\partial }_t^{\beta }{u}_{\Pi }=\left[-{\left(-\Delta \right)}^s{u}_{\Pi }+q\Delta u_{\Pi }+L_S\left({∥u_{\Pi }∥}_{H^{-\kappa }}\right)\Pi \zeta \left(u_{\Pi }\right)\right]\hfill \\ +\mu {\displaystyle \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}}{\displaystyle \sum _{j=1}^{d-1}}{\eta }_m\Pi \left({\omega }_{m,j}·\nabla u_{\Pi }\right){\partial }_t{W}^{m,j},\hfill \\ u_{\Pi }\left(0\right)=\Pi u_0,\hfill \end{array}\right.$$

(7)

with initial condition $u_0\in L^2\left({\mathbb{T}}^d\right)$, possesses a weak solution with global existence and uniqueness, which satisfies

$$\underset{t\in \left[0,T\right]}{sup}{∥u_{\Pi }\left(t\right)∥}_{L^2}^2\le C\left({∥u_0∥}_{L^2}^2,T\right);$$

(8)

here, $C\left({∥u_0∥}_{L^2}^2,T\right)$ denotes a constant associated with ${∥u_0∥}_{L^2}^2,T$.

Proof. 

Set $u_{\Pi }$ as the solution of the system (7); by utilizing Lemma 1, we state

$$\begin{array}{c}{\partial }_t^{\beta }{∥u_{\Pi }∥}_{L^2}^2\le 2{\int }_{{\mathbb{T}}^d}{u}_{\Pi }{\partial }_t^{\beta }{u}_{\Pi }dx=-2{∥{\Lambda }^s{u}_{\Pi }∥}_{L^2}^2-2q{∥\nabla u_{\Pi }∥}_{L^2}^2+2L_S\left({∥u_{\Pi }∥}_{H^{-\kappa }}\right)〈\zeta \left(u_{\Pi }\right),u_{\Pi }〉\hfill \\ +2\mu {\displaystyle \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}}{\displaystyle \sum _{j=1}^{d-1}}{\eta }_m\Pi \left({\omega }_{m,j}·\nabla u_{\Pi }\right){\partial }_t{W}^{m,j};\hfill \end{array}$$

since $\Pi$ is an orthogonal projection and ${\omega }_{m,j}$ is divergencefree, the stochastic term of the above inequality vanishes. Thus, it can be simplified to

$${\partial }_t^{\beta }{∥u_{\Pi }∥}_{L^2}^2\le 2L_S\left({∥u_{\Pi }∥}_{H^{-\kappa }}\right)〈\zeta \left(u_{\Pi }\right),u_{\Pi }〉-2{∥{\Lambda }^s{u}_{\Pi }∥}_{L^2}^2.$$

Combining the assumption (H2) with Young’s inequality and Equation (4), we can conclude

$$\begin{array}{cc}\hfill 2L_S\left({∥u_{\Pi }∥}_{H^{-\kappa }}\right)〈\zeta \left(u_{\Pi }\right),u_{\Pi }〉& \le 2C_1\left(1+S^{a_2\frac{s}{s+\kappa }}\right)\left(1+{∥u_{\Pi }∥}_{H^s}^{{\lambda }_2+a_2\frac{\kappa }{s+\kappa }}\right)\hfill \\ & \le C_2+{∥u_{\Pi }∥}_{L^2}^2+{∥{\Lambda }^s{u}_{\Pi }∥}_{L^2}^2,\hfill \end{array}$$

where $C_2=2\left[1+S^{\frac{a_{2}s}{s+\kappa }}+C_1{\left(1+S^{\frac{a_{2}s}{s+\kappa }}\right)}^{\frac{2}{2-{\lambda }_2-a_2\frac{\kappa }{s+\kappa }}}\right]$. Given the non-negativity of ${∥{\Lambda }^s{u}_{\Pi }∥}_{L^2}^2$, it is evident that

$${\partial }_t^{\beta }{∥u_{\Pi }∥}_{L^2}^2\le {∥u_{\Pi }∥}_{L^2}^2+C_3,$$

by applying Lemma 2 from [21], we can conclude that the inequality (8) holds.

Subsequently, we consider the uniqueness of the weak solution. Given two different solutions $u^{\prime },u^{\prime \prime }$ for the problem (7) with the same complex Brownian motion $W^{m,j}$ and initial function, let $U=u^{\prime }-u^{\prime \prime }$, then for any test function $\psi \in H^s\left({\mathbb{T}}^d\right)$, one has the following identity:

$$\begin{array}{c}〈U,\psi 〉=〈U_0,\psi 〉-\frac{1}{\Gamma \left(\beta \right)}{\int }_0^t{\left(t-\tau \right)}^{\beta -1}〈{\Lambda }^{s}U,{\Lambda }^s\psi 〉d\tau +\frac{1}{\Gamma \left(\beta \right)}{\int }_0^{t}q{\left(t-\tau \right)}^{\beta -1}〈\nabla U,\nabla \psi 〉d\tau \hfill \\ -\frac{\mu }{\Gamma \left(\beta \right)}{\displaystyle \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}}{\displaystyle \sum _{j=1}^d}{\eta }_m{\int }_0^t{\left(t-\tau \right)}^{\beta -1}〈U,{\omega }_{m,j}·\nabla \psi 〉d{W}_{\tau }^{m,j}\hfill \\ +\frac{1}{\Gamma \left(\beta \right)}{\int }_0^t{\left(t-\tau \right)}^{\beta -1}〈L_S\left({∥u^{\prime }∥}_{H^{-\kappa }}\right)\zeta \left(u^{\prime }\right)-L_S\left({∥u^{\prime \prime }∥}_{H^{-\kappa }}\right)\zeta \left(u^{\prime \prime }\right),\psi 〉d\tau ;\hfill \end{array}$$

similar to the previous argument, we can infer that

$$\begin{array}{c}{\partial }_t^{\beta }{∥U∥}_{L^2}^2\le -2{∥{\Lambda }^{s}U∥}_{L^2}^2+2〈L_S\left({∥u^{\prime }∥}_{H^{-\kappa }}\right)\zeta \left(u^{\prime }\right)-L_S\left({∥u^{\prime \prime }∥}_{H^{-\kappa }}\right)\zeta \left(u^{\prime \prime }\right),U〉\hfill \\ \le 2〈\left(L_S\left({∥u^{\prime }∥}_{H^{-\kappa }}\right)-L_S\left({∥u^{\prime \prime }∥}_{H^{-\kappa }}\right)\right)\zeta \left(u^{\prime }\right)+L_S\left({∥u^{\prime \prime }∥}_{H^{-\kappa }}\right)\left(\zeta \left(u^{\prime }\right)-\zeta \left(u^{\prime \prime }\right)\right),U〉-2{∥{\Lambda }^{s}U∥}_{L^2}^2\hfill \\ \le C_4{∥U∥}_{L^2}^2\left(1+{∥u^{\prime }∥}_{H^s}^2+{∥u^{\prime \prime }∥}_{H^s}^2\right),\hfill \end{array}$$

and the constants $C_4>0\left(k=1,\dots ,4\right)$ involved in the above inequality are independent of time. The last step can be derived from the hypothesis (H3) and the Lagrange mean value theorem. Recalling the fact that ${∥U_0∥}_{L^2}={∥{u^{\prime }}_0-{u^{\prime \prime }}_0∥}_{L^2}=0$, then we can establish the uniqueness of the solution using the Gronwall inequality. Therefore, the proof is complete. □

Notice the global existence of the weak solution mentioned in Lemma 2, which is equivalent to the local existence of the solution to the system (1) without a cut-off function. This also implies the boundedness of the solution sequence $\left\{u_{\Pi }\left(t\right)\right\}$. The convergence of the sequence is further investigated for the purpose of deriving the limit of the nonlinear term. To facilitate the subsequent discussion, we introduce an important lemma, which will be applied in the relevant compactness theorem.

Lemma 3

([13]).  Set $\mathcal{X}$ as the topology $C\left(\left[0,T\right];L^2\right)\cap C^{\gamma }\left(\left[0,T\right];H^{-\alpha }\right)\cap L^2\left(0,T;H^s\right)$; $\mathcal{Y}$ stands for the topology $L^p\left(0,T;L^2\right)\cap C\left(\left[0,T\right];H^{-\delta }\right)\cap L^2\left(0,T;H^{s-\delta }\right)$, where $\alpha ,\delta ,s,\gamma >0$ and p is finite, then the the embedding $X\hookrightarrow Y$ is said to be compact.

In conjunction with Lemma 3, one has the following estimates on the topology $\mathcal{X}$.

Lemma 4.

For the time-fractional stochastic differential equation with the initial value function $u_{\Pi }\left(0\right)=\Pi u_0$, shaped as

$${\partial }_t^{\beta }{u}_{\Pi }=\left[-{\left(-\Delta \right)}^s{u}_{\Pi }+q\Delta u_{\Pi }+L_S\left({∥u_{\Pi }∥}_{H^{-\kappa }}\right)\Pi \zeta \left(u_{\Pi }\right)\right]+\mu \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}\sum _{j=1}^{d-1}{\eta }_m\Pi \left({\omega }_{m,j}·\nabla u_{\Pi }\right){\partial }_t{W}^{m,j},$$

its solution satisfies

$$\mathbb{E}\left[{\left({∥u_{\Pi }∥}_{L^2{H}^s}+{∥u_{\Pi }∥}_{L^{\infty }{L}^2}+{∥u_{\Pi }∥}_{C^{\lambda }{H}^{-\alpha }}\right)}^2\right]<\infty .$$

(9)

Proof. 

For any $u\in H^s$, it is obvious that $\Delta u\in H^{s-2}$, and $H^{s-2}$ is a subspace of $H^{-s}$. It follows from the triangle inequality that

$$\begin{array}{c}\frac{1}{{\Gamma }^2\left(\beta \right)}{\int }_0^t{∥{\left(t-\tau \right)}^{\beta -1}\left[-{\left(-\Delta \right)}^{s}u+q\Delta u+L_S\left({∥u∥}_{H^{-\kappa }}\right)\zeta \left(u\right)\right]∥}_{H^{-s}}^{2}d\tau \hfill \\ \le \frac{C_1}{{\Gamma }^2\left(\beta \right)}{\int }_0^t{\left(t-\tau \right)}^{2\left(\beta -1\right)}\left(1+q^2\right){∥u∥}_{H^s}^{2}d\tau +\frac{1}{{\Gamma }^2\left(\beta \right)}{\int }_0^T{\left(t-\tau \right)}^{2\left(\beta -1\right)}{∥\zeta \left(u\right)∥}_{H^{-s}}^{2}d\tau ;\hfill \end{array}$$

by employing Hölder’s inequality, it fulfills

$$\frac{1}{{\Gamma }^2\left(\beta \right)}{\int }_0^t{\left(t-\tau \right)}^{2\beta -2}\left(1+q^2\right){∥u∥}_{H^s}^{2}d\tau \le C_2{∥u∥}_{L^2{H}^s}^2,$$

and combining the assumption (H1), we obtain

$$\frac{1}{{\Gamma }^2\left(\beta \right)}{\int }_0^t{\left(t-\tau \right)}^{2\left(\beta -1\right)}{∥\zeta \left(u\right)∥}_{H^{-s}}^{2}d\tau \le C_3\left(1+{∥u∥}_{L^2{H}^s}^2\right)\left(1+{∥u∥}_{L^{\infty }{L}^2}^{2a_1}\right).$$

To simplify the subsequent discussion, let us denote the following:

$$G_1\left(t\right)=\frac{1}{\Gamma \left(\beta \right)}{\int }_0^t{\left(t-\tau \right)}^{\beta -1}\left[-{\left(-\Delta \right)}^{s}u+q\Delta u+L_S\left({∥u∥}_{H^{-\kappa }}\right)\zeta \left(u\right)\right]d\tau ,$$

and

$$G_2\left(t\right)=\frac{\mu }{\Gamma \left(\beta \right)}{\int }_0^t\sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}\sum _{j=1}^{d-1}{\eta }_m{\omega }_{m,j}{\left(t-\tau \right)}^{\beta -1}·\nabla ud{W}_{\tau }^{m,j}.$$

Consequently, together with the above results and the conditions (H1)–(H4), we assert that

$${∥G_1\left(t\right)∥}_{C^{1/2}{H}^{-s}}\le C_4\left(1+{∥u∥}_{L^{\infty }{L}^2}^{a_1}\right)\left(1+{∥u∥}_{L^2{H}^s}\right);$$

moreover,

$$\left[〈G_2,e^{2\pi im·x}〉\right]\left(t\right)\le \frac{2{\mu }^2{∥\eta ∥}_{{\ell }^{\infty }}^2}{{\Gamma }^2\left(\beta \right)}{\left|\left|\Pi \left(\nabla e^{2\pi im·x}\right)\right|\right|}_{L^{\infty }}^2{\int }_0^t{\left(t-\tau \right)}^{2\left(\beta -1\right)}{\left|\left|u_{\Pi }\right|\right|}_{L^2}^{2}d\tau .$$

Since the arbitrary order derivatives of $e^{2\pi im·x}$ exist, incorporating the Burkholder–Davis–Gundy inequality [22], one has the estimate $\mathbb{E}\left[{∥G_2\left(t\right)∥}_{H^{-s}}^2\right]\le C_5{∥\eta ∥}_{{\ell }^{\infty }}^2{t}^{2\beta -1}$, where $C_k>0\left(k=1,2,3,4,5\right)$ are constants independent of time. Recalling the boundedness of the solution stated in Lemma 2, the inequality (9) is clearly satisfied. This completes the proof. □

Recalling the condition (H4), we made a hypothesis about the boundedness of the solution to the fractional deterministic Equation (3) in finite time. For the purpose of deriving the boundedness of the solution to the problem (6), we consider a specific sequence and discuss the convergence relationship between the solutions of the problem (3) and (6). The following conclusion can be drawn:

Lemma 5.

Take a sequence ${\left\{{\eta }^S\right\}}_{S\ge 1}\subset {\ell }^2$ such that, for $S\to \infty$, ${∥{\eta }^S∥}_{{\ell }^{\infty }}/\phantom{{∥{\eta }^S∥}_{{\ell }^{\infty }}{∥{\eta }^S∥}_{{\ell }^2}}{∥{\eta }^S∥}_{{\ell }^2}\to 0$. The set of bounded, closed, and convex initial functions ${\left\{u_0^S\right\}}_S\in L^2\left({\mathbb{T}}^d\right)$ weakly converges to $u_0$. On the topology $\mathcal{Y}$, the identity:

$${\partial }_t^{\beta }u=\left[-{\left(-\Delta \right)}^{s}u+q\Delta u+L_S\left({∥u∥}_{H^{-\kappa }}\right)\zeta \left(u\right)\right]+\mu \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}\sum _{j=1}^{d-1}{\eta }_m^S{\omega }_{m,j}·\nabla u{\partial }_t{W}^{m,j},$$

(10)

with initial value $u_0^S$, has a weak solution and converges in probability to the solution of the equation:

$${\partial }_t^{\beta }u=\left[-{\left(-\Delta \right)}^{s}u+q\Delta u+L_S\left({∥u∥}_{H^{-\kappa }}\right)\zeta \left(u\right)\right]$$

(11)

with initial value $u_0$.

Proof. 

Let us use the symbol $u^S$ to denote the solution of the system (10), and denote the measure of ${\left\{u^S\right\}}_S$ by ${\left\{u^S\right\}}_S$. Similar to the proof of Lemma 3 and Corollary 3.5 in [19], it is straightforward to demonstrate that the subsequence ${\left\{{\rho }^{S_i}\right\}}_{S_i\ge 1}\subset {\left\{{\rho }^S\right\}}_{S\ge 1}$ is compact in $L^2\left(0,T;L^2\right)$ and $C\left(\left[0,T\right],H^{-\delta }\right)$, then by the Prokhorov theorem [23], there is a subsequence ${\left\{{\rho }^{S_i}\right\}}_{S_i\ge 1}$ that weakly converges to a probability measure $\rho$. Additionally, it follows from the Skorokhod representation theorem [24] that a new probability space $\left(\tilde{\Omega },\tilde{\mathcal{F}},\tilde{P}\right)$ exists, on which we can select a random variable sequence ${\left\{{\rho }^{S_i}\right\}}_{S_i\ge 1}$ with the same distribution as ${\left\{{\tilde{u}}^{S_i}\right\}}_{S_i\ge 1}$ and $\tilde{u}$ distributed identically with $\rho$, satisfying $P\left({lim}_{i\to \infty }{\tilde{u}}^{S_i}=\tilde{u}\right)=1$.

Besides, we prove that $\tilde{u}$ is a weak solution to the deterministic Equation (11). Similar to the previous statement, on $\left(\tilde{\Omega },\tilde{\mathcal{F}},\tilde{P}\right)$, there exists a mutually independent set of complex Brownian motions ${\left\{{\tilde{W}}_{S_i}^{m,j}\right\}}_{m,j}$, $\left(u^{S_i},W^{m,j}\right)$ having the same distribution as $\left({\tilde{u}}^{S_i},{\tilde{W}}_{S_i}^{m,j}\right)$; moreover, ${\tilde{W}}_{S_i}^{m,j}$ almost surely converges to ${\tilde{W}}^{m,j}$ on $C\left(\left[0,T\right];\mathbb{C}\right)$. Therefore, for any $\psi \in C^{\infty }\left({\mathbb{T}}^d\right)$, we have

$$\begin{array}{c}〈{\tilde{u}}^{S_i},\psi 〉=〈u_0^{S_i},\psi 〉-\frac{1}{\Gamma \left(\beta \right)}{\int }_0^t\left[〈{\left(t-\tau \right)}^{\beta -1}{\Delta }^s{\tilde{u}}^{S_i},\psi 〉+q〈{\left(t-\tau \right)}^{\beta -1}\Delta {\tilde{u}}^{S_i},\psi 〉\right]d\tau \hfill \\ +\frac{1}{\Gamma \left(\beta \right)}{\int }_0^t{L}_S\left({∥{\tilde{u}}^{S_i}∥}_{H^{-\kappa }}\right)〈{\left(t-\tau \right)}^{\beta -1}\zeta \left({\tilde{u}}^{S_i}\right),\psi 〉d\tau \hfill \\ -\frac{\mu }{\Gamma \left(\beta \right)}{\displaystyle \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}}{\displaystyle \sum _{j=1}^{d-1}}{\eta }_m^{S_i}{\int }_0^t〈{\left(t-\tau \right)}^{\beta -1}{\tilde{u}}^{S_i}\left(\tau \right),{\omega }_{m,j}·\nabla \psi 〉d{W}_{\tau }^{m,j};\hfill \end{array}$$

for the last term of the above inequality, we can derive the expectation that it satisfies the estimate

$$\begin{array}{c}\tilde{\mathbb{E}}{\left(\frac{\mu }{\Gamma \left(\beta \right)}{\displaystyle \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}}{\displaystyle \sum _{j=1}^{d-1}}{\eta }_m^{S_i}{\int }_0^t〈{\left(t-\tau \right)}^{\beta -1}{\tilde{u}}^{S_i},{\omega }_{m,j}·\nabla \phi 〉d{W}_{\tau }^{m,j}\right)}^2\hfill \\ \le \frac{C{\mu }^2{∥\eta ∥}_{{\ell }^{\infty }}^2}{{\Gamma }^2\left(\beta \right){∥\eta ∥}_{{\ell }^2}^2}\tilde{\mathbb{E}}{\int }_0^t{\displaystyle \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}}{\displaystyle \sum _{j=1}^{d-1}}{\left|〈{\left(t-\tau \right)}^{\beta -1}{\tilde{u}}^{S_i},{\omega }_{m,j}·\nabla \phi 〉\right|}^{2}d\tau ;\hfill \end{array}$$

(12)

here, $\tilde{\mathbb{E}}$ stands for the expectation of the probability space $\left(\tilde{\Omega },\tilde{\mathcal{F}},\tilde{P}\right)$. Combining this with the definition of the vector field ${\omega }_{m,j}$, we have

$$\sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}{\sum _{j=1}^{d-1}\left|〈{\left(t-\tau \right)}^{\beta -1}{\tilde{u}}^{S_i},{\omega }_{m,j}·\nabla \phi 〉\right|}^2\le {∥\nabla \phi ∥}_{L^{\infty }}^2{∥{\left(t-\tau \right)}^{\beta -1}{\tilde{u}}^{S_i}∥}_{L^2}^2;$$

recalling the sequence ${\left\{{\eta }^{S_i}\right\}}_{i\ge 1}$, which satisfies ${∥{\eta }^{S_i}∥}_{{\ell }^{\infty }}/\phantom{{∥{\eta }^{S_i}∥}_{{\ell }^{\infty }}{∥{\eta }^{S_i}∥}_{{\ell }^2}}{∥{\eta }^{S_i}∥}_{{\ell }^2}\to 0$ for $i\to \infty$ and the boundedness of ${∥{\tilde{u}}^{S_i}∥}_{L^2}$ obtained from the previous discussion and Hölder’s inequality, we assert that the inequality (12) tends to 0 as $i\to \infty$. Since and ${\tilde{u}}^{S_i}$ almost surely converge to $\tilde{u}$ and $u_0^S$ weakly converges to $u_0$, the following identity holds:

$$\begin{array}{c}〈\tilde{u},\psi 〉=〈u_0,\psi 〉-\frac{1}{\Gamma \left(\beta \right)}{\int }_0^t{\left(t-\tau \right)}^{\beta -1}\left[〈{\Lambda }^s\tilde{u},{\Lambda }^s\psi 〉+q〈\nabla \tilde{u},\nabla \psi 〉\right]d\tau \hfill \\ +\frac{1}{\Gamma \left(\beta \right)}{\int }_0^t{\left(t-\tau \right)}^{\beta -1}{L}_S\left({∥\tilde{u}∥}_{H^{-\kappa }}\right)〈\zeta \left(\tilde{u}\right),\psi 〉d\tau ;\hfill \end{array}$$

this shows that $\tilde{u}$ is a weak solution of the problem (11). □

Remark 3.

The proofs of the existence and uniqueness of weak solutions are similar to our preprint [25], but we have improved the content and structure of Section 3 and cited important conclusions of [13,19] in the proofs of Lemmas 4 and 5. In addition, we further elucidated the convergence of sequences and variables in order to make the exposition more concise and logically rigorous.

4. Delayed Blow-Up

This section provides the proof for the main theorems.

4.1. Proof of Theorem 1

According to the assumption (H4), it can be inferred that ${∥u\left(u_0\right)∥}_{L^{\infty }\left(0,T;L^2\right)}$ no more than $S-1$. Recalling Lemma 5, for any $\theta >0$, we obtain

$$P\left({∥u^S\left(u_0,{\eta }^S\right)-u\left(u_0\right)∥}_{C\left(\left[0,T\right];H^{-\kappa }\right)}\le \theta \right)\ge 1-\theta ;$$

following from the property of norm, obviously,

$${∥u^S\left(u_0,{\eta }^S\right)∥}_{C\left(\left[0,T\right];H^{-\kappa }\right)}-{∥u\left(u_0\right)∥}_{C\left(\left[0,T\right];H^{-\kappa }\right)}\le {∥u^S\left(u_0,{\eta }^S\right)-u\left(u_0\right)∥}_{C\left(\left[0,T\right];H^{-\kappa }\right)};$$

hence, it is evident that ${∥u^S\left(u_0,{\eta }^S\right)∥}_{C\left(\left[0,T\right];H^{-\kappa }\right)}<S$, namely

$$P\left({∥u^S\left(u_0,{\eta }^S\right)∥}_{C\left(\left[0,T\right];H^{-\kappa }\right)}<S\right)\ge P\left({∥u^S\left(u_0,{\eta }^S\right)-u\left(u_0\right)∥}_{C\left(\left[0,T\right];H^{-\kappa }\right)}\le \theta \right)\ge 1-\theta .$$

In other words, on the interval $\left[0,T\right]$, the truncated function $L_S\left({∥u^S\left(u_0,{\eta }^S\right)∥}_{H^{-\kappa }}\right)$ takes the value of 1. This also implies that the solutions of the system (1) remain bounded within the finite time T; that is, the solution does not blow-up within the interval $\left[0,T\right]$.

4.2. Proof of Theorem 2

Under the assumption (H5), we assume the existence of sufficiently large constants C and $\lambda$, which cause the norm of the solution to fulfil the decay inequality ${∥u\left(u_0\right)∥}_{L^2}\le C{∥u_0∥}_{L^2}{e}^{-\lambda t}$. Thus, for any $\theta >0$, we say

$${∥u\left(u_0\right)∥}_{L^2\left(T-1,T;L^2\right)}\le {\left[{\int }_{T-1}^T{\left(K{∥u_0∥}_{L^2}{e}^{-\lambda t}\right)}^{2}dt\right]}^{1/2}\le \frac{K^2{∥u_0∥}_{L^2}^2}{2\lambda }{e}^{-2\lambda \left(T-1\right)}\le \frac{\theta }{2}.$$

From the above discussion, we obtain

$$P\left({∥u^S\left(u_0,\eta \right)-u\left(u_0\right)∥}_{L^2\left(0,T;L^2\right)}\le \theta \right)\ge 1-\theta ,$$

then we define the event $\mathbb{I}=\left\{{∥u^S\left(u_0,\eta \right)-u\left(u_0\right)∥}_{L^2\left(0,T;L^2\right)}\le \theta \right\}$. This also indicates that the probability of event $\mathbb{I}$ is no less than $1-\theta$. Combining this with the triangle inequality, it is evident that

$$\begin{array}{cc}\hfill {∥u^S\left(u_0,\eta \right)∥}_{L^2\left(T-1,T;L^2\right)}& \le {∥u^S\left(u_0,\eta \right)-u\left(u_0\right)∥}_{L^2\left(T-1,T;L^2\right)}+{∥u\left(u_0\right)∥}_{L^2\left(T-1,T;L^2\right)}\hfill \\ & \le \frac{\theta }{2}+\frac{\theta }{2}=\theta .\hfill \end{array}$$

Similar to Theorem 1.6 in [19], there exists $t\left(i\right)\in \left[T-1,T\right]$, which allows ${∥u^S\left(u_0,\eta \right)∥}_{L^2}$ to be taken as small enough, and $i\in \mathbb{I}$. With the condition (H6), choosing ${∥u^S\left(u_0,\eta \right)∥}_{L^2}$ as the initial data of the system (1), the solution exists globally; this means that the solution does not blow up in infinite time.

5. Example

Here, we illustrate the conclusions of this paper with an example.

Example 1.

For $d=3$, $\Omega ={\mathbb{T}}^3$, $s=1$, the function $\zeta \left(u\right)=-\nabla ·\left[u{\nabla }^{-1}\left(u-u_{\Omega }\right)\right]$ fulfills the assumptions (H1)–(H3) and $u_{\Omega }={\int }_{\Omega }u\left(x,t\right)dx$. Moreover, for initial condition $u_0\in L^2\left(\Omega \right)$ satisfying

$${∥u_0∥}_{L^2}^2\le S<\infty ,$$

then the assumption (H4) holds.

Proof. 

First, we discuss the assumptions (H1)–(H3). For $s=1$, we have

$$\begin{array}{c}{∥\zeta \left(u_1\right)-\zeta \left(u_2\right)∥}_{H^{-1}}={∥-\nabla ·\left[u_1{\nabla }^{-1}\left(u_1-{u_1}_{\Omega }\right)\right]+\nabla ·\left[u_2{\nabla }^{-1}\left(u_2-u_{2\Omega }\right)\right]∥}_{H^{-1}}\hfill \\ \le {∥u_2{\nabla }^{-1}\left(u_2-u_{2\Omega }\right)-u_1{\nabla }^{-1}\left(u_1-{u_1}_{\Omega }\right)∥}_{L^2}\hfill \\ \le {∥\left(u_2-u_1\right){\nabla }^{-1}\left(u_2-u_{2\Omega }\right)+u_1{\nabla }^{-1}\left(u_2-u_{2\Omega }\right)-u_1{\nabla }^{-1}\left(u_1-{u_1}_{\Omega }\right)∥}_{L^2}\hfill \\ \le {∥\left(u_2-u_1\right){\nabla }^{-1}\left(u_2-u_{2\Omega }\right)+u_1{\nabla }^{-1}\left(u_2-u_1+{u_1}_{\Omega }-u_{2\Omega }\right)∥}_{L^2}\hfill \\ \le {∥\left(u_2-u_1\right){\nabla }^{-1}\left(u_2-u_{2\Omega }\right)∥}_{L^2}+{∥u_1{\nabla }^{-1}\left(u_2-u_1+{u_1}_{\Omega }-u_{2\Omega }\right)∥}_{L^2}\hfill \\ \le {∥u_2-u_1∥}_{L^2}{∥{\nabla }^{-1}\left(u_2-u_{2\Omega }\right)∥}_{L^{\infty }}+{∥u_1∥}_{L^4}{∥{\nabla }^{-1}\left(u_2-u_1\right)∥}_{L^4};\hfill \end{array}$$

applying the Sobolev embedding theorem, it fulfills

$$\begin{array}{c}{∥\zeta \left(u_1\right)-\zeta \left(u_2\right)∥}_{H^{-1}}\le C_1{∥u_2-u_1∥}_{L^2}{∥{\nabla }^{-1}{u}_2∥}_{H^{\frac{7}{4}}}+C_2{∥u_1∥}_{H^{\frac{3}{4}}}{∥{\nabla }^{-1}\left(u_2-u_1\right)∥}_{H^{\frac{3}{4}}}\hfill \\ \le C_3{∥u_2-u_1∥}_{L^2}\left(1+{∥u_2∥}_{H^{\frac{3}{4}}}+{∥u_1∥}_{H^{\frac{3}{4}}}\right);\hfill \end{array}$$

the above-mentioned $C_i\left(i=1,2,3\right)$ are positive constants independent of time. Set $u_2=0$; we state

$${∥\zeta \left(u_1\right)∥}_{H^{-1}}\le C\left(1+{∥u_1∥}_{H^1}\right)\left(1+{∥u_1∥}_{L^2}\right);$$

the above inequality also demonstrates the continuity of the mapping $\zeta :H^{\frac{3}{4}}\to H^{-1}$. Namely, the condition (H1) holds with $a_1=1$ and ${\lambda }_1=\frac{1}{2}$.

For (H2), we have

$$\begin{array}{c}\left|〈\zeta \left(u\right),u〉\right|=\left|〈-\nabla ·\left[u{\nabla }^{-1}\left(u-u_{\Omega }\right)\right],u〉\right|\hfill \\ =\left|〈-\nabla u·{\nabla }^{-1}\left(u-u_{\Omega }\right),u〉+〈u\left(u-u_{\Omega }\right),u〉\right|\hfill \\ \le C{∥u∥}_{L^3}^3;\hfill \end{array}$$

the last step is obtained from the fact $〈u,v·\nabla u〉=-\frac{1}{2}〈u^2,\nabla ·u〉$. Then, according to the interpolation inequality and Sobolev embedding theorem, one can further obtain

$$\left|〈\zeta \left(u\right),u〉\right|\le C{∥u∥}_{L^2}^{3/2}{∥u∥}_{H^1}^{3/2}\le C\left(1+{∥u∥}_{L^2}^{3/2}\right)\left(1+{∥u∥}_{H^1}^{3/2}\right),$$

which indicates the validity of the condition (H2), and $a_2=\frac{3}{2},{\lambda }_2=\frac{3}{2}$.

For (H3), it is obvious that

$$\left|〈\zeta \left(u_1\right)-\zeta \left(u_2\right),u_1-u_2〉\right|\le {∥\zeta \left(u_1\right)-\zeta \left(u_2\right)∥}_{H^{-1}}{∥u_1-u_2∥}_{H^1},$$

combining the results of condition (H1), we say

$$\begin{array}{c}\left|〈\zeta \left(u_1\right)-\zeta \left(u_2\right),u_1-u_2〉\right|\le {∥\zeta \left(u_1\right)-\zeta \left(u_2\right)∥}_{H^{-1}}{∥u_1-u_2∥}_{H^1}\hfill \\ \le C{∥u_2-u_1∥}_{L^2}{∥u_1-u_2∥}_{H^1}\left(1+{∥u_2∥}_{H^1}+{∥u_1∥}_{H^1}\right);\hfill \end{array}$$

namely, the assumption (H3) holds for $a_3=1$, ${\lambda }_3=1$, and $\eta =1$.

Next, we discuss the assumption (H4). For the system

$${\partial }_t^{\beta }u=\left(1+q\right)\Delta u-\nabla ·\left[u{\nabla }^{-1}\left(u-u_{\Omega }\right)\right],$$

we recall the conclusion of Remark 1; it satisfies

$${\partial }_t^{\beta }{∥u∥}_{L^2}^2\le -2\left(2C_{2}q-1\right){∥u∥}_{L^2}^2+C_1\left[1+{∥u∥}_{L^2}^{a_4}\right],$$

where $a_4=\frac{2a_2}{2-{\lambda }_2}$. Set $\chi =2\left(2C_{2}q-1\right)$; we apply the Riemann–Liouville fractional integral $I_t^{\beta }$ to the above inequality and combine Theorem 2.4 in [26]; for the initial condition ${∥u_0∥}_{L^2}^2\le S<\infty$, one can choose the suitable noise intensity q to make $\chi$ large enough, such that

$${∥u\left(x,t\right)∥}_{L^2}^2<C\left(S,q\right),$$

where $C\left(S,q\right)$ is a constant associated with S and q. Namely, the assumption (H4) holds. □

Remark 4.

The well-known time-fractional Keller–Segel system is shaped as

$$\left\{\begin{array}{c}{\partial }_t^{\beta }u=\Delta u-\nabla ·\left(u\nabla v\right),x\in \Omega ,t>0,\hfill \\ -\Delta v=u-{\int }_{\Omega }u\left(x,t\right)dx,x\in \Omega ,t>0;\hfill \end{array}\right.$$

the blow-up of the solutions to the above model has been widely discussed, and many important research results have been obtained; more conclusions about the finite time blow-up of the solutions can be found in [27,28,29]. Example 1 and Theorem 1 indicate that, with the addition of multiplicative noise, which satisfies the certain conditions to the time-fractional Keller–Segel system, the solution of the system:

$${\partial }_t^{\beta }u=\Delta u-\nabla ·\left[u{\nabla }^{-1}\left(u-u_{\Omega }\right)\right]+\mu \sum _{m\in {\mathbb{Z}}^d\setminus \left\{0\right\}}{\sum }_{j=1}^{d-1}{\eta }_m{\omega }_{m,j}\left(x\right)\circ {\partial }_t{W}^{m,j},$$

remains bounded in finite time. Namely, multiplicative noise can effectively delay the blow-up of the solution.

6. Discussion

This paper primarily discusses the existence, uniqueness, and delayed blow-up of the weak solution to a class of time-fractional nonlinear stochastic differential equations driven by multiplicative noise. The Galerkin method was employed to project the original stochastic differential equation onto a finite-dimensional subspace. Truncation and orthogonalization operations were performed on the problem (1), and by utilizing inequalities and properties related to fractional-order operators, we provided a priori estimates for the weak solution. Based on these derivations, the existence and uniqueness were derived. Subsequently, by investigating the convergence relationship between the solutions of the time-fractional stochastic system and those of the time-fractional deterministic system, the main conclusions of this study were proven. Theorem 1 can be viewed as a generalization of Theorem 1.4 in [13] to fractional-order system. This theorem demonstrates that the solution of the problem (1) is bounded within the interval $\left[0,T\right]$, indicating that no blow-up occurs in finite time. Additionally, further assumptions were proposed to verify that the lifespan of the solution can be prolonged to infinite, which also indicates that the noise perturbation postpones the blow-up.