L1 Scheme for Semilinear Stochastic Subdiffusion with Integrated Fractional Gaussian Noise

Abstract

This paper considers a numerical method for solving the stochastic semilinear subdiffusion equation which is driven by integrated fractional Gaussian noise and the Hurst parameter $H\in (1/2,1)$. The finite element method is employed for spatial discretization, while the L1 scheme and Lubich’s first-order convolution quadrature formula are used to approximate the Caputo time-fractional derivative of order $\alpha \in (0,1)$ and the Riemann–Liouville time-fractional integral of order $\gamma \in (0,1)$, respectively. Using the semigroup approach, we establish the temporal and spatial regularity of the mild solution to the problem. The fully discrete solution is expressed as a convolution of a piecewise constant function with the inverse Laplace transform of a resolvent-related function. Based on the Laplace transform method and resolvent estimates, we prove that the proposed numerical scheme has the optimal convergence order $O({\tau }^{min\{H+\alpha +\gamma -1-\epsilon ,\alpha \}}),\epsilon >0$. Numerical experiments are presented to validate these theoretical convergence orders and demonstrate the effectiveness of this method.

1. Introduction

Stochastic time-fractional subdiffusion equations frequently arise when modeling heat conduction in materials with thermal memory under the influence of noise, as well as in the study of particle motion in viscoelastic fluids (see, e.g., [1,2,3,4]). To accurately describe random phenomena exhibiting long-range dependence, fractional noise becomes essential. Fractional noise is a zero-mean Gaussian process that is white in time, characterized by the Hurst index $H\in (0,1)$. When $H>1/2$, it exhibits a positive correlation, reflecting a persistent autocorrelation structure. The increase in fractional Brownian motion (fBm) demonstrates this long-range dependence, meaning that the correlation between distant observations decays slowly over time, leading to persistent behavior. This property is particularly relevant for modeling natural phenomena where past events significantly influence future outcomes, such as in financial markets, climate studies, and fluid dynamics. Moreover, $H>1/2$ implies a positive autocorrelation, indicating that an increase in one observation is likely to be followed by another increase, whereas a decrease is likely to be followed by another decrease. This positive correlation is often observed in systems with trends or cycles, making fBm with $H\in (1/2,1)$ a suitable choice for capturing such behavior [5].

In this work, we consider a numerical method for solving the following stochastic semilinear subdiffusion problem that is driven by integrated fractional noise with $0<\alpha <1,0\le \gamma \le 1$ and $1/2<H<1$:

$${}_0{}^{C}D_t^{\alpha }u(t)+Au(t)=f(u(t))+{}_0{}^{R}I_t^{\gamma }{\displaystyle \frac{d{W}^H(t)}{dt}},\mathrm{for}0<t\le T,\mathrm{with}u(0)=u_0,$$

(1)

where $A=-\mathsf{\Delta }$ with the domain $\mathcal{D}(A)=H_0^1(D)\cap H^2(D)$ and $D\subset {\mathbb{R}}^d,d=1,2,3$ is some regular domain with a smooth boundary. Here, ${}_0{}^{C}D_t^{\alpha }v(t)$ and ${}_0{}^{R}I_t^{\gamma }v(t)$ denote the left-sided Caputo fractional derivative and Riemann–Liouville fractional integral, respectively; see [6,7,8]. Additionally, $W^H(t)$ is the fractional Brownian motion seen with the Hurst parameter $H\in (1/2,1)$. The fractional noise ${\displaystyle \frac{d{W}^H(t)}{dt}}$ is a formal derivative of a centered Gaussian random field $W^H(t)$, which is defined on a complete filtered probability space $(\mathsf{\Omega },\mathcal{F},\mathcal{P},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0})$; see Section 2 for details. Here, $u_0\in L^2(D)$ is the initial value and $f:\mathbb{R}\to \mathbb{R}$ is a nonlinear function that is specified in Section 2.

In order to clarify the physical motivation of Model (1), we briefly introduce one of its typical applications; see [2] for details. Let the functions $u(t,x)$, $\tilde{Q}(t,x)$, and $F(t,x)$ represent body temperature, energy, and flux density, respectively. In a homogeneous medium with constants $\sigma$ and a, where $\sigma ,a>0$, we know that

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial \tilde{Q}(t,x)}{\partial t}}=-div\overrightarrow{F}(t,x),\hfill \\ \tilde{Q}(t,x)=\sigma u(t,x),\hfill \\ \overrightarrow{F}(t,x)=-a\nabla u(t,x),\hfill \end{array}\right.$$

which implies

$$\sigma {\displaystyle \frac{\partial u(t,x)}{\partial t}}=a\mathsf{\Delta }u.$$

But in some practical inhomogeneous medium environments, such as the heat conduction in materials with thermal memory subject to fractional noise, we need to express the energy term as

$$\tilde{Q}(t,x)={\int }_0^t{k}_1(t-r)u(r,x)dr+{\int }_0^t{k}_2(t-r)(b(u(r,x))+{\dot{W}}^H(r,x))dr,$$

(2)

where $k_1(t)=\mathsf{\Gamma }{(1-\alpha )}^{-1}{t}^{-\alpha }$, $k_2(t)=-\mathsf{\Gamma }{(\gamma +1)}^{-1}{t}^{\gamma }$. Differentiating (2), we obtain Model (1); that is,

$$\begin{array}{cc}\hfill -div\overrightarrow{F}& ={\displaystyle \frac{1}{\mathsf{\Gamma }(1-\alpha )}}{\displaystyle \frac{\partial }{\partial t}}{\int }_0^t{(t-r)}^{-\alpha }u(r,x)dr-{\displaystyle \frac{1}{\mathsf{\Gamma }(\gamma +1)}}{\displaystyle \frac{\partial }{\partial t}}{\int }_0^t{(t-r)}^{\gamma }(b(u(r,x))+{\dot{W}}^H(r,x))dr\hfill \\ \hfill & ={}_0{}^{C}D_t^{\alpha }u(t,x)-{\displaystyle \frac{1}{\mathsf{\Gamma }(\gamma )}}{\displaystyle \frac{\partial }{\partial t}}{\int }_0^t{\int }_0^{\tau }{(\tau -r)}^{\gamma }(b(u(r,x))+{\dot{W}}^H(r,x))drd\tau \hfill \\ \hfill & {}_0{}^{C}D_t^{\alpha }u(t,x)-{}_0{}^{R}I_t^{\gamma }(b(u(t,x))+{\dot{W}}^H(t,x)),\hfill \end{array}$$

which describes the heat conduction in a non-homogeneous medium subject to Gaussian noise with the Hurst index $H\in (1/2,1)$.

In recent years, significant advancements have been made in understanding the existence, uniqueness, and regularity of time-fractional stochastic partial differential equations. Key contributions include the works of Chen et al. [1], Mijena and Nane [3,4], Liu et al. [9], Anh et al. [10], Kang et al. [11], and Moulay et al. [12]. Alongside these theoretical developments, a growing body of research has focused on using numerical methods to solve time-fractional stochastic partial differential equations. Jin et al. [13] studied the strong and weak convergences of a fully discrete scheme to solve stochastic linear subdiffusion driven by additive noise using a semigroup approach. Gunzburger et al. [14,15] explored the time discretization and finite element method for approximating stochastic integral–differential equations driven by white noise. Wu et al. [16] analyzed a fully discrete scheme and proved their error estimates for approximating linear stochastic subdiffusion problems driven by integrated white noise. Kang et al. [11] examined the existence, uniqueness, and regularity of semilinear stochastic time–space-fractional partial differential equations. Hu et al. [17] investigated the use of the L1 scheme and weak convergence for solving stochastic semilinear subdiffusion problems with additive noise. Giordano [18] studied quasi-linear parabolic equations influenced by additive Gaussian noise that is white in time and fractional in space and characterized by a Hurst index of $H\in (0,1)$. Cao et al. [19] considered spatial semidiscretization for solving linear stochastic evolution equations driven by fractional noise with an $H\in (0,1/2)$. Deng et al. [20] explored a semidiscretization scheme for approximating semilinear stochastic wave equations driven by fractional noise with an $H\in (1/2,1)$. Additionally, Deng et al. [21] presented a unified convergence analysis for fractional diffusion equations driven by fractional Gaussian noise with an $H\in (0,1)$. For numerical methods used for stochastic parabolic partial differential equations, we refer to Dai et al. [22], Liu [23], Yan [24], Kruse [25], Jentzen and Kloeden [26], Chen et al. [27], and the references therein.

Despite these advancements, relatively little attention has been paid to the theory and numerical methods used for solving stochastic semilinear subdiffusion equations driven by integrated fractional noise with an $H\in (1/2,1)$. This problem presents two significant challenges: (1) the time-fractional derivative and time-fractional integration result in a solution operator lacking the semigroup property and (2) the interplay between nonlinear terms and fractional noise adds significant complexity to the numerical analysis.

In this work, we address these challenges by developing a numerical method for solving semilinear stochastic subdiffusion problems driven by integrated fractional noise. We first establish the time and space regularity of the mild solution using the semigroup approach and the properties of the Mittag-Leffler function.

We assume that the noise satisfies

$$\Vert A^{{\displaystyle \frac{\sigma -\kappa }{2}}}{\Vert }_{{\mathcal{L}}_2^0}<\infty ,-1\le \kappa -\sigma \le 1,$$

(3)

where $\kappa$ is defined by [13]

$$\kappa =\left\{\begin{array}{cc}\hfill & 2,1/2<\gamma \le 1,\hfill \\ \hfill & 2-\epsilon ,\gamma =1/2,\hfill \\ \hfill & 2-{\displaystyle \frac{1-2\gamma +\epsilon }{\alpha }},0\le \gamma <1/2,0<\epsilon <1,\hfill \end{array}\right.$$

(4)

and $\sigma \in [0,2]$. Here, ${\mathcal{L}}_2^0$ denotes the space of the Hilbert–Schmidt operators introduced in Section 2 below.

Let $u(t)$ be the solution to (1). We obtain the following spatial regularity:

$${\Vert u(t)\Vert }_{L^2(\mathsf{\Omega },{\dot{H}}^{\sigma })}\le C.$$

As for the temporal regularity, we denote it as follows:

$$\theta =(1-{\displaystyle \frac{\kappa -\sigma }{2}})\alpha +\gamma ,$$

(5)

and, with $\epsilon >0$, we obtain the following:

• If $0<\kappa -\sigma \le 1$, then for $\theta >{\displaystyle \frac{1}{2}}$,

  $$\Vert u(t_2)-u(t_1){\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C{(t_2-t_1)}^{\nu \alpha /2}{\Vert u_0\Vert }_{L^2(\mathsf{\Omega },{\dot{\mathbb{H}}}^{\nu })} +C{(t_2-t_1)}^{min\{\alpha ,\theta +H-1-\epsilon \}}.$$
• If $0\le \sigma -\kappa \le 1$, then for $\alpha +\gamma >{\displaystyle \frac{1}{2}}$,

  $$\Vert u(t_2)-u(t_1){\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C{(t_2-t_1)}^{\nu \alpha /2}{\Vert u_0\Vert }_{L^2(\mathsf{\Omega },{\dot{\mathbb{H}}}^{\nu })} +C{(t_2-t_1)}^{min\{\alpha ,\alpha +\gamma +H-1-\epsilon \}}.$$

Let $S_h\subset H_0^1(D)$ denote the linear finite element space. The finite element method is used to find $u_h(t)\in S_h$ such that with $u_h(0)=P_h{u}_0$,

$${}_0{}^{C}D_t^{\alpha }{u}_h(t)+A_h{u}_h(t)=P_{h}f(u_h(t))+P_h{}_0{}^{R}I_t^{\gamma }{\displaystyle \frac{d{W}^H(t)}{dt}},\mathrm{for}0<t\le T,$$

where $P_h:L^2(D)\to S_h$ is the $L^2$ projection operator and $A_h:S_h\to S_h$ is the discrete Laplacian operator. Let $0\le \mu \le 2$. Then, there is a positive constant C such that the following conclusions hold:

• For $0<\kappa -\sigma \le min\{1,\mu \}$ and $0<\epsilon <1$, we obtain

  $$\Vert u(t)-u_h{(t)\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C{h}^2{t}^{-\alpha {\displaystyle \frac{2-\nu }{2}}}{\Vert u_0\Vert }_{L^2(\mathsf{\Omega },{\dot{\mathbb{H}}}^{\nu })} +C{h}^{min\{2-\epsilon ,\mu -(\kappa -\sigma )\}},t\in [0,T],$$

  where $\alpha +\gamma -{\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha \mu }{2}}>0$.
• For $0\le \sigma -\kappa \le min\{1,\mu \}$, it holds that

  $$\Vert u(t)-u_h{(t)\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C{h}^2{t}^{-\alpha {\displaystyle \frac{2-\nu }{2}}}{\Vert u_0\Vert }_{L^2(\mathsf{\Omega },{\dot{\mathbb{H}}}^{\nu })} +C{h}^{min\{2-\epsilon ,\mu \}},t\in [0,T],$$

  where $\alpha +\gamma -{\displaystyle \frac{1}{2}}-\alpha (\mu -(\sigma -\kappa ))/2>0$ and $\alpha +\gamma -{\displaystyle \frac{\alpha \mu }{2}}+2H-{\displaystyle \frac{3}{2}}>0$.

Let $0=t_0<t_1<t_2<\cdots <t_N=T$ be a partition of $[0,T]$ and $\tau$ be the time step size. We shall approximate the time fractional derivative at $t=t_n$ by using the L1 scheme, with $0<\alpha <1$:

$${}_0{}^{C}D_t^{\alpha }u(t_n)={\tau }^{-\alpha }\sum _{i=0}^n{w}_{n-i}^{(\alpha )}u(t_i)+O({\tau }^{2-\alpha }),$$

where $w_i,i=1,2,\cdots$ are defined as follows (see [28]):

$$w_i^{(\alpha )}=\left\{\begin{array}{cc}\hfill & 1,\mathrm{for}i=0,\hfill \\ \hfill & -2i^{1-\alpha }+{(i-1)}^{1-\alpha }+{(i+1)}^{1-\alpha },\mathrm{for}i=,1,2,\cdots ,n-1,\hfill \\ \hfill & {(i-1)}^{1-\alpha }-i^{1-\alpha },\mathrm{for}i=n.\hfill \end{array}\right.$$

(6)

Further, we approximate the Riemann–Liouville fractional integral by using the first order convolution quadrature formula; that is,

$${}_0{}^{R}I_t^{-\gamma }u(t_n)={\tau }^{\gamma }\sum _{i=0}^n{w}_{n-i}^{(-\gamma )}u(t_i)+O(\tau ),$$

where $w_i^{(-\gamma )},i=0,1,\dots$ are generated by ${(1-\xi )}^{-\gamma }$, as seen in [29]:

$${(1-\xi )}^{-\gamma }=\sum _{i=0}^{\infty }{w}_i^{(-\gamma )}{\xi }^i.$$

(7)

As for the noise term $g(t)={\displaystyle \frac{d{W}^H(t)}{dt}}$, we approximate it at $t=t_n$ by using the Euler method, with $g^0=0$:

$${\displaystyle \frac{d{W}^H(t_n)}{dt}}\approx (W^H(t_n)-W^H(t_{n-1}))/\tau =g^n.$$

Let $u_h^n,F^n,g^n$ denote the approximate solutions of $u(t),f(u(t)),$ and $g(t)$ at $t=t_n$, respectively. We can therefore define the following fully discretized scheme:

$${\tau }^{-\alpha }\sum _{i=0}^n{w}_{n-i}^{(\alpha )}{u}_h^i+A_h{u}_h^n=P_h{F}^n+{\tau }^{\gamma }\sum _{i=0}^n{w}_{n-i}^{(-\gamma )}{P}_h{g}^i,$$

(8)

where $F^n=f(u_h(t_{n-1}))(F^0=0),g^n={\displaystyle \frac{W^H(t_n)-W^H(t_{n-1})}{\tau }},$ and $(g^0=0)$.

Following the approach of Gunzburger et al. [14], we introduce piecewise constant functions to handle the nonlinear terms and noise. This allows the fully discrete solution to be expressed as the convolution of a piecewise constant function using the inverse Laplace transform of a resolvent-related function, i.e.:

$$u_h^n={\int }_0^{t_n}{\overline{E}}_{\tau ,h}(t_n-s)P_h\overline{F}(s)ds+{\int }_0^{t_n}{\tilde{E}}_{\tau ,h}(t_n-s)P_h{\overline{\partial }}_{\tau }{W}^H(s)ds,$$

where the approximate operators ${\overline{E}}_{\tau ,h}$ and ${\tilde{E}}_{\tau ,h}$ are defined in (62) and (63).

The optimal time discretization convergence order is established using the Laplace transform method and the resolvent estimates; thus, for $\epsilon >0$, ${\displaystyle \frac{1}{2}}<H<1,0<\alpha <1$, $0\le \gamma \le 1$, and ${\displaystyle \frac{1}{2}}<\theta <2$, we obtain the following:

• If $0<\kappa -\sigma \le 1$, then

  $$\Vert u_h(t_n)-u_h^n{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C{\tau }^{min\{H+\theta -1-\epsilon ,\alpha \}}.$$
• If $0\le \sigma -\kappa \le 1$, then

  $$\Vert u_h(t_n)-u_h^n{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C{\tau }^{min\{H+\alpha +\gamma -1-\epsilon ,\alpha \}},$$

  where $\alpha +\gamma >{\displaystyle \frac{1}{2}}$.

Remark 1.

Let $\mu =2$ and $0<\epsilon <1$.

(i) When $0<\kappa -\sigma \le 1$, we obtain

$$\Vert u(t)-u_h{(t)\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C{h}^2{t}^{-\alpha {\displaystyle \frac{2-\nu }{2}}}{\Vert u_0\Vert }_{L^2(\mathsf{\Omega },{\dot{\mathbb{H}}}^{\nu })} +C{h}^{min\{2-\epsilon ,2-(\kappa -\sigma )\}},t\in [0,T],$$

where $\gamma >{\displaystyle \frac{1}{2}}$.

(ii) When $0\le \sigma -\kappa \le 1$, it holds that

$$\Vert u(t)-u_h{(t)\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C{h}^2{t}^{-\alpha {\displaystyle \frac{2-\nu }{2}}}{\Vert u_0\Vert }_{L^2(\mathsf{\Omega },{\dot{\mathbb{H}}}^{\nu })} +C{h}^{2-\epsilon },t\in [0,T],$$

where $\gamma -{\displaystyle \frac{1}{2}}+\alpha (\sigma -\kappa )/2>0$ and $\gamma +2H-{\displaystyle \frac{3}{2}}>0$.

Remark 2.

When $\sigma =\kappa$ and $H={\displaystyle \frac{1}{2}}$, we find, with $\alpha +\gamma >{\displaystyle \frac{1}{2}}$, the following:

(i) When $\gamma =0,$

$$\Vert u_h(t_n)-u_h^n{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C\tau t_n^{-1+{\displaystyle \frac{\nu \alpha }{2}}}{\Vert u_0\Vert }_{L^2(\mathsf{\Omega },{\dot{\mathbb{H}}}^{\nu })} +C{\tau }^{\alpha -{\displaystyle \frac{1}{2}}-\epsilon };$$

(ii) When $\gamma >0,$

$$\Vert u_h(t_n)-u_h^n{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C\tau t_n^{-1+{\displaystyle \frac{\nu \alpha }{2}}}{\Vert u_0\Vert }_{L^2(\mathsf{\Omega },{\dot{\mathbb{H}}}^{\nu })} +C{\tau }^{min\{\alpha +\gamma -{\displaystyle \frac{1}{2}}-\epsilon ,\alpha \}},$$

which is consistent with the error estimate of [27].

Remark 3.

When $\sigma =\kappa$ and $H>{\displaystyle \frac{1}{2}}$, we find, with $\alpha +\gamma >{\displaystyle \frac{1}{2}}$, the following:

(i) When $\gamma =0,$

$$\Vert u_h(t_n)-u_h^n{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C\tau t_n^{-1+{\displaystyle \frac{\nu \alpha }{2}}}+C{\tau }^{min\{\alpha +H-1-\epsilon ,\alpha \}};$$

(ii) When $\gamma >0,$

$$\Vert u_h(t_n)-u_h^n{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C\tau t_n^{-1+{\displaystyle \frac{\nu \alpha }{2}}}+C{\tau }^{min\{\alpha +\gamma +H-1-\epsilon ,\alpha \}}.$$

This paper is organized as follows: Section 2 introduces some key notations and assumptions regarding the nonlinear term f and the noise seen and establishes the spatial and temporal regularity of the mild solution. Section 3 presents the finite element scheme and derives the error estimates for the semidiscrete case. Section 4 extends the analysis to the fully discrete setting, proving the error estimates using the convolution representation of the fully discrete solution. Finally, Section 5 provides some numerical simulations that validate and support our theoretical analysis.

We use C to denote a positive constant independent of the functions and parameters concerned, but it is not necessarily the same when it occurs in different places. We use c to denote a particular positive constant independent of the functions and parameters concerned.

2. Preliminaries and Notations

Let $\mathbb{H}=L_2(\mathcal{D})$ be a real separable Hilbert space with the usual inner product $(·,·)$ and norm $\Vert ·\Vert$. Let $(\mathsf{\Omega },\mathcal{F},\mathcal{P},{\left\{{\mathcal{F}}_t\right\}}_{t\ge 0})$ be a filtered probability space and $\mathcal{L}(\mathbb{H})$ be the space of bounded linear operators from $\mathbb{H}$ to $\mathbb{H}$. The fractional noise has the following Fourier expansion:

$$W^H(t)=\sum _{l=1}^{\infty }{\gamma }_l^{1/2}{\beta }_l^H(t)e_l,$$

(9)

where ${\beta }_l^H(t),l=1,2,\cdots$ are real-valued mutually independent fractional Brownian motion ($fBm$) values with the Hurst parameter $H\in (1/2,1)$, and ${\{{\gamma }_j,e_j\}}_{j=1}^{\infty }$ are the eigenpairs of the operator $A=-\mathsf{\Delta }$, with $\mathcal{D}(A)=H_0^1(D)\cap H^2(D)$. The covariance operator $Q\in \mathcal{L}(\mathbb{H})$ and Q is a self-adjoint positive semidefinite operator. We use ${\mathcal{L}}_2^0=HS(Q^{1/2}(\mathbb{H}),\mathbb{H})$ to denote the space of Hilbert–Schmidt operators from $Q^{1/2}(\mathbb{H})$ to $\mathbb{H}$, equipped are with the inner product and norm:

$$\langle {\psi }_1,{\psi }_2\rangle =\sum _{j=1}^{\infty }({\psi }_1{Q}^{1/2}{e}_j,{\psi }_2{Q}^{1/2}{e}_j){,\Vert \psi \Vert }_{L_2^0}^2=\sum _{j=1}^{\infty }{\Vert \psi Q^{1/2}{e}_j\Vert }^2<\infty ,\mathrm{for}{\psi }_1,{\psi }_2,\psi \in {\mathcal{L}}_2^0,$$

where $HS$ denotes the Hilbert–Schmidt operator norm.

Let ${\dot{H}}^s, s\in \mathbb{R}$ be a Sobolev space defined by

$${\dot{H}}^s=D(A^{{\displaystyle \frac{s}{2}}})=\{v\in \mathbb{H}:\Vert A^{{\displaystyle \frac{s}{2}}}v{\Vert }^2 =\sum _{j=1}^{\infty }{\lambda }_j^s{(v,e_j)}^2<\infty \},$$

with the norm ${|v|}_s^2={\Vert A^{{\displaystyle \frac{s}{2}}}\Vert }^2={\sum }_{j=1}^{\infty }{\lambda }_j^s{(v,e_j)}^2\mathrm{for}v\in {\dot{H}}^s$, where ${\{{\lambda }_j,e_j\}}_{j=1}^{\infty }$ are the eigenpairs of $A=-\mathsf{\Delta }$, with $\mathcal{D}(A)={\dot{H}}^1(D)\cap H^2(D)$. For simplicity, we assume that A and Q have the same eigenfunctions $e_j,j=1,2,\cdots$.

Assumption 1.

When $\theta \in (\pi /2,\pi )$, the operator A satisfies the following resolvent estimate:

$${\Vert (zI+A)}^{-1}{\Vert \le C|z|}^{-1}\mathrm{for}z\in {\Sigma }_{\theta }=\{z\ne 0:|argz|<\theta \},$$

(10)

which implies that when $0<\alpha <1$ (see Yan et al. [28]),

$$\Vert {(z^{\alpha }I+A)}^{-1}{\Vert \le C|z|}^{-\alpha },\forall z\in {\Sigma }_{\theta }=\{z\ne 0:|argz|<\theta \}.$$

(11)

Assumption 2

([11]). The nonlinear function f satisfies

$$\Vert f(u)\Vert \le C(1+\Vert u\Vert ),\Vert f(u)-f(v)\Vert \le C\Vert u-v\Vert ,\forall u,v\in L^2(D).$$

The conditions placed on f can ensure the existence and uniqueness of $u(t)\in C([0,T],L^2(\mathsf{\Omega },\mathbb{H}))$.

Assumption 3

([13]). $\alpha \in (0,1),\gamma \in [0,1],and\alpha +\gamma >1/2.$

Assumption 4.

For $0\le \sigma \le 2,0\le \kappa \le 2,-1\le \kappa -\sigma \le 1$, we assume

$$\Vert A^{{\displaystyle \frac{\sigma -\kappa }{2}}}{\Vert }_{{\mathcal{L}}_2^0}<\infty ,$$

(12)

where κ is defined by (4), which ensures that ${\int }_0^t{\Vert A^{{\displaystyle \frac{\kappa }{2}}}\tilde{E}(s)\Vert }^{2}ds<\infty$, where $\tilde{E}(t)$ is the Mittag-Leffler function defined in (14) below.

Below we introduce some useful lemmas.

Lemma 1

([30,31]). For $H\in (1/2,1)$ and $g(t),h(t)\in L^2([0,T])$, it holds that

$$\begin{array}{cc}\hfill & \mathbb{E}\left({\int }_0^{t}g(s)d{\beta }_k^H(s){\int }_0^{t}h(s)d{\beta }_k^H(s)\right)=H(2H-1){\int }_0^t{\int }_0^{t}g(s)h(r){|r-s|}^{2H-2}drds.\hfill \end{array}$$

Lemma 2.

For $H\in (1/2,1)$ and $g(t),h(t)\in {\mathcal{L}}_2^0$, it holds that

$$\mathbb{E}\left({\int }_0^{t}h(s)d{W}^H(s),{\int }_0^{t}g(s)d{W}^H(s)\right)=H(2H-1){\int }_0^t{\int }_0^t\langle h(s),g(r)\rangle {|r-s|}^{2H-2}drds.$$

Proof. 

From the definition of $W^H(t)$ in (9), we obtain

$$\mathbb{E}\left({\int }_0^{t}h(s)d{W}^H(s),{\int }_0^{t}g(s)d{W}^H(s)\right)=\sum _{l=1}^{\infty }\mathbb{E}\left({\int }_0^{t}h(s){\gamma }_l^{1/2}d{\beta }_l^H(s)·{\int }_0^{t}g(s){\gamma }_l^{1/2}d{\beta }_l^H(s)\right).$$

By applying Lemma 1 and the definition of the inner product in ${\mathcal{L}}_2^0$, we obtain

$$\begin{array}{cc}\hfill & \sum _{l=1}^{\infty }\mathbb{E}\left({\int }_0^{t}h(s){\gamma }_l^{1/2}d{\beta }_l^H(s)·{\int }_0^{t}g(s){\gamma }_l^{1/2}d{\beta }_l^H(s)\right)\hfill \\ \hfill & =H(2H-1){\int }_0^t{\int }_0^t\sum _{l=1}^{\infty }(h(s){\gamma }_l^{1/2}{e}_l(x),g(r){\gamma }_l^{1/2}{e}_l(x)){|r-s|}^{2H-2}drds\hfill \\ \hfill & =H(2H-1){\int }_0^t{\int }_0^t\langle h(s),g(r)\rangle {|r-s|}^{2H-2}drds,\hfill \end{array}$$

which completes the proof of Lemma 2. □

Lemma 3

([32]). Let $\alpha \in (0,1)$. Suppose that y is non-negative and satisfies the inequality

$${}_0{}^{C}D_t^{\alpha }y(t)\le \beta y(t)+\sigma (t),$$

where the function $\sigma \in L^{\infty }(0,T)$ and the constant $\beta >0$. Then,

$$y(t)\le C(y(0)+\Vert \sigma {\Vert }_{L^{\infty }(0,T)}).$$

We denote $F(t)=f(u(t)),g(t)={\displaystyle \frac{d{W}^H(t)}{dt}}$, and then (1) can be formally written as

$${}_0{}^{C}D_t^{\alpha }u(t)+Au(t)=F(t)+{}_0{}^{R}I_t^{\gamma }g(t),\mathrm{for}0<t\le T,\mathrm{with}u(0)=u_0.$$

(13)

Taking the Laplace transform of (13), we obtain

$$z^{\alpha }\widehat{u}(z)-z^{\alpha -1}{u}_0+A\widehat{u}(z)=\widehat{F}(z)+z^{-\gamma }\widehat{g}(z),$$

which implies that

$$\widehat{u}(z)=z^{\alpha -1}{(z^{\alpha }+A)}^{-1}{u}_0+{(z^{\alpha }+A)}^{-1}\widehat{F}(z)+z^{-\gamma }{(z^{\alpha }+A)}^{-1}\widehat{g}(z).$$

Using the inverse Laplace transform, we obtain

$$u(t)=E(t)u_0+{\int }_0^t\overline{E}(t-s)f(u(s))ds+{\int }_0^t\tilde{E}(t-s)d{W}^H(s),$$

(14)

where

$$\begin{array}{cc}\hfill & E(t)={\displaystyle \frac{1}{2\pi i}}{\int }_{\mathsf{\Gamma }}{e}^{zt}{z}^{\alpha -1}{(z^{\alpha }+A)}^{-1}dz,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & \overline{E}(t)={\displaystyle \frac{1}{2\pi i}}{\int }_{\mathsf{\Gamma }}{e}^{zt}{(z^{\alpha }+A)}^{-1}dz,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & \tilde{E}(t){\displaystyle \frac{1}{2\pi i}}{\int }_{\mathsf{\Gamma }}{e}^{zt}{(z^{\alpha }+A)}^{-1}{z}^{-\gamma }dz.\hfill \end{array}$$

(17)

Here, $\mathsf{\Gamma }=\{z:arg(z)=\theta ,\theta \in (\pi /2,\pi ),\Im (z)$ increases from −∞ to ∞}.

According to the resolvent estimate (11) and interpolation theory, we can easily show that

$$\Vert A^r{(z^{\alpha }+A)}^{-1}{\Vert \le C|z|}^{\alpha (r-1)},r\in [0,1].$$

(18)

From (18), and similar to the estimates of Lemma 4.1 in [13], the above Mittag-Leffler functions have the following estimates when $p\ge 0$ and $q\ge 0$:

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{d^l}{dt}}E(t)v{|}_p\le C{t}^{-\alpha {\displaystyle \frac{p-q}{2}}-l}{|v|}_q,0\le p-q\le 2,l=0,1,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{d^l}{dt}}\overline{E}(t)v{|}_p\le C{t}^{-\alpha {\displaystyle \frac{p-q}{2}}+\alpha -1-l}{|v|}_q,0\le p-q\le 2,l=0,1,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{d^l}{dt}}\tilde{E}(t)v{|}_p\le C{t}^{-\alpha {\displaystyle \frac{p-q}{2}}+\alpha +\gamma -1-l}{|v|}_q,0\le p-q\le 2,l=0,1.\hfill \end{array}$$

(21)

Next, we shall consider the spatial and temporal regularities of the solution $u(t)$. First we well show its spatial regularity.

Theorem 1.

Let Assumptions 2–4 hold. Thus, we assume $u_0\in L^2(\mathsf{\Omega },{\dot{H}}^q)$ when $q\in [0,2]$. Then, there exists a positive constant C such that, with $0\le q\le \sigma \le 2$,

$${\Vert u(t)\Vert }_{L^2(\mathsf{\Omega },{\dot{H}}^{\sigma })}\le C.$$

Proof. 

Simple calculation gives us

$$\begin{array}{cc}\hfill {\Vert u(t)\Vert }_{L^2(\mathsf{\Omega },{\dot{H}}^{\sigma })}& \le \Vert E(t)u_0{\Vert }_{L^2(\mathsf{\Omega },{\dot{H}}^{\sigma })}+{\Vert {\int }_0^t\overline{E}(t-s)f(u(s))ds\Vert }_{L^2(\mathsf{\Omega },{\dot{H}}^{\sigma })}\hfill \\ \hfill & +\Vert {\int }_0^t\tilde{E}(t-s)d{W}^H{(s)\Vert }_{L^2(\mathsf{\Omega },{\dot{H}}^{\sigma })}\hfill \\ \hfill & =I+II+III.\hfill \end{array}$$

As for I, using (19) and with $l=0$, $p=q=\sigma$, we find that

$$I\le C\Vert u_0{\Vert }_{L^2(\mathsf{\Omega },{\dot{H}}^{\sigma })}.$$

(22)

As for $II$, the estimate (20), with $l=0$, and Assumption 2 lead to

$$II\le C{\int }_0^t{(t-s)}^{\alpha (1-{\displaystyle \frac{\sigma }{2}})-1}{(1+\Vert u(s)\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})})ds\le C,$$

(23)

where we require $\sigma <2$.

When $\sigma =2$, we obtain, from (20), $l=0,p=2,q=0$, and the fact that ${\displaystyle \frac{dE(t)}{dt}}=-A{E}_{\alpha ,\alpha }(-t^{\alpha }A)=-\overline{E}(t)$, the following:

$$\begin{array}{cc}\hfill II& = \Vert {\int }_0^{t}A\overline{E}{(t-s)f(u(s))ds\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & \le \Vert {\int }_0^{t}A\overline{E}{(t-s)[f(u(s)-f(u(t))]ds\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} + {\Vert {\int }_0^{t}A\overline{E}(t-s)f(u(t))ds\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})},\hfill \\ \hfill & \le C{\int }_0^t{(t-s)}^{-1}{\Vert u(s)-u(t)\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}ds + \Vert {\int }_0^{t}A\overline{E}(s)ds\Vert \underset{0\le t\le T}{sup}{\Vert f(u(t))\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})},\hfill \\ \hfill & \le C{\int }_0^t{(t-s)}^{-1+{\beta }_1}ds+\Vert E(t)-E(0)\Vert \le C.\hfill \end{array}$$

(24)

Here, ${\beta }_1>0$ comes from the time regularity in Theorem 2.

As for $III$, using Lemma 2, we obtain

$$\begin{array}{cc}\hfill II{I}^2& =\mathbb{E} \Vert {\int }_0^t{A}^{{\displaystyle \frac{\sigma }{2}}}\tilde{E}(t-s)d{W}^H(s){\Vert }^2\hfill \\ \hfill & =H(2H-1){\int }_0^t{\int }_0^t\langle A^{{\displaystyle \frac{\sigma }{2}}}\tilde{E}(t-s),A^{{\displaystyle \frac{\sigma }{2}}}\tilde{E}(t-r)\rangle {|r-s|}^{2H-2}drds.\hfill \end{array}$$

From the symmetry of the integral region and integrand function, we find that

$$II{I}^2=2H(2H-1){\int }_0^t{\int }_s^t\langle A^{{\displaystyle \frac{\sigma }{2}}}\tilde{E}(t-s),A^{{\displaystyle \frac{\sigma }{2}}}\tilde{E}(t-r)\rangle {|r-s|}^{2H-2}drds.$$

By using Assumption 4 about fractional noise, (21) when $l=0$, and the variable transformation $r-s=\overline{r}$, we obtain

$$\begin{array}{cc}\hfill II{I}^2& \le 2H(2H-1){\int }_0^t{\int }_s^t\Vert A^{{\displaystyle \frac{\sigma }{2}}}\tilde{E}{(t-s)\Vert }_{{\mathcal{L}}_2^0}\Vert A^{{\displaystyle \frac{\sigma }{2}}}\tilde{E}{(t-r)\Vert }_{{\mathcal{L}}_2^0}{|r-s|}^{2H-2}drds\hfill \\ \hfill & =2H(2H-1){\int }_0^t{\Vert A^{{\displaystyle \frac{\sigma }{2}}}\tilde{E}(t-s)\Vert }_{{\mathcal{L}}_2^0}·\left({\int }_0^{t-s}{\Vert A^{{\displaystyle \frac{\sigma }{2}}}\tilde{E}(t-s-r)\Vert }_{{\mathcal{L}}_2^0}{r}^{2H-2}dr\right)ds\hfill \\ \hfill & \le C{\int }_0^t\left(\Vert A^{{\displaystyle \frac{\kappa }{2}}}\tilde{E}(t-s)\Vert ·{\int }_0^{t-s}\Vert A^{{\displaystyle \frac{\kappa }{2}}}\tilde{E}(t-s-r)\Vert r^{2H-2}dr\right)ds{\Vert A^{{\displaystyle \frac{\sigma -\kappa }{2}}}\Vert }_{{\mathcal{L}}_2^0}^2.\hfill \end{array}$$

Further, using Cauchy–Schwarz inequality, it holds that

$$\begin{array}{cc}\hfill II{I}^2& \le C{\left({\int }_0^t{\Vert A^{{\displaystyle \frac{\kappa }{2}}}\tilde{E}(t-s)\Vert }^{2}ds\right)}^{1/2}·{\left({\int }_0^t{\left({\int }_0^{t-s}{(t-r-s)}^{-\alpha {\displaystyle \frac{\kappa }{2}}+\alpha +\gamma -1}{r}^{2H-2}dr\right)}^{2}ds\right)}^{1/2}\hfill \\ \hfill & \le C{\left({\int }_0^t{(t-s)}^{2(-\alpha {\displaystyle \frac{\kappa }{2}}+\alpha +\gamma +2H-2)}ds\right)}^{1/2}\le C.\hfill \end{array}$$

(25)

Here, the definitions of $\kappa$ and $H\in (1/2,1)$ ensure the integrals ${\int }_0^t{\Vert A^{{\displaystyle \frac{\kappa }{2}}}\tilde{E}(t-s)\Vert }^{2}ds$ and ${\int }_0^t{(t-s)}^{2(-\alpha {\displaystyle \frac{\kappa }{2}}+\alpha +\gamma +2H-2)}ds$ are finite.

The proof of Theorem 1 is complete. □

Next, we provide the temporal regularity of solution $u(t)$.

Theorem 2.

Let $0<\alpha <1,0\le \gamma \le 1,1/2<H<1$. Let Assumptions 2–4 hold. We assume that $\theta =(1-{\displaystyle \frac{\kappa -\sigma }{2}})\alpha +\gamma$, as given by (5), and $u_0\in L^2(\mathsf{\Omega },{\dot{H}}^{\nu })(0\le \nu \le 2)$. Then, for $\epsilon >0$ and $\le t_1<t_2\le T$ , we know the following:

(i) If $0<\kappa -\sigma \le 1$, then for $\theta >{\displaystyle \frac{1}{2}}$,

$$\Vert u(t_2)-u(t_1){\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C{(t_2-t_1)}^{\nu \alpha /2}{\Vert u_0\Vert }_{L^2(\mathsf{\Omega },{\dot{\mathbb{H}}}^{\nu })} +C{(t_2-t_1)}^{min\{\alpha ,\theta +H-1-\epsilon \}}.$$

(ii) If $0\le \sigma -\kappa \le 1$, then for $\alpha +\gamma >{\displaystyle \frac{1}{2}}$,

$$\Vert u(t_2)-u(t_1){\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C{(t_2-t_1)}^{\nu \alpha /2}{\Vert u_0\Vert }_{L^2(\mathsf{\Omega },{\dot{\mathbb{H}}}^{\nu })} +C{(t_2-t_1)}^{min\{\alpha ,\alpha +\gamma +H-1-\epsilon \}}.$$

Proof. 

First, we divide $\Vert u(t_2)-u(t_1){\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}$ into three parts:

$$\begin{array}{cc}\hfill & \Vert u(t_2)-u(t_1){\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & \le \Vert (E(t_2)-E(t_1))u_0{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}+{\Vert {\int }_0^{t_2}\overline{E}(s)f(u(t_2-s))ds-{\int }_0^{t_1}\overline{E}(s)f(u(t_1-s))ds\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & + \Vert {\int }_0^{t_2}\tilde{E}(t_2-s)d{W}^H(s)-{\int }_0^{t_1}\tilde{E}(t_1-s)d{W}^H(s){\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & =I+II+III.\hfill \end{array}$$

(26)

As for I, from the fact that ${\displaystyle \frac{dE(t)}{dt}}=-A\overline{E}(t)$ and (20) is true, we find that

$$\begin{array}{cc}\hfill I& = \Vert {\int }_{t_1}^{t_2}{E}^{\prime }(t)u_0{dt\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\le C{\int }_{t_1}^{t_2}\Vert A^{1-\nu /2}\overline{E}(t)\Vert dt\Vert u_0{\Vert }_{L^2(\mathsf{\Omega },{\dot{H}}^{\nu })}\hfill \\ \hfill & \le C{\int }_{t_1}^{t_2}{t}^{{\displaystyle \frac{\alpha \nu }{2}}-1}dt\Vert u_0{\Vert }_{L^2(\mathsf{\Omega },{\mathbb{H}}^{\nu })}\le C{(t_2-t_1)}^{{\displaystyle \frac{\alpha \nu }{2}}}{\Vert u_0\Vert }_{L^2(\mathsf{\Omega },{\mathbb{H}}^{\nu })}.\hfill \end{array}$$

(27)

By using Assumption 2, (20), $l=0$, and $p=q=0$, we obtain

$$\begin{array}{cc}\hfill II& \le \Vert {\int }_0^{t_1}\overline{E}(s)[f(u(t_2-s))-f(u(t_1-s))]{ds\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}+{\Vert {\int }_{t_1}^{t_2}\overline{E}(s)f(u(t_2-s))ds\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & \le {\int }_0^{t_1}\Vert \overline{E}(s)\Vert \Vert u(t_2-s)-u(t_1-s){\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}ds+{\int }_{t_1}^{t_2}\Vert \overline{E}(s)\Vert ds\underset{0\le s\le T}{sup}{\Vert f(u(s))\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & \le C{\int }_0^{t_1}{s}^{\alpha -1}{\Vert u(t_2-s)-u(t_1-s)\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}ds+C{(t_2-t_1)}^{\alpha }.\hfill \end{array}$$

(28)

As for $III$, we find that

$$\begin{array}{cc}\hfill III& ={∥{\int }_0^{t_2}\tilde{E}(t_2-s)d{W}^H(s)-{\int }_0^{t_1}\tilde{E}(t_1-s)d{W}^H(s)∥}_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & \le {∥{\int }_0^{t_1}[\tilde{E}(t_2-s)-\tilde{E}(t_1-s)]d{W}^H(s)∥}_{L^2(\mathsf{\Omega },\mathbb{H})}+{∥{\int }_{t_1}^{t_2}\tilde{E}(t_2-s)d{W}^H(s)∥}_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & =II{I}_1+II{I}_2.\hfill \end{array}$$

(29)

For $II{I}_2$, by applying Lemma 2 and Assumption 4, we obtain

$$\begin{array}{cc}\hfill II{I}_2^2& ={∥{\int }_{t_1}^{t_2}\tilde{E}(t_2-s)d{W}^H(s)∥}_{L^2(\mathsf{\Omega },\mathbb{H})}^2=\mathbb{E}{∥{\int }_{t_1}^{t_2}\tilde{E}(t_2-s)d{W}^H(s)∥}^2\hfill \\ \hfill & =H(2H-1){\int }_{t_1}^{t_2}{\int }_{t_1}^{t_2}\langle \tilde{E}(t_2-s),\tilde{E}(t_2-r)\rangle {|r-s|}^{2H-2}drds\hfill \\ \hfill & =2H(2H-1){\int }_{t_1}^{t_2}{\int }_s^{t_2}\langle \tilde{E}(t_2-s),\tilde{E}(t_2-r)\rangle {(r-s)}^{2H-2}drds\hfill \\ \hfill & \le C{\int }_{t_1}^{t_2}\Vert A^{{\displaystyle \frac{\kappa -\sigma }{2}}}\tilde{E}(t_2-s)\Vert ·\left({\int }_s^{t_2}\Vert A^{{\displaystyle \frac{\kappa -\sigma }{2}}}\tilde{E}(t_2-r)\Vert {(r-s)}^{2H-2}dr\right)ds{\Vert A^{{\displaystyle \frac{-(\kappa -\sigma )}{2}}}\Vert }_{{\mathcal{L}}_2^0}^2.\hfill \end{array}$$

Case 1. If $\sigma <\kappa$, then by employing (21), $p=\kappa -\sigma ,q=0$, and the variable transformation $r-s=\overline{r}$, we find that

$$\begin{array}{cc}\hfill II{I}_2^2& \le C{\int }_{t_1}^{t_2}\Vert A^{{\displaystyle \frac{\kappa -\sigma }{2}}}\tilde{E}(t_2-s)\Vert ·\left({\int }_0^{t_2-s}\Vert A^{{\displaystyle \frac{\kappa -\sigma }{2}}}\tilde{E}(t_2-s-r)\Vert r^{2H-2}dr\right)ds{\Vert A^{-{\displaystyle \frac{\kappa -\sigma }{2}}}\Vert }_{{\mathcal{L}}_2^0}^2\hfill \\ \hfill & \le C{\int }_{t_1}^{t_2}{(t_2-s)}^{\theta -1}·\left({\int }_0^{t_2-s}{(t_2-s-r)}^{\theta -1}{r}^{2H-2}dr\right)ds,\hfill \end{array}$$

where $\theta =(1-{\displaystyle \frac{\kappa -\sigma }{2}})\alpha +\gamma$. Further, by using the variable transformation $r=(t_2-s)\overline{r}$ and the characteristics of the function $B(·,·)$, when $\theta +H>1$, we obtain

$$\begin{array}{cc}\hfill II{I}_2^2& \le C{\int }_{t_1}^{t_2}{(t_2-s)}^{\theta -1}{(t_2-s)}^{\theta +2H-2}·{\int }_0^1{\overline{r}}^{\theta -1}{\overline{r}}^{2H-2}d\overline{r}ds\le C{\int }_{t_1}^{t_2}{(t_2-s)}^{2\theta +2H-3}ds\hfill \\ \hfill & \le C{(t_2-t_1)}^{2\theta +2H-2}.\hfill \end{array}$$

(30)

Case 2. If $\sigma \ge \kappa$, then by employing (21) and $l=0,p=q=0$, and similar to the discussion of case 1, we find that when $\alpha +\gamma +H>1$,

$$\begin{array}{cc}\hfill II{I}_2^2& \le C{\int }_{t_1}^{t_2}{(t_2-s)}^{\alpha +\gamma -1}·({\int }_0^{t_2-s}{(t_2-s-r)}^{\alpha +\gamma -1}{r}^{2H-2}dr)ds\hfill \\ \hfill & \le C{\int }_{t_1}^{t_2}{(t_2-s)}^{2(\alpha +\gamma )+2H-3}ds\le C{(t_2-t_1)}^{2(\alpha +\gamma )+2H-2}.\hfill \end{array}$$

(31)

Thus, for $\sigma <\kappa$,

$$II{I}_2^2\le C{(t_2-t_1)}^{min\{2\theta +2H-2,2\}},$$

and for $\sigma \ge \kappa$,

$$II{I}_2^2\le C{(t_2-t_1)}^{min\{2(\alpha +\gamma )+2H-2,2\}}.$$

Next, we turn to the estimation of $II{I}_1$. Let $g(s)=\tilde{E}(t_2-s)-\tilde{E}(t_1-s)$. We thus obtain

$$\begin{array}{cc}II{I}_1^2\hfill & =\Vert {\int }_0^{t_1}[\tilde{E}(t_2-s)-\tilde{E}(t_1-s)]d{W}^H(s){\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}^2\hfill \\ \hfill & \le CH(2H-1){\int }_0^{t_1}\Vert A^{{\displaystyle \frac{\kappa -\sigma }{2}}}g(s)\Vert ·\left[{\int }_s^{t_1}\Vert A^{{\displaystyle \frac{\kappa -\sigma }{2}}}g(r)\Vert {(r-s)}^{2H-2}dr\right]ds{\Vert A^{-{\displaystyle \frac{\kappa -\sigma }{2}}}\Vert }_{{\mathcal{L}}_2^0}^2.\hfill \end{array}$$

Note that for $\sigma <\kappa$,

$$\begin{array}{cc}\Vert A^{{\displaystyle \frac{\kappa -\sigma }{2}}}g(r)\Vert \hfill & \le {\displaystyle \frac{1}{2\pi i}}{\int }_{\mathsf{\Gamma }}|e^{z(t_1-r)}||e^{z(t_2-t_1)}-1|\Vert A^{{\displaystyle \frac{\kappa -\sigma }{2}}}{(z^{\alpha }+A)}^{-1}{\Vert |z|}^{-\gamma }d|z|\hfill \\ \hfill & \le C{(t_2-t_1)}^a{\int }_{\mathsf{\Gamma }}|e^{z(t_1-r)}{||z|}^{a-\theta }d|z|,\hfill \end{array}$$

(32)

where $\theta =\alpha (1-{\displaystyle \frac{\sigma -\kappa }{2}})+\gamma$.

Case 1. If $\theta +H>2$, i.e., $\theta >2-H>1$, by choosing $a=1$, we obtain the following:

$$\begin{array}{cc}II{I}_1^2\hfill & \le C{(t_2-t_1)}^2{\int }_0^{t_1}{(t_1-s)}^{\theta -2}{\int }_s^{t_1}{(t_1-r)}^{\theta -2}{(r-s)}^{2H-2}drds\hfill \\ \hfill & \le C{(t_2-t_1)}^2{\int }_0^{t_1}{(t_1-s)}^{\theta -2}{\int }_0^{t_1-s}{(t_1-s-r)}^{\theta -2}{r}^{2H-2}drds\hfill \\ \hfill & \le C{(t_2-t_1)}^2{\int }_0^{t_1}{(t_1-s)}^{2\theta +2H-5}ds\le C{(t_2-t_1)}^2.\hfill \end{array}$$

Case 2. If $1<\theta +H\le 2$, by choosing $a=\theta +H-1-\epsilon$ and $\epsilon >0$, we derive the following:

$$\begin{array}{cc}\hfill II{I}_1^2& \le C{(t_2-t_1)}^{2\theta +2H-2-2\epsilon }{\int }_0^{t_1}{(t_1-s)}^{-H+\epsilon }{\int }_s^{t_1}{(t_1-r)}^{-H+\epsilon }{(r-s)}^{2H-2}drds\hfill \\ \hfill & \le C{(t_2-t_1)}^{2\theta +2H-2-2\epsilon }{\int }_0^{t_1}{(t_1-s)}^{-H+\epsilon }{\int }_0^{t_1-s}{(t_1-s-r)}^{-H+\epsilon }{r}^{2H-2}drds\hfill \\ \hfill & \le C{(t_2-t_1)}^{2\theta +2H-2-2\epsilon }{\int }_0^{t_1}{(t_1-s)}^{2\epsilon -1}ds\le C{(t_2-t_1)}^{2\theta +2H-2-2\epsilon }.\hfill \end{array}$$

Therefore, for $\sigma <\kappa$,

$$II{I}_1^2\le C{(t_2-t_1)}^{min\{2\theta +2H-2-2\epsilon ,2\}},$$

(33)

and, similarly, for $\sigma \ge \kappa$,

$$II{I}_1^2\le C{(t_2-t_1)}^{min\{2(\alpha +\gamma )+2H-2-2\epsilon ,2\}}.$$

(34)

By combining the above estimates and using a continuous Gronwall inequality in Lemma 3, we complete the proof of Theorem 2. □

3. Space Discretization

In this section, we shall consider the space discretization of (1). Let $S_h\subset H_0^1$ be the piecewise linear finite element space defined upon the triangulation ${\mathcal{T}}_h$ of the domain $D\subset {\mathbb{R}}^d,d\le 3$ with a smooth boundary. Let $P_h:\mathbb{H}\to S_h$ denote the $L^2$ projection defined by

$$(P_{h}v,\chi )=(v,\chi ),\forall \chi \in S_h.$$

(35)

Let $A_h:S_h\to S_h$ denote the discrete analogue of the elliptic operator $A:\mathcal{D}(A)\subset H\to H$, which is defined by

$$(A_h\psi ,\chi )=(\nabla \psi ,\nabla \chi ),\forall \chi \in S_h.$$

(36)

The Ritz projection $R_h:{\dot{H}}^1\to S_h$ is defined by

$$(\nabla R_{h}v,\chi )=(\nabla v,\nabla \chi ),\forall \chi \in S_h.$$

(37)

It is well known that [6]

$$A_h{R}_{h}v=P_{h}Av,\forall v\in \mathcal{D}(A).$$

(38)

Therefore, following error estimates hold [6]:

$$\Vert v-P_{h}v\Vert +h\Vert \nabla (v-P_{h}v)\Vert \le C{h}^2{|v|}_2,\forall v\in H_0^1(D)\cap H^2(D),$$

(39)

$$\Vert v-R_{h}v\Vert +h\Vert \nabla (v-R_{h}v)\Vert \le C{h}^2{|v|}_2,\forall v\in H_0^1(D)\cap H^2(D),$$

(40)

These are used to derive the spatial error estimates of the solution operator approximations in Lemma 5 below, which is accomplished via interpolation.

The spatial semidiscrete problem of (1) is to find the solution $u_h(t)\in S_h$ such that

$$u_h(t)=E_h(t)P_h{u}_0+{\int }_0^t{\overline{E}}_h(t-s)P_{h}f(u_h(s))ds+{\int }_0^t{\tilde{E}}_h(t-s)P_{h}d{W}^H(s),$$

(41)

where

$$\begin{array}{cc}\hfill & E_h(t)={\displaystyle \frac{1}{2\pi i}}{\int }_{\mathsf{\Gamma }}{e}^{zt}{(z^{\alpha }+A_h)}^{-1}{z}^{\alpha -1}dz,\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill & {\overline{E}}_h(t)={\displaystyle \frac{1}{2\pi i}}{\int }_{\mathsf{\Gamma }}{e}^{zt}{(z^{\alpha }+A_h)}^{-1}dz,\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & {\tilde{E}}_h(t)={\displaystyle \frac{1}{2\pi i}}{\int }_{\mathsf{\Gamma }}{e}^{zt}{(z^{\alpha }+A_h)}^{-1}{z}^{-\gamma }dz.\hfill \end{array}$$

(44)

These operators have the similar smoothing properties to $E(t)$, $\overline{E}(t),$ and $\tilde{E}(t)$, which are defined in (15)–(17). Similar to the proofs of Theorems 1 and 2, one can obtain the same regularity of $u_h$ in space and time.

Lemma 4

([32]). Let $\alpha \in (0,1)$. Let $\omega (t)$ be absolutely continuous on $[0,T]$. Then,

$$\omega (t){}_0{}^{C}D_t^{\alpha }\omega (t)\ge {\displaystyle \frac{1}{2}}{\omega }^2(t).$$

We denote $F_h(t)=E(t)-E_h(t)P_h$, ${\overline{F}}_h(t)=\overline{E}(t)-{\overline{E}}_h(t)P_h$, ${\tilde{F}}_h(t)=\tilde{E}(t)-{\tilde{E}}_h(t)P_h$. Using the arguments in [8,25], we can prove Lemma 5.

Lemma 5.

Let $0\le \nu \le \mu \le 2$ and $v\in {\dot{\mathbb{H}}}^{\nu }$. It holds for $t>0$ that

$$\begin{array}{cc}\hfill \Vert F_h(t)v\Vert & \le C{h}^{\mu }{t}^{-\alpha {\displaystyle \frac{\mu -\nu }{2}}}{|v|}_{\nu },\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill \Vert {\overline{F}}_h(t)v\Vert & \le C{h}^{\mu }{t}^{-\alpha {\displaystyle \frac{\mu -\nu }{2}}+\alpha -1}{|v|}_{\nu },\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill \Vert {\tilde{F}}_h(t)v\Vert & \le C{h}^{\mu }{t}^{-\alpha {\displaystyle \frac{\mu -\nu }{2}}+\alpha +\gamma -1}{|v|}_{\nu }.\hfill \end{array}$$

(47)

By applying Lemma 5 and the arguments in [13], we can deduce Lemma 6.

Lemma 6.

Let $0\le \rho \le 1$, $0\le \rho \le \mu \le 2$; when $t>0$, we obtain

$$\begin{array}{cc}\hfill & \Vert A^{{\displaystyle \frac{\rho }{2}}}{\tilde{F}}_h(t)v\Vert \le C{h}^{\mu -\rho }{t}^{-\alpha {\displaystyle \frac{\mu -0}{2}}+(\alpha +\gamma -1)}\Vert v\Vert ,\forall v\in \mathbb{H},\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill & {\int }_0^t{\Vert A^{{\displaystyle \frac{\rho }{2}}}{\tilde{F}}_h(s)\Vert }^{2}ds\le C{h}^{2(\mu -\rho )},\mathrm{for}\alpha +\gamma -{\displaystyle \frac{1}{2}}-{\displaystyle \frac{\mu \alpha }{2}}>0,\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill & {\int }_0^t\Vert {\tilde{F}}_h{(s)v\Vert }^{2}ds\le C{h}^{2\mu }{|v|}_{\rho },\mathrm{for}\alpha +\gamma -{\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha (\mu -\rho )}{2}}>0,\forall v\in {\dot{\mathbb{H}}}^{\rho }.\hfill \end{array}$$

(50)

Proof. 

Case 1. $\rho =0$. On the one hand, by estimating $\Vert {(z^{\alpha }+A)}^{-1}-{(z^{\alpha }+A_h)}^{-1}{P}_h\Vert \le C{h}^2$ [8], we find that

$$\begin{array}{cc}\Vert {\tilde{F}}_h(t)\Vert \hfill & = \Vert {\displaystyle \frac{1}{2\pi i}}{\int }_{\mathsf{\Gamma }}{e}^{zt}{z}^{-\gamma }[{(z^{\alpha }+A)}^{-1}-{(z^{\alpha }+A_h)}^{-1}{P}_h]dz\Vert \hfill \\ \hfill & \le C{\int }_{\mathsf{\Gamma }}|e^{zt}{||z|}^{-\gamma }\Vert {(z^{\alpha }+A)}^{-1}-{(z^{\alpha }+A_h)}^{-1}{P}_h\Vert d|z|\hfill \\ \hfill & \le C{h}^2{\int }_{\mathsf{\Gamma }}|e^{zt}{||z|}^{-\gamma }d|z|\le C{h}^2{t}^{\gamma -1}.\hfill \end{array}$$

(51)

On the other hand, the application of the regularity estimate $\tilde{E}(t),{\tilde{E}}_h(t)$ yields

$$\Vert {\tilde{F}}_h(t)\Vert \le \Vert \tilde{E}(t)\Vert + \Vert {\tilde{E}}_h(t)P_h\Vert \le C{t}^{\alpha +\gamma -1}.$$

(52)

By interpolation, it holds that

$$\Vert {\tilde{F}}_h(t)\Vert \le C{h}^{\mu }{t}^{-\alpha {\displaystyle \frac{\mu -0}{2}}+(\alpha +\gamma -1)},0\le \mu \le 2.$$

(53)

Case 2. $\rho =1$. On the one hand, by repeating the proof of (51) and using the following estimate

$$\Vert A^{1/2}({(z^{\alpha }+A)}^{-1}-{(z^{\alpha }+A_h)}^{-1}{P}_h)\Vert \le Ch,$$

we obtain $\Vert A^{1/2}{\tilde{F}}_h(t)\Vert \le Ch{t}^{\gamma -1}$.

On the other hand, repeating the proof of (52) yields

$$\Vert A^{1/2}{\tilde{F}}_h(t)\Vert \le C{t}^{{\displaystyle \frac{\alpha }{2}}+\gamma -1}.$$

By interpolation, it holds that

$$\Vert A^{1/2}{\tilde{F}}_h(t)\Vert \le C{h}^{\mu -1}{t}^{-\alpha {\displaystyle \frac{\mu -0}{2}}+(\alpha +\gamma -1)},1\le \mu \le 2.$$

(54)

Thus, by repeated interpolation between $\rho =0$ and $\rho =1$, we arrive at

$$\Vert A^{{\displaystyle \frac{\rho }{2}}}{\tilde{F}}_h(t)\Vert \le C{h}^{\mu -\rho }{t}^{-\alpha {\displaystyle \frac{\mu -0}{2}}+(\alpha +\gamma -1)},0\le \rho \le \mu \le 2.$$

(55)

The proof of (48) is complete.

From (48) in Lemma 6 and (47) in Lemma 5, we can easily find the remaining inequalities (49) and (50). □

Theorem 3.

Let $u(t)$ and $u_h(t)$ be the solutions defined in (14) and (41), respectively. Let Assumptions 2–4 be valid. Let $u_0\in L^2(\mathsf{\Omega },{\dot{\mathbb{H}}}^{\nu })$ with $\nu \in [0,2]$. Let $0\le \mu \le 2$. Then, there is a positive constant C such that for $0<\kappa -\sigma \le min\{1,\mu \}$ and $0<\epsilon <1$,

$$\Vert u(t)-u_h{(t)\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C{h}^2{t}^{-\alpha {\displaystyle \frac{2-\nu }{2}}}{\Vert u_0\Vert }_{L^2(\mathsf{\Omega },{\dot{\mathbb{H}}}^{\nu })} +C{h}^{min\{2-\epsilon ,\mu -(\kappa -\sigma )\}},t\in [0,T],$$

where $\alpha +\gamma -{\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha \mu }{2}}>0$. Moreover, when $0\le \sigma -\kappa \le min\{1,\mu \}$, it holds that

$$\Vert u(t)-u_h{(t)\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C{h}^2{t}^{-\alpha {\displaystyle \frac{2-\nu }{2}}}{\Vert u_0\Vert }_{L^2(\mathsf{\Omega },{\dot{\mathbb{H}}}^{\nu })} +C{h}^{min\{2-\epsilon ,\mu \}},t\in [0,T],$$

where $\alpha +\gamma -{\displaystyle \frac{1}{2}}-\alpha (\mu -(\sigma -\kappa ))/2>0$ and $\alpha +\gamma -{\displaystyle \frac{\alpha \mu }{2}}+2H-{\displaystyle \frac{3}{2}}>0$.

Proof. 

By introducing the auxiliary process

$${\tilde{u}}_h(t)=E_h(t)P_h{u}_0+{\int }_0^t{\overline{E}}_h(t-s)P_{h}f(u(s))ds+{\int }_0^t{\tilde{E}}_h(t-s)P_{h}d{W}^H(s),$$

(56)

we find that

$$u(t)-u_h(t)=(u(t)-{\tilde{u}}_h(t))+({\tilde{u}}_h(t)-u_h(t)).$$

We first estimate $u(t)-{\tilde{u}}_h(t)$. One may write this as follows:

$$\begin{array}{cc}\hfill u(t)-{\tilde{u}}_h(t)& =F_h(t)u_0+{\int }_0^t{\overline{F}}_h(t-s)f(u(t))ds\hfill \\ \hfill & +{\int }_0^t{\overline{F}}_h(t-s)(f(u(s))-f(u(t)))ds+{\int }_0^t{\tilde{F}}_h(t-s)d{W}^H(s)\hfill \\ \hfill & =I_1+I_2+I_3+I_4.\hfill \end{array}$$

For $I_1$, we obtain, using (45) and $\mu =2$,

$$\Vert I_1{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} = \Vert F_h(t)u_0{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\le C{h}^2{t}^{-\alpha {\displaystyle \frac{2-\nu }{2}}}{\Vert u_0\Vert }_{L^2(\mathsf{\Omega },{\dot{H}}^{\nu })}.$$

For $I_2$, we obtain, by applying (46), $\mu =2-\epsilon ,$ and $\nu =0$,

$$\begin{array}{cc}\hfill \Vert I_2{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}& =\Vert {\int }_0^t{\overline{F}}_h(t-s)f(u(t))ds{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & \le C{h}^{2-\epsilon }{\int }_0^t{(t-s)}^{{\displaystyle \frac{\alpha \epsilon }{2}}-1}ds\underset{0\le s\le T}{sup}{\Vert f(u(s))\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\le C{h}^{2-\epsilon }.\hfill \end{array}$$

For $I_3$, we obtain, using the Assumption about f, (46), $\mu =2,$ and $\nu =0$,

$$\begin{array}{cc}\hfill \Vert I_3{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}& =\Vert {\int }_0^t{\overline{F}}_h(t-s)(f(u(s))-f(u(t)))ds{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & \le C{h}^2{\int }_0^t{(t-s)}^{-1}{(t-s)}^{\beta }ds\le C{h}^2.\hfill \end{array}$$

As for $I_4$, we find that

$$\begin{array}{cc}\hfill \Vert I_4{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}^2& = \Vert {\int }_0^t{\tilde{F}}_h(t-s)d{W}^H{(s)\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}^2\hfill \\ \hfill & =2H(2H-1){\int }_0^t{\int }_s^t<{\tilde{F}}_h(t-s),{\tilde{F}}_h(t-r)>{(r-s)}^{2H-2}drds\hfill \\ \hfill & \le C{\int }_0^t\Vert A^{{\displaystyle \frac{\kappa -\sigma }{2}}}{\tilde{F}}_h(t-s)\Vert ·\left({\int }_s^t\Vert A^{{\displaystyle \frac{\kappa -\sigma }{2}}}{\tilde{F}}_h(t-r)\Vert {(r-s)}^{2H-2}dr\right)ds{\Vert A^{{\displaystyle \frac{\sigma -\kappa }{2}}}\Vert }_{{\mathcal{L}}_2^0}^2\hfill \\ \hfill & \le C{\int }_0^t\Vert A^{{\displaystyle \frac{\kappa -\sigma }{2}}}{\tilde{F}}_h(t-s)\Vert ·\left({\int }_0^{t-s}\Vert A^{{\displaystyle \frac{\kappa -\sigma }{2}}}{\tilde{F}}_h(t-r-s)\Vert r^{2H-2}dr\right)ds.\hfill \end{array}$$

Through the use of the Cauchy–Schwarz inequality, we deduce that

$$\Vert I_4{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}^2 \le C{\left({\int }_0^t{\Vert A^{{\displaystyle \frac{\kappa -\sigma }{2}}}{\tilde{F}}_h(t-s)\Vert }^{2}ds\right)}^{{\displaystyle \frac{1}{2}}}{\left[{\int }_0^t{\left({\int }_0^{t-s}\Vert A^{{\displaystyle \frac{\kappa -\sigma }{2}}}{\tilde{F}}_h(t-s-r)\Vert r^{2H-2}dr\right)}^{2}ds\right]}^{{\displaystyle \frac{1}{2}}}.$$

Case 1. If $0<\kappa -\sigma \le min\{1,\mu \}$, then from (48) and (49) in Lemma 6 and $\rho =\kappa -\sigma$, it holds that

$$\begin{array}{cc}\hfill \Vert I_4{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}^2& \le C{h}^{2(\mu -(\kappa -\sigma ))}{\left[{\int }_0^t{\left({\int }_0^{t-s}{(t-r-s)}^{-\alpha \mu /2+\alpha +\gamma -1}{r}^{2H-2}dr\right)}^{2}ds\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le C{h}^{2(\mu -(\kappa -\sigma ))}{\left({\int }_0^t{(t-s)}^{-\alpha \mu +2(\alpha +\gamma +2H-2)}ds\right)}^{{\displaystyle \frac{1}{2}}}\le C{h}^{2(\mu -(\kappa -\sigma ))},\hfill \end{array}$$

where $\alpha +\gamma -{\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha \mu }{2}}>0$.

Case 2. If $0\le \sigma -\kappa \le min\{1,\mu \}$, then by applying (48), where $\rho =0$, and (50), where $\rho =\sigma -\kappa$, to Lemma 6, we arrive at

$$\begin{array}{cc}\hfill \Vert I_4{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}^2& \le C{h}^{2\mu }{\left[{\int }_0^t{\left({\int }_0^{t-s}{(t-r-s)}^{-\alpha {\displaystyle \frac{\mu }{2}}+\alpha +\gamma -1}{r}^{2H-2}dr\right)}^{2}ds\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le C{h}^{2\mu }{\left({\int }_0^t{(t-s)}^{-\alpha \mu +2(\alpha +\gamma +2H-2)}ds\right)}^{{\displaystyle \frac{1}{2}}}\le C{h}^{2\mu },\hfill \end{array}$$

where $\alpha +\gamma -{\displaystyle \frac{1}{2}}-\alpha (\mu -(\sigma -\kappa ))/2>0$ and $\alpha +\gamma -{\displaystyle \frac{\alpha \mu }{2}}+2H-{\displaystyle \frac{3}{2}}>0$.

We next estimate $\tilde{e}(t)={\tilde{u}}_h(t)-u_h(t)$, which satisfies

$${}_0{}^{C}D_t^{\alpha }\tilde{e}(t)+A_h\tilde{e}(t)=P_h[f(u(t))-f(u_h(t))],\mathrm{for}0<t\le T,\mathrm{with}\tilde{e}(0)=0.$$

(57)

By multiplying both sides by $\tilde{e}(t)$, we arrive at, based on the Assumption about f, the Cauchy–Schwarz inequality

$$\begin{array}{cc}\hfill ({}_0{}^{C}D_t^{\alpha }\tilde{e}(s),\tilde{e}(s))+(\nabla \tilde{e}(s),\nabla \tilde{e}(s))& =(f({\tilde{u}}_h(s))-f(u_h(s)),\tilde{e}(s))+(f(u(s))-f({\tilde{u}}_h(s)),\tilde{e}(s))\hfill \\ \hfill & \le C\Vert \tilde{e}{(s)\Vert }^2+C\Vert A^{-1/2}(f(u(s))-f({\tilde{u}}_h(s))\Vert ·\Vert \nabla \tilde{e}(s)\Vert \hfill \\ \hfill & \le C\Vert \tilde{e}{(s)\Vert }^2+C\Vert u(s)-{\tilde{u}}_h{(s)\Vert }^2+1/2{\Vert \nabla \tilde{e}(s)\Vert }^2.\hfill \end{array}$$

The application of Lemma 4 yields

$${\displaystyle \frac{1}{2}}{}_0{}^{C}D_t^{\alpha }\Vert \tilde{e}{(s)\Vert }^2+{\displaystyle \frac{1}{2}}\Vert \nabla \tilde{e}{(s)\Vert }^2\le C\Vert \tilde{e}{(s)\Vert }^2+C{\Vert u(s)-{\tilde{u}}_h(s)\Vert }^2.$$

Using Lemma 3, we find that

$$\Vert \tilde{e}{(t)\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\le C{\Vert u(t)-{\tilde{u}}_h(t)\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})},$$

which completes the proof of Theorem 3. □

4. Time Discretization

In this section, we shall consider the time discretization scheme (8). Since the homogeneous problem of Model (1) has been studied in [33], here we only need to discuss the numerical schemes and error estimates for the inhomogeneous problem of Model (1), i.e., $u_0=0$.

Taking the discrete Laplace transform of both sides of (8), we obtain

$${\tau }^{-\alpha }\sum _{n=1}^{\infty }\left(\sum _{i=0}^n{w}_{n-i}^{(\alpha )}{u}_h^i\right){\xi }^n+\sum _{n=1}^{\infty }(A_h{u}_h^n){\xi }^n=\sum _{n=1}^{\infty }{P}_h{F}^n{\xi }^n+{\tau }^{\gamma }\sum _{n=1}^{\infty }\left(\sum _{i=0}^n{w}_{n-i}^{(-\gamma )}{P}_h{g}^i\right){\xi }^n.$$

(58)

We can write out the discrete Laplace transform of the sequences

$${\left\{{\omega }_n^{(\alpha )}\right\}}_{n=0}^{\infty },{\left\{{\omega }_n^{(-\gamma )}\right\}}_{n=0}^{\infty },{\left\{u_h^n\right\}}_{n=0}^{\infty },{\left\{F^n\right\}}_{n=0}^{\infty },{\left\{g^n\right\}}_{n=0}^{\infty },$$

as

$$\begin{array}{cc}\hfill & {\tilde{w}}^{(\alpha )}(\xi )=\sum _{n=0}^{\infty }{w}_n^{(\alpha )}{\xi }^n,{\tilde{w}}^{(-\gamma )}(\xi )=\sum _{n=0}^{\infty }{w}_n^{(-\gamma )}{\xi }^n,\tilde{u}(\xi )=\sum _{n=0}^{\infty }{u}_h^n{\xi }^n,\hfill \\ \hfill & \tilde{F}(\xi )=\sum _{n=0}^{\infty }{F}^n{\xi }^n,\tilde{g}(\xi )=\sum _{n=0}^{\infty }{g}^n{\xi }^n,\hfill \end{array}$$

respectively. We then arrive at

$$({\tau }^{-\alpha }{\tilde{w}}^{(\alpha )}(\xi )+A_h)\tilde{u}(\xi )=P_h\tilde{F}(\xi )+{\tau }^{\gamma }{\tilde{w}}^{(-\gamma )}(\xi )P_h\tilde{g}(\xi ),$$

i.e.,

$$\tilde{u}(\xi )={({\tau }^{-\alpha }{\tilde{w}}^{(\alpha )}(\xi )+A_h)}^{-1}{P}_h\tilde{F}(\xi )+{({\tau }^{-\alpha }{\tilde{w}}^{(\alpha )}(\xi )+A_h)}^{-1}{\tau }^{\gamma }{\tilde{w}}^{(-\gamma )}(\xi )P_h\tilde{g}(\xi ).$$

By using the inverse discrete Laplace transform, we find that

$$\begin{array}{cc}\hfill u_h^n=& {\displaystyle \frac{\tau }{2\pi i}}{\int }_{{\mathsf{\Gamma }}_{\tau }}{e}^{z{t}_n}[{({\tau }^{-\alpha }{\tilde{w}}^{(\alpha )}(e^{-z\tau })+A_h)}^{-1}\tilde{F}(e^{-z\tau })\hfill \\ \hfill & +{({\tau }^{-\alpha }{\tilde{w}}^{(\alpha )}(e^{-z\tau })+A_h)}^{-1}{\tau }^{\gamma }{\tilde{w}}^{(-\gamma )}(e^{-z\tau })P_h\tilde{g}(e^{-z\tau })]dz,\hfill \end{array}$$

where ${\mathsf{\Gamma }}_{\tau }=\{z:z\in \mathsf{\Gamma },|\Im z|\le {\displaystyle \frac{\pi }{\tau }}\}$.

By denoting ${\tau }^{-\alpha }{\tilde{w}}^{(\alpha )}(e^{-z\tau })=z_1^{\alpha },{\tau }^{\gamma }{\tilde{w}}^{(-\gamma )}(e^{-z\tau })=z_2^{-\gamma }$, which are suitable approximations of $z^{\alpha },z^{-\gamma }$ for $z\in {\mathsf{\Gamma }}_{\tau }$, respectively, we then obtain

$$u_h^n={\displaystyle \frac{\tau }{2\pi i}}{\int }_{{\mathsf{\Gamma }}_{\tau }}{e}^{z{t}_n}[{(z_1^{\alpha }+A_h)}^{-1}{P}_h\tilde{F}(e^{-z\tau })+{(z_1^{\alpha }+A_h)}^{-1}{z}_2^{-\gamma }{P}_h\tilde{g}(e^{-z\tau })]dz.$$

(59)

We will show that $u_h^n$ can be expressed as a convolution of the piecewise constant functions $\overline{F}(t)$ and $\overline{g}(t)$. To obtain this, we first introduce the following piecewise constant functions, $\overline{F}(t),\overline{g}(t),t>0$, as defined by ( $\overline{F}(0)=\overline{g}(0)=0$):

$$\overline{F}(t)=\left\{\begin{array}{cc}\hfill & F^j=f(u(t_{j-1})),t\in (t_{j-1},t_j],j=1,2,\dots ,N,\hfill \\ \hfill & 0,t>T=t_N,\hfill \end{array}\right.$$

(60)

and

$$\overline{g}(t)=\left\{\begin{array}{cc}\hfill & g^j,t\in (t_{j-1},t_j],j=1,2,\dots ,N,\hfill \\ \hfill & 0,t>T=t_N.\hfill \end{array}\right.$$

(61)

Similar to the proof of Lemma 2.1 in Wu et al. [16], $u_h^n$ in (59) takes the following form:

$$u_h^n={\int }_0^{t_n}{\overline{E}}_{\tau ,h}(t_n-s)P_h\overline{F}(s)ds+{\int }_0^{t_n}{\tilde{E}}_{\tau ,h}(t_n-s)P_h\overline{g}(s)ds.$$

Here,

$${\overline{E}}_{\tau ,h}(t)={\displaystyle \frac{1}{2\pi i}}{\int }_{{\mathsf{\Gamma }}_{\tau }}{e}^{zt}{(z_1^{\alpha }+A_h)}^{-1}{\displaystyle \frac{z\tau }{e^{z\tau }-1}}dz,$$

(62)

and

$${\tilde{E}}_{\tau ,h}(t)={\displaystyle \frac{1}{2\pi i}}{\int }_{{\mathsf{\Gamma }}_{\tau }}{e}^{zt}{(z_1^{\alpha }+A_h)}^{-1}{z}_2^{-\gamma }{\displaystyle \frac{z\tau }{e^{z\tau }-1}}dz.$$

(63)

Let

$${\overline{\partial }}_{\tau }{W}^H(t)=\left\{\begin{array}{cc}\hfill & 0,t=0,\hfill \\ \hfill & g^j,t\in (t_{j-1},t_j],j=1,2,\dots ,N,\hfill \\ \hfill & 0,t>T=t_N.\hfill \end{array}\right.$$

(64)

Then, $u_h^n$ can also be written as

$$u_h^n={\int }_0^{t_n}{\overline{E}}_{\tau ,h}(t_n-s)P_h\overline{F}(s)ds+{\int }_0^{t_n}{\tilde{E}}_{\tau ,h}(t_n-s)P_h{\overline{\partial }}_{\tau }{W}^H(s)ds.$$

(65)

Next, we introduce some lemmas which will be used in the error estimate of the time discretization.

Lemma 7

([13,15]). Let $z_1$ and $z_2$ be as defined above; then, we find obtain

$$\begin{array}{cc}\hfill z_1\sim z,& \forall z\in {\mathsf{\Gamma }}_{\tau },\hfill \\ \hfill |z-z_1|\le C{\tau }^{2-\alpha }{|z|}^{3-\alpha },& \forall z\in {\mathsf{\Gamma }}_{\tau },\hfill \\ \hfill z_2\sim z,& \forall z\in {\mathsf{\Gamma }}_{\tau },\hfill \\ \hfill |z-z_2{|\le C\tau |z|}^2,& \forall z\in {\mathsf{\Gamma }}_{\tau }.\hfill \end{array}$$

Here, $z_1\sim z$ (similarly to $z_2\sim z$) means that there exist constants $c_1>0$ and $c_2>0$ such that

$$c_1|z_1|\le |z|\le c_2|z_1|,\forall z\in {\mathsf{\Gamma }}_{\tau }.$$

Lemma 8.

Denote $\widehat{{\overline{E}}_{\tau }}(z)={(z_1^{\alpha }+A)}^{-1}{\displaystyle \frac{z\tau }{e^{z\tau }-1}}$, $\widehat{{\tilde{E}}_{\tau }}(z)={(z_1^{\alpha }+A)}^{-1}{z}_2^{-\gamma }{\displaystyle \frac{z\tau }{e^{z\tau }-1}}$. Then, for $0\le p\le 2$, we know that

$$\begin{array}{cc}\hfill & \Vert \widehat{\overline{E}}(z)-\widehat{{\overline{E}}_{\tau }}{(z)\Vert \le C\tau |z|}^{1-\alpha },\hfill \\ \hfill & |\widehat{\tilde{E}}(z)-\widehat{{\tilde{E}}_{\tau }}{(z)|}_p\le C\tau {|z|}^{1-\gamma -\alpha (1-{\displaystyle \frac{p}{2}})}.\hfill \end{array}$$

Proof. 

We only need to prove the second inequality as the first inequality is a special case of the second inequality. First, we know that

$$\begin{array}{cc}\hfill & |\widehat{\tilde{E}}(z)-\widehat{{\tilde{E}}_{\tau }}{(z)|}_p\hfill \\ \hfill & = \Vert A^{{\displaystyle \frac{p}{2}}}[{(z^{\alpha }+A)}^{-1}{z}^{-\gamma }-{(z_1^{\alpha }+A)}^{-1}{z}_2^{-\gamma }{\displaystyle \frac{z\tau }{e^{z\tau }-1}}]\Vert \hfill \\ \hfill & \le \Vert A^{{\displaystyle \frac{p}{2}}}[{(z^{\alpha }+A)}^{-1}-{(z_1^{\alpha }+A)}^{-1}]z^{-\gamma }{\displaystyle \frac{z\tau }{e^{z\tau }-1}}\Vert + \Vert A^{{\displaystyle \frac{p}{2}}}{(z_1^{\alpha }+A)}^{-1}(z^{-\gamma }-z_2^{-\gamma }){\displaystyle \frac{z\tau }{e^{z\tau }-1}}\Vert \hfill \\ \hfill & + \Vert A^{{\displaystyle \frac{p}{2}}}{(z_1^{\alpha }+A)}^{-1}{z}_2^{-\gamma }(1-{\displaystyle \frac{z\tau }{e^{z\tau }-1}})\Vert \hfill \\ \hfill & =I_1+I_2+I_3.\hfill \end{array}$$

Next, we estimate $I_1,I_2,I_3$ one by one. Note that by using (11), we can easily obtain

$$\Vert A^r{(z^{\alpha }+A)}^{-1}{\Vert \le C|z|}^{\alpha (r-1)},\forall 0\le r\le 1.$$

(66)

For $I_1$, by using (11), (66), the mean value theorem, and Lemma 7, it holds that

$$\begin{array}{cc}\hfill I_1& \le C\Vert A^{{\displaystyle \frac{p}{2}}}{(z^{\alpha }+A)}^{-1}{(z_1^{\alpha }+A)}^{-1}(z^{\alpha }-z_1^{\alpha }){\Vert |z|}^{-\gamma }\hfill \\ \hfill & \le {C|z|}^{\alpha ({\displaystyle \frac{p}{2}}-1)}|z_1{|}^{-\alpha }{|z|}^{\alpha -1}|z-z_1{||z|}^{-\gamma }\le C\tau {|z|}^{1-\gamma -\alpha (1-{\displaystyle \frac{p}{2}})}.\hfill \end{array}$$

(67)

Due to the fact that $|1-{\displaystyle \frac{z\tau }{e^{z\tau }-1}}|\le C\tau |z|$, similar to the estimate of $I_1$, we derive

$$I_2 \le C|z_1{|}^{\alpha ({\displaystyle \frac{p}{2}}-1)}{|z|}^{-\gamma -1}|z-z_2{| \le C\tau |z|}^{1-\gamma -\alpha (1-{\displaystyle \frac{p}{2}})},$$

(68)

and

$$I_3 \le C\tau |z_1{|}^{\alpha ({\displaystyle \frac{p}{2}}-1)}|z_2{|}^{-\gamma }|z|={C\tau |z|}^{1-\gamma -\alpha (1-{\displaystyle \frac{p}{2}})}.$$

(69)

From (67)–(69), we obtain $|\widehat{\tilde{E}}(z)-\widehat{{\tilde{E}}_{\tau }}{(z)|}_p\le C\tau {|z|}^{1-\gamma -\alpha (1-{\displaystyle \frac{p}{2}})},0\le p\le 2.$ □

Theorem 4.

Assume that Assumptions 2–4 are satisfied and set $u_0=0$. Let $u(t)$ and $u_h^n$ be the solution to (1) and (8), respectively. Let $0\le \mu \le 2$ be used for Theorem 3. Let $\theta =(1-{\displaystyle \frac{\kappa -\sigma }{2}})\alpha +\gamma$, and then we can obtain the following estimates for $\epsilon >0$, ${\displaystyle \frac{1}{2}}<H<1,0<\alpha <1$, $0\le \gamma \le 1$, and ${\displaystyle \frac{1}{2}}<\theta <2$:

Case 1. If $0<\kappa -\sigma \le min\{1,\mu \}$, then

$$\Vert u(t_n)-u_h^n{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C{h}^{min\{2-\epsilon ,\mu -(\kappa -\sigma )\}}+C{\tau }^{min\{H+\theta -1-\epsilon ,\alpha \}},$$

where $\alpha +\gamma -{\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha \mu }{2}}>0$.

Case 2. If $0\le \sigma -\kappa \le min\{1,\mu \}$, then

$$\Vert u(t_n)-u_h^n{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le C{h}^{min\{2-\epsilon ,\mu \}}+C{\tau }^{min\{H+\alpha +\gamma -1-\epsilon ,,\alpha \}},$$

where $\alpha +\gamma -{\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha (\mu -(\sigma -\kappa ))}{2}}>0,\alpha +\gamma -{\displaystyle \frac{\alpha \mu }{2}}+2H-{\displaystyle \frac{3}{2}}>0$.

Proof. 

Note that

$$\Vert u(t_n)-u_h^n{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})} \le \Vert u(t_n)-u_h(t_n){\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}+{\Vert u_h(t_n)-u_h^n\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}.$$

The error estimate $\Vert u(t_n)-u_h(t_n){\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}$ has been proved in Theorem 3, so we only need to prove $\Vert u_h(t_n)-u_h^n{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}$.

By subtracting (65) from (41) when $u_0=0$, we find that

$$\begin{array}{cc}\hfill & \Vert u_h(t_n)-u_h^n{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\le {∥{\int }_0^{t_n}[{\overline{E}}_h(t_n-s)P_{h}f(u_h(s))-{\overline{E}}_{\tau ,h}(t_n-s)P_h\overline{F}(s)]ds∥}_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & +{∥{\int }_0^{t_n}{\tilde{E}}_h(t_n-s)P_{h}d{W}^H(s)-{\int }_0^{t_n}{\tilde{E}}_{\tau ,h}(t_n-s)P_h{\overline{\partial }}_{\tau }{W}^H(s)ds∥}_{L^2(\mathsf{\Omega },\mathbb{H})}=G+J.\hfill \end{array}$$

As for G, we obtain

$$\begin{array}{cc}\hfill G& \le \Vert {\int }_0^{t_n}\left[{\displaystyle \frac{1}{2\pi i}}{\int }_{\mathsf{\Gamma }}{e}^{z(t_n-s)}{(z^{\alpha }+A_h)}^{-1}dz-{\displaystyle \frac{1}{2\pi i}}{\int }_{{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-s)}{(z_1^{\alpha }+A_h)}^{-1}{\displaystyle \frac{z\tau }{e^{z\tau }-1}}dz\right]\hfill \\ \hfill & ·P_{h}f(u_h(s))ds{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}+{∥\sum _{i=1}^n{\int }_{t_{i-1}}^{t_i}{\overline{E}}_{\tau ,h}(t_n-s)P_h(f(u_h(s))-f(u_h(t_{i-1})))ds∥}_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & \le C{∥{\int }_0^{t_n}{\int }_{\mathsf{\Gamma }/{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-s)}{(z^{\alpha }+A_h)}^{-1}{P}_{h}f(u_h(s))dzds∥}_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & +C{∥{\int }_0^{t_n}{\int }_{{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-s)}[{(z^{\alpha }+A_h)}^{-1}-{(z_1^{\alpha }+A_h)}^{-1}{\displaystyle \frac{z\tau }{e^{z\tau }-1}}]P_{h}f(u_h(s))dzds∥}_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & +C{∥\sum _{i=1}^n{\int }_{t_{i-1}}^{t_i}{\overline{E}}_{\tau ,h}(t_n-s)P_h(f(u_h(s))-f(u_h(t_{i-1})))ds∥}_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & =G_1+G_2+G_3.\hfill \end{array}$$

By simple calculation, we can derive that

$$\begin{array}{ccc}\hfill G_1& \le C{\int }_0^{t_n}{\int }_{{\displaystyle \frac{1}{\tau }}}^{\infty }|e^{zs}{||z|}^{-\alpha }d|z|ds\underset{0\le s\le T}{sup}{\Vert f(u_h(s))\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill & \hfill \le C{\int }_{{\displaystyle \frac{1}{\tau }}}^{\infty }{|z|}^{-\alpha -1}d|z|\le C{\tau }^{\alpha }.\end{array}$$

(70)

Similarly, by using Lemma 8, we obtain

$$\begin{array}{cc}\hfill G_2& \le C{\int }_0^{t_n}{\int }_0^{{\displaystyle \frac{1}{\tau }}}|e^{zs}|\Vert [{(z^{\alpha }+A_h)}^{-1}-{(z_1^{\alpha }+A_h)}^{-1}{\displaystyle \frac{z\tau }{e^{z\tau }-1}}]\Vert d|z|ds\underset{0\le s\le T}{sup}{\Vert f(u_h(s))\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & \le C\tau {\int }_0^{t_n}{\int }_0^{{\displaystyle \frac{1}{\tau }}}|e^{zs}{||z|}^{1-\alpha }d|z|ds\le C\tau {\int }_0^{{\displaystyle \frac{1}{\tau }}}{|z|}^{-\alpha }d|z|\le C{\tau }^{\alpha }.\hfill \end{array}$$

(71)

By applying Theorem 2, Lemma 7, and Assumption 2 we obtain

$$\begin{array}{cc}\hfill G_3& \le C\sum _{i=1}^n{\int }_{t_{i-1}}^{t_i}{\int }_{{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-s)}\Vert {(z_1^{\alpha }+A_h)}^{-1}\Vert |{\displaystyle \frac{z\tau }{e^{z\tau }-1}}|d|z|\Vert u_h(s)-u_h(t_{i-1}){\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}ds\hfill \\ \hfill & \le C{\tau }^{\beta }{\int }_0^{t_n}{\int }_{{\mathsf{\Gamma }}_{\tau }}|e^{zs}{||z|}^{-\alpha }d|z|ds\le C{\tau }^{\beta }{\int }_0^{t_n}{s}^{\alpha -1}ds\le C{\tau }^{\beta },\hfill \end{array}$$

(72)

where $\beta$ is the index of time regularity.

As for J, we split this into three parts.

$$\begin{array}{cc}\hfill J& \le \Vert {\int }_0^{t_n}[{\tilde{E}}_h(t_n-s)-{\tilde{E}}_{\tau ,h}(t_n-s)]P_{h}d{W}^H(s){\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & +\Vert {\int }_0^{t_n}{\tilde{E}}_{\tau ,h}(t_n-s)P_h(d{W}^H(s)-{\overline{\partial }}_{\tau }{W}^H(s)ds){\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & \le C\Vert {\int }_0^{t_n}{\int }_{\mathsf{\Gamma }/{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-s)}\widehat{{\tilde{E}}_h}(z)P_{h}dzd{W}^H(s){\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & +C\Vert {\int }_0^{t_n}{\int }_{{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-s)}(\widehat{{\tilde{E}}_h}(z)-\widehat{{\tilde{E}}_{\tau ,h}}(z))P_{h}dzd{W}^H(s){\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & +\Vert {\int }_0^{t_n}{\tilde{E}}_{\tau ,h}(t_n-s)P_h(d{W}^H(s)-{\overline{\partial }}_{\tau }{W}^H(s)ds){\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\hfill \\ \hfill & =J_1+J_2+J_3.\hfill \end{array}$$

(73)

As for $J_1$, by applying Lemma 2, Assumption 4, and $1/2<H<1$, we find that

$$\begin{array}{cc}\hfill J_1^2& =C\mathbb{E}{∥{\int }_0^{t_n}{\int }_{\mathsf{\Gamma }/{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-s)}\widehat{{\tilde{E}}_h}(z)dz{P}_{h}d{W}^H(s)∥}^2\hfill \\ \hfill & =CH(2H-1){\int }_0^{t_n}{\int }_0^{t_n}\langle {\int }_{\mathsf{\Gamma }/{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-s)}\widehat{{\tilde{E}}_h}(z)dz,{\int }_{\mathsf{\Gamma }/{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-r)}\widehat{{\tilde{E}}_h}(z)dz\rangle {|r-s|}^{2H-2}drds\hfill \\ \hfill & =2CH(2H-1){\int }_0^{t_n}{\int }_s^{t_n}\langle {\int }_{\mathsf{\Gamma }/{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-s)}\widehat{{\tilde{E}}_h}(z)dz,{\int }_{\mathsf{\Gamma }/{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-r)}\widehat{{\tilde{E}}_h}(z)dz\rangle {|r-s|}^{2H-2}drds\hfill \\ \hfill & \le C{\int }_0^{t_n}{\int }_s^{t_n}∥{\int }_{\mathsf{\Gamma }/{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-s)}{A}^{{\displaystyle \frac{\kappa -\sigma }{2}}}\widehat{{\tilde{E}}_h}(z)dz∥·∥{\int }_{\mathsf{\Gamma }/{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-r)}{A}^{{\displaystyle \frac{\kappa -\sigma }{2}}}\widehat{{\tilde{E}}_h}(z)dz∥\hfill \\ \hfill & ·{(r-s)}^{2H-2}drds·{\Vert A^{{\displaystyle \frac{\sigma -\kappa }{2}}}\Vert }_{{\mathcal{L}}_2^0}^2.\hfill \end{array}$$

By using (11) and letting $\theta =\alpha (1-{\displaystyle \frac{\kappa -\sigma }{2}})+\gamma$, we can obtain the following estimates for $0<\kappa -\sigma \le 1$:

Case 1. $1<\theta +H\le 2$.

$$\begin{array}{cc}\hfill J_1^2& \le C{\int }_0^{t_n}{\int }_{\mathsf{\Gamma }/{\mathsf{\Gamma }}_{\tau }}|e^{z(t_n-s)}{||z|}^{-\theta }d|z|\left({\int }_s^{t_n}{\int }_{\mathsf{\Gamma }/{\mathsf{\Gamma }}_{\tau }}|e^{z(t_n-r)}{||z|}^{-\theta }d|z|{(r-s)}^{2H-2}dr\right)ds\hfill \\ \hfill & =C{\int }_0^{t_n}{\int }_{\mathsf{\Gamma }/{\mathsf{\Gamma }}_{\tau }}|e^{z(t_n-s)}{||z|}^{-\theta }d|z|\left({\int }_{\mathsf{\Gamma }/{\mathsf{\Gamma }}_{\tau }}{\int }_0^{t_n-s}|e^{z(t_n-s-r)}{||z|}^{-\theta }{r}^{2H-2}drd|z|\right)ds\hfill \\ \hfill & \le C{\int }_0^{t_n}\left({\int }_{{\displaystyle \frac{1}{\tau }}}^{\infty }|e^{z(t_n-s)}{||z|}^{-\theta }d|z|·{\int }_{{\displaystyle \frac{1}{\tau }}}^{\infty }|e^{z(t_n-s)}{||z|}^{-\theta }{|z|}^{1-2H}d|z|\right)ds\hfill \\ \hfill & \le C{\tau }^{2H-1}{\int }_0^{t_n}{\left({\int }_{{\displaystyle \frac{1}{\tau }}}^{\infty }|e^{zs}{||z|}^{-\theta }d|z|\right)}^{2}ds\hfill \\ \hfill & =C{\tau }^{2H-1}{\int }_0^{t_n}{\left({\int }_{{\displaystyle \frac{1}{\tau }}}^{\infty }|e^{zs}{||z|}^{-\theta }{|z|}^{-b_1}{|z|}^{b_1}d|z|\right)}^{2}ds\hfill \\ \hfill & \le C{\tau }^{2H-1+2b_1}{\int }_0^{t_n}{\left({\int }_{{\displaystyle \frac{1}{\tau }}}^{\infty }|e^{zs}{||z|}^{-\theta +b_1}d|z|\right)}^{2}ds\hfill \\ \hfill & \le C{\tau }^{2H-1+2b_1}{\int }_0^{t_n}{s}^{2(\theta -b_1-1)}ds.\hfill \end{array}$$

We choose $0<b_1=\theta -{\displaystyle \frac{1}{2}}-\epsilon <1-\epsilon (0<\epsilon <{\displaystyle \frac{1}{2}})$, which ensures that

$${\int }_0^{\infty }|e^{zs}{||z|}^{-\theta +b_1}d|z|<\infty ,{\int }_0^{t_n}{s}^{2(\theta -b_1-1)}ds<\infty .$$

Thus, we obtain

$$J_1\le C{\tau }^{\theta +H-1-\epsilon }.$$

(74)

Case 2. $\theta +H>2$.

We note that $1<\theta <2$; thus,

$$\begin{array}{cc}\hfill J_1^2& \le C{\tau }^2{\int }_0^{t_n}{\int }_{{\displaystyle \frac{1}{\tau }}}^{\infty }|e^{z(t_n-s)}{||z|}^{-a}d|z|·{\int }_s^{t_n}{\int }_{{\displaystyle \frac{1}{\tau }}}^{\infty }|e^{z(t_n-r)}{||z|}^{-a}d|z|{(r-s)}^{2H-2}drds\hfill \\ \hfill & \le C{\tau }^2{\int }_0^{t_n}{(t_n-s)}^{a-1}{\int }_s^{t_n}{(t_n-r)}^{a-1}{(r-s)}^{2H-2}drds\hfill \\ \hfill & \le C{\tau }^2{\int }_0^{t_n}{(t_n-s)}^{2H+2a-3}ds\le C{\tau }^2,\hfill \end{array}$$

where $\theta =a+1,0<a<1$.

Thus, with $0<\kappa -\sigma \le 1$, we obtain

$$J_1\le C{\tau }^{min\{\theta +H-1-\epsilon ,1\}}.$$

(75)

Similarly, we also can prove this inequality with $0\le \sigma -\kappa \le 1$:

$$J_1\le C{\tau }^{min\{\alpha +\gamma +H-1-\epsilon ,1\}}.$$

(76)

Next, we estimate $J_2$. Let $g_{\tau ,h}(z)=\widehat{{\tilde{E}}_h}(z)-\widehat{{\tilde{E}}_{\tau ,h}}(z)$. Following Lemma 2, Assumption 4, and Lemma 8, we find that, for $0<\kappa -\sigma \le 1$,

$$\begin{array}{cc}\hfill J_2^2& =C\mathbb{E}\Vert {\int }_0^{t_n}{\int }_{{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-s)}{g}_{\tau ,h}(z)dzd{W}^H(s){\Vert }^2\hfill \\ \hfill & =2CH(2H-1){\int }_0^{t_n}{\int }_s^{t_n}\langle {\int }_{{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-s)}{g}_{\tau ,h}(z)dz,{\int }_{{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-r)}{g}_{\tau ,h}(z)dz\rangle {|r-s|}^{2H-2}drds\hfill \\ \hfill & \le C{\int }_0^{t_n}\left[\Vert {\int }_{{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-s)}{g}_{\tau ,h}(z)dz{\Vert }_{{\mathcal{L}}_2^0}·{\int }_s^{t_n}\Vert {\int }_{{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-r)}{g}_{\tau ,h}(z)dz{\Vert }_{{\mathcal{L}}_2^0}{(r-s)}^{2H-2}dr\right]ds\hfill \\ \hfill & \le C{\int }_0^{t_n}\left[\Vert {\int }_{{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-s)}{A}^{{\displaystyle \frac{\kappa -\sigma }{2}}}{g}_{\tau ,h}(z)dz\Vert ·{\int }_s^{t_n}\Vert {\int }_{{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-r)}{g}_{\tau ,h}(z)dz\Vert {(r-s)}^{2H-2}dr\right]ds\hfill \\ \hfill & \Vert A^{{\displaystyle \frac{\sigma -\kappa }{2}}}{\Vert }_{{\mathcal{L}}_2^0}^2\hfill \\ \hfill & \le C{\tau }^2{\int }_0^{t_n}\left[{\int }_{{\mathsf{\Gamma }}_{\tau }}|e^{z(t_n-s)}{||z|}^{1-\theta }d|z|·{\int }_s^{t_n}{\int }_{{\mathsf{\Gamma }}_{\tau }}|e^{z(t_n-r)}{||z|}^{1-\theta }d|z|{(r-s)}^{2H-2}dr\right]ds.\hfill \end{array}$$

(77)

Case 1. $\theta +H>2$.

It holds that since $1<\theta <2$,

$$\begin{array}{cc}\hfill J_2^2& \le C{\tau }^2{\int }_0^{t_n}\left[{\int }_0^{{\displaystyle \frac{1}{\tau }}}|e^{z(t_n-s)}{||z|}^{1-\theta }d|z|·{\int }_s^{t_n}{\int }_0^{{\displaystyle \frac{1}{\tau }}}|e^{z(t_n-r)}{||z|}^{1-\theta }d|z|{(r-s)}^{2H-2}dr\right]ds\hfill \\ \hfill & \le C{\tau }^2{\int }_0^{t_n}{(t_n-s)}^{\theta -2}{\int }_0^{t_n-s}{(t_n-r)}^{\theta -2}{(r-s)}^{2H-2}drds\hfill \\ \hfill & \le C{\tau }^2{\int }_0^{t_n}{(t_n-s)}^{2\theta +2H-5}ds\le C{\tau }^2.\hfill \end{array}$$

(78)

Case 2. $\theta +H\le 2$.

(i) If $\theta \ge 1$, we obtain

$$\begin{array}{cc}\hfill J_2^2& \le C{\tau }^2{\int }_0^{t_n}\left[{\int }_0^{{\displaystyle \frac{1}{\tau }}}|e^{z(t_n-s)}{||z|}^{1-\theta }d|z|\right]·{\int }_s^{t_n}{(t_n-r)}^{\theta -2}{(r-s)}^{2H-2}drds\hfill \\ \hfill & \le C{\tau }^2{\int }_0^{{\displaystyle \frac{1}{\tau }}}{\int }_0^{t_n}|e^{z(t_n-s)}|{(t_n-s)}^{\theta +2H-3}{ds|z|}^{1-\theta }d|z|\hfill \\ \hfill & \le C{\tau }^2{\int }_0^{{\displaystyle \frac{1}{\tau }}}{|z|}^{3-2\theta -2H}d|z|\le C{\tau }^{2\theta +2H-2}.\hfill \end{array}$$

(79)

(ii) If $\theta <1$, we obtain

$$\begin{array}{cc}\hfill J_2^2& \le C{\tau }^{2\theta }{\int }_0^{t_n}\left[{\int }_{{\mathsf{\Gamma }}_{\tau }}|e^{z(t_n-s)}|d|z|\right]·{\int }_{{\mathsf{\Gamma }}_{\tau }}|e^{z(t_n-s)}{||z|}^{1-2H}d|z|ds\hfill \\ \hfill & \le C{\tau }^{2\theta }{\int }_0^{{\displaystyle \frac{1}{\tau }}}{\int }_0^{t_n}|e^{z(t_n-s)}|{(t_n-s)}^{2H-2}dsd|z|\hfill \\ \hfill & \le C{\tau }^{2\theta }{\int }_0^{{\displaystyle \frac{1}{\tau }}}{|z|}^{1-2H}d|z|\le C{\tau }^{2\theta +2H-2}.\hfill \end{array}$$

(80)

Combining Estimates (78)–(80), for $0<\kappa -\sigma \le 1$, we derive that

$$J_2\le \left\{\begin{array}{cc}\hfill & C{\tau }^{\theta +H-1},1<\theta +H\le 2,\hfill \\ \hfill & C\tau ,\theta +H>2.\hfill \end{array}\right.$$

(81)

i.e.,

$$J_2\le C{\tau }^{min\{\theta +H-1,1\}}.$$

And, for $0\le \sigma -\kappa \le 1$,

$$J_2\le C{\tau }^{min\{\alpha +\gamma +H-1,1\}}.$$

(82)

Finally we turn to estimate $J_3$. From the definition of ${\overline{\partial }}_{\tau }{W}^H(t)$, it holds that

$$\begin{array}{cc}\hfill J_3^2& =\mathbb{E} \Vert {\int }_0^{t_n}{\tilde{E}}_{\tau ,h}(t_n-s)P_h(d{W}^H(s)-{\overline{\partial }}_{\tau }{W}^H(s)ds){\Vert }^2\hfill \\ \hfill & =\mathbb{E} \Vert \sum _{i=1}^n{\int }_{t_{i-1}}^{t_i}{\tilde{E}}_{\tau ,h}(t_n-s)P_{h}d{W}^H(s)-\sum _{i=1}^n{\int }_{t_{i-1}}^{t_i}{\tilde{E}}_{\tau ,h}(t_n-\overline{s})\left({\displaystyle \frac{1}{\tau }}{\int }_{t_{i-1}}^{t_i}{P}_{h}d{W}^H(s)\right)d\overline{s}{\Vert }^2\hfill \\ \hfill & =\mathbb{E} \Vert \sum _{i=1}^n{\int }_{t_{i-1}}^{t_i}\left[{\tilde{E}}_{\tau ,h}(t_n-s)-{\displaystyle \frac{1}{\tau }}{\int }_{t_{i-1}}^{t_i}{\tilde{E}}_{\tau ,h}(t_n-\overline{s})d\overline{s}\right]P_{h}d{W}^H(s){\Vert }^2\hfill \\ \hfill & \le C\mathbb{E} \Vert {\int }_0^{t_n}{\displaystyle \frac{1}{\tau }}\sum _{i=1}^n{\chi }_{(t_{i-1},t_i]}(s){\int }_{t_{i-1}}^{t_i}\left[{\tilde{E}}_{\tau ,h}(t_n-s)-{\tilde{E}}_{\tau ,h}(t_n-\overline{s})\right]d\overline{s}d{W}^H(s){\Vert }^2.\hfill \end{array}$$

Further, by using Lemma 2, we find that

$$\begin{array}{cc}\hfill J_3^2& \le C{\int }_0^{t_n}{\int }_0^{t_n}\langle {\displaystyle \frac{1}{\tau }}\sum _{i=1}^n{\chi }_{(t_{i-1},t_i]}(s){\int }_{t_{i-1}}^{t_i}[{\tilde{E}}_{\tau ,h}(t_n-s)-{\tilde{E}}_{\tau ,h}(t_n-\overline{s})]d\overline{s},\hfill \\ \hfill & {\displaystyle \frac{1}{\tau }}\sum _{i=1}^n{\chi }_{(t_{i-1},t_i]}(r){\int }_{t_{i-1}}^{t_i}[{\tilde{E}}_{\tau ,h}(t_n-r)-{\tilde{E}}_{\tau ,h}(t_n-\overline{s})]d\overline{s}{\rangle |r-s|}^{2H-2}drds\hfill \\ \hfill & \le C{\int }_0^{t_n}{\int }_s^{t_n}\Vert {\displaystyle \frac{1}{\tau }}\sum _{i=1}^n{\chi }_{(t_{i-1},t_i]}(s){\int }_{t_{i-1}}^{t_i}[{\tilde{E}}_{\tau ,h}(t_n-s)-{\tilde{E}}_{\tau ,h}(t_n-\overline{s})]d\overline{s}{\Vert }_{{\mathcal{L}}_2^0}\hfill \\ \hfill & ·\Vert {\displaystyle \frac{1}{\tau }}\sum _{i=1}^n{\chi }_{(t_{i-1},t_i]}(r){\int }_{t_{i-1}}^{t_i}[{\tilde{E}}_{\tau ,h}(t_n-r)-{\tilde{E}}_{\tau ,h}(t_n-\overline{s})]d\overline{s}{\Vert }_{{\mathcal{L}}_2^0}{|r-s|}^{2H-2}drds.\hfill \\ \hfill & \le C{\int }_0^{t_n}{\int }_s^{t_n}\Vert {\displaystyle \frac{1}{\tau }}\sum _{i=1}^n{\chi }_{(t_{i-1},t_i]}(s){\int }_{t_{i-1}}^{t_i}{A}^{{\displaystyle \frac{\kappa -\sigma }{2}}}[{\tilde{E}}_{\tau ,h}(t_n-s)-{\tilde{E}}_{\tau ,h}(t_n-\overline{s})]d\overline{s}\Vert \hfill \\ \hfill & ·\Vert {\displaystyle \frac{1}{\tau }}\sum _{i=1}^n{\chi }_{(t_{i-1},t_i]}(r){\int }_{t_{i-1}}^{t_i}{A}^{{\displaystyle \frac{\kappa -\sigma }{2}}}[{\tilde{E}}_{\tau ,h}(t_n-r)-{\tilde{E}}_{\tau ,h}(t_n-\overline{s})]d\overline{s}\Vert {|r-s|}^{2H-2}drds.\hfill \\ \hfill & \Vert A^{{\displaystyle \frac{\sigma -\kappa }{2}}}{\Vert }_{{\mathcal{L}}_2^0}^2\hfill \\ \hfill & \le C{\int }_0^{t_n}{\int }_s^{t_n}\Vert {\displaystyle \frac{1}{\tau }}\sum _{i=1}^n{\chi }_{(t_{i-1},t_i]}(s){\int }_{t_{i-1}}^{t_i}{A}^{{\displaystyle \frac{\kappa -\sigma }{2}}}[{\tilde{E}}_{\tau ,h}(t_n-s)-{\tilde{E}}_{\tau ,h}(t_n-\overline{s})]d\overline{s}\Vert \hfill \\ \hfill & ·\Vert {\displaystyle \frac{1}{\tau }}\sum _{i=1}^n{\chi }_{(t_{i-1},t_i]}(r){\int }_{t_{i-1}}^{t_i}{A}^{{\displaystyle \frac{\kappa -\sigma }{2}}}[{\tilde{E}}_{\tau ,h}(t_n-r)-{\tilde{E}}_{\tau ,h}(t_n-\overline{s})]d\overline{s}\Vert {(r-s)}^{2H-2}drds.\hfill \end{array}$$

(83)

By using (11), Lemma 7, and the fact that $|{\displaystyle \frac{z\tau }{e^{z\tau }-1}}|\le C$, we can easily obtain

$$\begin{array}{cc}\hfill & |{\tilde{E}}_{\tau }(t_n-s)-{\tilde{E}}_{\tau }(t_n-\overline{s}){|}_{\kappa -\sigma }\hfill \\ \hfill & =\Vert A^{{\displaystyle \frac{\kappa -\sigma }{2}}}\left[{\displaystyle \frac{1}{2\pi i}}{\int }_{{\mathsf{\Gamma }}_{\tau }}{e}^{z(t_n-s)}{(z_{\tau }^{\alpha }+A)}^{-1}{z}_{\tau }^{-\gamma }{\displaystyle \frac{z\tau }{e^{z\tau }-1}}dz-{\displaystyle \frac{1}{2\pi i}}{\int }_{{\mathsf{\Gamma }}_{\tau }}{e}^{(t_n-\overline{s})}{(z_{\tau }^{\alpha }+A)}^{-1}{z}_{\tau }^{-\gamma }{\displaystyle \frac{z\tau }{e^{z\tau }-1}}dz\right]\Vert \hfill \\ \hfill & \le C{\int }_{{\mathsf{\Gamma }}_{\tau }}|e^{z(t_n-s)}||e^{z(s-\overline{s})}-1|·\Vert A^{{\displaystyle \frac{\kappa -\sigma }{2}}}{(z_{\tau }^{\alpha }+A)}^{-1}{z}_{\tau }^{-\gamma }{\displaystyle \frac{z\tau }{e^{z\tau }-1}}\Vert d|z|\hfill \\ \hfill & \le C\tau {\int }_{{\mathsf{\Gamma }}_{\tau }}|e^{z(t_n-s)}{||z|}^{\alpha ({\displaystyle \frac{\kappa -\sigma }{2}}-1)-\gamma +1}d|z|,\hfill \end{array}$$

(84)

and, similarly,

$$|{\tilde{E}}_{\tau }(t_n-r)-{\tilde{E}}_{\tau }(t_n-\overline{s}){|}_{\kappa -\sigma }\le C\tau {\int }_{{\mathsf{\Gamma }}_{\tau }}|e^{z(t_n-r)}{||z|}^{\alpha ({\displaystyle \frac{\kappa -\sigma }{2}}-1)-\gamma +1}d|z|.$$

(85)

Then, by substituting (84) and (85) into (83), we obtain

$$J_3^2\le C{\tau }^2{\int }_0^{t_n}\left[{\int }_{{\mathsf{\Gamma }}_{\tau }}|e^{z(t_n-s)}{||z|}^{1-\theta }d|z|·{\int }_s^{t_n}{\int }_{{\mathsf{\Gamma }}_{\tau }}|e^{z(t_n-r)}{||z|}^{1-\theta }d|z|{(r-s)}^{2H-2}dr\right]ds.$$

(86)

Thus, from the estimate of (77), we can obtain the estimate of (86).

By combining the above estimates, we complete the proof of Theorem 4. □

By applying Theorem 4 and the fully discrete error estimate for the homogeneous equation corresponding to Model (1) ([13]), we obtain the following Theorem 5 for (1).

Theorem 5.

Let Assumptions 2–4 hold and $u_0\ne 0$. Let $u(t)$ and $u_h^n$ be the mild solution and fully discrete solution to Model (1), respectively. Let $0\le \mu \le 2$. If we let $\theta =(1-{\displaystyle \frac{\kappa -\sigma }{2}})\alpha +\gamma$, $u_0\in L^2(\mathsf{\Omega },{\dot{\mathbb{H}}}^{\nu })$ when $\nu \in [0,2]$, then we have the following estimates for $\epsilon >0$, ${\displaystyle \frac{1}{2}}<H<1,0<\alpha <1$, $0\le \gamma \le 1$, ${\displaystyle \frac{1}{2}}<\theta <2$:

Case 1. If $0<\kappa -\sigma \le min\{1,\mu \}$, then

$$\begin{array}{cc}\hfill \Vert u(t_n)-u_h^n{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}& \le C\tau t_n^{-1+{\displaystyle \frac{\nu }{2}}\alpha }+C{h}^2{t}_n^{-\alpha +{\displaystyle \frac{\nu }{2}}\alpha }+C{h}^{min\{2-\epsilon ,\mu -(\kappa -\sigma )\}}\hfill \\ \hfill & +C{\tau }^{min\{H+\theta -1-\epsilon ,\alpha \}},\hfill \end{array}$$

where $\alpha +\gamma -{\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha \mu }{2}}>0$.

Case 2. If $0\le \sigma -\kappa \le min\{1,\mu \}$, then

$$\begin{array}{cc}\hfill \Vert u(t_n)-u_h^n{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}& \le C\tau t_n^{-1+{\displaystyle \frac{\nu }{2}}\alpha }+C{h}^2{t}_n^{-\alpha +{\displaystyle \frac{\nu }{2}}\alpha }+C{h}^{min\{2-\epsilon ,\mu \}}\hfill \\ \hfill & +C{\tau }^{min\{H+\alpha +\gamma -1-\epsilon ,\alpha \}},\hfill \end{array}$$

where $\alpha +\gamma -{\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha (\mu -(\sigma -\kappa ))}{2}}>0,\alpha +\gamma -{\displaystyle \frac{\alpha \mu }{2}}+2H-{\displaystyle \frac{3}{2}}>0$.

5. Numerical Simulations

In this section, we will consider the numerical simulations required for solving (1) when $0<\alpha <1$, $0\le \gamma \le 1$:

$$\left\{\begin{array}{cc}\hfill & {}_0{}^{C}D_t^{\alpha }u(t,x)-{\displaystyle \frac{{\partial }^{2}u(t,x)}{\partial x^2}}=f(u(t,x))+{}_0{}^{R}I_t^{\gamma }{\displaystyle \frac{d{W}_H(t)}{dt}},0<x<1,0<t<T,\hfill \\ \hfill & u(t,0)=u(t,1)=0,0<t<T,\hfill \\ \hfill & u(0,x)=0,0<x<1,\hfill \end{array}\right.$$

(87)

where $f(u(t,x))=sin(u(t,x))$ and

$${\displaystyle \frac{d{W}^H(t)}{dt}}=\sum _{k=1}^{\infty }{\gamma }_k^{1/2}{\phi }_k(x){\displaystyle \frac{d{\beta }_k^H(t)}{dt}},$$

where ${\beta }_k^H(t)$ represents fractional Brownian motion with the Hurst index $H\in (1/2,1)$. It is straightforward to verify that $f(u)=sin(u)$ satisfies the nonlinear function Assumption 2.

Let the eigenfunctions of the operator $A=-{\displaystyle \frac{{\partial }^2}{\partial x^2}}$ in $\mathbb{H}$ be given by

$${\phi }_k(x)=\sqrt{2}sin(k\pi x),k=1,2,\dots$$

with the domain $\mathcal{D}(A)=H_0^1(0,1)\cap H^2(0,1)$. The eigenvalues are defined as ${\gamma }_k=k^{-m}$ for $m\ge 0$. If $m=0$, then $W^H(t)$ corresponds to white noise. If $m>1$, then $W^H(t)$ corresponds to smooth noise, since ${\sum }_{k=1}^{\infty }{\gamma }_k={\sum }_{k=1}^{\infty }{k}^{-m}<\infty$.

Let $0=t_0<t_1<t_2<\cdots <t_N=T$ be a uniform partition of $[0,T]$ with a time step size $\tau$. We define the following time discretization scheme for (87) when $t=t_n$:

$${\tau }^{-\alpha }\sum _{k=0}^n{\omega }_{n-k}^{(\alpha )}{u}^k+A{u}^n=F^n+{\tau }^{\gamma }\sum _{k=0}^n{\omega }_{n-k}^{(-\gamma )}{g}^k,$$

(88)

where the coefficients ${\omega }_j^{(\alpha )},j=0,1,\dots ,n$ are generated using the L1 scheme in (6) and ${\omega }_j^{(-\gamma )},j=0,1,\dots ,n$ are generated with Lubich’s convolution quadrature formula from (7). Here, $g^k$ is defined as

$$g^k={\displaystyle \frac{W^H(t_k)-W^H(t_{k-1})}{\tau }},$$

where $W^H(t)$ is generated using the MATLAB function fbmid.m from MathWorks. Let $0=x_0<x_1<\cdots <x_M=1$ be a uniform partition of $[0,1]$ with the space step size h. We discretize the spatial variable using the linear finite element method to solve (88).

In our numerical simulations, we set $T=t_N=1$ and $h=2^{-6}$. Since the exact solution of (87) is not available, we compute a reference solution $u(t_N)$ using $\tau =2^{-7}$. We then approximate the solutions $u^N$ using the time steps $\tau =2^{-5},2^{-4},2^{-3},2^{-2}$ to calculate the time convergence orders.

According to Theorem 4, the time convergence order follows one of two cases, with $\epsilon \in (0,1/2)$ and $\theta =\left(1-{\displaystyle \frac{\kappa -\sigma }{2}}\right)\alpha +\gamma$:

Case 1: When $\sigma <\kappa$,

$$\Vert u(t_n)-u_h^n{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\le C{\tau }^{min\{H+\theta -1-\epsilon ,\alpha \}}.$$

Case 2: When $\sigma \ge \kappa$,

$$\Vert u(t_n)-u_h^n{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}\le C{\tau }^{min\{H+\alpha +\gamma -1-\epsilon ,\alpha \}}.$$

In particular, when $\sigma \ge \kappa$, i.e., the trace class case, and $H\approx 1/2$, we obtain

$$\begin{array}{cc}\hfill \Vert u(t_n)-u_h^n{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}& \le C{\tau }^{min\{(\alpha +\gamma -1/2)+(H-1/2-ϵ),\alpha \}}\hfill \\ \hfill & \approx C{\tau }^{min\{(\alpha +\gamma -1/2-ϵ),\alpha \}}.\hfill \end{array}$$

For example, when $\alpha =0.9$ and $\gamma =0$, the convergence order is $min(\alpha +\gamma -1/2,\alpha )=min(0.4,0.9)=0.4$. When $\alpha =0.9$ and $\gamma =0.5$, the convergence order is $min(\alpha +\gamma -1/2,\alpha )=min(0.9,0.9)=0.9$. These results are verified in

Table 1. Convergence orders for trace class noise with $\kappa =\sigma$ and $H=1/2$.

$\mathit{\alpha }$	$\mathit{\gamma }\setminus \mathit{\tau }$	$2^{-2}$	$2^{-3}$	$2^{-4}$	$2^{-5}$	Orders (Average)
0.3	0.8	1.82 $\times {10}^{-2}$	1.17 $\times {10}^{-2}$	8.40 $\times {10}^{-3}$	4.75 $\times {10}^{-3}$	0.64 (0.60)
0.5	0.5	3.35 $\times {10}^{-2}$	2.47 $\times {10}^{-2}$	1.89 $\times {10}^{-2}$	1.13 $\times {10}^{-2}$	0.52 (0.50)
0.7	0.2	2.55 $\times {10}^{-2}$	1.77 $\times {10}^{-2}$	1.35 $\times {10}^{-2}$	1.08 $\times {10}^{-2}$	0.47 (0.40)

. All computations were implemented in MATLAB R2024b on a laptop equipped with an Intel Core i5-8250U CPU (8th Generation, 1.6 GHz base frequency, up to 3.4 GHz), running on an Acer Aspire 5 model with a Windows 10 operating system.

When $\sigma \ge \kappa$, i.e., the trace class case, and $H\in (1/2,1)$, we obtain

$$\begin{array}{cc}\hfill \Vert u(t_n)-u_h^n{\Vert }_{L^2(\mathsf{\Omega },\mathbb{H})}& \le C{\tau }^{min\{(\alpha +\gamma -1/2)+(H-1/2-ϵ),\alpha \}}.\hfill \end{array}$$

For example, when $\alpha =0.9$, $\gamma =0$, and $H=0.6$, the convergence order is $min((\alpha +\gamma -1/2)+(H-1/2),\alpha )=min(0.5,0.9)=0.5$. When $\alpha =0.9$, $\gamma =0$, and $H=0.8$, the convergence order is $min((\alpha +\gamma -1/2)+(H-1/2),\alpha )=min(0.7,0.9)=0.7$. These results can be observed in

Table 2. Convergence orders for smooth noise with $\sigma >\kappa$ and $\gamma =0$.

$\mathit{\alpha }$	$\mathit{H}\setminus \mathit{\tau }$	$2^{-2}$	$2^{-3}$	$2^{-4}$	$2^{-5}$	Orders (Average)
0.6	0.6	2.66 $\times {10}^{-1}$	2.29 $\times {10}^{-1}$	2.03 $\times {10}^{-1}$	1.35 $\times {10}^{-1}$	0.32 (0.20)
0.8	0.8	2.82 $\times {10}^{-2}$	1.76 $\times {10}^{-2}$	1.26 $\times {10}^{-2}$	8.19 $\times {10}^{-3}$	0.60 (0.60)
0.9	0.9	1.26 $\times {10}^{-2}$	6.64 $\times {10}^{-3}$	4.03 $\times {10}^{-3}$	2.07 $\times {10}^{-3}$	0.84 (0.80)

.

In Table 1, we set $\kappa =\sigma$ and $H=1/2$. The table presents the average convergence orders for different values of $\alpha$ and $\gamma$, confirming their agreement with our theoretical predictions.

In Table 2, we set $\sigma >\kappa$ and $\gamma =0$, and a comparison of convergence orders for different $\alpha$ and H again confirms their consistency with our theoretical expectations.

Conclusions

In this work, we introduced a new fully discrete scheme to approximate a stochastic semilinear fractional subdiffusion equation driven by integrated fractional Gaussian noise. The temporal and spatial regularity of the mild solution were proven using the semigroup approach. The finite element method was employed for spatial discretization, while the L1 scheme and Lubich’s first-order convolution quadrature formula were used to approximate the Caputo time-fractional derivative and the Riemann–Liouville time-fractional integral, respectively. The error estimates of the proposed scheme are proved in detail and explicitly show how the error depends on the parameters $\alpha$, $\gamma$, and H.