Almost Automorphic Strong Oscillation in Time-Fractional Parabolic Equations

Abstract

This paper gives some results on almost automorphic strong solutions to time-fractional partial differential equations by employing a mix o thef Galerkin method, Fourier series, and Picard iteration. As an application, the existence, uniqueness, and global Mittag–Leffler convergence of almost automorphic strong solution are discussed to a concrete time-fractional parabolic equations. To the best of our knowledge, this is the first study on almost automorphic strong solutions on this subject.

1. Introduction

Let ${\mathbf{R}}^d$ be d-dimensional Euclidian space, ${\mathbf{R}}_0=[0,+\infty )$, ${\mathbf{R}}_{+}=(0,+\infty )$; ${\mathbf{Z}}_{+}=\{1,2,\dots \}$; and $\mathbf{C}$ be the field of complex numbers. Let $\mathsf{\Omega }$ be a bounded open domain in ${\mathbf{R}}^d$ with a smooth boundary $\partial \mathsf{\Omega }$, $\overline{\mathsf{\Omega }}:=\mathsf{\Omega }\cup \partial \mathsf{\Omega }$, and let $\mathbb{H}$ denote some Hilbert space. This paper is devoted to discussing time-fractional partial differential equations in the form of

$$\left\{\begin{array}{cc}{{}^{c}D}_{0\ast }^{\alpha }\mathbf{u}(x,t)=\mathcal{A}\mathbf{u}(x,t)+F(\mathbf{u}(x,t),x,t),\hfill & \mathrm{i}\mathrm{n}\mathsf{\Omega }\times {\mathbf{R}}_{+},\hfill \\ \mathbf{u}(x,t)=0,\hfill & \mathrm{i}\mathrm{n}\partial \mathsf{\Omega }\times {\mathbf{R}}_0\hfill \end{array}\right.$$

(1)

with $\mathbf{u}(x,0)=\phi \left(x\right)\in \mathbb{H}$ for $x\in \overline{\mathsf{\Omega }}$, where $\mathbf{u}(t,·):{\mathbf{R}}_0\to \mathbb{H}$; ${{}^{c}D}_{0\ast }^{\alpha }$ is the Caputo fractional derivative from initial point 0 and $0<\alpha \le 1$; $\mathcal{A}$ is an $\mathbb{H}$-valued continuous linear operator; $F(\mathbf{u},·,t):\mathbb{H}\times {\mathbf{R}}_{+}\to \mathbb{H}$ for each $(\mathbf{u},t)\in \mathbb{H}\times {\mathbf{R}}_{+}$ and it meets the Caratheodory condition, i.e., it is continuous with respect to the first variable and measurable with respect to the other variables.

Bochner, in a study on differential geometry [1], firstly proposed the conception of almost automorphy [2], which is a generalization to almost periodicity [3] and periodicity [4]. The importance of almost automorphy in qualitative studies of dynamics is further illustrated by the fact that the conception of almost automorphy proves essential in the study of almost automorphic dynamics for a class of ordinary, parabolic, and generalized differential equations. First, in terms of the nature of the lift from the coefficient space to the solution space, the dynamics are usually closed, not in the category of almost periodicity, but in the category of almost automorphy. Second, the emergence of almost automorphic dynamics shows the main difference between periodic systems and almost periodic systems. For example, in a monotonic dynamical system, the “lift” from the period coefficient can never be almost automorphy. Another important implication of the study of almost automorphic dynamics is its connection with Levitan N-almost periodicity. Because almost automorphic functions are essentially N-almost periodic, the current study of almost automorphic dynamics is closely related to N-almost periodic cases, please see refs. [3,5,6,7,8,9] for more details.

1.1. Almost Automorphic Mild Solutions

During the last two decades, numerous scholars have focused on investigating almost automorphic dynamics of evolution equations described by

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\mathbf{u}\left(t\right)=\mathcal{A}\mathbf{u}\left(t\right)+F(\mathbf{u}\left(t\right),t),\forall t\in (0,T],}\hfill \\ \mathbf{u}\left(0\right)={\mathbf{u}}_0,\hfill \end{array}\right.$$

(2)

where $\mathbf{u}:[0,T]\to X$ for some constant $T>0, X$ is a finite dimensional Banach space, and $\mathcal{A}$ is a bounded linear operator on X. Under some suitable assumptions, Equation (2) can be transformed into the parabolic equations with the Dirichlet boundary condition in the form of

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{\partial }{\partial t}\mathbf{u}(x,t)=\mathsf{\Delta }\mathbf{u}(x,t)+F(\mathbf{u}(x,t),x,t),}\hfill & \mathrm{i}\mathrm{n}\mathsf{\Omega }\times (0,T],\hfill \\ \mathbf{u}(x,t)=0,\hfill & \mathrm{i}\mathrm{n}\partial \mathsf{\Omega }\times [0,T],\hfill \\ \mathbf{u}(x,0)=\phi \left(x\right),\hfill & \mathrm{i}\mathrm{n}\overline{\mathsf{\Omega }},\hfill \end{array}\right.\end{array}$$

(3)

where $\mathbf{u}:\overline{\mathsf{\Omega }}\times [0,T]\to \mathbf{R}$, and $\mathsf{\Delta }={\displaystyle \sum _{i=1}^d}\frac{{\partial }^2}{\partial x_i^2}$ denotes the Laplacian operator.

By employing a mix of the semigroup theory and fixed point theorems, almost automorphic (mild) solutions of Equations (2) and (3) have been widely discussed by a large number of scholars [10,11,12,13,14,15,16,17,18,19] during the last two decades.

Definition 1. 

A continuous function $\mathbf{u}:[0,T]\to X$ is said to be a mild solution of Equation (2) if

$$\mathbf{u}\left(t\right)=\mathcal{U}\left(t\right)\mathbf{u}\left(0\right)+{\int }_0^t\mathcal{U}(t,s)F(\mathbf{u}\left(s\right),s)\mathrm{d}s,\forall t\in [0,T],$$

where $\mathcal{U}(·,·)$ is an exponentially stable evolution family on X.

Researchers [10,11,12] have obtained the existence of (compact) almost automorphic mild solutions to Equations (2) and (3) by supposing $\mathcal{A}$ is the infinitesimal generator of a $C_0$ semigroup. Researchers [13,14] have used the extrapolation methods to study the existence and uniqueness of ($\mu$ pseudo) almost automorphic mild solution to the boundary system depicted by

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\mathbf{u}\left(t\right)={\mathcal{A}}_{*}\mathbf{u}\left(t\right)+F(\mathbf{u}\left(t\right),t),}\hfill \\ B\left(t\right)\mathbf{u}\left(t\right)=G\left(\mathbf{u}\right(t),t),\forall t\in \mathbf{R},\hfill \end{array}\right.$$

(4)

where ${\mathcal{A}}_{*}$ is a densely defined linear operator on some Banach space X. Additionally, the results in [13,14] can be adopted to obtain the existence of a unique almost automorphic mild solution to Equation (3). So, letting $\mathcal{A}:\mathbb{H}\to 2^{\mathbb{H}}$ be a multivalued maximal monotone operator on some Hilbert space $\mathbb{H}$, Es-sebbar et al. [15] studied the existence of compact almost automorphic weak solutions for some differential inclusion. For more details on the studies of almost automorphy for Equations (2) and (3), please see $C^{\left(n\right)}$-almost automorphic mild solutions [16], almost automorphic mild solutions in neutral functional differential equations [17], and Fréchet spaces [18], square-mean weighted pseudo almost automorphic mild solutions with stochastic factors [19]. The interest in studying fractional order differential equations is strongly motivated by the knowledge that they have been demonstrated to be a valuable facility for modernizing a variety of processes in different aspects of science, physics, business, and technology [20,21,22]. It has a wide range of possible usages in physical activities such as seismic nonlinear vibrations, porous media seepage, and fluid dynamics traffic models. Resembling the studies of almost automorphic mild solutions to integer-order evolution Equation (2), some scholars concentrated on investigating almost automorphic dynamics of fractional evolution equations. For instance, Wang and Xia [23] obtained the existence and uniqueness of a $(\mu ,\nu )$-pseudo almost automorphic mild solution to fractional evolution equations with Caputo operators. Chen et al. studied almost automorphic mild solutions to fractional evolution equations with Riemann–Liouville operators [24]. Authors [25,26] have discussed (weighted pseudo) almost automorphic mild solutions of fractional evolution equations with higher orders. Pardoa and Lizama [27] considered (weighted pseudo) almost automorphic mild solutions to fractional evolution equations with two different fractional orders. In addition, a (pseudo) almost automorphic mild solution for stochastic fractional evolution equations was reported in the literature [28,29].

It should be noted that a mild solution of (fractional) parabolic or evolution equations is not a real sense of a solution to the corresponding parabolic or evolution equations. In allusion to integer-order evolution Equation (2), it possesses a classical solution $\mathbf{u}\in \mathrm{C}([0,T],D\left(\mathcal{A}\right))\cap {\mathrm{C}}^1([0,T],X)$ if $\mathbf{u}$ a mild solution of Equation (2), ${\mathbf{u}}_0\in D\left(\mathcal{A}\right)$ and $F(·,·)$ are local and global Lipschitz continuous with respect to the first and second variables, respectively [30]. Here, $D\left(\mathcal{A}\right)$ is a Banach space endowed with the graph norm ${\parallel \mathbf{u}\parallel }_{D\left(\mathcal{A}\right)}={\parallel \mathcal{A}\mathbf{u}\parallel }_X$, $\forall \mathbf{u}\in D\left(\mathcal{A}\right)$. However, it is different for fractional evolution equations. In [31], Wang et al. considered the Cauchy problem of a fractional evolution equation with the Caputo operator as described below

$$\left\{\begin{array}{c}{{}^{c}D}_{0\ast }^{\alpha }\mathbf{u}\left(t\right)=\mathcal{A}\mathbf{u}\left(t\right)+F\left(\mathbf{u}\left(t\right)\right),\forall t\in (0,T],\hfill \\ \mathbf{u}\left(0\right)={\mathbf{u}}_0,\hfill \end{array}\right.$$

(5)

where $\alpha \in (0,1],\mathbf{u}:{\mathbf{R}}_0\to X$, and $\mathcal{A}$ is an almost sectorial operator defined in Definition 1.1 in ref. [31]. In view of Theorem 5.2 in ref. [31], Equation (5) admits that $D\left(\mathcal{A}\right)$-smooth solutions requiring a few extremely strict conditions for F, e.g., $F(·)$ needs to fill the Lipschitz condition on $D\left(\mathcal{A}\right)$. For example, taking $\mathcal{A}=\mathsf{\Delta }$, and $D\left(\mathcal{A}\right)=C_0^2(\overline{\mathsf{\Omega }},\mathbf{R})$, (correspondingly, classical solution) or $D\left(\mathcal{A}\right)={\mathrm{W}}_0^{1,2}(\mathsf{\Omega },\mathbf{R})\cap {\mathrm{W}}^{2,2}(\mathsf{\Omega },\mathbf{R})$ (correspondingly, strong solution), then Equation (5) is turned into the time-fractional parabolic equations below

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{{}^{c}D}_{0\ast }^{\alpha }\mathbf{u}(x,t)=\mathsf{\Delta }\mathbf{u}(x,t)+F\left(\mathbf{u}(x,t)\right),\hfill & \mathrm{in}\mathsf{\Omega }\times (0,T],\hfill \\ \mathbf{u}(x,t)=0,\hfill & \mathrm{in}\partial \mathsf{\Omega }\times [0,T],\hfill \\ \mathbf{u}(x,0)=\phi \left(x\right),\hfill & \mathrm{in}\overline{\mathsf{\Omega }},\hfill \end{array}\right.\end{array}$$

(6)

where $\mathbf{u}:\overline{\mathsf{\Omega }}\times [0,T]\to \mathbf{R}$. The definitions of the spaces $C_0^2,{\mathrm{W}}_0^{1,2}, \mathrm{and}{\mathrm{W}}^{2,2}$ can be found in a book [32]. By observing Example 6.2 in ref. [31], Equation (6) admits a $D\left(\mathcal{A}\right)$-smooth solution u if u is a mild solution; ${\mathbf{u}}_0\in D\left(\mathcal{A}\right)$ and $F^{\prime },F^{″}$ are continuously differentiable functions satisfying the Lipschitz condition. Here, $F^{\prime }\mathrm{and} F^{″}$ stand for the first and second (weak) derivatives for $D\left(\mathcal{A}\right)=C_0^2(\overline{\mathsf{\Omega }},\mathbf{R})\mathrm{and}D\left(\mathcal{A}\right)={\mathrm{W}}_0^{1,2}(\mathsf{\Omega },\mathbf{R})\cap {\mathrm{W}}^{2,2}(\mathsf{\Omega },\mathbf{R})$, respectively.

Throughout history, almost all existing studies in this field concentrated on the examination of almost automorphic mild solutions to time-fractional parabolic equations by utilizing the theory of operator semigroups and fixed point theorems [23,24,25,26,27,28,29]. Motivated by this, we investigated the existence and uniqueness of an almost automorphic strong solution of time-fractional partial differential Equation (1) in some suitable function space by using a mix of the Galerkin method and Picard iteration.

1.2. Main Results

Assume that $\mathbb{H}$, ${\mathbb{H}}_1$ and ${\mathbb{H}}_2$ are reflexible and separable Hilbert spaces; $\mathbb{X}$ denotes some dense linear space in $\mathbb{H}$ and $\mathrm{supp}\eta \subset \mathsf{\Omega }$, $\forall \eta \in \mathbb{X}$; $\mathbb{X}\subseteq {\mathbb{H}}_2\subseteq {\mathbb{H}}_1\subseteq \mathbb{H}$; ${\mathbb{H}}_1\hookrightarrow \mathbb{H}$ is compact. Furthermore, $\mathbb{H}$ fills Clarkson’s condition $\left(\mathrm{C}\right)$ in Theorem 3. Define $\mathcal{G}:{\mathbb{H}}_2\to \mathbb{H}$ by $\mathcal{G}\mathbf{u}\left(t\right):=\mathcal{A}\mathbf{u}\left(t\right)+\mathcal{F}\left(\mathbf{u}\right(t\left)\right)$ and $\mathcal{F}\left(\mathbf{u}\right(t\left)\right):=F\left(\mathbf{u}\right(t),·,t)$, $\forall \mathbf{u}\left(t\right):=\mathbf{u}(t,·)\in {\mathbb{H}}_2$ for each $t\in {\mathbf{R}}_0$. Equation (1) is formulated by the operator equation as described below

$$\left\{\begin{array}{cc}{{}^{c}D}_{0\ast }^{\alpha }\mathbf{u}\left(t\right)=\mathcal{G}\mathbf{u}\left(t\right),\hfill & \mathrm{in}{\mathbf{R}}_{+},\hfill \\ \mathbf{u}\left(0\right)=\phi \in \mathbb{H},\hfill & \hfill \end{array}\right.$$

(7)

where $\mathbf{u}:{\mathbf{R}}_0\to \mathbb{H}$.

Definition 2. 

$\mathbf{u}\in {\mathrm{L}}_{\mathrm{loc}}^2({\mathbf{R}}_{+},{\mathbb{H}}_2)\cap {\mathrm{L}}_{\mathrm{loc}}^{\infty }({\mathbf{R}}_{+},\mathbb{H})$is a global weak solution of Equations (1) or (7) if

$$\begin{array}{ccc}\hfill {\langle \mathbf{u}\left(t\right),\mathbf{v}\rangle }_{\mathbb{H}}& =& {\langle \phi ,\mathbf{v}\rangle }_{\mathbb{H}}+\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}{\langle \mathcal{G}\mathbf{u}\left(s\right),\mathbf{v}\rangle }_{\mathbb{H}}\mathrm{d}s,\forall \mathbf{v}\in \mathbb{H},t\in {\mathbf{R}}_{+}.\hfill \end{array}$$

From Definition 2 and item (4) in Lemma 1, if the three items below hold,

(1)

${\langle \mathcal{G}\mathbf{u}(·),\mathbf{v}\rangle }_{\mathbb{H}}\in {\mathrm{L}}^{\infty }(I_T,\mathbb{H})$ or ${\langle \mathcal{G}\mathbf{u}(·),\mathbf{v}\rangle }_{\mathbb{H}}\in \mathrm{C}({\overline{I}}_T,\mathbb{H})$, $\forall \mathbf{v}\in \mathbb{H}$.

(2)

${{}^{c}D}_{0\ast }^{\alpha }{\langle \mathbf{u}(·),\mathbf{v}\rangle }_{\mathbb{H}}={\langle {{}^{c}D}_{0\ast }^{\alpha }\mathbf{u}(·),\mathbf{v}\rangle }_{\mathbb{H}},$$\forall \mathbf{v}\in \mathbb{H}.$

(3)

${{}^{c}D}_{0\ast }^{\alpha }\mathbf{u}(·)\in \mathbb{H}$.

then

$$\left\{\begin{array}{cc}{〈{{}^{c}D}_{0\ast }^{\alpha }\mathbf{u}\left(t\right)-\mathcal{G}\mathbf{u}\left(t\right),\mathbf{v}〉}_{\mathbb{H}}=0,\hfill & \mathrm{in}{\mathbf{R}}_{+},\hfill \\ {\langle \mathbf{u}\left(0\right),\mathbf{v}\rangle }_{\mathbb{H}}={\langle \phi ,\mathbf{v}\rangle }_{\mathbb{H}},\hfill & \forall \mathbf{v}\in \mathbb{H}.\hfill \end{array}\right.$$

(8)

Therefore, the global strong solution of Equations (1) or (7) ought to be defined by

$$\mathbf{u}\in \mathcal{L}({\mathbf{R}}_0,\mathbb{H}):={\mathrm{L}}_{\mathrm{loc}}^{\infty }({\mathbf{R}}_{+},{\mathbb{H}}_1)\cap {\mathrm{L}}_{\mathrm{loc}}^2({\mathbf{R}}_{+},{\mathbb{H}}_2)\cap {\mathrm{W}}_{\mathrm{loc}}^{\alpha ,2}({\mathbf{R}}_{+},\mathbb{H})$$

satisfies Equation (8).

The spaces mentioned here are introduced in Section 2.

Definition 3. 

Operator $\mathcal{G}:{\mathbb{H}}_2\to \mathbb{H}$ is said to be weakly continuous if ${\eta }_n\rightharpoonup {\eta }_0\in {\mathbb{H}}_2$ in $\mathbb{H}$ for arbitrary $\left\{{\eta }_n\right\}\subseteq {\mathbb{H}}_2$, then

$$\underset{n\to \infty }{lim}{\langle \mathcal{G}{\eta }_n,\zeta \rangle }_{\mathbb{H}}={\langle \mathcal{G}{\eta }_0,\zeta \rangle }_{\mathbb{H}},\forall \zeta \in \mathbb{H}.$$

Some necessary assumptions are listed here:

$\left({\mathcal{H}}_1\right)$

There exists an orthonormal basis $\{e_k\in \mathbb{X}:k\in {\mathbf{Z}}_{+}\}$ of $\mathbb{H}$ satisfying $\mathcal{A}{e}_k=-{\lambda }_k{e}_k$, $\forall k\in {\mathbf{Z}}_{+}$. Here, $\{{\lambda }_k:k\in {\mathbf{Z}}_{+}\}$ is the real eigenvalue sequence with $0<{\lambda }_1\le {\lambda }_2\le \cdots \le {\lambda }_k\le \cdots$, ${lim}_{k\to \infty }{\lambda }_k=\infty$.

$\left({\mathcal{H}}_2\right)$

$\mathcal{A}:{\mathbb{H}}_2\to \mathbb{H}$ is a weakly continuous operator meeting

$${\langle \mathcal{A}\eta ,\zeta \rangle }_{\mathbb{H}}=-a_1{\langle \eta ,\zeta \rangle }_{{\mathbb{H}}_1},\forall \eta \in {\mathbb{H}}_2,\zeta \in {\mathbb{H}}_1;$$

$$a_3{\langle \eta ,\eta \rangle }_{{\mathbb{H}}_2}-a_2{\langle \eta ,\eta \rangle }_{\mathbb{H}}\le {\langle \mathcal{A}\eta ,\mathcal{A}\eta \rangle }_{\mathbb{H}}\le a_4{\langle \eta ,\eta \rangle }_{{\mathbb{H}}_2},\forall \eta \in {\mathbb{H}}_2,$$

where $a_i>0$ is constant for $i=1,2,3,4$.

$\left({\mathcal{H}}_3\right)$

$F(0,·,t)\in {\mathrm{L}}_{\mathrm{loc}}^{\infty }({\mathbf{R}}_{+},\mathbb{H})$ with respect to $t\in {\mathbf{R}}_{+}$ and $F(·,·,·)$ fulfils the Lipschitz condition on $\mathbb{H}$ with respect to the first variable with a Lipschitz constant $L_F>0$, i.e.,

$$\parallel F(\eta ,·,·)-F(\zeta ,·,·){\parallel }_{\mathbb{H}}\le L_F{\parallel \eta -\zeta \parallel }_{\mathbb{H}},\forall \eta ,\zeta \in \mathbb{H}.$$

$\left({\mathcal{H}}_4\right)$

There exists $\delta >0$ such that

$${\langle {{}^{c}D}_{0\ast }^{\alpha }\eta ,\eta \rangle }_{\mathbb{H}}\ge \delta {{}^{c}D}_{0\ast }^{\alpha }{\langle \eta ,\eta \rangle }_{\mathbb{H}},\forall \eta \in {\mathrm{H}}^{\alpha }({\mathbf{R}}_0,\mathbb{H})\cap {\mathrm{W}}_{\mathrm{loc}}^{\alpha ,2}({\mathbf{R}}_{+},\mathbb{H}).$$

Furthermore,

$${{}^{c}D}_{0\ast }^{\alpha }{\langle \eta ,\zeta \rangle }_{\mathbb{H}}={\langle {{}^{c}D}_{0\ast }^{\alpha }\eta ,\zeta \rangle }_{\mathbb{H}},\forall \eta \in {\mathrm{W}}_{\mathrm{loc}}^{\alpha ,2}({\mathbf{R}}_{+},\mathbb{H}),\zeta \in \mathbb{H}.$$

Let us introduce some brief notations, which are frequently utilized in the following. Let

$${\mathcal{J}}_s^{\alpha }\left(t\right)={(t-s)}^{\alpha -1},{\mathcal{E}}_t^{\alpha }\left(a\right)={\mathbf{E}}_{\alpha }(-a{t}^{\alpha }),{\mathcal{E}}_{t,s}^{\alpha ,\beta }\left(a\right)={(t-s)}^{\beta -1}{\mathbf{E}}_{\alpha ,\beta }(-a{(t-s)}^{\alpha }),$$

where $s,t\in \mathbf{R}$ with $s\le t$, $\alpha ,\beta \in {\mathbf{R}}_0$ and $a\in \mathbf{R}$.

First, we obtain the existence result of a unique continuous global strong solution to Equation (1) as below.

Theorem 1. 

Let $\phi \in {\mathbb{H}}_1$ and $\left({\mathcal{H}}_1\right)$–$\left({\mathcal{H}}_4\right)$ hold. Then, the following assertions hold:

(i) 

Equation (1) possesses at least one global strong solution ${\mathbf{u}}_0\in \mathcal{L}({\mathbf{R}}_0,\mathbb{H})$.

(ii) 

${\mathbf{u}}_0\in {\mathrm{H}}^{\alpha }({\mathbf{R}}_0,\mathbb{H})$ can be expressed by the Fourier series below

$${\mathbf{u}}_0\left(t\right)=\sum _{k=1}^{\infty }\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\phi }_k+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left({\mathbf{u}}_0\left(s\right)\right)\mathrm{d}s\right]e_k,$$

(9)

where ${\phi }_k={\langle \phi ,e_k\rangle }_{\mathbb{H}}$ and ${\mathcal{F}}_k\left({\mathbf{u}}_0\left(t\right)\right)={\langle \mathcal{F}\left({\mathbf{u}}_0\left(t\right)\right),e_k\rangle }_{\mathbb{H}},k=1,2,\dots ,t\in {\mathbf{R}}_0$.

(iii) 

The energy estimation

$$\underset{t\in {\mathbf{R}}_0}{max}{\parallel {\mathbf{u}}_0\left(t\right)\parallel }_{\mathbb{H}}\le \frac{{\lambda }_1\phi +F_{\infty }}{{\lambda }_1-L_F}.$$

holds if ${\lambda }_1>L_F$ and $F_{\infty }:=\mathrm{ess}{sup}_{t\in {\mathbf{R}}_0}{\parallel F(0,·,t)\parallel }_{\mathbb{H}}<\infty$.

(iv) 

${\mathbf{u}}_0$ is a unique solution in $\mathrm{C}({\mathbf{R}}_0,\mathbb{H})$ solving Equation (1).

(v) 

Equation (1) is global Mittag–Leffler convergent in the sense of strong solutions if $a_0>L_F$, where $a_0:={inf}_{\eta \in {\mathbb{H}}_2}\frac{-{\langle A\eta ,\eta \rangle }_{\mathbb{H}}}{{\langle \eta ,\eta \rangle }_{\mathbb{H}}}$.

Remark 1. 

In the proof of Theorem 1, the standard Galerkin method is utilized. In the literature [33], it was employed to investigate the existence of solutions of integer-order partial differential equations, e.g., parabolic equations, Navier–Stokes equations, etc. We discuss the existence of solutions of time-fractional partial differential equations via this method. As we know, some important properties with respect to integer-order derivatives are invalid for fractional derivatives, e.g., ${\left(f^2\right)}^{\prime }=2f{f}^{\prime }$ for $f\in {\mathrm{C}}^1$. Superadding the complexity of nonlocal calculus (e.g., items (4) and (5) in Lemma 1), this greatly increases the difficulty of the study.

Next, the main existence result is achieved as follows.

Theorem 2. 

Let $\phi \in {\mathbb{H}}_1$, $\left({\mathcal{H}}_1\right)$–$\left({\mathcal{H}}_6\right)$ hold. Then, the following assertions hold.

(i) 

${\xi }_{*}\in \mathbb{AAA}({\mathbf{R}}_0,\mathbb{H})\cap \mathcal{L}({\mathbf{R}}_0,\mathbb{H})\cap {\mathrm{H}}^{\alpha }({\mathbf{R}}_0,\mathbb{H})$ is a unique asymptotically almost automorphic strong solution solving Equation (1), which possesses the Fourier series from (25).

(ii) 

Equation (1) is global Mittag–Leffler convergent in the sense of strong solutions.

Remark 2. 

Refs. [23,24,25,26,27,28,29] investigated the almost automorphic mild solutions of time-fractional parabolic equations by employing the theory of operator semigroups and fixed point theorems. Based on the Fourier series expression in Equation (23), we are the first to investigate the almost automorphic strong solutions of time-fractional partial differential equations. Apparently, our study improves and extends the corresponding results in refs. [23,24,25,26,27,28,29].

1.3. Structure of This Paper

In Section 2, some essential function spaces are introduced, and a novel nonlocal Sobolev space is set up. In Section 3, the existence of a unique global Mittag–Leffler convergent strong solution to Equation (1) is discussed by means of the Galerkin method. In Section 4, the existence of a unique global Mittag–Leffler convergent asymptotically almost automorphic strong solution to Equation (1) is studied by an approach of Picard iteration. As an application, a nonlinear time-fractional parabolic equation is considered in Section 5.

2. The Setup of Function Spaces

Let $I_T=(0,T)$, ${\overline{I}}_T=[0,T]$ for some real constant $T>0$, $\mathbb{K}$ be a subset of some finite dimension metric space, and $\mathbb{V}$ be some finite or infinite dimension metric space with the distance ${\parallel ·\parallel }_{\mathbb{V}}$.

• ${\mathrm{C}}^m\left(\mathbb{K}\right):={\mathrm{C}}^m(\mathbb{K},\mathbb{V})$ is the space of all m-order continuously differential mappings from $\mathbb{K}$ to $\mathbb{V}$, endowed with the norm ${\parallel ·\parallel }_{{\mathrm{C}}^m}$, $\forall m\in {\mathbf{Z}}_{+}$ or $m=\infty$.
• ${\mathrm{L}}^p\left(\mathbb{K}\right):={\mathrm{L}}^p(\mathbb{K},\mathbb{V})$ is a collection of all measurable mappings $f:\mathbb{K}\to \mathbb{V}$ with a finite norm ${\parallel ·\parallel }_{{\mathrm{L}}^p}$, where $p\in [1,\infty ]$ and

  $${\parallel f\parallel }_{{\mathrm{L}}^p}=\left\{\begin{array}{cc}{\displaystyle {\left[{\int }_{\mathbb{K}}{\parallel f\left(t\right)\parallel }_{\mathbb{V}}^p\mathrm{d}t\right]}^{\frac{1}{p}},}\hfill & \mathrm{if}p\in [1,\infty );\hfill \\ \mathrm{ess}\underset{t\in \mathbb{K}}{sup}{\parallel f\left(t\right)\parallel }_{\mathbb{V}},\hfill & \mathrm{if}p=\infty .\hfill \end{array}\right.$$
  Usually, ${\mathrm{L}}^p(\mathbb{K},\mathbb{V})$ is called the Lebesgue space. Additionally, ${\mathrm{L}}_{\mathrm{loc}}^p({\mathbf{R}}_{+},\mathbb{V})$ represents the local Lebesgue space defined by ${\mathrm{L}}_{\mathrm{loc}}^p({\mathbf{R}}_{+},\mathbb{V})=\left\{f\in {\mathrm{L}}^p(I_T,\mathbb{V}):\forall T>0\right\}.$
• ${\mathrm{AC}}^n\left({\overline{I}}_T\right):={\mathrm{AC}}^n({\overline{I}}_T,\mathbb{V})$ shows a family of all $(n-1)$-order continuously differential functions $f:{\overline{I}}_T\to \mathbb{V}$, and $f^{(n-1)}$ is absolutely continuous.
• ${\mathrm{H}}^{\gamma }\left({\overline{I}}_T\right):={\mathrm{H}}^{\gamma }({\overline{I}}_T,\mathbb{V})$ denotes the Hölder space by

  $${\mathrm{H}}^{\gamma }({\overline{I}}_T,\mathbb{V})=\left\{f\in \mathrm{C}({\overline{I}}_T,\mathbb{V}): \parallel f\left(t\right)-f\left(s\right){\parallel }_{\mathbb{V}}\le h{|t-s|}^{\gamma },\forall s,t\in {\overline{I}}_T\right\},$$
  where $h>0$ is called the Hölder constant and $\gamma \in (0,1]$. ${\mathrm{H}}^{\gamma }\left({\overline{I}}_T\right)$ is a Banach space equipped with the norm ${\parallel f\parallel }_{{\mathrm{H}}^{\gamma }}={\parallel f\parallel }_{\mathrm{C}}+{\left[f\right]}_{{\mathrm{H}}^{\gamma }}$, where

  $${\left[f\right]}_{{\mathrm{H}}^{\gamma }}=\underset{s,t\in {\overline{I}}_T,s\ne t}{sup}\frac{{\parallel f\left(t\right)-f\left(s\right)\parallel }_{\mathbb{V}}}{{|t-s|}^{\gamma }},\forall f\in {\mathrm{H}}^{\gamma }\left({\overline{I}}_T\right).$$

For any $f\in \mathrm{L}(I_T,\mathbb{V})$, the Riemann-Liouville fractional integral and derivative of order $\alpha \in (n-1,n](n\in {\mathbf{Z}}_{+})$ from initial point $t=0$ are, respectively, defined as

$$J_0^{\alpha }f\left(t\right)=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}f\left(s\right)\mathrm{d}s,\forall t\in I_T,$$

$$D_0^{\alpha }f\left(t\right)=\frac{1}{\mathsf{\Gamma }(n-\alpha )}\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}{\int }_0^t{(t-s)}^{n-\alpha -1}f\left(s\right)\mathrm{d}s,\forall t\in I_T,$$

where $\mathsf{\Gamma }(·)$ represents the Gamma function.

If $f\in {\mathrm{AC}}^n(I_T,\mathbb{V})$ and $\mathbb{V}$ is a reflexible Banach space, then $f^{(n-1)}$ is differential a.e. in $I_T$ by Theorem 1.2.9 in ref. [34]. The Caputo fractional derivative is given by

$${{}^{c}D}_0^{\alpha }f\left(t\right)=\frac{1}{\mathsf{\Gamma }(n-\alpha )}{\int }_0^t{(t-s)}^{n-\alpha -1}{f}^{\left(n\right)}\left(s\right)\mathrm{d}s,\forall t\in I_T,$$

where $\alpha \in (n-1,n]$, $n\in {\mathbf{Z}}_{+}$.

The definition of ${{}^{c}D}_0^{\alpha }$ is limited in the space of absolutely continuous functions, i.e., ${\mathrm{AC}}^n(I_T,\mathbb{V})$, which is rigorous compared with the Riemann–Liouville sense as described below. As a result, the Riemann–Liouville operator is adopted to define the Caputo fractional derivative as

$${{}^{c}D}_{0\ast }^{\alpha }f\left(t\right)=D_0^{\alpha }\left[f\left(t\right)-\sum _{k=0}^{n-1}\frac{t^k}{k!}\underset{t\to 0^{+}}{lim}{f}^{\left(k\right)}\left(t\right)\right],\forall t\in I_T,$$

whenever the right side of ${{}^{c}D}_{0\ast }^{\alpha }$ exists, where $\alpha \in (n-1,n]$, $n\in {\mathbf{Z}}_{+}$.

The readers can find more of the excellent properties for the fractional calculus in refs. [35,36,37,38,39]. Here, we list several useful items for the subsequent segments.

Lemma 1. 

(1)

$J_0^{\alpha }$ is bounded from ${\mathrm{L}}^p\left(I_T\right)$ to ${\mathrm{L}}^p\left(I_T\right)$ for $p\in [1,\infty ]$ and $\mathrm{Re}\left(\alpha \right)>0$.

(2)

${{}^{c}D}_{0\ast }^{\alpha }f\left(t\right)=D_0^{\alpha }(f\left(t\right)-f\left(0\right))$ if $\alpha \in (0,1]$.

(3)

${{}^{c}D}_{0\ast }^{\alpha }f\left(t\right)={{}^{c}D}_0^{\alpha }f\left(t\right)$ if $f\in {\mathrm{AC}}^n\left({\overline{I}}_T\right)$ for $\alpha \in (n-1,n]$ and $n\in {\mathbf{Z}}_{+}$.

(4)

${{}^{c}D}_{0\ast }^{\alpha }{J}_0^{\alpha }f\left(t\right)=f\left(t\right)$ if $f\in {\mathrm{L}}^{\infty }\left(I_T\right)$ or $f\in \mathrm{C}\left({\overline{I}}_T\right)$ for $\alpha >0$.

(5)

${\displaystyle J_0^{\alpha }{{}^{c}D}_{0\ast }^{\alpha }f\left(t\right)=f\left(t\right)-\sum _{k=0}^{n-1}\frac{f^{\left(k\right)}\left(0\right)t^k}{k!}}$ if $f\in {\mathrm{AC}}^n\left({\overline{I}}_T\right)$ or $f\in {\mathrm{C}}^n\left({\overline{I}}_T\right)$ for $\alpha \in (n-1,n]$ and $n\in {\mathbf{Z}}_{+}$.

(6)

The Cauchy problem

$${{}^{c}D}_{0\ast }^{\alpha }y\left(t\right)=\mu y\left(t\right)+g\left(t\right),y\left(0\right)=y_0\in \mathbf{R},\alpha \in (0,1],$$

has a unique solution given by

$$y\left(t\right)=y_0{\mathbf{E}}_{\alpha }\left(\mu t^{\alpha }\right)+{\int }_0^t{(t-s)}^{\alpha -1}{\mathbf{E}}_{\alpha ,\alpha }\left(\mu {(t-s)}^{\alpha }\right)g\left(s\right)\mathrm{d}s,\forall t>0.$$

Here, we use the Mittag–Leffler functions defined by

$${\mathbf{E}}_{\alpha }\left(z\right)=\sum _{k=0}^{\infty }\frac{z^k}{\mathsf{\Gamma }(\alpha k+1)},{\mathbf{E}}_{\alpha ,\beta }\left(z\right)=\sum _{k=0}^{\infty }\frac{z^k}{\mathsf{\Gamma }(\alpha k+\beta )},z,\beta \in \mathbf{C},\mathrm{Re}\left(\alpha \right)>0.$$

(7)

${\mathbf{E}}_{\alpha }\left(0\right)={\mathbf{E}}_{\alpha ,\alpha }\left(0\right)=0$, ${\mathbf{E}}_{\alpha }\left(t\right)$ and ${\mathbf{E}}_{\alpha ,\alpha }\left(t\right)$ are increasing with respect to $t\in \mathbf{R}$ for $\alpha \in (0,1]$.

(8)

${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left[t^{\alpha }{\mathbf{E}}_{\alpha ,\alpha +1}\left(\mu t^{\alpha }\right)\right]=t^{\alpha -1}{\mathbf{E}}_{\alpha ,\alpha }\left(\mu t^{\alpha }\right)}$ for $\alpha ,\mu ,t\in \mathbf{C}$.

(9)

$\mu t^{\alpha }{\mathbf{E}}_{\alpha ,\alpha +1}\left(\mu t^{\alpha }\right)={\mathbf{E}}_{\alpha }\left(\mu t^{\alpha }\right)-1$ for $\alpha \in (0,1]$, $\mu \in \mathbf{R}$ and $t\in \mathbf{R}$, please see ref. [38] for details.

(10)

Let $\alpha >0$, $T>0$ be a constant, and a and b be nonnegative constants. If y is nonnegative and locally integrable for $t\in [0,T]$ satisfying

$$y\left(t\right)\le a+b{\int }_0^t{(t-s)}^{\alpha -1}y\left(s\right)\mathrm{d}s,\forall t\in [0,T],$$

then $y\left(t\right)\le a{\mathbf{E}}_{\alpha }\left(b\mathsf{\Gamma }\left(\alpha \right)t^{\alpha }\right)$ for $t\in [0,T]$. This is the Gronwall inequality in fractional calculus.

(11)

Let $y:{\overline{I}}_T\to \mathbf{R}$ be measurable and meet

$$y\left(t\right)\le {\mathbf{E}}_{\alpha }(-a{t}^{\alpha })b+{\int }_0^t{(t-s)}^{\alpha -1}{\mathbf{E}}_{\alpha ,\alpha }(-a{(t-s)}^{\alpha })[c+dy\left(s\right)]\mathrm{d}s,\forall t\in I_T,$$

where $b\in \mathbf{R}$, $a>0$, $c\ge 0$ and $d\in (0,a)$. Then,

$$y\left(t\right)\le {\mathbf{E}}_{\alpha }\left((-a+d)t^{\alpha }\right)b+c{\int }_0^t{(t-s)}^{\alpha -1}{\mathbf{E}}_{\alpha ,\alpha }\left((-a+d){(t-s)}^{\alpha }\right)\mathrm{d}s,\forall t\in I_T.$$

Please see ref. [39] for details.

Define

$${\mathrm{W}}^{\alpha ,p}(I_T,\mathbb{H})=\left\{f\in {\mathrm{L}}^p(I_T,\mathbb{H}):{{}^{c}D}_{0\ast }^{\alpha }f\mathrm{exists}\mathrm{and}\mathrm{belongs}\mathrm{to}{\mathrm{L}}^p(I_T,\mathbb{H})\right\}$$

and

$${\mathrm{W}}_{\mathrm{loc}}^{\alpha ,p}({\mathbf{R}}_{+},\mathbb{H})=\left\{f\in {\mathrm{W}}^{\alpha ,p}(I_T,\mathbb{H}):\forall T>0\right\}$$

for $\alpha \in (0,1]$ and $p\in [1,\infty )$. For any $f\in {\mathrm{W}}^{\alpha ,p}(I_T,\mathbb{H})$, it is endowed with the norm

$${\parallel f\parallel }_{\alpha ,p}={\left[{\parallel f\parallel }_{{\mathrm{L}}^p}^p+{\parallel {{}^{c}D}_{0\ast }^{\alpha }f\parallel }_{{\mathrm{L}}^p}^p\right]}^{\frac{1}{p}}.$$

Therem 3. 

${\mathrm{W}}^{\alpha ,p}(I_T,\mathbb{H})$ is a Banach space endowed with the norm ${\parallel ·\parallel }_{\alpha ,p}$ for $1\le p<\infty$. Furthermore, it is reflexible for $2\le p<\infty$ if $\mathbb{H}$ satisfies

(C)

The Clarkson inequality holds for $\mathbb{H}$:

$${\left({\parallel \mathbf{u}+\mathbf{v}\parallel }_{\mathbb{H}}^{r^{\prime }}+{\parallel \mathbf{u}-\mathbf{v}\parallel }_{\mathbb{H}}^{r^{\prime }}\right)}^{\frac{1}{r^{\prime }}}\le 2^{\frac{1}{r^{\prime }}}{\left({\parallel \mathbf{u}\parallel }_{\mathbb{H}}^r+{\parallel \mathbf{v}\parallel }_{\mathbb{H}}^r\right)}^{\frac{1}{r}},\forall \mathbf{u},\mathbf{v}\in \mathbb{H},$$

where $r\in (1,2]$, $\frac{1}{r}+\frac{1}{r^{\prime }}=1$.

Proof. 

Let $\left\{f_n\right\}$ be a Cauchy sequence belonging to ${\mathrm{W}}^{\alpha ,p}(I_T,\mathbb{H})$. Then $\left\{f_n\right\}$ and $\left\{{{}^{c}D}_{0\ast }^{\alpha }{f}_n\right\}$ are Cauchy sequences in ${\mathrm{L}}^p(I_T,\mathbb{H})$. By the completeness of ${\mathrm{L}}^p(I_T,\mathbb{H})$, there must exist ${\mathbf{f}}_0,{\mathbf{f}}_{\alpha }\in {\mathrm{L}}^p(I_T,\mathbb{H})$ such that

$$\parallel f_n-{\mathbf{f}}_0{\parallel }_{{\mathrm{L}}^p}\to 0,{\parallel {{}^{c}D}_{0\ast }^{\alpha }{f}_n-{\mathbf{f}}_{\alpha }\parallel }_{{\mathrm{L}}^p}\to 0,\mathrm{as}n\to \infty .$$

Noting that $J_0^{1-\alpha }$ is a continuous operator from ${\mathrm{L}}^p(I_T,\mathbb{H})$ to ${\mathrm{L}}^p(I_T,\mathbb{H})$ (see item (1) in Lemma 1), we have

$${\int }_{\epsilon }^t{{}^{c}D}_{0\ast }^{\alpha }{f}_n\left(s\right)\mathrm{d}s={\int }_{\epsilon }^t{D}_0^{\alpha }(f_n\left(s\right)-f_n\left(0\right))\mathrm{d}s=J_0^{1-\alpha }(f_n\left(t\right)-f_n\left(0\right))-J_0^{1-\alpha }(f_n\left(\epsilon \right)-f_n\left(0\right))$$

tends to

$${\int }_{\epsilon }^t{\mathbf{f}}_{\alpha }\left(s\right)\mathrm{d}s=J_0^{1-\alpha }({\mathbf{f}}_0\left(t\right)-{\mathbf{f}}_0\left(0\right))-J_0^{1-\alpha }({\mathbf{f}}_0\left(t\right)-{\mathbf{f}}_0\left(0\right)){|}_{t=\epsilon }$$

in ${\mathrm{L}}^p(I_T,\mathbb{H})$, as $n\to \infty$, $\epsilon \in (0,t)$ is a constant, $\forall t\in I_T$. Performing the differential operation in the above equality, we obtain

$${\mathrm{L}}^p(I_T,\mathbb{H})\ni {\mathbf{f}}_{\alpha }\left(t\right)=\frac{\mathrm{d}}{\mathrm{d}t}{J}_0^{1-\alpha }({\mathbf{f}}_0\left(t\right)-{\mathbf{f}}_0\left(0\right))={{}^{c}D}_{0\ast }^{\alpha }{\mathbf{f}}_0\left(t\right),\forall t\in I_T.$$

So, ${\mathbf{f}}_0$ is the limitation of the Cauchy sequence of $\left\{f_n\right\}$ in ${\mathrm{W}}^{\alpha ,p}(I_T,\mathbb{H})$, and ${\mathrm{W}}^{\alpha ,p}(I_T,\mathbb{H})$ is a Banach space endowed with the norm ${\parallel ·\parallel }_{\alpha ,p}$.

Next, the proof of reflexibility to ${\mathrm{W}}^{\alpha ,p}(I_T,\mathbb{H})$ is divided into two steps, as described.

Step 1. Here, we need the following Clarkson inequality of Lebesgue–Bochner spaces ${\mathrm{L}}^p(I_T,\mathbb{H})$ from the literature [40,41].

$${\left({\parallel f+g\parallel }_{{\mathrm{L}}^p}^p+{\parallel f-g\parallel }_{{\mathrm{L}}^p}^p\right)}^{\frac{1}{p}}\le 2^{\frac{1}{p}}{\left({\parallel f\parallel }_{{\mathrm{L}}^p}^{p^{\prime }}+{\parallel g\parallel }_{{\mathrm{L}}^p}^{p^{\prime }}\right)}^{\frac{1}{p^{\prime }}},\forall f,g\in {\mathrm{L}}^p(I_T,\mathbb{H})$$

(10)

holds for $p\in [r^{\prime },+\infty )\subseteq [2,\infty )$ if and only if (C) holds.

Clarkson’s inequality also holds in ${\mathrm{W}}^{\alpha ,p}(I_T,\mathbb{H})$, i.e.,

$${\parallel f+g\parallel }_{\alpha ,p}^p{+\parallel f-g\parallel }_{\alpha ,p}^p\le 2^{p-1}{(\parallel f\parallel }_{\alpha ,p}^p+{\parallel g\parallel }_{\alpha ,p}^p),\forall f,g\in {\mathrm{W}}^{\alpha ,p}(I_T,\mathbb{H}).$$

(11)

Using (10) leads to

$$\begin{array}{cc}& {\parallel f+g\parallel }_{\alpha ,p}^p+{\parallel f-g\parallel }_{\alpha ,p}^p\hfill \\ \hfill =& \left[{\parallel f+g\parallel }_{{\mathrm{L}}^p}^p+{\parallel f-g\parallel }_{{\mathrm{L}}^p}^p\right]+\left[\parallel {{}^{c}D}_{0\ast }^{\alpha }{(f+g)\parallel }_{{\mathrm{L}}^p}^p+{\parallel {{}^{c}D}_{0\ast }^{\alpha }(f-g)\parallel }_{{\mathrm{L}}^p}^p\right]\hfill \\ \hfill \le & 2{\left[{\parallel f\parallel }_{{\mathrm{L}}^p}^{p^{\prime }}+{\parallel g\parallel }_{{\mathrm{L}}^p}^{p^{\prime }}\right]}^{\frac{p}{p^{\prime }}}+2{\left[\parallel {{}^{c}D}_{0\ast }^{\alpha }{f\parallel }_{{\mathrm{L}}^p}^{p^{\prime }}+{\parallel {{}^{c}D}_{0\ast }^{\alpha }g\parallel }_{{\mathrm{L}}^p}^{p^{\prime }}\right]}^{\frac{p}{p^{\prime }}}\hfill \\ \hfill \le & 2^{\frac{p}{p^{\prime }}}{(\parallel f\parallel }_{\alpha ,p}^p+{\parallel g\parallel }_{\alpha ,p}^p),\forall f,g\in {\mathrm{W}}^{\alpha ,p}(I_T,\mathbb{H})\hfill \end{array}$$

.

Noting that $\frac{p}{p^{\prime }}=p-1$, so (11) is valid.

Step 2.${\mathrm{W}}^{\alpha ,p}(I_T,\mathbb{H})$ is uniformly convex. Let $\nu \in (0,1)$, ${\parallel f\parallel }_{\alpha ,p}\le 1$, ${\parallel g\parallel }_{\alpha ,p}\le 1$, and ${\parallel f-g\parallel }_{\alpha ,p}>\nu$ for some $f,g\in {\mathrm{W}}^{\alpha ,p}(I_T,\mathbb{H})$. According to (11), we obtain

$$\parallel \frac{f+g}{2}{\parallel }_{\alpha ,p} \le \sqrt[p]{1-\frac{{\nu }^p}{2^p}}=1-\delta ,$$

(12)

where $\delta :=1-\sqrt[p]{1-\frac{{\nu }^p}{2^p}}$. Inequality (12) implies ${\mathrm{W}}^{\alpha ,2}(I_T,\mathbb{H})$ is uniformly convex. By the Milman–Pettis theorem in Theorem 3.31 of ref. [32], ${\mathrm{W}}^{\alpha ,p}(I_T,\mathbb{H})$ is reflexible for p ≥ 2. This achieves the proof. □

Remark 3. 

In Theorem 3, we only verify the reflexibility of ${\mathrm{W}}^{\alpha ,p}(I_T,\mathbb{H})$ for $\alpha \in (0,1]$ and $p\in [2,\infty )$. For $\alpha \in (1,\infty )$ or $p\in [1,2)$, interested readers may study this further. In refs. [42,43], Riemann–Liouville fractional Sobolev spaces were introduced, and some embedding properties were achieved. Note that there are few studies focusing on the reflexibility of nonlocal Sobolev spaces, Theorem 3 fills this gap.

3. Proof of Theorem 1

Proof of Theorem 1. 

(i)Denote

$${\mathbb{K}}_n=\mathrm{span}\{e_1,\dots ,e_n\},{\tilde{\mathbb{K}}}_n=\left\{\sum _{k=1}^n{\varpi }_k\left(t\right)e_k:{\varpi }_k\in {\mathrm{L}}_{\mathrm{loc}}\left({\mathbf{R}}_{+}\right)\right\},n\in {\mathbf{Z}}_{+}.$$

Consider the following nonlocal differential system:

$$\left\{\begin{array}{cc}{{}^{c}D}_{0\ast }^{\alpha }{\mathbf{w}}_k^n\left(t\right)={\langle \mathcal{G}{\mathbf{u}}_n\left(t\right),e_k\rangle }_{\mathbb{H}}={\langle \mathcal{A}{\mathbf{u}}_n\left(t\right),e_k\rangle }_{\mathbb{H}}+{\langle \mathcal{F}\left({\mathbf{u}}_n\left(t\right)\right),e_k\rangle }_{\mathbb{H}},\hfill & \mathrm{in}{\mathbf{R}}_{+},\hfill \\ {\mathbf{w}}_k^n\left(0\right)={\langle \phi ,e_k\rangle }_{\mathbb{H}},\hfill & 1\le k\le n,\hfill \end{array}\right.$$

(13)

where ${\displaystyle {\mathbf{u}}_n\left(t\right)=\sum _{k=1}^n{\mathbf{w}}_k^n\left(t\right)e_k}$ for all $t\in {\overline{I}}_T$ and $n\in {\mathbf{Z}}_{+}$. By employing the work in ref. [44], Equation (13) possesses a unique solution

$$\{{\mathbf{w}}_1^n\left(t\right),\dots ,{\mathbf{w}}_n^n\left(t\right)\},\forall t\in {\overline{I}}_T,n\in {\mathbf{Z}}_{+},$$

which belongs to the space ${\mathrm{H}}^{\alpha }\left({\overline{I}}_T\right)$.

In light of Lemma A2, it follows from Equation (13) that

$$\left\{\begin{array}{c}{\langle {\mathbf{u}}_n\left(t\right),\mathbf{v}\rangle }_{\mathbb{H}}={\langle \phi ,\mathbf{v}\rangle }_{\mathbb{H}}+J_0^{\alpha }\left[{\langle \mathcal{A}{\mathbf{u}}_n\left(t\right),\mathbf{v}\rangle }_{\mathbb{H}}+{\langle \mathcal{F}\left({\mathbf{u}}_n\left(t\right)\right),\mathbf{v}\rangle }_{\mathbb{H}}\right],\forall \mathbf{v}\in {\mathbb{K}}_n,\hfill \\ J_0^{\alpha }{\langle {{}^{c}D}_{0\ast }^{\alpha }{\mathbf{u}}_n\left(t\right),\mathbf{v}\rangle }_{\mathbb{H}}=J_0^{\alpha }\left[{\langle \mathcal{A}{\mathbf{u}}_n\left(t\right),\mathbf{v}\rangle }_{\mathbb{H}}+{\langle \mathcal{F}\left({\mathbf{u}}_n\left(t\right)\right),\mathbf{v}\rangle }_{\mathbb{H}}\right],\forall \mathbf{v}\in {\tilde{\mathbb{K}}}_n,t\in {\overline{I}}_T\backslash \left\{0\right\}.\hfill \end{array}\right.$$

(14)

By $\left({\mathcal{H}}_2\right)$, $\{e_k:k\in {\mathbf{Z}}_{+}\}$ is an orthogonal set of ${\mathbb{H}}_1$. From Lemma A1, we have

$$\begin{array}{ccc}\hfill J_0^{\alpha }{〈{{}^{c}D}_{0\ast }^{\alpha }{\mathbf{u}}_n\left(t\right),{\mathbf{u}}_n\left(t\right)〉}_{{\mathbb{H}}^{\prime }}& =& J_0^{\alpha }{〈{{}^{c}D}_{0\ast }^{\alpha }\sum _{k=1}^n{\mathbf{w}}_k^n\left(t\right)e_k,\sum _{k=1}^n{\mathbf{w}}_k^n\left(t\right)e_k〉}_{{\mathbb{H}}^{\prime }}\hfill \\ & =& J_0^{\alpha }\sum _{k=1}^n{\mathbf{w}}_k^n\left(t\right){{}^{c}D}_{0\ast }^{\alpha }{\mathbf{w}}_k^n\left(t\right){\langle e_k,e_k\rangle }_{{\mathbb{H}}^{\prime }}\hfill \\ & \ge & \frac{1}{2}{J}_0^{\alpha }\sum _{k=1}^n{{}^{c}D}_{0\ast }^{\alpha }{\left[{\mathbf{w}}_k^n\left(t\right)\right]}^2{\langle e_k,e_k\rangle }_{{\mathbb{H}}^{\prime }}\hfill \\ & =& \frac{1}{2}{J}_0^{\alpha }{{}^{c}D}_{0\ast }^{\alpha }{\langle {\mathbf{u}}_n\left(t\right),{\mathbf{u}}_n\left(t\right)\rangle }_{{\mathbb{H}}^{\prime }},\forall t\in {\overline{I}}_T\backslash \left\{0\right\}.\hfill \end{array}$$

(15)

Here, ${\mathbb{H}}^{\prime }=\mathbb{H}\mathrm{or}{\mathbb{H}}_1$.

Let $\mathbf{v}={\mathbf{u}}_n\left(t\right)$ in the second equation in Equation (14); using (15), Lemma A2, $\left({\mathcal{H}}_1\right)$, $\left({\mathcal{H}}_3\right)$, and Schwarz and Young inequalities yields

$$\begin{array}{ccc}\hfill \frac{1}{2}{\parallel {\mathbf{u}}_n\left(t\right)\parallel }_{\mathbb{H}}^2& \le & \frac{1}{2}{\parallel {\mathbf{u}}_n\left(0\right)\parallel }_{\mathbb{H}}^2+J_0^{\alpha }{\langle {{}^{c}D}_{0\ast }^{\alpha }{\mathbf{u}}_n\left(t\right),{\mathbf{u}}_n\left(t\right)\rangle }_{\mathbb{H}}\hfill \\ & =& \frac{1}{2}{\parallel {\phi }_n\parallel }_{\mathbb{H}}^2+J_0^{\alpha }\left[-\sum _{k=1}^n{\lambda }_k{\left[{\mathbf{w}}_k^n\left(t\right)\right]}^2+{\langle \mathcal{F}\left({\mathbf{u}}_n\left(t\right)\right),{\mathbf{u}}_n\left(t\right)\rangle }_{\mathbb{H}}\right]\hfill \\ & \le & \frac{1}{2}{\parallel {\phi }_n\parallel }_{\mathbb{H}}^2+J_0^{\alpha }\left[-{\lambda }_1\parallel {\mathbf{u}}_n{\left(t\right)\parallel }_{\mathbb{H}}^2+\parallel \mathcal{F}\left({\mathbf{u}}_n\left(t\right)\right){\parallel }_{\mathbb{H}}{\parallel {\mathbf{u}}_n\left(t\right)\parallel }_{\mathbb{H}}\right]\hfill \\ & \le & \frac{1}{2}{\parallel {\phi }_n\parallel }_{\mathbb{H}}^2+J_0^{\alpha }\left[-\left({\lambda }_1-\frac{1}{2}\right)\parallel {\mathbf{u}}_n{\left(t\right)\parallel }_{\mathbb{H}}^2+\frac{1}{2}{\parallel \mathcal{F}\left({\mathbf{u}}_n\left(t\right)\right)\parallel }_{\mathbb{H}}^2\right]\hfill \\ & \le & \frac{1}{2}{\parallel {\phi }_n\parallel }_{\mathbb{H}}^2+J_0^{\alpha }\left[-\left({\lambda }_1-\frac{1}{2}-L_F^2\right)\parallel {\mathbf{u}}_n{\left(t\right)\parallel }_{\mathbb{H}}^2+{\parallel \mathcal{F}\left(0\right)\parallel }_{\mathbb{H}}^2\right]\hfill \\ & \le & \frac{1}{2}\parallel {\phi }_n{\parallel }_{\mathbb{H}}^2+\frac{T^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}{\parallel \mathcal{F}\left(0\right)\parallel }_{{\mathrm{L}}^{\infty }}^2+d_1{J}_0^{\alpha }\parallel {\mathbf{u}}_n{\left(t\right)\parallel }_{\mathbb{H}}^2,\forall t\in {\overline{I}}_T\backslash \left\{0\right\},\hfill \end{array}$$

(16)

where $d_1:=|{\lambda }_1-\frac{1}{2}-L_F^2|\mathrm{and}{\phi }_n={\displaystyle \sum _{k=1}^n}{\langle \phi ,e_k\rangle }_{\mathbb{H}}{e}_k\mathrm{for\; all}n\in {\mathbf{Z}}_{+}$. According to the Bessel inequality and item (10) in Lemma 1, (16) leads to

$${\parallel {\mathbf{u}}_n\left(t\right)\parallel }_{\mathbb{H}}^2\le \left[{\parallel \phi \parallel }_{\mathbb{H}}^2+\frac{2T^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}{\parallel \mathcal{F}\left(0\right)\parallel }_{{\mathrm{L}}^{\infty }}^2\right]{\mathbf{E}}_{\alpha }\left(2d_1{T}^{\alpha }\right):=M_1^T,\forall t\in {\overline{I}}_T.$$

(17)

Due to $\left({\mathcal{H}}_2\right)$, ${\langle \eta ,e_k\rangle }_{\mathbb{H}}=\frac{a_1}{{\lambda }_k}{\langle \eta ,e_k\rangle }_{{\mathbb{H}}_1}$ for all $\eta \in {\mathbb{H}}_1$. As a consequence,

$${\langle {\phi }_n,{\phi }_n\rangle }_{{\mathbb{H}}_1}=\sum _{k=1}^n|{\langle \phi ,e_k\rangle }_{\mathbb{H}}{|}^2{\langle e_k,e_k\rangle }_{{\mathbb{H}}_1}=\sum _{k=1}^n\frac{a_1}{{\lambda }_k}|{\langle \phi ,e_k\rangle }_{{\mathbb{H}}_1}{|}^2=\sum _{k=1}^n|{\langle \phi ,e_k^{\prime }\rangle }_{{\mathbb{H}}_1}{|}^2\le {\parallel \phi \parallel }_{{\mathbb{H}}_1}^2.$$

Here, $\left\{e_k^{\prime }=\frac{e_k}{\parallel e_k{\parallel }_{{\mathbb{H}}_1}}\right\}$ is an orthonormal set in ${\mathbb{H}}_1$. Let $\mathbf{v}=\mathcal{A}{\mathbf{u}}_n\left(t\right)$ in the second equation in Equation (14); from (15), we have

$$\begin{array}{ccc}\hfill \frac{1}{2}{\parallel {\mathbf{u}}_n\left(t\right)\parallel }_{{\mathbb{H}}_1}^2& \le & \frac{1}{2}{\parallel {\mathbf{u}}_n\left(0\right)\parallel }_{{\mathbb{H}}_1}^2+J_0^{\alpha }{\langle {{}^{c}D}_{0\ast }^{\alpha }{\mathbf{u}}_n\left(t\right),{\mathbf{u}}_n\left(t\right)\rangle }_{{\mathbb{H}}_1}\hfill \\ & =& \frac{1}{2}{\parallel {\mathbf{u}}_n\left(0\right)\parallel }_{{\mathbb{H}}_1}^2-\frac{1}{a_1}{J}_0^{\alpha }{\langle {{}^{c}D}_{0\ast }^{\alpha }{\mathbf{u}}_n\left(t\right),\mathcal{A}{\mathbf{u}}_n\left(t\right)\rangle }_{\mathbb{H}}\hfill \\ & =& \frac{1}{2}{\parallel {\phi }_n\parallel }_{{\mathbb{H}}_1}^2-\frac{1}{a_1}{J}_0^{\alpha }{\langle \mathcal{G}{\mathbf{u}}_n\left(t\right),\mathcal{A}{\mathbf{u}}_n\left(t\right)\rangle }_{\mathbb{H}}\hfill \\ & =& \frac{1}{2}{\parallel {\phi }_n\parallel }_{{\mathbb{H}}_1}^2-\frac{1}{a_1}{J}_0^{\alpha }\left[\parallel \mathcal{A}{\mathbf{u}}_n{\left(t\right)\parallel }_{\mathbb{H}}^2+{\langle \mathcal{F}\left({\mathbf{u}}_n\left(t\right)\right),\mathcal{A}{\mathbf{u}}_n\left(t\right)\rangle }_{\mathbb{H}}\right]\hfill \\ & \le & \frac{1}{2}{\parallel {\phi }_n\parallel }_{{\mathbb{H}}_1}^2-\frac{1}{a_1}{J}_0^{\alpha }\left[\parallel \mathcal{A}{\mathbf{u}}_n{\left(t\right)\parallel }_{\mathbb{H}}^2-\parallel \mathcal{F}\left({\mathbf{u}}_n\left(t\right)\right){\parallel }_{\mathbb{H}}{\parallel \mathcal{A}{\mathbf{u}}_n\left(t\right)\parallel }_{\mathbb{H}}\right]\hfill \\ & \le & \frac{1}{2}{\parallel {\phi }_n\parallel }_{{\mathbb{H}}_1}^2+\frac{1}{a_1}{J}_0^{\alpha }\left[-\frac{1}{2}\parallel \mathcal{A}{\mathbf{u}}_n{\left(t\right)\parallel }_{\mathbb{H}}^2+\frac{1}{2}{\parallel \mathcal{F}\left({\mathbf{u}}_n\left(t\right)\right)\parallel }_{\mathbb{H}}^2\right]\hfill \\ & \le & \frac{1}{2}{\parallel {\phi }_n\parallel }_{{\mathbb{H}}_1}^2+\frac{1}{a_1}{J}_0^{\alpha }\left[-\frac{a_3}{2}\parallel {\mathbf{u}}_n{\left(t\right)\parallel }_{{\mathbb{H}}_2}^2+\left(\frac{a_2}{2}+L_F^2\right)\parallel {\mathbf{u}}_n{\left(t\right)\parallel }_{\mathbb{H}}^2+{\parallel \mathcal{F}\left(0\right)\parallel }_{\mathbb{H}}^2\right]\hfill \\ & \le & -\frac{a_3}{2a_1}{J}_0^{\alpha }\parallel {\mathbf{u}}_n{\left(t\right)\parallel }_{{\mathbb{H}}_2}^2+\frac{1}{2}\parallel {\phi }_n{\parallel }_{{\mathbb{H}}_1}^2+\frac{T^{\alpha }{\parallel \mathcal{F}\left(0\right)\parallel }_{{\mathrm{L}}^{\infty }}^2}{a_1\mathsf{\Gamma }(\alpha +1)}+\frac{(a_2+2L_F^2)T^{\alpha }{M}_1^T}{2a_1\mathsf{\Gamma }(\alpha +1)}\hfill \end{array}$$

is equivalent to

$$a_1\parallel {\mathbf{u}}_n{\left(t\right)\parallel }_{{\mathbb{H}}_1}^2+a_3{J}_0^{\alpha }{\parallel {\mathbf{u}}_n\left(t\right)\parallel }_{{\mathbb{H}}_2}^2\le a_1{\parallel \phi \parallel }_{{\mathbb{H}}_1}^2+\frac{2T^{\alpha }{\parallel \mathcal{F}\left(0\right)\parallel }_{{\mathrm{L}}^{\infty }}^2}{\mathsf{\Gamma }(\alpha +1)}+\frac{(a_2+2L_F^2)T^{\alpha }{M}_1^T}{\mathsf{\Gamma }(\alpha +1)}$$

for all $t\in {\overline{I}}_T$. (17) and $\phi \in {\mathbb{H}}_1$ lead to

$$\parallel {\mathbf{u}}_n{\left(t\right)\parallel }_{{\mathbb{H}}_1}^2\le {\parallel \phi \parallel }_{{\mathbb{H}}_1}^2+\frac{2T^{\alpha }{\parallel \mathcal{F}\left(0\right)\parallel }_{{\mathrm{L}}^{\infty }}^2}{a_1\mathsf{\Gamma }(\alpha +1)}+\frac{(a_2+2L_F^2)T^{\alpha }{M}_1^T}{a_1\mathsf{\Gamma }(\alpha +1)}:=M_2^T$$

(18)

and

$$\frac{1}{T^{1-\alpha }\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t\parallel {\mathbf{u}}_n{\left(s\right)\parallel }_{{\mathbb{H}}_2}^2\mathrm{d}s\le J_0^{\alpha }{\parallel {\mathbf{u}}_n\left(t\right)\parallel }_{{\mathbb{H}}_2}^2\le \frac{a_1{M}_2^T}{a_3}:=M_3^T,\forall t\in {\overline{I}}_T.$$

(19)

Third, let $\mathbf{v}={{}^{c}D}_{0\ast }^{\alpha }{\mathbf{u}}_n\left(t\right)$ in the second equation in Equation (14), which gives

$$\begin{array}{ccc}\hfill J_0^{\alpha }\parallel {{}^{c}D}_{0\ast }^{\alpha }{\mathbf{u}}_n\left(t\right){\parallel }_{\mathbb{H}}^2& =& J_0^{\alpha }\left[{\langle \mathcal{A}{\mathbf{u}}_n\left(t\right),{{}^{c}D}_{0\ast }^{\alpha }{\mathbf{u}}_n\left(t\right)\rangle }_{\mathbb{H}}+{\langle \mathcal{F}\left({\mathbf{u}}_n\left(t\right)\right),{{}^{c}D}_{0\ast }^{\alpha }{\mathbf{u}}_n\left(t\right)\rangle }_{\mathbb{H}}\right]\hfill \\ & \le & J_0^{\alpha }\left[\parallel \mathcal{A}{\mathbf{u}}_n\left(t\right){\parallel }_{\mathbb{H}}\parallel {{}^{c}D}_{0\ast }^{\alpha }{\mathbf{u}}_n\left(t\right){\parallel }_{\mathbb{H}}+\parallel \mathcal{F}\left({\mathbf{u}}_n\left(t\right)\right){\parallel }_{\mathbb{H}}\parallel {{}^{c}D}_{0\ast }^{\alpha }{\mathbf{u}}_n\left(t\right){\parallel }_{\mathbb{H}}\right]\hfill \\ & \le & J_0^{\alpha }\left[\frac{1}{2}\parallel {{}^{c}D}_{0\ast }^{\alpha }{\mathbf{u}}_n\left(t\right){\parallel }_{\mathbb{H}}^2+4\parallel \mathcal{A}{\mathbf{u}}_n\left(t\right){\parallel }_{\mathbb{H}}^2+4\parallel \mathcal{F}\left({\mathbf{u}}_n\left(t\right)\right){\parallel }_{\mathbb{H}}^2\right],\hfill \end{array}$$

which induces

$$\begin{array}{ccc}\hfill \frac{1}{2T^{1-\alpha }\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t\parallel {{}^{c}D}_{0\ast }^{\alpha }{\mathbf{u}}_n\left(s\right){\parallel }_{\mathbb{H}}^2\mathrm{d}s& \le & J_0^{\alpha }\left[4a_4{\parallel {\mathbf{u}}_n\left(t\right)\parallel }_{{\mathbb{H}}_2}^2+4\parallel \mathcal{F}\left({\mathbf{u}}_n\left(t\right)\right){\parallel }_{\mathbb{H}}^2\right]\hfill \\ & \le & 4a_4{M}_3^T+\frac{32T^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\left[L_F^2{M}_1^T+{\parallel \mathcal{F}\left(0\right)\parallel }_{{\mathrm{L}}^{\infty }}^2\right]\hfill \end{array}$$

(20)

for all $t\in {\overline{I}}_T$.

Inequalities (17)–(20) indicate that the solution of Equation (13) can be extended to ${\mathbf{R}}_{+}$, and ${\left\{{\mathbf{u}}_n\right\}}_{n\in {\mathbf{Z}}_{+}}$ is bounded in ${\mathrm{L}}_{\mathrm{loc}}^{\infty }({\mathbf{R}}_{+},{\mathbb{H}}_1)\cap {\mathrm{L}}_{\mathrm{loc}}^2({\mathbf{R}}_{+},{\mathbb{H}}_2)\cap {\mathrm{W}}_{\mathrm{loc}}^{\alpha ,2}({\mathbf{R}}_{+},\mathbb{H})$. Thus, ${\left\{{\mathbf{u}}_n\right\}}_{n\in {\mathbf{Z}}_{+}}$ possesses a subsequence (still denoted by ${\left\{{\mathbf{u}}_n\right\}}_{n\in {\mathbf{Z}}_{+}}$) such that

$${\mathbf{u}}_n\rightharpoonup {\mathbf{u}}_0\mathrm{in}{\mathrm{L}}_{\mathrm{loc}}^2({\mathbf{R}}_{+},{\mathbb{H}}_2)\cap {\mathrm{W}}_{\mathrm{loc}}^{\alpha ,2}({\mathbf{R}}_{+},\mathbb{H}),{\mathbf{u}}_n\stackrel{*}{\rightharpoonup }{\mathbf{u}}_0\mathrm{in}{\mathrm{L}}_{\mathrm{loc}}^{\infty }({\mathbf{R}}_{+},{\mathbb{H}}_1),$$

as $n\to \infty$. From the compactness of ${\mathbb{H}}_1\hookrightarrow \mathbb{H}$, we can choose a subsequence of ${\left\{{\mathbf{u}}_n\right\}}_{n\in {\mathbf{Z}}_{+}}$ (still expressed by itself) such that ${\mathbf{u}}_n\left(t\right)\to {\mathbf{u}}_0\left(t\right)$ in $\mathbb{H}$ for each $t\in {\mathbf{R}}_{+}$. Thus,

$$\left|{〈\mathcal{F}\left({\mathbf{u}}_n\left(t\right)\right)-\mathcal{F}\left({\mathbf{u}}_0\left(t\right)\right),\mathbf{v}〉}_{\mathbb{H}}\right|\le \parallel \mathcal{F}\left({\mathbf{u}}_n\left(t\right)\right)-\mathcal{F}\left({\mathbf{u}}_0\left(t\right)\right){\parallel }_{\mathbb{H}}{\parallel \mathbf{v}\parallel }_{\mathbb{H}}$$

$$\le L_F\parallel {\mathbf{u}}_n\left(t\right)-{\mathbf{u}}_0{\left(t\right)\parallel }_{\mathbb{H}}{\parallel \mathbf{v}\parallel }_{\mathbb{H}}$$

tends to 0 as $n\to \infty$ for each $t\in {\mathbf{R}}_{+}$. Letting $n\to \infty$ in the first equation in Equation (14) and recalling the Lebesgue dominated convergence theorem, we obtain from $\left({\mathcal{H}}_2\right)$ that

$$\begin{array}{c}\hfill {\langle {\mathbf{u}}_0\left(t\right),\mathbf{v}\rangle }_{\mathbb{H}}={\langle \phi ,\mathbf{v}\rangle }_{\mathbb{H}}+J_0^{\alpha }{〈\mathcal{A}{\mathbf{u}}_0\left(t\right)+\mathcal{F}\left({\mathbf{u}}_0\left(t\right)\right),\mathbf{v}〉}_{\mathbb{H}},\forall \mathbf{v}\in {\mathbb{K}}_n,t\in {\mathbf{R}}_{+}.\end{array}$$

(21)

According to $\left({\mathcal{H}}_2\right)$ and $\left({\mathcal{H}}_3\right)$, we readily obtain ${〈\mathcal{A}{\mathbf{u}}_0\left(t\right)+\mathcal{F}\left({\mathbf{u}}_0\left(t\right)\right),\mathbf{v}〉}_{\mathbb{H}}\in {\mathrm{L}}_{\mathrm{loc}}^{\infty }\left({\mathbf{R}}_{+}\right)$, $\forall \mathbf{v}\in {\mathbb{K}}_n$, $t\in {\mathbf{R}}_{+}.$ From item (4) in Lemma 1, $\left({\mathcal{H}}_4\right)$ and the density of ${\bigcup }_{n=1}^{\infty }{\mathbb{K}}_n$ in $\mathbb{H}$, we have

$$\left\{\begin{array}{cc}{\langle {{}^{c}D}_{0\ast }^{\alpha }{\mathbf{u}}_0\left(t\right),\mathbf{v}\rangle }_{\mathbb{H}}={〈\mathcal{A}{\mathbf{u}}_0\left(t\right)+\mathcal{F}\left({\mathbf{u}}_0\left(t\right)\right),\mathbf{v}〉}_{\mathbb{H}},\hfill & \mathrm{in}{\mathbf{R}}_{+},\hfill \\ {\langle {\mathbf{u}}_0\left(0\right),\mathbf{v}\rangle }_{\mathbb{H}}={\langle \phi ,\mathbf{v}\rangle }_{\mathbb{H}},\hfill & \mathbf{v}\in \mathbb{H},\hfill \end{array}\right.$$

which implies that ${\mathbf{u}}_0\in \mathcal{L}({\mathbf{R}}_0,\mathbb{H})$ is a global strong solution of Equation (1).

(ii)${\mathbf{u}}_0\in {\mathrm{H}}^{\alpha }({\mathbf{R}}_0,\mathbb{H})$ is verified. To this end, we first deduce that ${\mathbf{u}}_0$ is bounded in $\mathbb{H}$ and ${\mathbb{H}}_1$. Because ${\mathbf{u}}_n\left(t\right)\to {\mathbf{u}}_0\left(t\right)$ in $\mathbb{H}$ for each $t\in {\mathbf{R}}_{+}$, then

$$\parallel {\mathbf{u}}_0{\left(t\right)\parallel }_{\mathbb{H}}\le \parallel {\mathbf{u}}_n{\left(t\right)\parallel }_{\mathbb{H}} + \parallel {\mathbf{u}}_n\left(t\right)-{\mathbf{u}}_0{\left(t\right)\parallel }_{\mathbb{H}}\le \sqrt{M_1^T}+{\parallel {\mathbf{u}}_n\left(t\right)-{\mathbf{u}}_0\left(t\right)\parallel }_{\mathbb{H}}\to \sqrt{M_1^T},$$

as $n\to \infty$, i.e., $\parallel {\mathbf{u}}_0{\left(t\right)\parallel }_{\mathbb{H}}^2\le M_1^T$, $\forall t\in {\overline{I}}_T.$ For another, by $\left({\mathcal{H}}_2\right)$, we acquire

$$\left|{\langle \mathcal{A}{\mathbf{u}}_m\left(t\right),{\mathbf{u}}_n\left(t\right)\rangle }_{\mathbb{H}}\right|=a_1|{\langle {\mathbf{u}}_m\left(t\right),{\mathbf{u}}_n\left(t\right)\rangle }_{{\mathbb{H}}_1}| \le a_1\parallel {\mathbf{u}}_m{\left(t\right)\parallel }_{{\mathbb{H}}_1}{\parallel {\mathbf{u}}_n\left(t\right)\parallel }_{{\mathbb{H}}_1}\le a_1{M}_2^T,$$

where $m,n\in {\mathbf{Z}}_{+}$, $t\in {\overline{I}}_T\backslash \left\{0\right\}$. Letting $n\to \infty$ and $m\to \infty$ successively in the above inequality, we have $\parallel {\mathbf{u}}_0{\left(t\right)\parallel }_{{\mathbb{H}}_1}^2=\frac{1}{a_1}\left|{\langle \mathcal{A}{\mathbf{u}}_0\left(t\right),{\mathbf{u}}_0\left(t\right)\rangle }_{\mathbb{H}}\right|\le M_2^T$ for all $t\in {\overline{I}}_T$.

Take arbitrary $t_0\in {\mathbf{R}}_{+}$ and $ϵ\in (0,1)$. By the boundedness of ${\mathbf{u}}_0$ in $\mathbb{H}$, there exists a constant $M_{\mathcal{F}}>0$ such that $\mathrm{ess}{sup}_{t\in I_T}{\parallel \mathcal{F}\left({\mathbf{u}}_0\left(t\right)\right)\parallel }_{\mathbb{H}}\le M_{\mathcal{F}}$. Without loss of generality, assume that $t_1$ is an arbitrary point in the neighborhood of $t_0$ with $0<t_1-t_0<{\left[\frac{\alpha \mathsf{\Gamma }\left(\alpha \right)ϵ}{2(a_1{M}_2^T+M_{\mathcal{F}}{M}_1^T)}\right]}^{\frac{1}{\alpha }}$. There exists $T>0$ such that $t_0,t_1\in I_T$. According to Equation (21) and $\left({\mathcal{H}}_2\right)$, we obtain

$$\begin{array}{ccc}\hfill \left|{\langle {\mathbf{u}}_0\left(t_1\right)-{\mathbf{u}}_0\left(t_0\right),\mathbf{v}\rangle }_{\mathbb{H}}\right|& \le & \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{t_0}^{t_1}{\mathcal{J}}_s^{\alpha }\left(t_1\right)|a_1{\langle {\mathbf{u}}_0\left(s\right),\mathbf{v}\rangle }_{{\mathbb{H}}_1}+{\langle \mathcal{F}\left({\mathbf{u}}_0\left(s\right)\right),\mathbf{v}\rangle }_{\mathbb{H}}|\mathrm{d}s\hfill \\ & & +\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{t_0}\left[{\mathcal{J}}_s^{\alpha }\left(t_0\right)-{\mathcal{J}}_s^{\alpha }\left(t_1\right)\right]|a_1{\langle {\mathbf{u}}_0\left(s\right),\mathbf{v}\rangle }_{{\mathbb{H}}_1}+{\langle \mathcal{F}\left({\mathbf{u}}_0\left(s\right)\right),\mathbf{v}\rangle }_{\mathbb{H}}|\mathrm{d}s\hfill \\ & \le & \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{t_0}^{t_1}{\mathcal{J}}_s^{\alpha }\left(t_1\right)\left(a_1\sqrt{M_2^T}{\parallel \mathbf{v}\parallel }_{{\mathbb{H}}_1}+M_{\mathcal{F}}{\parallel \mathbf{v}\parallel }_{\mathbb{H}}\right)\mathrm{d}s\hfill \\ & & +\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{t_0}\left[{\mathcal{J}}_s^{\alpha }\left(t_0\right)-{\mathcal{J}}_s^{\alpha }\left(t_1\right)\right]\left(a_1\sqrt{M_2^T}{\parallel \mathbf{v}\parallel }_{{\mathbb{H}}_1}+M_{\mathcal{F}}{\parallel \mathbf{v}\parallel }_{\mathbb{H}}\right)\mathrm{d}s\hfill \\ & \le & \frac{2{(t_1-t_0)}^{\alpha }-t_1^{\alpha }+t_0^{\alpha }}{\alpha \mathsf{\Gamma }\left(\alpha \right)}\left(a_1\sqrt{M_2^T}{\parallel \mathbf{v}\parallel }_{{\mathbb{H}}_1}+M_{\mathcal{F}}{\parallel \mathbf{v}\parallel }_{\mathbb{H}}\right),\forall \mathbf{v}\in \mathbb{H}.\hfill \end{array}$$

Taking $\mathbf{v}={\mathbf{u}}_0\left(t_1\right)-{\mathbf{u}}_0\left(t_0\right)\in \mathbb{H}$ for fixed t0, t1 results in

$$\parallel {\mathbf{u}}_0\left(t_1\right)-{\mathbf{u}}_0\left(t_0\right){\parallel }_{\mathbb{H}}^2 \le \frac{2}{\alpha \mathsf{\Gamma }\left(\alpha \right)}\left(a_1{M}_2^T+M_{\mathcal{F}}{M}_1^T\right){(t_1-t_0)}^{\alpha }<ϵ.$$

$\frac{2}{\alpha \mathsf{\Gamma }\left(\alpha \right)}\left(a_1{M}_2^T+M_{\mathcal{F}}{M}_1^T\right)$ is independent of t0, t1. Then, ${\mathbf{u}}_0\in {\mathrm{H}}^{\alpha }({\mathbf{R}}_{+},\mathbb{H}).$

Taking $t_0=0$ and $t_1$ the same as before, we have

$$\begin{array}{ccc}\hfill \left|{\langle {\mathbf{u}}_0\left(t_1\right)-{\mathbf{u}}_0\left(0\right),\mathbf{v}\rangle }_{\mathbb{H}}\right|& \le & \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^{t_1}{\mathcal{J}}_s^{\alpha }\left(t_1\right)|a_1{\langle {\mathbf{u}}_0\left(s\right),\mathbf{v}\rangle }_{{\mathbb{H}}_1}+{\langle \mathcal{F}\left({\mathbf{u}}_0\left(s\right)\right),\mathbf{v}\rangle }_{\mathbb{H}}|\mathrm{d}s\hfill \\ & \le & \frac{t_1^{\alpha }}{\alpha \mathsf{\Gamma }\left(\alpha \right)}\left(a_1\sqrt{M_2^T}{\parallel \mathbf{v}\parallel }_{{\mathbb{H}}_1}+M_{\mathcal{F}}{\parallel \mathbf{v}\parallel }_{\mathbb{H}}\right),\forall \mathbf{v}\in \mathbb{H}.\hfill \end{array}$$

Likewise, taking $\mathbf{v}={\mathbf{u}}_0\left(t_1\right)-{\mathbf{u}}_0\left(0\right)\in \mathbb{H}$ leads to

$$\parallel {\mathbf{u}}_0\left(t_1\right)-{\mathbf{u}}_0\left(0\right){\parallel }_{\mathbb{H}}^2 \le \frac{t_1^{\alpha }}{\alpha \mathsf{\Gamma }\left(\alpha \right)}\left(a_1{M}_2^T+M_{\mathcal{F}}{M}_1^T\right)<ϵ.$$

So ${\mathbf{u}}_0\in {\mathrm{H}}^{\alpha }({\mathbf{R}}_0,\mathbb{H})$.

For each $t\in {\mathbf{R}}_{+}$, suppose that ${\mathbf{u}}_0\left(t\right)={\sum }_{k=1}^{\infty }{u}_k\left(t\right)e_k.$ Consider the partial sum ${\mathbf{u}}_0^n\left(t\right)={\sum }_{k=1}^n{u}_k\left(t\right)e_k,$$\forall t\in {\mathbf{R}}_{+},n=1,2,\dots .$ Thus, $\parallel {\mathbf{u}}_0^n{\left(t\right)\parallel }_{\mathbb{H}}^2\le M_1^T$ and ${\left\{{\mathbf{u}}_0^n\left(t\right)\right\}}_{n\in {\mathbf{Z}}_{+}}$ converges to ${\mathbf{u}}_0\left(t\right)$ in $\mathbb{H}$ uniformly for $t\in {\overline{I}}_T$. By the weak continuousness of $\mathcal{A}$, we have

$${\langle \mathcal{A}{\mathbf{u}}_0\left(t\right),e_k\rangle }_{\mathbb{H}}=\underset{n\to \infty }{lim}{\langle \mathcal{A}{\mathbf{u}}_0^n\left(t\right),e_k\rangle }_{\mathbb{H}}=-{\lambda }_k{u}_k\left(t\right),k=1,2,\dots ,$$

where $t\in I_T$. From Equation (21), we have

$$u_k\left(t\right)={\phi }_k+J_0^{\alpha }\left[-{\lambda }_k{u}_k\left(t\right)+{\mathcal{F}}_k\left({\mathbf{u}}_0\left(t\right)\right)\right],$$

where ${\phi }_k={\langle \phi ,e_k\rangle }_{\mathbb{H}}$ and ${\mathcal{F}}_k\left({\mathbf{u}}_0\left(t\right)\right)={\langle \mathcal{F}\left({\mathbf{u}}_0\left(t\right)\right),e_k\rangle }_{\mathbb{H}}$, $k=1,2,\dots$, $t\in I_T$. By the boundedness of ${\mathbf{u}}_0$ in $\mathbb{H}$, we obtain $-{\lambda }_k{u}_k\left(t\right)+{\mathcal{F}}_k\left({\mathbf{u}}_0\left(t\right)\right)\in {\mathrm{L}}_{\mathrm{loc}}^{\infty }\left({\mathbf{R}}_{+}\right)$ in a breeze and

$${{}^{c}D}_{0\ast }^{\alpha }{u}_k\left(t\right)=-{\lambda }_k{u}_k\left(t\right)+{\mathcal{F}}_k\left({\mathbf{u}}_0\left(t\right)\right),u_k\left(0\right)={\phi }_k,$$

which is equal to

$$u_k\left(t\right)={\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\phi }_k+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left({\mathbf{u}}_0\left(s\right)\right)\mathrm{d}s,$$

where $t\in {\mathbf{R}}_0$, $k=1,2,\dots$.

(iii) For $t\in {\overline{I}}_T$, it follows from Minkowski, Hölder, and Bessel inequalities and items $\left(7\right)$–$\left(9\right)$ in Lemma 1 that

$$\begin{array}{ccc}\hfill \underset{t\in {\overline{I}}_T}{max}{\parallel {\mathbf{u}}_0^n\left(t\right)\parallel }_{\mathbb{H}}& =& \underset{t\in {\overline{I}}_T}{max}{\left\{\sum _{k=1}^n{\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\phi }_k+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left({\mathbf{u}}_0\left(s\right)\right)\mathrm{d}s\right]}^2\right\}}^{\frac{1}{2}}\hfill \\ & \le & \underset{t\in {\overline{I}}_T}{max}{\left\{\sum _{k=1}^n{\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\phi }_k\right]}^2\right\}}^{\frac{1}{2}}+\underset{t\in {\overline{I}}_T}{max}{\left\{\sum _{k=1}^n{\left[{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left({\mathbf{u}}_0\left(s\right)\right)\mathrm{d}s\right]}^2\right\}}^{\frac{1}{2}}\hfill \\ & \le & {\parallel \phi \parallel }_{\mathbb{H}}+\underset{t\in {\overline{I}}_T}{max}{\left\{\sum _{k=1}^n\left[{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right)\mathrm{d}s{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k^2\left({\mathbf{u}}_0\left(s\right)\right)\mathrm{d}s\right]\right\}}^{\frac{1}{2}}\hfill \\ & \le & {\parallel \phi \parallel }_{\mathbb{H}}+\underset{t\in {\overline{I}}_T}{max}{\left\{{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1\right)\mathrm{d}s{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1\right){\mathcal{F}}^2\left({\mathbf{u}}_0\left(s\right)\right)\mathrm{d}s\right\}}^{\frac{1}{2}}\hfill \\ & \le & {\parallel \phi \parallel }_{\mathbb{H}}+\frac{1}{{\lambda }_1}\left[L_F\underset{t\in {\overline{I}}_T}{max}{\parallel {\mathbf{u}}_0\left(t\right)\parallel }_{\mathbb{H}}+F_{\infty }\right],\forall n\in {\mathbf{Z}}_{+}.\hfill \end{array}$$

The desired conclusion holds by letting $T\to \infty$ in the above inequality.

(iv) Here, the method of reduction to absurdity should be employed. Suppose that $\tilde{\mathbf{u}}\in \mathrm{C}({\mathbf{R}}_0,\mathbb{H})$ is another solution of Equation (13). Let $\tilde{\mathbf{u}}\left(t\right)={\sum }_{k=1}^{\infty }{\tilde{u}}_k\left(t\right)e_k$ and ${\tilde{\mathbf{u}}}^n\left(t\right)={\sum }_{k=1}^n{\tilde{u}}_k\left(t\right)e_k$, where ${\tilde{u}}_k\left(t\right)={\langle \tilde{\mathbf{u}}\left(t\right),e_k\rangle }_{\mathbb{H}}$, $k=1,2,\dots$, $t\in {\mathbf{R}}_0$. Owing to $\tilde{\mathbf{u}}\in \mathrm{C}({\mathbf{R}}_0,\mathbb{H})$, $\left\{{\tilde{\mathbf{u}}}^n\left(t\right)\right\}$ converges to $\tilde{\mathbf{u}}\left(t\right)$ in $\mathbb{H}$ for $t\in {\mathbf{R}}_0$. Similar to $u_k\left(t\right)$, ${\tilde{u}}_k\left(t\right)$ can be expressed by

$${\tilde{u}}_k\left(t\right)={\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\phi }_k+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left(\tilde{\mathbf{u}}\left(s\right)\right)\mathrm{d}s,$$

where $t\in {\mathbf{R}}_0$, $k=1,2,\dots$. As a consequence, from items $\left(8\right)$–$\left(9\right)$ in Lemma 1, we find that

$$\begin{array}{ccc}\hfill \parallel {\mathbf{u}}_0^n\left(t\right)-{\tilde{\mathbf{u}}}^n{\left(t\right)\parallel }_{\mathbb{H}}^2& =& \sum _{k=1}^n{\left[u_k\left(t\right)-{\tilde{u}}_k\left(t\right)\right]}^2\hfill \\ & =& \sum _{k=1}^n{\left[{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right)\left({\mathcal{F}}_k\left({\mathbf{u}}_0\left(s\right)\right)-{\mathcal{F}}_k\left(\tilde{\mathbf{u}}\left(s\right)\right)\right)\mathrm{d}s\right]}^2\hfill \\ & \le & \sum _{k=1}^n{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right)\mathrm{d}s{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\left[{\mathcal{F}}_k\left({\mathbf{u}}_0\left(s\right)\right)-{\mathcal{F}}_k\left(\tilde{\mathbf{u}}\left(s\right)\right)\right]}^2\mathrm{d}s\hfill \\ & \le & {\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1\right)\mathrm{d}s{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1\right)\sum _{k=1}^n{\left[{\mathcal{F}}_k\left({\mathbf{u}}_0\left(s\right)\right)-{\mathcal{F}}_k\left(\tilde{\mathbf{u}}\left(s\right)\right)\right]}^2\mathrm{d}s\hfill \\ & \le & \frac{1}{{\lambda }_1}{\int }_0^t{\mathcal{J}}_s^{\alpha }\left(t\right)\parallel \mathcal{F}\left({\mathbf{u}}_0\left(s\right)\right)-\mathcal{F}\left(\tilde{\mathbf{u}}\left(s\right)\right){\parallel }_{\mathbb{H}}^2\mathrm{d}s\hfill \\ & \le & \frac{L_F^2}{{\lambda }_1}{\int }_0^t{\mathcal{J}}_s^{\alpha }\left(t\right){\parallel {\mathbf{u}}_0\left(s\right)-\tilde{\mathbf{u}}\left(s\right)\parallel }_{\mathbb{H}}^2\mathrm{d}s,\forall t\in {\mathbf{R}}_0,n=1,2,\dots .\hfill \end{array}$$

The continuity of ${\langle ·\rangle }_{\mathbb{H}}$ with respect to ${\parallel ·\parallel }_{\mathbb{H}}$ and letting $n\to \infty$ in the above inequality induces

$$\parallel {\mathbf{u}}_0\left(t\right)-\tilde{\mathbf{u}}{\left(t\right)\parallel }_{\mathbb{H}}^2\le \frac{L_F^2}{{\lambda }_1}{\int }_0^t{\mathcal{J}}_s^{\alpha }\left(t\right){\parallel {\mathbf{u}}_0\left(s\right)-\tilde{\mathbf{u}}\left(s\right)\parallel }_{\mathbb{H}}^2\mathrm{d}s,\forall t\in {\mathbf{R}}_0.$$

By the Gronwall inequality in item (10) in Lemma 1, $\parallel {\mathbf{u}}_0\left(t\right)-\tilde{\mathbf{u}}{\left(t\right)\parallel }_{\mathbb{H}}=0$ for all $t\in {\mathbf{R}}_0$.

(v) Let $\widehat{\mathbf{u}}$ and $\breve{\mathbf{u}}$ be any two strong solutions of Equation (1) with initial values $\widehat{\mathbf{u}}(x,0)=\widehat{\phi }\left(x\right)$ and $\breve{\mathbf{u}}(x,0)=\breve{\phi }\left(x\right)$ for $x\in \overline{\mathsf{\Omega }},$ respectively. Then, $\widehat{\mathbf{u}}$ and $\breve{\mathbf{u}}$ belong to $\mathcal{L}({\mathbf{R}}_0,\mathbb{H})\cap {\mathrm{H}}^{\alpha }({\mathbf{R}}_0,\mathbb{H})$. Then

$$\left\{\begin{array}{cc}{{}^{c}D}_{0\ast }^{\alpha }\rho (x,t)=\mathcal{A}\rho (x,t)+\mathbb{F}(\rho (x,t),x,t),\hfill & \mathrm{in}\mathsf{\Omega }\times {\mathbf{R}}_{+},\hfill \\ \rho (x,t)=0,\hfill & \mathrm{in}\partial \mathsf{\Omega }\times {\mathbf{R}}_0,\hfill \\ \rho (x,0)=\varrho \left(x\right),\hfill & \mathrm{in}\overline{\mathsf{\Omega }},\hfill \end{array}\right.$$

(22)

where $\rho (x,t)=\widehat{\mathbf{u}}(x,t)-\breve{\mathbf{u}}(x,t)$, $\mathbb{F}(\rho (x,t),x,t)=F(\widehat{\mathbf{u}}(x,t),x,t)-F(\breve{\mathbf{u}}(x,t),x,t)$, $\varrho \left(x\right)=\widehat{\phi }\left(x\right)-\breve{\phi }\left(x\right)$, $\forall (x,t)\in \overline{\mathsf{\Omega }}\times {\mathbf{R}}_0$.

From the first part of Equation (22), we have

$${〈{{}^{c}D}_{0\ast }^{\alpha }\rho (x,t),\rho (x,t)〉}_{\mathbb{H}}={〈\mathcal{A}\rho (x,t),\rho (x,t)〉}_{\mathbb{H}}+{〈\mathbb{F}\left(\rho \right(x,t),x,t),\rho (x,t)〉}_{\mathbb{H}},$$

which contains from $\left({\mathcal{H}}_4\right)$ that

$$\begin{array}{ccc}\hfill \delta {{}^{c}D}_{0\ast }^{\alpha }\parallel \rho (x,t){\parallel }_{\mathbb{H}}^2& \le & -a_0\parallel \rho (x,t){\parallel }_{\mathbb{H}}^2+\parallel \mathbb{F}(\rho (x,t),x,t){\parallel }_{\mathbb{H}}\parallel \rho (x,t){\parallel }_{\mathbb{H}}\hfill \\ & \le & -(a_0-L_F)\parallel \rho (x,t){\parallel }_{\mathbb{H}}^2,\hfill \end{array}$$

which induces

$$\parallel \rho (x,t){\parallel }_{\mathbb{H}}^2 \le \parallel \varrho \left(x\right){\parallel }_{\mathbb{H}}^2{\mathbf{E}}_{\alpha }\left[-\frac{(a_0-L_F)}{\delta }{t}^{\alpha }\right],\forall (x,t)\in \overline{\mathsf{\Omega }}\times {\mathbf{R}}_0.$$

This achieves the proof. □

4. Almost Autormorphic Strong Solution

Here, the conception of almost automorphic vector valued functions should to be introduced; please see [45] for more details.

An $\mathbb{H}$-valued function $f\in \mathrm{C}(\mathbf{R},\mathbb{H})$ is called an almost automorphic function if for every sequence ${\left\{{\overline{t}}_p\right\}}_{p\in {\mathbf{Z}}_{+}}$, there exists a subsequence ${\left\{t_p\right\}}_{p\in {\mathbf{Z}}_{+}}\subseteq {\left\{{\overline{t}}_p\right\}}_{p\in {\mathbf{Z}}_{+}}$ and a function $\tilde{f}:\mathbf{R}\to \mathbb{H}$ such that

$$\tilde{f}\left(t\right)=\underset{p\to \infty }{lim}f(t+t_p),\underset{p\to \infty }{lim}\tilde{f}(t-t_p)=f\left(t\right)$$

are well-defined in $\mathbb{H}$ for each $t\in \mathbf{R}$. The set of all these $\mathbb{H}$-valued functions is represented by $\mathbb{AA}(\mathbf{R},\mathbb{H})$. Let $\mathbb{B}\subseteq \mathbb{H}$ be an arbitrary bounded set. A mapping $f=f(\eta ,t):\mathbb{H}\times \mathbf{R}\to \mathbb{H}$, $\forall (\eta ,t)\in \mathbb{H}\times \mathbf{R}$ is said to be almost automorphic with respect to variable t when $f(\eta ,t)$ is almost automorphic with respect to variable t uniformly for $\eta \in \mathbb{B}$. The set of the whole such mappings is denoted by $\mathbb{AA}(\mathbb{H}\times \mathbf{R},\mathbb{H})$.

Let $\aleph ({\mathbf{R}}_0,\mathbb{H})=\{\psi \in \mathrm{C}({\mathbf{R}}_0,\mathbb{H}):{lim}_{t\to \infty }\parallel \psi \left(t\right){\parallel }_{\mathbb{H}}=0\}$. An $\mathbb{H}$-valued continuous function $f:{\mathbf{R}}_0\to \mathbb{H}$ is called an asymptotically almost automorphic function if there exist $\varphi \in \mathbb{AA}(\mathbf{R},\mathbb{H})$ and $\psi \in \aleph ({\mathbf{R}}_0,\mathbb{H})$ satisfying $f=\varphi +\psi$. The set composed by all these functions is represented by $\mathbb{AAA}({\mathbf{R}}_0,\mathbb{H})$. In particular, $\mathbb{AAA}({\mathbf{R}}_0,\mathbb{H})$ is a Banach space endowed with the norm

$${\parallel f\parallel }_{\mathbb{AAA}}=\underset{t\in \mathbf{R}}{sup}{\parallel \varphi \parallel }_{\mathbb{H}}+\underset{t\in {\mathbf{R}}_0}{sup}{\parallel \psi \parallel }_{\mathbb{H}},$$

where $f=\varphi +\psi \in \mathbb{AAA}({\mathbf{R}}_0,\mathbb{H})$, $\varphi \in \mathbb{AA}(\mathbf{R},\mathbb{H})$ and $\psi \in \aleph ({\mathbf{R}}_0,\mathbb{H})$.

Let $\aleph (\mathbb{H}\times {\mathbf{R}}_0,\mathbb{H})$ be the collection of the whole jointly continuous mappings $\psi :\mathbb{H}\times {\mathbf{R}}_0\to \mathbb{H}$ satisfying ${lim}_{t\to \infty }{\parallel \psi (\eta ,t)\parallel }_{\mathbb{H}}=0$ uniformly for $\eta \in \mathbb{B}$. A mapping $f=f(\eta ,t):\mathbb{H}\times {\mathbf{R}}_0\to \mathbb{H}$, $\forall (\eta ,t)\in \mathbb{H}\times {\mathbf{R}}_0$ is said to be asymptotically almost automorphic with respect to variable t in the case where $f(\eta ,t)$ is asymptotically almost automorphic with respect to variable t uniformly for $\eta \in \mathbb{B}$. That is, $f=\varphi +\psi$, where the principal term $\varphi \in \mathbb{AA}(\mathbb{H}\times \mathbf{R},\mathbb{H})$ and $\psi \in \aleph (\mathbb{H}\times {\mathbf{R}}_0,\mathbb{H})$. The set of the whole such asymptotically almost automorphic mappings is represented by $\mathbb{AAA}(\mathbb{H}\times {\mathbf{R}}_0,\mathbb{H})$.

The method of Picard iteration is employed here. Based on Equation (9), a Picard iterative sequence is constructed as

$$\left\{\begin{array}{cc}{\xi }_0\left(t\right)=\phi \in \mathbb{H},\hfill & \hfill \\ {\displaystyle {\xi }_{m+1}\left(t\right)=\sum _{k=1}^{\infty }\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\phi }_k+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left({\xi }_m\left(s\right)\right)\mathrm{d}s\right]e_k,}\hfill & \hfill \end{array}\right.$$

(23)

where ${\phi }_k={\langle \phi ,e_k\rangle }_{\mathbb{H}}$ and ${\mathcal{F}}_k\left({\xi }_m\right)={\langle \mathcal{F}\left({\xi }_m\right),e_k\rangle }_{\mathbb{H}}$, $k=1,2,\dots$, $m=0,1,\dots ,$$t\in {\mathbf{R}}_0$.

Proposition 1. 

Let $\left({\mathcal{H}}_1\right)$ and $\left({\mathcal{H}}_3\right)$ hold. $\{{\xi }_m:m=0,1,\dots \}\subseteq \mathbb{H}$ is well-defined, $\mathbb{H}$-bounded and $\mathbb{H}$-continuous if the condition below holds:

• $\left({\mathcal{H}}_5\right) {\lambda }_1>L_F$.

Proof. 

According to Equation (23) and by using Minkowski and Hölder inequalities, we have from items (7)–(9) in Lemma 1 that

$$\begin{array}{ccc}\hfill \parallel {\xi }_1{\left(t\right)\parallel }_{\mathbb{H}}^2& =& \underset{K\to \infty }{lim}\sum _{k=1}^K{\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\phi }_k+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left(\phi \right)\mathrm{d}s\right]}^2\hfill \\ & \le & \underset{K\to \infty }{lim}{\left\{{\left[\sum _{k=1}^K{\left({\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\phi }_k\right)}^2\right]}^{\frac{1}{2}}+{\left[\sum _{k=1}^K{\left({\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left(\phi \right)\mathrm{d}s\right)}^2\right]}^{\frac{1}{2}}\right\}}^2\hfill \\ & \le & \underset{K\to \infty }{lim}{\left\{{\mathcal{E}}_t^{\alpha }\left({\lambda }_1\right){\left[\sum _{k=1}^K{\phi }_k^2\right]}^{\frac{1}{2}}+{\left[\sum _{k=1}^K{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1\right)\mathrm{d}s{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1\right){\mathcal{F}}_k^2\left(\phi \right)\mathrm{d}s\right]}^{\frac{1}{2}}\right\}}^2\hfill \\ & \le & {\left\{{\mathcal{E}}_t^{\alpha }\left({\lambda }_1\right){\parallel \phi \parallel }_{\mathbb{H}}+{\left[{\mathcal{E}}_{t,0}^{\alpha ,\alpha +1}\left({\lambda }_1\right){\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1\right){\parallel \mathcal{F}\left(\phi \right)\parallel }_{\mathbb{H}}^2\mathrm{d}s\right]}^{\frac{1}{2}}\right\}}^2\hfill \\ & \le & {\left\{{\mathcal{E}}_t^{\alpha }\left({\lambda }_1\right){\parallel \phi \parallel }_{\mathbb{H}}+\frac{[1-{\mathcal{E}}_t^{\alpha }\left({\lambda }_1\right)]L_F{\parallel \phi \parallel }_{\mathbb{H}}}{{\lambda }_1}+\frac{[1-{\mathcal{E}}_t^{\alpha }\left({\lambda }_1\right)]F_{\infty }}{{\lambda }_1}\right\}}^2\hfill \\ & \le & {\left[{\parallel \phi \parallel }_{\mathbb{H}}+\frac{F_{\infty }}{{\lambda }_1}\right]}^2\le {\left[{\parallel \phi \parallel }_{\mathbb{H}}+\frac{F_{\infty }}{{\lambda }_1-L_F}\right]}^2,\forall t\in {\mathbf{R}}_0.\hfill \end{array}$$

Meanwhile, ${\xi }_1(·)\in \mathbb{H}$ is well-defined. Similarly,

$$\begin{array}{ccc}\hfill \parallel {\xi }_2{\left(t\right)\parallel }_{\mathbb{H}}^2& =& \underset{K\to \infty }{lim}\sum _{k=1}^K{\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\phi }_k+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left({\xi }_1\left(t\right)\right)\mathrm{d}s\right]}^2\hfill \\ & \le & {\left\{{\mathcal{E}}_t^{\alpha }\left({\lambda }_1\right){\parallel \phi \parallel }_{\mathbb{H}}+{\left[{\mathcal{E}}_{t,0}^{\alpha ,\alpha +1}\left({\lambda }_1\right){\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1\right){\parallel \mathcal{F}\left({\xi }_1\left(t\right)\right)\parallel }_{\mathbb{H}}^2\mathrm{d}s\right]}^{\frac{1}{2}}\right\}}^2\hfill \\ & \le & {\left\{{\mathcal{E}}_t^{\alpha }\left({\lambda }_1\right){\parallel \phi \parallel }_{\mathbb{H}}+\frac{[1-{\mathcal{E}}_t^{\alpha }\left({\lambda }_1\right)]L_F{\parallel \phi \parallel }_{\mathbb{H}}}{{\lambda }_1}+\frac{[1-{\mathcal{E}}_t^{\alpha }\left({\lambda }_1\right)]}{{\lambda }_1}\left(1+\frac{L_F}{{\lambda }_1-L_F}\right)F_{\infty }\right\}}^2\hfill \\ & \le & {\left\{{\parallel \phi \parallel }_{\mathbb{H}}+\left(1+\frac{L_F}{{\lambda }_1-L_F}\right)\frac{F_{\infty }}{{\lambda }_1}\right\}}^2\le {\left\{{\parallel \phi \parallel }_{\mathbb{H}}+\frac{F_{\infty }}{{\lambda }_1-L_F}\right\}}^2,\forall t\in {\mathbf{R}}_0.\hfill \end{array}$$

By the method of mathematical induction, we obtain

$$\parallel {\xi }_m{\left(t\right)\parallel }_{\mathbb{H}}^2\le {\left\{{\parallel \phi \parallel }_{\mathbb{H}}+\sum _{k=0}^{m-1}{\left(\frac{L_F}{{\lambda }_1}\right)}^k\frac{F_{\infty }}{{\lambda }_1}\right\}}^2\le {\left\{{\parallel \phi \parallel }_{\mathbb{H}}+\frac{F_{\infty }}{{\lambda }_1-L_F}\right\}}^2,$$

where $t\in {\mathbf{R}}_0,m=3,4,\dots .$ So, ${\xi }_m(·)\in \mathbb{H}$ is well-defined, and $\parallel {\xi }_m{\left(t\right)\parallel }_{\mathbb{H}}\le {\parallel \phi \parallel }_{\mathbb{H}}+\frac{F_{\infty }}{{\lambda }_1-L_F}$ for all $t\in {\mathbf{R}}_0$, $m=0,1,\dots$. The boundedness is verified.

Because $\left\{{\xi }_m\right\}$ is bounded in $\mathbb{H}$, there exists constant ${\mathcal{F}}_{\infty }>0$ so that $\mathrm{ess}{sup}_{t\in {\mathbf{R}}_0}{\parallel \mathcal{F}\left({\xi }_m\left(t\right)\right)\parallel }_{\mathbb{H}}<{\mathcal{F}}_{\infty }$ for all $m=0,1,\dots$. For any $ϵ\in (0,1)$, there exists a large integer $K_0>0$ such that ${\lambda }_{K_0+1}ϵ>\sqrt{6}{\mathcal{F}}_{\infty }$. Without loss of generality, taking arbitrary $t_0,t_1\in {\mathbf{R}}_0$ meeting $t_1\ge t_0$, $1-{\mathcal{E}}_{t_1-t_0}^{\alpha }\left({\lambda }_1\right)<\frac{{\lambda }_1ϵ}{\sqrt{6}{\mathcal{F}}_{\infty }}$ and

$$\underset{k=1,2,\dots ,K_0}{max}\left\{1-{\mathcal{E}}_{t_1-t_0}^{\alpha }\left({\lambda }_k\right)+{\mathcal{E}}_{t_1}^{\alpha }\left({\lambda }_k\right)-{\mathcal{E}}_{t_0}^{\alpha }\left({\lambda }_k\right)\right\}<\frac{{\lambda }_1^2{ϵ}^2}{6{\mathcal{F}}_{\infty }^2}.$$

Given iterative sequence (23) and the Hölder inequality, we calculate

$$\begin{array}{ccc}\hfill \parallel {\xi }_m\left(t_1\right)-{\xi }_m\left(t_0\right){\parallel }_{\mathbb{H}}^2& =& \underset{K\to \infty }{lim}\sum _{k=1}^K[{\int }_0^{t_1}{\mathcal{E}}_{t_1,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left({\xi }_{m-1}\left(s\right)\right)\mathrm{d}s\hfill \\ & & -{\int }_0^{t_0}{\mathcal{E}}_{t_0,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left({\xi }_{m-1}\left(s\right)\right)\mathrm{d}s{]}^2\hfill \\ & \le & 2\underset{K\to \infty }{lim}\sum _{k=1}^K{\left[{\int }_{t_0}^{t_1}{\mathcal{E}}_{t_1,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left({\xi }_{m-1}\left(s\right)\right)\mathrm{d}s\right]}^2\hfill \\ & & +2\underset{K\to \infty }{lim}\sum _{k=1}^K{\left[{\int }_0^{t_0}\left[{\mathcal{E}}_{t_1,s}^{\alpha ,\alpha }\left({\lambda }_k\right)-{\mathcal{E}}_{t_0,s}^{\alpha ,\alpha }\left({\lambda }_k\right)\right]{\mathcal{F}}_k\left({\xi }_{m-1}\left(s\right)\right)\mathrm{d}s\right]}^2\hfill \\ & \le & 2\underset{K\to \infty }{lim}{\int }_{t_0}^{t_1}{\mathcal{E}}_{t_1,s}^{\alpha ,\alpha }\left({\lambda }_1\right)\mathrm{d}s{\int }_{t_0}^{t_1}{\mathcal{E}}_{t_1,s}^{\alpha ,\alpha }\left({\lambda }_1\right)\sum _{k=1}^K{\mathcal{F}}_k^2\left({\xi }_{m-1}\left(s\right)\right)\mathrm{d}s\hfill \\ & & +2\underset{K\to \infty }{lim}\sum _{k=1}^K{\int }_0^{t_0}\left[{\mathcal{E}}_{t_0,s}^{\alpha ,\alpha }\left({\lambda }_k\right)-{\mathcal{E}}_{t_1,s}^{\alpha ,\alpha }\left({\lambda }_k\right)\right]\mathrm{d}s\hfill \\ & & \times {\int }_0^{t_0}\left[{\mathcal{E}}_{t_0,s}^{\alpha ,\alpha }\left({\lambda }_k\right)-{\mathcal{E}}_{t_1,s}^{\alpha ,\alpha }\left({\lambda }_k\right)\right]{\mathcal{F}}_k^2\left({\xi }_{m-1}\left(s\right)\right)\mathrm{d}s\hfill \\ & =& {\mathcal{I}}_1+{\mathcal{I}}_2,\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill {\mathcal{I}}_1& :=& 2\underset{K\to \infty }{lim}{\int }_{t_0}^{t_1}{\mathcal{E}}_{t_1,s}^{\alpha ,\alpha }\left({\lambda }_1\right)\mathrm{d}s{\int }_{t_0}^{t_1}{\mathcal{E}}_{t_1,s}^{\alpha ,\alpha }\left({\lambda }_1\right)\sum _{k=1}^K{\mathcal{F}}_k^2\left({\xi }_{m-1}\left(s\right)\right)\mathrm{d}s\hfill \\ & \le & \frac{2{\mathcal{F}}_{\infty }^2{\left[1-{\mathcal{E}}_{t_1-t_0}^{\alpha }\left({\lambda }_1\right)\right]}^2}{{\lambda }_1^2}<\frac{{ϵ}^2}{3}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\mathcal{I}}_2& :=& 2\underset{K\to \infty }{lim}\sum _{k=1}^K{\int }_0^{t_0}\left[{\mathcal{E}}_{t_0,s}^{\alpha ,\alpha }\left({\lambda }_k\right)-{\mathcal{E}}_{t_1,s}^{\alpha ,\alpha }\left({\lambda }_k\right)\right]\mathrm{d}s\hfill \\ & & \times {\int }_0^{t_0}\left[{\mathcal{E}}_{t_0,s}^{\alpha ,\alpha }\left({\lambda }_k\right)-{\mathcal{E}}_{t_1,s}^{\alpha ,\alpha }\left({\lambda }_k\right)\right]{\mathcal{F}}_k^2\left({\xi }_{m-1}\left(s\right)\right)\mathrm{d}s\hfill \\ & \le & 2\sum _{k=1}^{K_0}{\int }_0^{t_0}\left[{\mathcal{E}}_{t_0,s}^{\alpha ,\alpha }\left({\lambda }_k\right)-{\mathcal{E}}_{t_1,s}^{\alpha ,\alpha }\left({\lambda }_k\right)\right]\mathrm{d}s{\int }_0^{t_0}{\mathcal{E}}_{t_0,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k^2\left({\xi }_{m-1}\left(s\right)\right)\mathrm{d}s\hfill \\ & & +2\underset{(K_0<)K\to \infty }{lim}\sum _{k=K_0+1}^K{\int }_0^{t_0}{\mathcal{E}}_{t_0,s}^{\alpha ,\alpha }\left({\lambda }_k\right)\mathrm{d}s{\int }_0^{t_0}{\mathcal{E}}_{t_0,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k^2\left({\xi }_{m-1}\left(s\right)\right)\mathrm{d}s\hfill \\ & \le & \frac{2}{{\lambda }_1}\underset{k=1,2,\dots ,K_0}{max}\left\{1-{\mathcal{E}}_{t_1-t_0}^{\alpha }\left({\lambda }_k\right)+{\mathcal{E}}_{t_1}^{\alpha }\left({\lambda }_k\right)-{\mathcal{E}}_{t_0}^{\alpha }\left({\lambda }_k\right)\right\}\hfill \\ & & \times {\int }_0^{t_0}{\mathcal{E}}_{t_0,s}^{\alpha ,\alpha }\left({\lambda }_1\right){\mathcal{F}}_{\infty }^2\mathrm{d}s+\frac{2}{{\lambda }_{K_0+1}}\underset{K\to \infty }{lim}{\int }_0^{t_0}{\mathcal{E}}_{t_0,s}^{\alpha ,\alpha }\left({\lambda }_{K_0+1}\right){\mathcal{F}}_{\infty }^2\mathrm{d}s\hfill \\ & \le & \frac{2{\mathcal{F}}_{\infty }^2}{{\lambda }_1^2}\times \frac{{\lambda }_1^2{ϵ}^2}{6{\mathcal{F}}_{\infty }^2}+\frac{2}{{\lambda }_{K_0+1}^2}{\mathcal{F}}_{\infty }^2\le \frac{{ϵ}^2}{3}+\frac{{ϵ}^2}{3}=\frac{2{ϵ}^2}{3}.\hfill \end{array}$$

In the above calculations, items (7)–(9) in Lemma 1 are employed.

Summarizing the above results, $\parallel {\xi }_m\left(t_1\right)-{\xi }_m\left(t_0\right){\parallel }_{\mathbb{H}}<ϵ$ for all $m=0,1,\dots$. Then, $\{{\xi }_m:m=0,1,\dots \}$ is $\mathbb{H}$-continuous. This provides the proof. □

Consider the partial sum of ${\xi }_m$ denoted by

$${\xi }_m^K\left(t\right)=\sum _{k=1}^K\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\phi }_k+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left({\xi }_{m-1}\left(s\right)\right)\mathrm{d}s\right]e_k\in \mathbb{H},$$

where $t\in {\mathbf{R}}_0,K,m=1,2,\dots .$ It follows that

$$\begin{array}{ccc}\hfill \parallel {\xi }_m\left(t\right)-{\xi }_m^K\left(t\right){\parallel }_{\mathbb{H}}^2& =& \sum _{k=K+1}^{\infty }{\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\phi }_k+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left({\xi }_{m-1}\left(s\right)\right)\mathrm{d}s\right]}^2\hfill \\ & \le & 2\sum _{k=K+1}^{\infty }{\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\phi }_k\right]}^2+2\sum _{k=K+1}^{\infty }{\left[{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left({\xi }_{m-1}\left(s\right)\right)\mathrm{d}s\right]}^2\hfill \\ & \le & 2\sum _{k=K+1}^{\infty }{\phi }_k^2+\frac{2{\mathcal{F}}_{\infty }^2}{{\lambda }_{K+1}^2}\to 0,\mathrm{as}K\to \infty ,\hfill \end{array}$$

(24)

which contains $\{{\xi }_m^K\left(t\right):K=1,2,\dots \}$, which converges to ${\xi }_m\left(t\right)$ in $\mathbb{H}$ uniformly for $t\in {\mathbf{R}}_0$ and $m=1,2,\dots$.

Proposition 2. 

$\{{\xi }_m:m=0,1,\dots \}\subseteq \mathbb{AAA}({\mathbf{R}}_0,\mathbb{H})$ if $\left({\mathcal{H}}_1\right)$, $\left({\mathcal{H}}_3\right)$, $\left({\mathcal{H}}_5\right)$ and the assumption below hold.

• $\left({\mathcal{H}}_6\right)$$F(\eta ,·,t)\in \mathbb{AAA}(\mathbb{H}\times {\mathbf{R}}_0,\mathbb{H})$ for $(\eta ,t)\in \mathbb{H}\times {\mathbf{R}}_0$, and the principal term of F meets the Lipschitz condition with respect to the variable $\eta \in \mathbb{H}$.

Proof. 

Owing to $\phi \in \mathbb{AAA}({\mathbf{R}}_0,\mathbb{H})$ and by Theorem 3.6 in ref. [45], $\mathcal{F}\left(\phi \right)=F(\phi ,·,t)\in \mathbb{AAA}(\mathbb{H}\times {\mathbf{R}}_0,\mathbb{H})$ for $(\phi ,t)\in \mathbb{H}\times {\mathbf{R}}_0$. Thus, $\mathcal{F}\left(\phi \right)=F(\phi ,·,t)={\mathcal{F}}_{\varphi }(\phi ,t)+{\mathcal{F}}_{\psi }(\phi ,t)$ for $(\phi ,t)\in \mathbb{H}\times {\mathbf{R}}_0$, where ${\mathcal{F}}_{\varphi }\in \mathbb{AA}(\mathbb{H}\times \mathbf{R},\mathbb{H})$ and ${\mathcal{F}}_{\psi }\in \aleph (\mathbb{H}\times {\mathbf{R}}_0,\mathbb{H})$.

If $m=1$, then ${\xi }_1^K\left(t\right)={\sum }_{i=1}^4{\xi }_{1,i}^K\left(t\right)$, where

$${\xi }_{1,1}^K\left(t\right)=\sum _{k=1}^K{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\phi }_k{e}_k,{\xi }_{1,2}^K\left(t\right)=\sum _{k=1}^K{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_{{\psi }_k}(\phi ,s)e_k\mathrm{d}s,$$

$${\xi }_{1,3}^K\left(t\right)=-\sum _{k=1}^K{\int }_{-\infty }^0{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_{{\varphi }_k}(\phi ,s)e_k\mathrm{d}s,{\xi }_{1,4}^K\left(t\right)=\sum _{k=1}^K{\int }_{-\infty }^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_{{\varphi }_k}(\phi ,s)e_k\mathrm{d}s,$$

where ${\mathcal{F}}_{{\varphi }_k}(\phi ,t)={\langle {\mathcal{F}}_{\varphi }(\phi ,t),e_k\rangle }_{\mathbb{H}}$ and ${\mathcal{F}}_{{\psi }_k}(\phi ,t)={\langle {\mathcal{F}}_{\psi }(\phi ,t),e_k\rangle }_{\mathbb{H}}$, $k=1,2,\dots ,K$, $K\in {\mathbf{Z}}_{+}\cup \{\infty \}$, $t\in {\mathbf{R}}_0$.

Similar to Proposition 1, $\{{\xi }_{1,i}^K:K\in {\mathbf{Z}}_{+}\cup \{\infty \}\}$ is bounded in $\mathbb{H}$, $i=1,2,3,4$. Similar to (24), $\left\{{\xi }_{1,i}^K\left(t\right)\right\}$ (respectively, $\left\{{\xi }_{1,4}^K\left(t\right)\right\})$ converges to ${\xi }_{1,i}^{\infty }\left(t\right)\in \mathbb{H}(\mathrm{respectively},{\xi }_{1,4}^{\infty }\left(t\right)\in \mathbb{H})$ in $\mathbb{H}$ uniformly for $t\in {\mathbf{R}}_0(\mathrm{respectively},t\in \mathbf{R})$, $i=1,2,3$. Furthermore, we compute

$$\underset{t\to \infty }{lim}\parallel {\xi }_{1,1}^{\infty }{\left(t\right)\parallel }_{\mathbb{H}}^2=\underset{t\to \infty }{lim}\underset{K\to \infty }{lim}\sum _{k=1}^K{\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right)\right]}^2{\phi }_k^2\le \underset{t\to \infty }{lim}{\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_1\right)\right]}^2{\parallel \phi \parallel }_{\mathbb{H}}^2=0,$$

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim}{\parallel {\xi }_{1,2}^{\infty }\left(t\right)\parallel }_{\mathbb{H}}^2\le & \underset{t\to \infty }{lim}\underset{K\to \infty }{lim}\frac{1}{{\lambda }_1}{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1\right)\sum _{k=1}^K{\mathcal{F}}_{{\psi }_k}^2(\phi ,s)\mathrm{d}s\hfill \\ \hfill \le & \underset{t\to \infty }{lim}\frac{1}{{\lambda }_1}{\int }_0^{\frac{t}{2}}{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1\right){\parallel {\mathcal{F}}_{\psi }(\phi ,s)\parallel }_{\mathbb{H}}^2\mathrm{d}s\hfill \\ & +\underset{t\to \infty }{lim}\frac{1}{{\lambda }_1}{\int }_{\frac{t}{2}}^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1\right){\parallel {\mathcal{F}}_{\psi }(\phi ,s)\parallel }_{\mathbb{H}}^2\mathrm{d}s\hfill \\ \hfill \le & \underset{t\to \infty }{lim}\frac{{\mathcal{E}}_{t/2}^{\alpha }\left({\lambda }_1\right)-{\mathcal{E}}_t^{\alpha }\left({\lambda }_1\right)}{{\lambda }_1^2}\underset{s\in {\mathbf{R}}_0}{sup}{\parallel {\mathcal{F}}_{\psi }(\phi ,s)\parallel }_{\mathbb{H}}^2\hfill \\ & +\underset{t\to \infty }{lim}\frac{1}{{\lambda }_1^2}\underset{s\in [\frac{t}{2},\infty )}{sup}\parallel {\mathcal{F}}_{\psi }{(\phi ,s)\parallel }_{\mathbb{H}}^2=0\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim}{\parallel {\xi }_{1,3}^{\infty }\left(t\right)\parallel }_{\mathbb{H}}^2\le & \underset{t\to \infty }{lim}\underset{K\to \infty }{lim}\frac{{\mathcal{E}}_t^{\alpha }\left({\lambda }_1\right)}{{\lambda }_1}{\int }_{-\infty }^0{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1\right){\displaystyle \sum _{k=1}^K}{\mathcal{F}}_{{\varphi }_k}^2(\phi ,s)\mathrm{d}s\hfill \\ \hfill \le & \underset{t\to \infty }{lim}\frac{{\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_1\right)\right]}^2}{{\lambda }_1^2}\underset{s\in {\mathbf{R}}_0}{sup}{\parallel {\mathcal{F}}_{\varphi }(\phi ,s)\parallel }_{\mathbb{H}}^2=0,\hfill \end{array}$$

which contain ${\xi }_{1,i}^{\infty }\in \aleph ({\mathbf{R}}_0,\mathbb{H}),i=1,2,3$.

Because ${\mathcal{F}}_{\varphi }\in \mathbb{AA}(\mathbb{H}\times \mathbf{R},\mathbb{H})$, for every sequence ${\left\{{\overline{t}}_p\right\}}_{p\in {\mathbf{Z}}_{+}}$, there exist a subsequence ${\left\{t_p\right\}}_{p\in {\mathbf{Z}}_{+}}\subseteq {\left\{{\overline{t}}_p\right\}}_{p\in {\mathbf{Z}}_{+}}$ and a function ${\tilde{\mathcal{F}}}_{\varphi }:\mathbb{H}\times \mathbf{R}\to \mathbb{H}$ such that

$${\tilde{\mathcal{F}}}_{\varphi }(\xi ,t)=\underset{p\to \infty }{lim}{\mathcal{F}}_{\varphi }(\xi ,t+t_p),\underset{p\to \infty }{lim}\widetilde{{\mathcal{F}}_{\varphi }}(\xi ,t-t_p)={\mathcal{F}}_{\varphi }(\xi ,t)$$

in $\mathbb{H}$ uniformly for $\xi \in \left\{\xi \in {\mathbb{H}:\parallel \xi \parallel }_{\mathbb{H}}\le {\parallel \phi \parallel }_{\mathbb{H}}+\frac{F_{\infty }}{{\lambda }_1-L_F}\right\}$ and for each $t\in \mathbf{R}$. Let

$${\tilde{\xi }}_{1,4}^K\left(t\right)=\sum _{k=1}^K{\int }_{-\infty }^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\tilde{\mathcal{F}}}_{{\varphi }_k}(\phi ,s)e_k\mathrm{d}s,$$

where ${\tilde{\mathcal{F}}}_{{\varphi }_k}(\phi ,t)={\langle {\tilde{\mathcal{F}}}_{\varphi }(\phi ,t),e_k\rangle }_{\mathbb{H}}$, $k=1,2,\dots ,K$, $K=1,2,\dots$, $t\in \mathbf{R}$. As a consequence,

$$\begin{array}{ccc}\hfill \parallel {\xi }_{1,4}^K(t+t_p)-{\tilde{\xi }}_{1,4}^K\left(t\right){\parallel }_{\mathbb{H}}^2& =& \sum _{k=1}^K{\left[{\int }_{-\infty }^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right)\left({\mathcal{F}}_{{\varphi }_k}(\phi ,s+t_p)-{\tilde{\mathcal{F}}}_{{\varphi }_k}(\phi ,s)\right)\mathrm{d}s\right]}^2\hfill \\ & \le & \frac{1}{{\lambda }_1}{\int }_{-\infty }^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1\right)\parallel {\mathcal{F}}_{\varphi }(\phi ,s+t_p)-{\tilde{\mathcal{F}}}_{\varphi }(\phi ,s){\parallel }_{\mathbb{H}}^2\mathrm{d}s,\hfill \end{array}$$

which induces, from the Lebesgue dominated convergence theorem:

$$\underset{p\to \infty }{lim}\parallel {\xi }_{1,4}^K(t+t_p)-{\tilde{\xi }}_{1,4}^K\left(t\right){\parallel }_{\mathbb{H}}=0\mathrm{for}K=1,2,\dots ,t\in \mathbf{R}.$$

Similarly, we readily verify ${lim}_{p\to \infty }\parallel {\tilde{\xi }}_{1,4}^K(t-t_p)-{\xi }_{1,4}^K\left(t\right){\parallel }_{\mathbb{H}}=0$ for $K=1,2,\dots$, $t\in \mathbf{R}$. Consequently, $\{{\xi }_{1,4}^K(·):K=1,2,\dots \}\subseteq \mathbb{AA}(\mathbf{R},\mathbb{H})$. Because $\{{\xi }_{1,4}^K(·):K=1,2,\dots \}$ uniformly converges to ${\xi }_{1,4}^{\infty }(·)$ in $\mathbb{H}$, then ${\xi }_{1,4}^{\infty }(·)\in \mathbb{AA}(\mathbf{R},\mathbb{H})$ according to Theorem 2.7 in ref. [45]. Consequently,

$${\xi }_1^K(·)={\xi }_{1,4}^K(·)+\sum _{i=1}^3{\xi }_{1,i}^K(·)\stackrel{\mathbb{H}}{⟶}{\xi }_1(·)={\xi }_{1,4}^{\infty }(·)+\sum _{i=1}^3{\xi }_{1,i}^{\infty }(·)\in \mathbb{AAA}({\mathbf{R}}_0,\mathbb{H})$$

as $K\to \infty$. By a mathematical inductive approach, we easily prove that $\{{\xi }_m:m=2,3,\dots \}\subseteq \mathbb{AAA}({\mathbf{R}}_0,\mathbb{H})$. This achieves the proof. □

Proposition 3. 

$\{{\xi }_m:m=0,1,\dots \}$ is a Cauchy sequence in $\mathbb{AAA}({\mathbf{R}}_0,\mathbb{H})$ under all the assumptions of Proposition 2.

Proof. 

Define an equivalent norm ${\parallel \eta \parallel }_{+}={sup}_{t\in {\mathbf{R}}_0}{\parallel \eta \left(t\right)\parallel }_{\mathbb{H}}$ of ${\parallel \eta \parallel }_{\mathbb{AAA}}$, $\forall \eta \in \mathbb{AAA}({\mathbf{R}}_0,\mathbb{H})$ (see page 34 in ref. [45]). By the partial sum of $\left\{{\xi }_m\right\}$, we compute

$$\begin{array}{ccc}\hfill \parallel {\xi }_{m+1}^K\left(t\right)-{\xi }_m^K\left(t\right){\parallel }_{\mathbb{H}}^2& =& \sum _{k=1}^K{\left[{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right)\left({\mathcal{F}}_k\left({\xi }_m\left(s\right)\right)-{\mathcal{F}}_k\left({\xi }_{m-1}\left(s\right)\right)\right)\mathrm{d}s\right]}^2\hfill \\ & \le & \frac{1}{{\lambda }_1}{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right)\parallel \mathcal{F}\left({\xi }_m\left(s\right)\right)-\mathcal{F}\left({\xi }_{m-1}\left(s\right)\right){\parallel }_{\mathbb{H}}^2\mathrm{d}s\hfill \\ & \le & \frac{L_F^2}{{\lambda }_1^2}\parallel {\xi }_m-{\xi }_{m-1}{\parallel }_{+}^2,\forall t\in {\mathbf{R}}_0,K=1,2,\dots .\hfill \end{array}$$

Noting that $\{{\xi }_m^K\left(t\right):K=1,2,\dots \}$ converges to ${\xi }_m\left(t\right)$ in $\mathbb{H}$ uniformly for $t\in {\mathbf{R}}_0$, then

$$\begin{array}{ccc}\hfill \parallel {\xi }_{m+1}\left(t\right)-{\xi }_m\left(t\right){\parallel }_{\mathbb{H}}& \le & \parallel {\xi }_{m+1}\left(t\right)-{\xi }_{m+1}^K\left(t\right){\parallel }_{\mathbb{H}}+\parallel {\xi }_{m+1}^K\left(t\right)-{\xi }_m^K\left(t\right){\parallel }_{\mathbb{H}}+\parallel {\xi }_m^K\left(t\right)-{\xi }_m\left(t\right){\parallel }_{\mathbb{H}}\hfill \\ & \le & \parallel {\xi }_{m+1}\left(t\right)-{\xi }_{m+1}^K\left(t\right){\parallel }_{\mathbb{H}}+\frac{L_F^2}{{\lambda }_1^2}\parallel {\xi }_m-{\xi }_{m-1}{\parallel }_{+}^2+\parallel {\xi }_m^K\left(t\right)-{\xi }_m\left(t\right){\parallel }_{\mathbb{H}}\hfill \\ & \to & \frac{L_F^2}{{\lambda }_1^2}\parallel {\xi }_m-{\xi }_{m-1}{\parallel }_{+}^2\mathrm{as}K\to \infty ,\hfill \end{array}$$

where $t\in {\mathbf{R}}_0,m=1,2,\dots$. This induces $\parallel {\xi }_{m+1}-{\xi }_m{\parallel }_{+} \le \frac{L_F}{{\lambda }_1}\parallel {\xi }_m-{\xi }_{m-1}{\parallel }_{+}$, $m=1,2,\dots$. So

$$\parallel {\xi }_{m+1}-{\xi }_m{\parallel }_{+} \le \frac{L_F^m}{{\lambda }_1^m}\parallel {\xi }_1-{\xi }_0{\parallel }_{+},\forall m=1,2,\dots .$$

For any $p\in {\mathbf{Z}}_{+}$, we have

$$\begin{array}{ccc}\hfill \parallel {\xi }_{m+p}-{\xi }_m{\parallel }_{+}& \le & \sum _{k=m}^{m+p-1}\parallel {\xi }_{k+1}-{\xi }_k{\parallel }_{+}\hfill \\ & \le & \sum _{k=m}^{m+p-1}\frac{L_F^k}{{\lambda }_1^k}\parallel {\xi }_1-{\xi }_0{\parallel }_{+} ={\left[\frac{L_F}{{\lambda }_1}\right]}^m\frac{{\lambda }_1\parallel {\xi }_1-{\xi }_0{\parallel }_{+}}{{\lambda }_1-L_F}\to 0,\hfill \end{array}$$

as $m\to \infty$. Hence, $\left\{{\xi }_m\right\}$ is a Cauchy sequence in $\mathbb{AAA}({\mathbf{R}}_0,\mathbb{H})$. This achieves the proof. □

By Proposition 3 and the completeness of $\mathbb{AAA}({\mathbf{R}}_0,\mathbb{H})$, there exists ${\xi }_{*}\in \mathbb{AAA}({\mathbf{R}}_0,\mathbb{H})$ such that $\left\{{\xi }_m\right\}$ converges to ${\xi }_{*}$ in $\mathbb{AAA}({\mathbf{R}}_0,\mathbb{H})$. Define

$${\tilde{\xi }}_{*}\left(t\right)=\sum _{k=1}^{\infty }\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\phi }_k+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left({\xi }_{*}\left(s\right)\right)\mathrm{d}s\right]e_k$$

for $t\in {\mathbf{R}}_0$. Similar to Proposition 1, ${\tilde{\xi }}_{*}(·)\in \mathbb{H}$ is well-defined. Let ${\tilde{\xi }}_{*}^K$ be the partial sum of ${\tilde{\xi }}_{*}$ denoted by

$${\tilde{\xi }}_{*}^K\left(t\right)=\sum _{k=1}^K\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\phi }_k+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left({\xi }_{*}\left(s\right)\right)\mathrm{d}s\right]e_k$$

for $t\in {\mathbf{R}}_0$, $K=1,2,\dots$.

Similar to (24), $\left\{{\tilde{\xi }}_{*}^K\left(t\right)\right\}$ converges to ${\tilde{\xi }}_{*}\left(t\right)$ in $\mathbb{H}$ uniformly for $t\in {\mathbf{R}}_0$. Then,

$$\begin{array}{ccc}\hfill \parallel {\xi }_m^K\left(t\right)-{\tilde{\xi }}_{*}^K\left(t\right){\parallel }_{\mathbb{H}}^2& =& \sum _{k=1}^K{\left[{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right)\left({\mathcal{F}}_k\left({\xi }_m\left(s\right)\right)-{\mathcal{F}}_k\left({\xi }_{*}\left(s\right)\right)\right)\mathrm{d}s\right]}^2\hfill \\ & \le & \frac{L_F^2}{{\lambda }_1^2}\parallel {\xi }_m-{\xi }_{*}{\parallel }_{+}^2\to 0,\mathrm{as}m\to \infty ,\hfill \end{array}$$

where $t\in {\mathbf{R}}_0,K=1,2,\dots .$ By the continuity of inner product ${\langle ·\rangle }_{\mathbb{H}}$ with respect to ${\parallel ·\parallel }_{\mathbb{H}}$ and letting $K\to \infty$ and $m\to \infty$ in the last inequality successively, $\parallel {\xi }_{*}\left(t\right)-{\tilde{\xi }}_{*}\left(t\right){\parallel }_{\mathbb{H}}=0$, $\forall t\in {\mathbf{R}}_0$. Hence,

$$\begin{array}{c}\hfill {\xi }_{*}\left(t\right)=\sum _{k=1}^{\infty }\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\phi }_k+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left({\xi }_{*}\left(s\right)\right)\mathrm{d}s\right]e_k,t\in {\mathbf{R}}_0.\end{array}$$

(25)

Proof of Theorem 2. 

(i) In accordance with Theorem 1, ${\mathbf{u}}_0$ is a unique solution of Equation (1) in $\mathrm{C}({\mathbf{R}}_0,\mathbb{H})$ filling the Fourier series (9). Resembling item (iv) in Theorem 1, we have ${\xi }_{*}={\mathbf{u}}_0$.

(ii) Suppose that $\widehat{\xi }$ and $\breve{\xi }$ are arbitrary two strong solutions of Equation (1) with initial values $\widehat{\xi }(x,0)=\widehat{\phi }\left(x\right)$ and $\breve{\xi }(x,0)=\breve{\phi }\left(x\right)$ for $x\in \overline{\mathsf{\Omega }},$ respectively. According to items (ii)–(iv) in Theorem 1, $\widehat{\xi }$ and $\breve{\xi }$ are $\mathbb{H}$-bounded and fulfill the Fourier series (25). Let ${\widehat{\xi }}^K$ and ${\breve{\xi }}^K$ be the partial sums of $\widehat{\xi }$ and $\breve{\xi }$ described by

$${\widehat{\xi }}^K\left(t\right)=\sum _{k=1}^K\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\widehat{\phi }}_k+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left(\widehat{\xi }\left(s\right)\right)\mathrm{d}s\right]e_k,$$

$${\breve{\xi }}^K\left(t\right)=\sum _{k=1}^K\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right){\breve{\phi }}_k+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right){\mathcal{F}}_k\left(\breve{\xi }\left(s\right)\right)\mathrm{d}s\right]e_k,$$

where ${\widehat{\phi }}_k={\langle \widehat{\phi },e_k\rangle }_{\mathbb{H}}$, ${\breve{\phi }}_k={\langle \breve{\phi },e_k\rangle }_{\mathbb{H}}$, $t\in {\mathbf{R}}_0$, $k=1,2,\dots ,K$, $K=1,2,\dots$. Resembling Proposition 1 and (24), $\left\{{\widehat{\xi }}^K\right\}$ and $\left\{{\breve{\xi }}^K\right\}$ are $\mathbb{H}$-bounded, $\mathbb{H}$-continuous and converge to $\widehat{\xi }$ and $\breve{\xi }$ in $\mathbb{H}$ uniformly for $t\in {\mathbf{R}}_0$, respectively.

Owing to ${\lambda }_1>L_F$, there exists an $ϵ>0$ small enough such that ${\lambda }_1^2>(1+ϵ)L_F^2$. We obtain

$$\begin{array}{ccc}\hfill \parallel {\widehat{\xi }}^K\left(t\right)-{\breve{\xi }}^K\left(t\right){\parallel }_{\mathbb{H}}^2& =& \sum _{k=1}^K{\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right)\left({\widehat{\phi }}_k-{\breve{\phi }}_k\right)+{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right)\left({\mathcal{F}}_k\left(\widehat{\xi }\left(s\right)\right)-{\mathcal{F}}_k\left(\breve{\xi }\left(s\right)\right)\right)\mathrm{d}s\right]}^2\hfill \\ & \le & \sum _{k=1}^K(1+{ϵ}^{-1}){\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_k\right)\right]}^2{\left({\widehat{\phi }}_k-{\breve{\phi }}_k\right)}^2\hfill \\ & & +\sum _{k=1}^K(1+ϵ){\left[{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_k\right)\left({\mathcal{F}}_k\left(\widehat{\xi }\left(s\right)\right)-{\mathcal{F}}_k\left(\breve{\xi }\left(s\right)\right)\right)\mathrm{d}s\right]}^2\hfill \\ & \le & {\left[{\mathcal{E}}_t^{\alpha }\left({\lambda }_1\right)\right]}^2(1+{ϵ}^{-1})\parallel \widehat{\phi }-\breve{\phi }{\parallel }_{\mathbb{H}}^2\hfill \\ & & +\frac{(1+ϵ)L_F^2}{{\lambda }_1}{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1\right)\parallel \widehat{\xi }\left(s\right)-\breve{\xi }\left(s\right){\parallel }_{\mathbb{H}}^2\mathrm{d}s,\hfill \end{array}$$

(26)

where $t\in {\mathbf{R}}_0,K=1,2,\dots .$ The above derivation uses the Bohr inequality ${(a+b)}^2\le (1+{ϵ}^{-1})a^2+(1+ϵ)b^2$ for $ϵ>0$, $a,b\in \mathbf{R}$. Letting $K\to \infty$ in (26) leads to

$$\begin{array}{ccc}\hfill \parallel \widehat{\xi }\left(t\right)-\breve{\xi }\left(t\right){\parallel }_{\mathbb{H}}^2& \le & {\mathcal{E}}_t^{\alpha }\left({\lambda }_1\right)(1+{ϵ}^{-1})\parallel \widehat{\phi }-\breve{\phi }{\parallel }_{\mathbb{H}}^2\hfill \\ & & +\frac{(1+ϵ)L_F^2}{{\lambda }_1}{\int }_0^t{\mathcal{E}}_{t,s}^{\alpha ,\alpha }\left({\lambda }_1\right)\parallel \widehat{\xi }\left(s\right)-\breve{\xi }\left(s\right){\parallel }_{\mathbb{H}}^2\mathrm{d}s\hfill \end{array}$$

(27)

for all $t\in {\mathbf{R}}_0$. By adopting item (11) in Lemma 1 to (27), we obtain

$$\parallel \widehat{\xi }\left(t\right)-\breve{\xi }\left(t\right){\parallel }_{\mathbb{H}}^2 \le {\mathbf{E}}_{\alpha }\left[-\left(\frac{{\lambda }_1^2-(1+ϵ)L_F^2}{{\lambda }_1}\right)t^{\alpha }\right](1+{ϵ}^{-1})\parallel \widehat{\phi }-\breve{\phi }{\parallel }_{\mathbb{H}}^2\to 0,$$

as $t\to \infty$. This achieves the proof. □

5. Time-Fractional Parabolic Equation

As an application, let us consider the nonlinear time-fractional parabolic equation with Dirichlet boundary condition as noted below:

$$\left\{\begin{array}{cc}{{}^{c}D}_{0\ast }^{\alpha }\mathbf{u}(x,t)=\kappa \mathsf{\Delta }\mathbf{u}(x,t)-\vartheta \left(x\right)\mathbf{u}(x,t)+F(\mathbf{u}(x,t),x,t),\hfill & \mathrm{in}\mathsf{\Omega }\times {\mathbf{R}}_{+},\hfill \\ \mathbf{u}(x,t)=0,\hfill & \mathrm{in}\partial \mathsf{\Omega }\times {\mathbf{R}}_0,\hfill \\ \mathbf{u}(x,0)=\phi \left(x\right),\hfill & \mathrm{in}\overline{\mathsf{\Omega }},\hfill \end{array}\right.$$

(28)

where $\mathbf{u}:\overline{\mathsf{\Omega }}\times {\mathbf{R}}_0\to \mathbf{R}$, $\mathsf{\Delta }={\sum }_{i=1}^d\frac{{\partial }^2}{\partial x_i^2}$ denotes the Laplacian operator, κ is a positive constant, and $\vartheta \in {\mathrm{C}}^{\infty }(\mathsf{\Omega },\mathbf{R})$ with ${\vartheta }_0:={inf}_{x\in \mathsf{\Omega }}\vartheta \left(x\right)>0$. The meanings of the other symbols in Equation (28) are the same as those in before.

Take the space sequence in Section 3 as follows:

$$\mathbb{H}={\mathrm{L}}^2(\mathsf{\Omega },\mathbf{R}),{\mathbb{H}}_1={\mathrm{W}}_0^{1,2}(\mathsf{\Omega },\mathbf{R}),{\mathbb{H}}_2={\mathrm{W}}_0^{1,2}(\mathsf{\Omega },\mathbf{R})\cap {\mathrm{W}}^{2,2}(\mathsf{\Omega },\mathbf{R}),\mathbb{X}={\mathrm{C}}_0^{\infty }(\mathsf{\Omega },\mathbf{R}).$$

Definition 4. 

$\mathbf{u}\in {\mathrm{L}}_{\mathrm{loc}}^{\infty }({\mathbf{R}}_{+},{\mathbb{H}}_1)\cap {\mathrm{L}}_{\mathrm{loc}}^2({\mathbf{R}}_{+},{\mathbb{H}}_2)\cap {\mathrm{W}}_{\mathrm{loc}}^{\alpha ,2}({\mathbf{R}}_{+},\mathbb{H})$ is aglobal strong solutionof Equation (28) if it satisfies

$$\left\{\begin{array}{cc}{〈{{}^{c}D}_{0\ast }^{\alpha }\mathbf{u}(x,t),\mathbf{v}〉}_{\mathbb{H}}={〈\kappa \mathsf{\Delta }\mathbf{u}(x,t)-\vartheta \left(x\right)\mathbf{u}(x,t),\mathbf{v}〉}_{\mathbb{H}}+{〈F\left(\mathbf{u}\right(x,t),x,t),\mathbf{v}〉}_{\mathbb{H}},\hfill & \mathit{in}\mathsf{\Omega }\times {\mathbf{R}}_{+},\hfill \\ {\langle \mathbf{u}(x,0),\mathbf{v}\rangle }_{\mathbb{H}}={\langle \phi ,\mathbf{v}\rangle }_{\mathbb{H}},\hfill & \mathit{in}\overline{\mathsf{\Omega }},\hfill \end{array}\right.$$

for all $\mathbf{v}\in \mathbb{H}$.

In the following, Theorems 1 and 2 are adopted to discuss the global strong solution and almost automorphic strong solution of Equation (28), respectively.

Let $\left\{{\lambda }_k\right\}$ denote the whole eigenvalues of $(\kappa \mathsf{\Delta }-\vartheta )$ with Dirichlet boundary condition and $\left\{e_k\right\}$ be the corresponding sequence of the eigenfunctions filling

$$\kappa \mathsf{\Delta }{e}_k-\vartheta (·)e_k=-{\lambda }_k{e}_k,e_k{|}_{\partial \mathsf{\Omega }}=0,k=1,2,\dots .$$

In accordance with Section 6.5 in ref. [46] or Theorem 7.22 in ref. [47], $\left\{{\lambda }_k\right\}$ has a finite multiplicity with ${\vartheta }_0<{\lambda }_1\le {\lambda }_2\le \cdots \le {\lambda }_k\le \cdots$, ${lim}_{k\to \infty }{\lambda }_k=\infty$. Furthermore, $\left\{e_k\right\}\subseteq {\mathrm{C}}^{\infty }(\mathsf{\Omega },\mathbf{R})$. So, $\left({\mathcal{H}}_1\right)$ holds.

If ${\eta }_n\rightharpoonup {\eta }_0\in {\mathbb{H}}_2={\mathrm{W}}_0^{1,2}(\mathsf{\Omega },\mathbf{R})\cap {\mathrm{W}}^{2,2}(\mathsf{\Omega },\mathbf{R})$ in $\mathbb{H}={\mathrm{L}}^2(\mathsf{\Omega },\mathbf{R})$ for arbitrary $\left\{{\eta }_n\right\}\subseteq {\mathbb{H}}_2$, then we obtain from the Green formula in ${\mathrm{W}}^{2,2}(\mathsf{\Omega },\mathbf{R})$ (see page 316 in ref. [32]) that

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim}{〈(\kappa \mathsf{\Delta }-\vartheta ){\eta }_n,\zeta 〉}_{\mathbb{H}}& =& \underset{n\to \infty }{lim}{\langle \kappa \mathsf{\Delta }{\eta }_n,\zeta \rangle }_{\mathbb{H}}-\underset{n\to \infty }{lim}{\langle \vartheta {\eta }_n,\zeta \rangle }_{\mathbb{H}}\hfill \\ & =& \underset{n\to \infty }{lim}{\langle \kappa {\eta }_n,\mathsf{\Delta }\zeta \rangle }_{\mathbb{H}}-\underset{n\to \infty }{lim}{\langle {\eta }_n,\vartheta \zeta \rangle }_{\mathbb{H}}\hfill \\ & =& {\langle \kappa {\eta }_0,\mathsf{\Delta }\zeta \rangle }_{\mathbb{H}}-{\langle {\eta }_0,\vartheta \zeta \rangle }_{\mathbb{H}}\hfill \\ & =& {\langle \kappa \mathsf{\Delta }{\eta }_0,\zeta \rangle }_{\mathbb{H}}-{\langle \vartheta {\eta }_0,\zeta \rangle }_{\mathbb{H}}\hfill \\ & =& {〈(\kappa \mathsf{\Delta }-\vartheta ){\eta }_0,\zeta 〉}_{\mathbb{H}},\forall \zeta \in {\mathbb{K}}_n,\hfill \end{array}$$

where ${\mathbb{K}}_n$ is defined the same as in Theorem 1. By the density of ${\mathbb{K}}_n$ in $\mathbb{H}$, $(\kappa \mathsf{\Delta }-\vartheta )$ is weakly continuous. Without loss of generality, the inner product and norm of ${\mathbb{H}}_1={\mathrm{W}}_0^{1,2}(\mathsf{\Omega },\mathbf{R})$ can be defined by

$${\langle \eta ,\zeta \rangle }_{{\mathbb{H}}_1}={\langle \nabla \eta ,\nabla \zeta \rangle }_{\mathbb{H}}+{\langle {\kappa }^{-1}\vartheta \eta ,\zeta \rangle }_{\mathbb{H}},{\parallel \eta \parallel }_{{\mathbb{H}}_1}=\sqrt{{\langle \eta ,\eta \rangle }_{{\mathbb{H}}_1}},\forall \eta ,\zeta \in {\mathbb{H}}_1.$$

Hence,

$${〈(\kappa \mathsf{\Delta }-\vartheta )\eta ,\zeta 〉}_{\mathbb{H}}={\langle \kappa \mathsf{\Delta }\eta ,\zeta \rangle }_{\mathbb{H}}-{\langle \vartheta \eta ,\zeta \rangle }_{\mathbb{H}}=-\kappa \left[{\langle \nabla \eta ,\nabla \zeta \rangle }_{\mathbb{H}}+{\langle {\kappa }^{-1}\vartheta \eta ,\zeta \rangle }_{\mathbb{H}}\right]=-\kappa {\langle \eta ,\zeta \rangle }_{{\mathbb{H}}_1}$$

for all $\eta \in {\mathbb{H}}_2,\zeta \in {\mathbb{H}}_1$. For another, there exists constants $c_1,c_2>0$ such that $c_2{\langle \eta ,\eta \rangle }_{{\mathbb{H}}_2}\le {\langle \mathsf{\Delta }\eta ,\mathsf{\Delta }\eta \rangle }_{\mathbb{H}}\le c_1{\langle \eta ,\eta \rangle }_{{\mathbb{H}}_2}$ for all $\eta \in {\mathbb{H}}_2$. Then,

$$\begin{array}{ccc}\hfill {〈(\kappa \mathsf{\Delta }-\vartheta )\eta ,(\kappa \mathsf{\Delta }-\vartheta )\eta 〉}_{\mathbb{H}}& =& {\kappa }^2{\langle \mathsf{\Delta }\eta ,\mathsf{\Delta }\eta \rangle }_{\mathbb{H}}+{\langle \vartheta \eta ,\vartheta \eta \rangle }_{\mathbb{H}}-2{\langle \kappa \mathsf{\Delta }\eta ,\vartheta \eta \rangle }_{\mathbb{H}}\hfill \\ & \le & 2{\kappa }^2{\langle \mathsf{\Delta }\eta ,\mathsf{\Delta }\eta \rangle }_{\mathbb{H}}+2{\vartheta }_{\infty }^2{\langle \eta ,\eta \rangle }_{\mathbb{H}}\hfill \\ & \le & 2max\{c_1{\kappa }^2,c_{\mathrm{sob}}^2{\vartheta }_{\infty }^2\}{\langle \eta ,\eta \rangle }_{{\mathbb{H}}_2},\hfill \end{array}$$

where ${\vartheta }_{\infty }:={sup}_{x\in \mathsf{\Omega }}\vartheta \left(x\right)$, $c_{\mathrm{sob}}$ is the Sobolev constant in the embedding theorem, and

$$\begin{array}{ccc}\hfill {〈(\kappa \mathsf{\Delta }-\vartheta )\eta ,(\kappa \mathsf{\Delta }-\vartheta )\eta 〉}_{\mathbb{H}}& \ge & {\kappa }^2{\langle \mathsf{\Delta }\eta ,\mathsf{\Delta }\eta \rangle }_{\mathbb{H}}+{\langle \vartheta \eta ,\vartheta \eta \rangle }_{\mathbb{H}}-2{\langle \kappa \mathsf{\Delta }\eta ,\vartheta \eta \rangle }_{\mathbb{H}}\hfill \\ & \ge & \frac{c_2{\kappa }^2}{2}{\langle \eta ,\eta \rangle }_{{\mathbb{H}}_2}-7{\vartheta }_{\infty }^2{\langle \eta ,\eta \rangle }_{\mathbb{H}},\forall \eta \in {\mathbb{H}}_2.\hfill \end{array}$$

In the above calculation, the classical inequality $ab\le \frac{1}{4}{a}^2+4b^2(a,b\in \mathbf{R})$ is employed. Taking $a_1=\kappa$, $a_2=7{\vartheta }_{\infty }^2$, $a_3=\frac{c_2{\kappa }^2}{2}$ and $a_4=2max\{c_1{\kappa }^2,c_{\mathrm{sob}}^2{\vartheta }_{\infty }^2\}$, $\left({\mathcal{H}}_2\right)$ holds.

Third, for $\eta \in {\mathrm{W}}_{\mathrm{loc}}^{\alpha ,2}({\mathbf{R}}_{+},\mathbb{H})$ and $\zeta \in \mathbb{H}$, we have from Fubini theorem and Lebesgue convergence theorem that

$$\begin{array}{ccc}\hfill {{}^{c}D}_{0\ast }^{\alpha }{\langle \eta ,\zeta \rangle }_{\mathbb{H}}& =& \frac{1}{\mathsf{\Gamma }(1-\alpha )}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_0^t{(t-s)}^{-\alpha }{\int }_{\mathsf{\Omega }}\left[\eta (x,t)-\eta (x,0)\right]\zeta \left(x\right)\mathrm{d}x\mathrm{d}s\hfill \\ & =& \frac{1}{\mathsf{\Gamma }(1-\alpha )}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{\mathsf{\Omega }}{\int }_0^t{(t-s)}^{-\alpha }\left[\eta (x,t)-\eta (x,0)\right]\mathrm{d}s\zeta \left(x\right)\mathrm{d}x\hfill \\ & =& {\int }_{\mathsf{\Omega }}{{}^{c}D}_{0\ast }^{\alpha }\eta (x,t)·\zeta \left(x\right)\mathrm{d}x\hfill \\ & =& {\langle {{}^{c}D}_{0\ast }^{\alpha }\eta ,\zeta \rangle }_{\mathbb{H}},\forall t>0.\hfill \end{array}$$

By Lemma A1 in Appendix A, we readily verify

$${\langle {{}^{c}D}_{0\ast }^{\alpha }\eta ,\eta \rangle }_{\mathbb{H}}\ge \frac{1}{2}{{}^{c}D}_{0\ast }^{\alpha }{\langle \eta ,\eta \rangle }_{\mathbb{H}},\forall \eta \in {\mathrm{H}}^{\alpha }({\mathbf{R}}_0,\mathbb{H})\cap {\mathrm{W}}_{\mathrm{loc}}^{\alpha ,2}({\mathbf{R}}_{+},\mathbb{H}).$$

Taking $\delta =\frac{1}{2}$, then $\left({\mathcal{H}}_4\right)$ holds.

Ultimately, we calculate

$$-{〈(\kappa \mathsf{\Delta }-\vartheta )\eta ,\eta 〉}_{\mathbb{H}}\ge {\vartheta }_0{\langle \eta ,\eta \rangle }_{\mathbb{H}}⟹\frac{-{〈(\kappa \mathsf{\Delta }-\vartheta )\eta ,\eta 〉}_{\mathbb{H}}}{{\langle \eta ,\eta \rangle }_{\mathbb{H}}}\ge {\vartheta }_0,\forall \eta \in {\mathrm{W}}_{\mathrm{loc}}^{\alpha ,2}({\mathbf{R}}_{+},\mathbb{H}).$$

In item (v) of Theorem 1, $a_0\ge {\vartheta }_0$.

Summarizing the above analyses and according to Theorems 1 and 2, we have

Theorem 4. 

Let $\phi \in {\mathbb{H}}_1$, $\left({\mathcal{H}}_3\right)$, and ${\vartheta }_0>L_F$ hold. Then, the following assertions hold:

(i) 

A unique global strong solution ${\mathbf{u}}^{\diamond }\in \mathcal{L}({\mathbf{R}}_0,\mathbb{H})\cap {\mathrm{H}}^{\alpha }({\mathbf{R}}_0,\mathbb{H})$ solving Equation (28), which possesses a Fourier series such as Equation (9).

(ii) 

${\mathbf{u}}^{\diamond }\in \mathbb{AAA}({\mathbf{R}}_0,\mathbb{H})$ if $\left({\mathcal{H}}_6\right)$ holds.

(iii) 

Equation (28) is global Mittag–Leffle- convergent in the sense of strong solutions.