On existence and regularity of a terminal value problem for the time fractional diffusion equation

Nonlinear diffusion equations, an important class of parabolic equations, come from many diffuse phenomena that appear widely in nature. They are proposed as mathematical models of physical problems in many areas, such as filtering, phase transition, biochemistry and dynamics of biological groups. Many new ideas and methods have been developed to consider some various kinds of nonlinear diffusion equations. We list some selected works in recent time, for example Caffarelli et al [1], Duzaar et al [2–4], Vazquez et al [5–8] and the references therein.

We present existence and regularity estimates for the solution to a final boundary value problem for a space-time fractional diffusion equation. Let D be an open and bounded domain in ${\mathbb{R}}^{k},\left(k\ge 1\right)$ with boundary ∂D. Given 0 < α < 1 and 0 < β ≤ 1, a forcing (or source) function F, we consider the final value problem for the time fractional diffusion equation

$$\begin{equation}{}^{\mathrm{c}}D_{t}^{\alpha }u\left(t,x\right)=-{\mathcal{L}}^{\beta }u\left(t,x\right)+F\left(t,x,u\left(t,x\right)\right),\quad \left(t,x\right)\in J{\times}D,\end{equation} \tag{ 1.1 }$$

with the boundary condition

$$\begin{equation}\mathcal{H}u\left(t,x\right)=0,\quad \left(t,x\right)\in J{\times}\partial D,\end{equation} \tag{ 1.2 }$$

and the final condition

$$\begin{equation}u\left(x,T\right)=\varphi \left(x\right),\quad x\in D,\end{equation} \tag{ 1.3 }$$

where φ is a given function. Here J is the interval (0, T ), the notation ${}^{\mathrm{c}}D_{t}^{\alpha }$ for 0 < α < 1 represents the left Caputo fractional derivative of order α which is defined by

$$\begin{equation*}{}^{\mathrm{c}}D_{t}^{\alpha }v\left(t\right){:=}{I}_{t}^{1-\alpha }{D}_{t}v\left(t\right),\quad t\ge 0,\end{equation*}$$

provided that ${I}_{t}^{\alpha }v\left(t\right){:=}{g}_{\alpha }\left(t\right)\star v\left(t\right)$, here ${g}_{\alpha }\left(t\right)=\frac{1}{{\Gamma}\left(\alpha \right)}{t}^{\alpha -1}$, t > 0, ⋆ denotes the convolution. For α = 1, we consider the usual time derivative $\frac{\partial u}{\partial t}$. The fractional power ${\mathcal{L}}^{\beta }\hspace{0.17em}0{< }\beta \le 1$ of the Laplacian operator $\mathcal{L}$ on D is defined by its spectrum. The symmetric uniformly elliptic operator is defined on the space L²(D) by

$$\begin{equation*}\mathcal{L}u\left(x\right)=-\sum _{i=1}^{k}\frac{\partial }{\partial {x}_{i}}\left(\sum _{j=1}^{k}{\mathcal{L}}_{ij}\left(x\right)\frac{\partial }{\partial {x}_{j}}u\left(x\right)\right)+b\left(x\right)u\left(x\right),\end{equation*}$$

provided that ${\mathcal{L}}_{ij}\in {C}^{1}\left(\overline{{\Omega}}\right)$, $b\in C\left(\overline{{\Omega}}\right)$, b(x) ≥ 0 for all $x\in \overline{{\Omega}}$, ${\mathcal{L}}_{ij}={\mathcal{L}}_{ji},1\le i,\;j\le k$, and ${\xi }^{T}\left[{\mathcal{L}}_{ij}\left(x\right)\right]\xi \ge {L}_{0}{\vert \xi \vert }^{2}$ for some L₀ > 0, $x\in \overline{{\Omega}}$, $\xi =\left({\xi }_{1},{\xi }_{2},\dots ,{\xi }_{k}\right)\in {\mathbb{R}}^{k}$. The equation (1.1) is equipped with $\mathcal{H}v=v\;\mathrm{o}\mathrm{r}\;\mathcal{H}v=\left(\to {L}\nabla v\right).\to {n}$, where $\to {L}={\left[{\mathcal{L}}_{ij}\left(x\right)\right]}_{i,j=1}^{k}$ is a k × k matrix and n is the outer normal vector of ∂D. Then the operator $\mathcal{L}$ is self-adjoint under this impedance boundary condition.

The time fractional reaction diffusion equation arises in describing 'memory' occurring in physics such as plasma turbulence [9] and it was introduced by Nigmatullin [10] to describe diffusion in media with fractal geometry, which is a special type of porous media and is applied in the flow in highly heterogeneous aquifer [11] and single-molecular protein dynamics [12]. In a physical model presented in [13], the fractional diffusion corresponds to a diverging jump length variance in the random walk, and a fractional time derivative arises when the characteristic waiting time diverges.

If the final condition (1.3) is replaced by the initial condition

$$\begin{equation}u\left(x,0\right)={u}_{0}\left(x\right),\quad x\in D\end{equation} \tag{ 1.4 }$$

then problem (1.1), (1.2), (1.4) is called a forward problem (or an initial value problem) for time-space fractional diffusion equations; for applications of this type of equation see [14] and for the abstract form of (1.1)–(1.4) see [15]. Carvalho et al [16] established a local theory of mild solutions for problem (1.1)–(1.4) where ${\mathcal{L}}^{\beta }$ is a sectorial (nonpositive) operator. Guswanto [17] studied the existence and uniqueness of a local mild solution for a class of initial value problems for nonlinear fractional evolution equations and the study of existence of initial value problems was considered by Warma et al [14]. A significant number of papers wew devoted to extend properties holding in the standard setting to the fractional one (see for example [18, 19, 20–22]).

Numerical approximation for solutions for problem (1.1)–(1.4) was studied by Jin et al [23, 24] and for other works on fractional diffusion see [25–28]. However, the literature on regularity of the initial value problem for fractional diffusion-wave equations is scarce; for the linear case see [26, 29–31], and for the nonlinear case see [14, 32–34]. Although there are many works on the direct problem, but the results on inverse problem for fractional diffusion are scarce. We can list some papers of Yamamoto and his group see [35–37, 38, 39, 40], of Kaltenbacher et al [41, 42], of Rundell et al [43, 44], of Janno see [45, 46], etc.

In practice, initial data of some problems may not be known since many phenomena cannot be measured at the initial time. Phenomena can be observed at a final time t = T, such as, in the image processing area. A picture is not processed at the capturing time t = 0. Instead, one wishes to recover the original information of the picture from its blurry form. Hence, inverse problems or terminal value problems or final value problems (IPs/FVPs), i.e., the fractional differential equations (FDEs) equipped with final value data, have been considered. IPs/FVPs are important in engineering in detecting the previous status of physical fields from its present information. If F = 0 in (1.1), Yamamoto et al [31] showed that problem (1.1)–(1.3) has a unique weak solution when φ ∈ H²(Ω). If b = 0 and F(u(x, t)) = F(x, t), Tuan et al [47] showed that problem (1.1)–(1.3) has a unique weak solution when φ ∈ H²(Ω) and F ∈ L^∞(0, T ; H²(Ω)), and other works on the homogeneous case for problem (1.1)–(1.3) can be found in [25, 47–49]. Wei and her group [62–52] studied some regularization methods for homogeneous backward problem and Yamamoto et al [53] considered a backward problem in time for a time-fractional partial differential equation in the one-dimensional case. When α = 1, systems (1.1)–(1.3) are reduced to the backward problem for classical reaction diffusion equations, and were studied in [54–56].

To the best of the authors' knowledge this is the first paper that analyzes problem (1.1)–(1.3). We present existence and uniqueness results and derive regularity estimates both in time and space. In what follows, we analyze the difficulties of this problem. By letting$u\left(t,\cdot \right)=\mathcal{O}\left(t\right)\left(\hspace{0.17em}f,\varphi \right)$, the solution operator $\mathcal{O}\left(0\right)$ is really not bounded in L²(D). Hence continuity of mild solutions does not hold at the initial time t = 0. In addition, since the fractional derivative is non-locally defined, if we put v(t) = u(T − t) then ${\left.{}^{\mathrm{c}}D_{t}^{\alpha }u\left(s\right)\right\vert }_{s=T-t}$ does not equal ${}^{\mathrm{c}}D_{t}^{\alpha }v\left(t\right)$, so the problem cannot be changed to an initial value problem. As a result we need new techniques to deal with the FVP (1.1)–(1.3). To the best of our knowledge, the work on the final value problem is still limited.

Our main results in this paper can be split into two parts, linear and nonlinear source functions. Linear models are sometimes good approximations of the real problems under consideration and provides mathematical tools needed to study nonlinear phenomena, especially for semi-linear and quasi-linear equations. In part 1, we consider the regularity property of the solution in the linear case F. We seek to address the following question: if the data is regular, how regular is the solution? Our task in this part is to find a suitable Banach space for the given data (φ, F ) in order to obtain regularity results for the corresponding solution. In part 2, we discuss existence, uniqueness and regularity for the solutions to (1.1)–(1.3) for the nonlinear problem. Our main motivation for deriving regularity results is that one needs it for a rigorous study of a numerical scheme to approximate the solution. To the best of our knowledge, regularity results on inverse initial value problems (final value problems) for fractional diffusion is still unavailable in the literature. For initial value problems, McLean et al [30], Jin et al [24] and Ahmad et al [34] considered existence and regularity results of the solution in C([0, T ]; L²(D)). However it seems the techniques in [24, 34] cannot be applied for our problems (it is impossible to apply some well-known fixed point theorems with some spaces in [24] for establishing unique solutions). To overcome this we need data φ in a suitable space and we will use a Picard iteration argument and then develop some new techniques to obtain existence and regularity of the solution.

The rest of this paper is organized as follows. In section 2, we give basic notations and preliminaries, and we propose a mild solution of our problem. In section 3, we give some regularity results of the linear inhomogeneous problem. Section 4 is devoted to existence and regularity for nonlinear problems and section 5 considers global existence.

2.1. Functional space

In this subsection, we introduce some functional spaces for solutions of FVP (1.1)–(1.3). By ${\left\{{m}_{j}\right\}}_{j\ge 1}$ and ${\left\{{e}_{j}\left(x\right)\right\}}_{j\ge 1}$, we denote the spectrum and sequence of eigenfunctions of $\mathcal{L}$ which satisfy ${e}_{j}\in \left\{v\in {H}^{2}\left(D\right):\mathcal{H}v=0\right\}$, $\mathcal{L}{e}_{j}\left(x\right)={m}_{j}{e}_{j}\left(x\right)$, 0 < m₁ ≤ m₂ ≤ ... ≤ m_j ≤ ..., and $\underset{j\to \infty }{\mathrm{lim}}{m}_{j}=\infty$. The sequence ${\left\{{e}_{j}\left(x\right)\right\}}_{j\ge 1}$ forms an orthonormal basis of the space L²(D). For a given real number p ≥ 0, the Hilbert scale space H^2p(D) is defined by

$$\begin{equation*}\left\{v\in {L}^{2}\left(D\right)\quad \mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}\quad {{\Vert}v{\Vert}}_{{H}^{2p}\left(D\right)}^{2}{:=}\sum _{j=1}^{\infty }{\left(v,{e}_{j}\right)}^{2}{m}_{j}^{2p}{< }\infty \right\},\end{equation*}$$

where (⋅, ⋅) is the usual inner product of L²(D). The fractional power ${\mathcal{L}}^{\beta }$, β ≥ 0, of the Laplacian operator $\mathcal{L}$ on D is defined by

$$\begin{equation}{\mathcal{L}}^{\beta }v\left(x\right){:=}\sum _{j=1}^{\infty }\left(v,{e}_{j}\right){m}_{j}^{\beta }{e}_{j}\left(x\right).\end{equation} \tag{ 2.1 }$$

Then, ${\left\{{m}_{j}^{\beta }\right\}}_{j\ge 1}$ is the spectrum of the operator ${\mathcal{L}}^{\beta }$. We denote by V_β the domain of ${\mathcal{L}}^{\beta }$, and then

$$\begin{equation*}{\mathbf{V}}_{\beta }=\left\{v\in {L}^{2}\left(D\right):{\Vert}{\mathcal{L}}^{\beta }v{\Vert}{< }\infty \right\}\end{equation*}$$

where ${\Vert}.{\Vert}$ is the usual norm of L²(D), and V_β is a Banach space with respect to the norm ${{\Vert}v{\Vert}}_{{\mathbf{V}}_{\beta }}={\Vert}{\mathcal{L}}^{\beta }v{\Vert}$. Moreover, the inclusion V_β ⊂ H^2β(D) holds for β > 0. We identify the dual space ${\left({L}^{2}\left(D\right)\right)}^{\prime }={L}^{2}\left(D\right)$ and define the domain ${\mathbf{V}}_{-\beta }{:=}D\left({\mathcal{L}}^{-\beta }\right)$ by the dual space of V_β, i.e., ${\mathbf{V}}_{-\beta }={\left({\mathbf{V}}_{\beta }\right)}^{\prime }$. Then, V_−β is a Hilbert space endowed with the norm

$$\begin{equation*}{{\Vert}v{\Vert}}_{{\mathbf{V}}_{-\beta }}{:=}{\left\{\sum _{j=1}^{\infty }{\left(v,{e}_{j}\right)}_{-\beta ,\beta }^{2}{m}_{j}^{-2\beta }\right\}}^{1/2},\end{equation*}$$

where (⋅, ⋅)_−β,β denotes the dual inner product between V_−β and V_β. We note that the Sobolev embedding V_β ↪ L²(D) ↪ V_−β holds for 0 < β < 1, and (,v)_−β,β = (, v), for ∈ L²(D), v ∈ V_β. Hence, we have

$$\begin{equation}{\left({e}_{i},{e}_{j}\right)}_{-\beta ,\beta }=\left({e}_{i},{e}_{j}\right)={\delta }_{ij}\end{equation} \tag{ 2.2 }$$

where δ_ij is the Kronecker delta for $i,j\in \mathbb{N}$, i, j ≥ 1. Moreover, for given p₁ ≥ 1 and 0 < η < 1, we denote by ${\mathcal{X}}_{{p}_{1},\eta }\left(J{\times}D\right)$ the set of all functions f from J to ${L}^{{p}_{1}}\left(D\right)$ such that

$$\begin{equation}{\left\vert \left\vert \left\vert \hspace{0.17em},f\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{{p}_{1},\eta }}{:=}\underset{0\le t\le T}{\text{ess}}\hspace{0.17em}{\text{sup}}{\int }_{0}^{t}{{\Vert}\hspace{0.17em}f\left(\tau ,\cdot \right){\Vert}}_{{p}_{1}}{\left(t-\tau \right)}^{\eta -1}\mathrm{d}\tau {< }\infty ,\end{equation} \tag{ 2.3 }$$

where ${{\Vert}.{\Vert}}_{{p}_{1}}$ is the norm of ${L}^{{p}_{1}}\left(D\right)$. Note that, for fixed t > 0, the Hölder's inequality shows that

$$\begin{equation*}{\int }_{0}^{t}{{\Vert}\hspace{0.17em}f\left(\tau ,\cdot \right){\Vert}}_{{p}_{1}}{\left(t-\tau \right)}^{\eta -1}\mathrm{d}\tau \le {\left[{\int }_{0}^{t}{{\Vert}\hspace{0.17em}f\left(\tau ,\cdot \right){\Vert}}_{{p}_{1}}^{{p}_{2}}\mathrm{d}\tau \right]}^{\frac{1}{{p}_{2}}}{\left[{\int }_{0}^{t}{\left(t-\tau \right)}^{\frac{{p}_{2}\left(\eta -1\right)}{{p}_{2}-1}}\mathrm{d}\tau \right]}^{\frac{{p}_{2}-1}{{p}_{2}}}.\end{equation*}$$

In the above inequality, we note that the function $\tau \to {\left(t-\tau \right)}^{\frac{{p}_{2}\left(\eta -1\right)}{{p}_{2}-1}}$ is integrable for ${p}_{2}{ >}\frac{1}{\eta }$. Therefore, if we let ${L}^{{p}_{2}}\left(0,T;{L}^{{p}_{1}}\left(D\right)\right)$, p₁, p₂ ≥ 1, be the space of all Bochner's measurable functions f from J to ${L}^{{p}_{1}}\left(D\right)$ such that

$$\begin{equation*}{{\Vert}\hspace{0.17em}f{\Vert}}_{{L}^{{p}_{2}}\left(0,T;{L}^{{p}_{1}}\left(D\right)\right)}{:=}{\left[{\int }_{0}^{t}{{\Vert}\hspace{0.17em}f\left(\tau ,\cdot \right){\Vert}}_{{p}_{1}}^{{p}_{2}}\mathrm{d}\tau \right]}^{\frac{1}{{p}_{2}}}{< }\infty ,\end{equation*}$$

then the following inclusion holds

$$\begin{equation}{L}^{{p}_{2}}\left(0,T;{L}^{{p}_{1}}\left(D\right)\right)\subset {\mathcal{X}}_{{p}_{1},\eta }\left(J{\times}D\right),\quad \mathrm{f}\mathrm{o}\mathrm{r}\hspace{0.17em}{p}_{2}{ >}\frac{1}{\eta },\end{equation} \tag{ 2.4 }$$

and there exists a positive constant C > 0 such that

$$\begin{equation}{\left\vert \left\vert \left\vert \hspace{0.17em},f\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{{p}_{1},\eta }}\le C{{\Vert}\hspace{0.17em}f{\Vert}}_{{L}^{{p}_{2}}\left(0,T;{L}^{{p}_{1}}\left(D\right)\right)},\end{equation} \tag{ 2.5 }$$

here, C depends only on p₂, η, and T. Moreover, for a given number s such that 0 < s < η, we have ${\mathcal{X}}_{{p}_{1},\eta -s}\left(J{\times}D\right)\subset {\mathcal{X}}_{{p}_{1},\eta }\left(J{\times}D\right)$ since ${\left\vert \left\vert \left\vert \hspace{0.17em},f\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{{p}_{1},\eta }}\le {T}^{s}{\left\vert \left\vert \left\vert \hspace{0.17em}f\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{{p}_{1},\eta -s}}$. Let B be a Banach space, and we denote by C([0, T ], B) the space of all continuous functions from [0, T ] to B endowed with the norm ${{\Vert}v{\Vert}}_{C\left(\left[0,T\right];B\right)}{:=}{\mathrm{sup}}_{0\le t\le T}{{\Vert}v\left(t\right){\Vert}}_{B}$, and by C^θ([0, T ], B), 0 < θ ≤ 1, the subspace of C([0, T ]; B) which includes all Hölder-continuous functions, and is equipped with the norm

$$\begin{equation*}{\left\vert \left\vert \left\vert v\right\vert \right\vert \right\vert }_{{C}^{\theta }\left(\left[0,T\right],B\right)}{:=}\underset{0\le {t}_{1}{< }{t}_{2}\le T}{\mathrm{sup}}\frac{{{\Vert}v\left({t}_{2}\right)-v\left({t}_{1}\right){\Vert}}_{B}}{{\vert {t}_{2}-{t}_{1}\vert }^{\theta }}.\end{equation*}$$

In some cases, a given function might not be continuous at t = 0. Hence, it is useful to consider the set $C\left(\left(0,T\right];B\right)$ which consists of all continuous functions from (0, T ] to B. We define by ${C}_{w}^{\rho }\left(\left(0,T\right];B\right)$ the weighted Banach space of all functions v in C((0, T ]; B) such that

$$\begin{equation*}{{\Vert}v{\Vert}}_{{C}_{w}^{\rho }\left(\left(0,T\right];B\right)}{:=}\underset{0{< }t\le T}{\mathrm{sup}}{t}^{\rho }{{\Vert}v\left(t\right){\Vert}}_{B}{< }\infty .\end{equation*}$$

Now, we discuss solutions of the FVP for the fractional ordinary equation

$$\begin{equation}{}^{\mathrm{c}}D_{t}^{\alpha }v\left(t\right)=g\left(t,v\left(t\right)\right)-mv\left(t\right),\quad t\in J,\quad \mathrm{a}\mathrm{n}\mathrm{d}\quad v\left(T\right)={v}_{T},\end{equation} \tag{ 2.6 }$$

where m, v_T are given real numbers. Here, we wish to find a representation formula for v in terms of the given function g and the final value data v_T. By writing ${}^{\mathrm{c}}D_{t}^{\alpha }={I}_{t}^{1-\alpha }{D}_{t}$, and applying the fractional integral ${I}_{t}^{\alpha }$ on both sides of equation (2.6), we obtain

$$\begin{equation*}v\left(t\right)=v\left(0\right)+{I}_{t}^{\alpha }\left[g\left(t,v\left(t\right)\right)-mv\left(t\right)\right].\end{equation*}$$

The Laplace transform yields that

$$\begin{equation*}\hat{v}=\frac{{\lambda }^{\alpha -1}}{{\lambda }^{\alpha }+m}v\left(0\right)+\frac{1}{{\lambda }^{\alpha }+m}\hat{g\left(v\right)},\end{equation*}$$

where $\hat{v}$ is the Laplace transform of v. Hence, the inverse Laplace transform implies

$$\begin{equation}v\left(t\right)=v\left(0\right){E}_{\alpha ,1}\left(-m{t}^{\alpha }\right)+g\left(t,v\left(t\right)\right)\star \left[{t}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-m{t}^{\alpha }\right)\right].\end{equation} \tag{ 2.7 }$$

Here, E_α,1 and E_α,α are the Mittag-Leffler functions which are generally defined by ${E}_{a,b}\left(z\right)={\sum }_{k=1}^{\infty }\frac{{z}^{k}}{{\Gamma}\left(ak+b\right)}$, for a > 0, $b\in \mathbb{R}$, $z\in \mathbb{C}$. Now, a representation of the solution of FVP (2.6) can be obtained by substituting t = T into (2.7), and using the final value data v(T ) = v_T, i.e.,

$$\begin{equation*}v\left(t\right)=g\left(t,v\left(t\right)\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-m{t}^{\alpha }\right)+\left[{v}_{T}-{\left.\left(g\left(r,v\left(r\right)\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-m{r}^{\alpha }\right)\right)\right\vert }_{r=T}\right]\frac{{E}_{\alpha ,1}\left(-m{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-m{T}^{\alpha }\right)},\end{equation*}$$

where

$$\begin{equation}{{\sim}{E}}_{\alpha ,\alpha }\left(-m{t}^{\alpha }\right){:=}{t}^{\alpha -1}{E}_{\alpha ,\alpha }\left(-m{t}^{\alpha }\right).\end{equation} \tag{ 2.8 }$$

2.2. Mild solutions of FVP (1.1)–(1.3) and unboundedness of solution operators

A representation of solutions and the definition of mild solutions are given in this subsection, and then we analyze the unboundedness of solution operators. By the definition (2.1) of ${\mathcal{L}}^{\beta }$, the identity ${\mathcal{L}}^{\beta }{e}_{j}\left(x\right)={m}_{j}^{\beta }{e}_{j}\left(x\right)$ holds. Hence, in view of the Fourier expansion $u\left(t,x\right)={\sum }_{j=1}^{\infty }{u}_{j}\left(t\right){e}_{j}\left(x\right)$, where u_j(t) = (u(t, ⋅), e_j), equation (1.1) can be rewritten as

$$\begin{equation*}\left({}^{\mathrm{c}}D_{t}^{\alpha }\sum _{j=1}^{\infty }{u}_{j}\left(t\right){e}_{j},{e}_{j}\right)=-\left({\mathcal{L}}^{\beta }\sum _{j=1}^{\infty }{u}_{j}\left(t\right){e}_{j},{e}_{j}\right)+\left(F\left(t,x,u\left(t,x\right)\right),{e}_{j}\right),\quad t\in J.\end{equation*}$$

This is equivalent to the equation

$$\begin{equation*}{}^{\mathrm{c}}D_{t}^{\alpha }{u}_{j}\left(t\right)={F}_{j}\left(t,u\left(t\right)\right)-{m}_{j}^{\beta }{u}_{j}\left(t\right),\quad {F}_{j}\left(t,u\left(t\right)\right)=\left(F\left(t,x,u\left(t,x\right)\right),{e}_{j}\right).\end{equation*}$$

By applying the method of solutions of FVPs for fractional ordinary equations in subsection 2.1, and using the final value data (1.3), we derive

$$\begin{align}\hfill & {u}_{j}\left(t\right)={F}_{j}\left(t,u\left(t\right)\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)\hfill \\ \hfill & +\left[{\varphi }_{j}-{\left.\left({F}_{j}\left(r,u\left(r\right)\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{r}^{\alpha }\right)\right)\right\vert }_{r=T}\right]\frac{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)},\hfill \end{align} \tag{ 2.9 }$$

where φ_j = (φ, e_j). Therefore, we obtain a spectral representation for u as follows:

$$\begin{align}\hfill u\left(t,x\right)& =\sum _{j=1}^{\infty }{F}_{j}\left(t,u\left(t\right)\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{t}^{\alpha }\right){e}_{j}\left(x\right)\hfill \\ \hfill & \quad +\sum _{j=1}^{\infty }\left[{\varphi }_{j}-{\left.\left({F}_{j}\left(r,u\left(r\right)\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{r}^{\alpha }\right)\right)\right\vert }_{r=T}\right]\frac{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}{e}_{j}\left(x\right).\hfill \end{align}$$

For g ∈ L²(0, T; L²(D)) and v ∈ L²(D), let us denote by ${\mathcal{O}}_{n}$, 1 ≤ n ≤ 3, the following operators

$$\begin{equation*}\left({\mathcal{O}}_{1}g\right)\left(t,x\right){:=}\sum _{j=1}^{\infty }{g}_{j}\left(t\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{t}^{\alpha }\right){e}_{j}\left(x\right),\quad \left({\mathcal{O}}_{2}\left(t\right)v\right)\left(x\right){:=}\sum _{j=1}^{\infty }{v}_{j}\frac{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}{e}_{j}\left(x\right),\end{equation*}$$

and $\left({\mathcal{O}}_{3}g\right)\left(t\right){:=}-{\mathcal{O}}_{2}\left(t\right)\left({\mathcal{O}}_{1}g\right)\left(T\right)$ on L²(D), for t ∈ J. Then, the solution u can be represented as

$$\begin{equation}u\left(t,x\right)=\left({\mathcal{O}}_{1}F\right)\left(t,x\right)+\left({\mathcal{O}}_{2}\left(t\right)\varphi \right)\left(x\right)+\left({\mathcal{O}}_{3}F\right)\left(t,x\right),\end{equation} \tag{ 2.10 }$$

for (t, x) ∈ J × D.

One of the most important things, when we consider the well-posedness of a PDE, is the boundedness of solution operators. Corresponding to the initial value problem (1.1), (1.2), (1.4), the solution operators are usually bounded in L²(D); see e.g., [27, 30, 33, 34, 57]. Unfortunately, some solution operators of FVP (1.1)–(1.3) are not bounded on L²(D) at t = 0. For this purpose, we recall that, for 0 < α < 1 and z < 0, there exist positive constants c_α, ${\hat{c}}_{\alpha }$ such that

$$\begin{equation}\frac{{c}_{\alpha }}{1+\vert z\vert }\le \vert {E}_{\alpha ,1}\left(z\right)\vert \le \frac{{\hat{c}}_{\alpha }}{1+\vert z\vert },\quad \vert {E}_{\alpha ,\alpha }\left(z\right)\vert \le \mathrm{min}\left\{\frac{{\hat{c}}_{\alpha }}{1+\vert z\vert },\frac{{\hat{c}}_{\alpha }}{1+{\vert z\vert }^{2}}\right\},\end{equation} \tag{ 2.11 }$$

see, for example [58, 59, 60]. Now, let v₀ be defined by ${v}_{0,j}{:=}\left({v}_{0},{e}_{j}\right)={j}^{-1/2}{m}_{j}^{-\beta }$, j ≥ 1. Then, it is easy to see that v₀ belongs to V_βγ for 0 ≤ γ < 1, and does not for γ ≥ 1. Using the inequalities (2.11), we have

$$\begin{align}\hfill {{\Vert}{\mathcal{O}}_{2}\left(0\right){v}_{0}{\Vert}}^{2}& =\sum _{j=1}^{\infty }\frac{{v}_{0,j}^{2}}{{E}_{\alpha ,1}^{2}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\ge {c}_{\alpha }^{-2}\sum _{j=1}^{\infty }{v}_{0,j}^{2}{\left(1+{m}_{j}^{\beta }{T}^{\alpha }\right)}^{2}\hfill \\ \hfill & \ge {c}_{\alpha }^{-2}{T}^{2\alpha }\sum _{j=1}^{\infty }{v}_{0,j}^{2}{m}_{j}^{2\beta }={c}_{\alpha }^{-2}{T}^{2\alpha }\sum _{j=1}^{\infty }{j}^{-1}=\infty ,\hfill \end{align}$$

which shows the unboundedness of ${\mathcal{O}}_{2}\left(0\right)$ on L²(D). Similarly, the unboundedness of ${\mathcal{O}}_{3}\left(0\right)$ on L²(D) can be shown.

In this section, we study the regularity of mild solutions of FVP (1.1)–(1.3) corresponding to the linear source function F, i.e., F(t, x, u(t, x)) = F(t, x) which does not include u. We will investigate the regularity of the following FVP

$$\begin{equation}\begin{cases}{}^{\mathrm{c}}D_{t}^{\alpha }u\left(t,x\right)=-{\mathcal{L}}^{\beta }u\left(t,x\right)+F\left(t,x\right),\hfill & \quad \left(t,x\right)\in J{\times}D,\hfill \\ \mathcal{H}u\left(t,x\right)=0,\hfill & \quad \left(t,x\right)\in J{\times}\partial D,\hfill \\ u\left(x,T\right)=\varphi \left(x\right),\hfill & \quad x\in D,\hfill \end{cases}\end{equation} \tag{ 3.1 }$$

where φ, F will be specified later. In order to consider this problem, it it necessary to give a definition of mild solutions based on (2.10) as follows.

Definition 3.1. If a function u belongs to L^p(0, T; L^q(D)), for some p, q ≥ 1, and satisfies the equation

$$\begin{equation}u\left(t,x\right)=\left({\mathcal{O}}_{1}F\right)\left(t,x\right)+\left({\mathcal{O}}_{2}\left(t\right)\varphi \right)\left(x\right)+\left({\mathcal{O}}_{3}F\right)\left(t,x\right),\end{equation} \tag{ 3.2 }$$

then u is said to be a mild solution of FVP (3.1).

In what follows, we introduce some assumptions on the final value data φ and the linear source function F.

• (R1) 0 < p, q < 1 such that p + q = 1;
• (R2) $0{< }r\le \frac{1-\alpha q}{\alpha q}$;
• (R3) $0{< }s{< }\mathrm{min}\left(\alpha q,1-\alpha q\right)$;
• (R4) $0{< }{p}^{\prime }\le p-\frac{s}{\alpha },\quad \quad \quad \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{q}^{\prime }=1-{p}^{\prime },\quad 0{< }r\le \frac{1-\alpha {q}^{\prime }}{\alpha {q}^{\prime }}$;
• (R5) $0\le \hat{q}\le \mathrm{min}\left(p,q,\frac{s}{\alpha }\right),\quad \hat{p}=1-\hat{q},\quad \hspace{0.17em}\hspace{0.17em}0{< }\hat{r}\le \frac{1-\alpha }{\alpha }$.

In the following lemma, we will show that solutions of FVP (3.1) must be bounded by a power function t^−αq, for some appropriate number q, i.e.,

$$\begin{equation*}{\Vert}u\left(t,\cdot \right){\Vert}< sim {t}^{-\alpha q},\end{equation*}$$

for all 0 < t ≤ T.

Lemma 3.2. Let p, q be defined by (R1), and u satisfies (3.2). If φ ∈ V_βp, and $F\in {\mathcal{X}}_{2,\alpha q}\left(J{\times}D\right)$, then there exists a constant C₀ > 0 such that

$$\begin{equation}{\Vert}u\left(t,\cdot \right){\Vert}\le {C}_{0}\left({{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}+{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q}}\right){t}^{-\alpha q}.\end{equation} \tag{ 3.3 }$$

Proof. The inequalities (2.11) shows that

$$\begin{equation}\;{E}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\left(t-\tau \right)}^{\alpha }\right)\le {\hat{c}}_{\alpha }{\left[1+{m}_{j}^{\beta }{\left(t-\tau \right)}^{\alpha }\right]}^{-p}\le {\hat{c}}_{\alpha }{m}_{j}^{-\beta p}{\left(t-\tau \right)}^{-\alpha p}.\end{equation} \tag{ 3.4 }$$

Combined with the definition $\left({\mathcal{O}}_{1}F\right)\left(t,x\right)={\sum }_{j=1}^{\infty }{F}_{j}\left(t\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{t}^{\alpha }\right){e}_{j}\left(x\right)$, we have that

$$\begin{align}\hfill {\Vert}\left({\mathcal{O}}_{1}F\right)\left(t,\cdot \right){\Vert}& \le {\int }_{0}^{t}{\Vert}\sum _{j=1}^{\infty },{F}_{j},\left(\tau \right),{{\sim}{E}}_{\alpha ,\alpha },\left(-{m}_{j}^{\beta }{\left(t-\tau \right)}^{\alpha }\right),{e}_{j}{\Vert}\mathrm{d}\tau \hfill \\ \hfill & ={\int }_{0}^{t}{\left\{\sum _{j=1}^{\infty }{F}_{j}^{2}\left(\tau \right){E}_{\alpha ,\alpha }^{2}\left(-{m}_{j}^{\beta }{\left(t-\tau \right)}^{\alpha }\right){\left(t-\tau \right)}^{2\alpha -2}\right\}}^{1/2}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }{\int }_{0}^{t}{\left\{\sum _{j=1}^{\infty }{F}_{j}^{2}\left(\tau \right){m}_{j}^{-2\beta p}{\left(t-\tau \right)}^{-2\alpha p}{\left(t-\tau \right)}^{2\alpha -2}\right\}}^{1/2}\mathrm{d}\tau .\hfill \end{align} \tag{ 3.5 }$$

Hence, we obtain the following estimate

$$\begin{equation}{\Vert}\left({\mathcal{O}}_{1}F\right)\left(t,\cdot \right){\Vert}\le \;{\hat{c}}_{\alpha }{m}_{1}^{-\beta p}{\int }_{0}^{t}{\Vert}F\left(\tau ,\cdot \right){\Vert}{\left(t-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \le {M}_{1}{t}^{-\alpha q}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q}},\end{equation} \tag{ 3.6 }$$

by noting (2.3) and letting ${M}_{1}={\hat{c}}_{\alpha }{m}_{1}^{-\beta p}{T}^{\alpha q}$. In addition, the norm ${\Vert}{\mathcal{O}}_{2}\left(t\right)\varphi {\Vert}$ can be estimated as

$$\begin{equation*}{\Vert}{\mathcal{O}}_{2}\left(t\right)\varphi {\Vert}=\;{\left\{\sum _{j=1}^{\infty }{\varphi }_{j}^{2}\frac{{E}_{\alpha ,1}^{2}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}^{2}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\right\}}^{1/2}\le {\hat{c}}_{\alpha }{c}_{\alpha }^{-1}{\left\{\sum _{j=1}^{\infty }{\varphi }_{j}^{2}{\left[\frac{1+{m}_{j}^{\beta }{T}^{\alpha }}{1+{m}_{j}^{\beta }{t}^{\alpha }}\right]}^{2}\right\}}^{1/2}.\end{equation*}$$

The ratio $\frac{1+{m}_{j}^{\beta }{T}^{\alpha }}{1+{m}_{j}^{\beta }{t}^{\alpha }}$ is clearly bounded by both $1+{m}_{j}^{\beta }{T}^{\alpha }$ and $\frac{{T}^{\alpha }}{{t}^{\alpha }}$. Moreover, the increasing property of the sequence ${\left\{{m}_{j}\right\}}_{j\ge 1}$ shows $1\le {m}_{1}^{-\beta }{m}_{j}^{\beta }$. Thus, we have $1+{m}_{j}^{\beta }{T}^{\alpha }\le \left({m}_{1}^{-\beta }+{T}^{\alpha }\right){m}_{j}^{\beta }$. By noting p + q = 1, one can deduce that the ratio is bounded by the product of T^αqt^−αq and ${\left(1+{m}_{j}^{\beta }{T}^{\alpha }\right)}^{p}$. Bring the above arguments together, and this leads to

$$\begin{equation}{\Vert}{\mathcal{O}}_{2}\left(t\right)\varphi {\Vert}\le \;{\hat{c}}_{\alpha }{c}_{\alpha }^{-1}{\left\{\sum _{j=1}^{\infty }{\varphi }_{j}^{2}\frac{{T}^{2\alpha q}}{{t}^{2\alpha q}}{\left(1+{m}_{j}^{\beta }{T}^{\alpha }\right)}^{2p}\right\}}^{1/2}\le {M}_{2}{t}^{-\alpha q}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}},\end{equation} \tag{ 3.7 }$$

where

$$\begin{equation*}{M}_{2}={\hat{c}}_{\alpha }{c}_{\alpha }^{-1}{T}^{\alpha q}{\left({m}_{1}^{-\beta }+{T}^{\alpha }\right)}^{p}.\end{equation*}$$

Now, we proceed to estimate ${\Vert}\left({\mathcal{O}}_{3}F\right)\left(t,\cdot \right){\Vert}$ by using the same techniques as in (3.5) and (3.7). As a consequence of

$$\begin{equation*}\left({\mathcal{O}}_{3}F\right)\left(t,x\right)=-{\mathcal{O}}_{2}\left(t\right)\left({\mathcal{O}}_{1}F\right)\left(T\right)\left(x\right)=-\sum _{j=1}^{\infty }{\left.\left({F}_{j}\left(r\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{r}^{\alpha }\right)\right)\right\vert }_{r=T}\frac{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}{e}_{j}\left(x\right),\end{equation*}$$

we can obtain the following estimates

$$\begin{align}\hfill {\Vert}\left({\mathcal{O}}_{3}F\right)\left(t,\cdot \right){\Vert}& \le {\int }_{0}^{T}{\Vert}\sum _{j=1}^{\infty },{F}_{j},\left(\tau \right),{E}_{\alpha ,\alpha },\left(-{m}_{j}^{\beta }{\left(T-\tau \right)}^{\alpha }\right),{\left(T-\tau \right)}^{\alpha -1},\frac{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)},{e}_{j}{\Vert}\mathrm{d}\tau \hfill \\ \hfill & ={\int }_{0}^{T}{\left\{\sum _{j=1}^{\infty }{F}_{j}^{2}\left(\tau \right){E}_{\alpha ,\alpha }^{2}\left(-{m}_{j}^{\beta }{\left(T-\tau \right)}^{\alpha }\right){\left(T-\tau \right)}^{2\alpha -2}\frac{{E}_{\alpha ,1}^{2}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}^{2}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\right\}}^{1/2}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }^{2}{c}_{\alpha }^{-1}{\int }_{0}^{T}{\left\{\sum _{j=1}^{\infty }{F}_{j}^{2}\left(\tau \right){m}_{j}^{-2\beta p}{\left(T-\tau \right)}^{2\alpha q-2}\frac{{T}^{2\alpha q}}{{t}^{2\alpha q}}{\left(1+{m}_{j}^{\beta }{T}^{\alpha }\right)}^{2p}\right\}}^{1/2}\mathrm{d}\tau .\hfill \end{align}$$

A simple computation shows that

$$\begin{equation}{\Vert}\left({\mathcal{O}}_{3}F\right)\left(t,\cdot \right){\Vert}\le \;{M}_{3}{t}^{-\alpha q}{\int }_{0}^{T}{\Vert}F\left(\tau ,\cdot \right){\Vert}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \le {M}_{3}{t}^{-\alpha q}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q}},\end{equation} \tag{ 3.8 }$$

where we let ${M}_{3}={\hat{c}}_{\alpha }{M}_{2}$. Finally, it follows from (3.6)–(3.8), and the identity (3.2) that

$$\begin{align}\hfill {\Vert}u\left(t,\cdot \right){\Vert}& \le {\Vert}\left({\mathcal{O}}_{3}F\right)\left(t,\cdot \right){\Vert}+{\Vert}{\mathcal{O}}_{2}\left(t\right)\varphi {\Vert}+{\Vert}\left({\mathcal{O}}_{3}F\right)\left(t,\cdot \right){\Vert} \hfill \\ \hfill & \le \left(\sum _{1\le n\le 3}{M}_{n}\right)\left({{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}+{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q}}\right){t}^{-\alpha q}.\hfill \end{align}$$

The inequality (3.3) is proved by letting ${C}_{0}=\sum _{1\le n\le 3}{M}_{n}.$□

Based on lemma 3.2, we consider existence, uniqueness, and regularity of solutions in the following theorem which is divided into two parts. In the first part, we obtain the existence and uniqueness of a mild solution in the space ${L}^{\frac{1}{\alpha q}-r}\left(0,T;{L}^{2}\left(D\right)\right)$ for some suitable numbers q, r and for the given assumptions on φ and F as in lemma 3.2. In the second part, we improve the smoothness of the mild solution by considering the spatial-fractional derivative ${\mathcal{L}}^{\beta \left(p-{p}^{\prime }\right)}$. It is very important to investigate the continuity of the mild solution. We first show that the mild solution is continuous from (0, T] to L²(D). Moreover, we establish the continuity on the closed interval [0, T] which corresponds to lower spatial-smoothness, V_−βq', for a relevant number q^'.

Theorem 3.3. 

• (a)  
  Let p, q, r be defined by (R1), (R2). If φ ∈ V_βp and $F\in {\mathcal{X}}_{2,\alpha q}\left(J{\times}D\right)$, then FVP (3.1) has a unique solution u in ${L}^{\frac{1}{\alpha q}-r}\left(0,T;{L}^{2}\left(D\right)\right)$. Moreover, there exists a positive constant C₁ such that

  $$\begin{equation}{{\Vert}u{\Vert}}_{{L}^{\frac{1}{\alpha q}-r}\left(0,T;{L}^{2}\left(D\right)\right)}\le {C}_{1}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}+{C}_{1}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q}}.\end{equation} \tag{ 3.9 }$$
• (b)  
  Let p, q, s, r, p', q' be defined by (R1), (R3), (R4). If φ ∈ V_βp, and $F\in {\mathcal{X}}_{2,\alpha q-s}\left(J{\times}D\right)$, then FVP (3.1) has a unique solution u such that

  $$\begin{equation*}\;u\in {L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}\right)\cap {C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)\cap {C}^{s}\left(\left[0,T\right];{\mathbf{V}}_{-\beta {q}^{\prime }}\right).\end{equation*}$$

  Moreover, there exists a positive constant C₂ such that

  $$\begin{align}\hfill & {{\Vert}u{\Vert}}_{{L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}\right)}+{{\Vert}u{\Vert}}_{{C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)}+{\left\vert \left\vert \left\vert u\right\vert \right\vert \right\vert }_{{C}^{s}\left(\left[0,T\right];{\mathbf{V}}_{-\beta {q}^{\prime }}\right)}\hfill \\ \hfill & \quad \quad \le {C}_{2}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}+{C}_{2}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}}.\hfill \end{align} \tag{ 3.10 }$$

Proof. The proof of part (a) can be easily obtained from lemma 3.2. Indeed, the inequality (3.3) leads to

$$\begin{equation*}{{\Vert}u{\Vert}}_{{L}^{\frac{1}{\alpha q}-r}\left(0,T;{L}^{2}\left(D\right)\right)}\le {C}_{0}\left({{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}+{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q}}\right){\left\{{\int }_{0}^{T}{t}^{-\alpha q\left(\frac{1}{\alpha q}-r\right)}\mathrm{d}t\right\}}^{\frac{\alpha q}{1-\alpha qr}}.\end{equation*}$$

Since $-\alpha q\left(\frac{1}{\alpha q}-r\right){ >}-1$, the integral in the above inequality exists, i.e., t^−αq belongs to ${L}^{\frac{1}{\alpha q}-r}\left(0,T;\mathbb{R}\right)$. Hence, FVP (3.1) has a solution u in ${L}^{\frac{1}{\alpha q}-r}\left(0,T;{L}^{2}\left(D\right)\right)$. The uniqueness of u depends on the uniqueness of the ODE (2.6). For the uniqueness of this ODE, see [61] (theorem 3.25). Moreover, the inequality (3.9) is derived by letting ${C}_{1}={C}_{0}{{\Vert}{t}^{-\alpha q}{\Vert}}_{{L}^{\frac{1}{\alpha q}-r}\left(0,T;\mathbb{R}\right)}$. Now, we proceed to prove part (b) which will be presented in the following steps.

Step 1: We prove $u\in {L}^{\frac{1}{\alpha q}-r}\left(0,T;{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}\right)$. Firstly, by the same argument as in the proof of (3.5), we derive the following chain of inequalities

$$\begin{align}\hfill {{\Vert}\left({\mathcal{O}}_{1}F\right)\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}}& \le {\int }_{0}^{t}{\Vert}{\mathcal{L}}^{\beta \left(p-{p}^{\prime }\right)},\sum _{j=1}^{\infty },{F}_{j},\left(\tau \right),{{\sim}{E}}_{\alpha ,\alpha },\left(-{m}_{j}^{\beta }{\left(t-\tau \right)}^{\alpha }\right),{e}_{j}{\Vert}\mathrm{d}\tau ,\hfill \\ \hfill & \le {\hat{c}}_{\alpha }{\int }_{0}^{t}{\left\{\sum _{j=1}^{\infty }{F}_{j}^{2}\left(\tau \right){m}_{j}^{-2\beta p}{\left(t-\tau \right)}^{-2\alpha p}{\left(t-\tau \right)}^{2\alpha -2}{m}_{j}^{2\beta \left(p-{p}^{\prime }\right)}\right\}}^{1/2}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }{m}_{1}^{-\beta {p}^{\prime }}{\int }_{0}^{t}{\Vert}F\left(\tau ,\cdot \right){\Vert}{\left(t-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \le {M}_{4}{t}^{-\alpha {q}^{\prime }}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}},\hfill \end{align} \tag{ 3.11 }$$

for ${M}_{4}={\hat{c}}_{\alpha }{m}_{1}^{-\beta {p}^{\prime }}{T}^{\alpha {q}^{\prime }+s}$, where the inequality ${\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q}}\le {T}^{s}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}}$ holds. Similarly, from ${{\Vert}{\mathcal{O}}_{2}\left(t\right)\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}}={\Vert}{\mathcal{L}}^{\beta \left(p-{p}^{\prime }\right)}{\mathcal{O}}_{2}\left(t\right)\varphi {\Vert}$ and the same way as in the proof of (3.7), we have

$$\begin{align}\hfill & {{\Vert}{\mathcal{O}}_{2}\left(t\right)\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}}\hfill \\ \hfill & \quad \quad \le \;{\hat{c}}_{\alpha }{c}_{\alpha }^{-1}{\left\{\sum _{j=1}^{\infty }{\varphi }_{j}^{2}\frac{{T}^{2\alpha {q}^{\prime }}}{{t}^{2\alpha {q}^{\prime }}}{\left(1+{m}_{j}^{\beta }{T}^{\alpha }\right)}^{2{p}^{\prime }}{m}_{j}^{2\beta \left(p-{p}^{\prime }\right)}\right\}}^{1/2}\le {M}_{5}{t}^{-\alpha {q}^{\prime }}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}},\hfill \end{align} \tag{ 3.12 }$$

where we let ${M}_{5}={\hat{c}}_{\alpha }{c}_{\alpha }^{-1}{T}^{\alpha {q}^{\prime }}{\left({m}_{1}^{-\beta }+{T}^{\alpha }\right)}^{{p}^{\prime }}$. Now, we will estimate the norm ${{\Vert}\left({\mathcal{O}}_{3}F\right)\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}}$ which will use the same estimate for the fraction $\frac{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}$ as in (3.12). Indeed, noting that ${\left(1+{m}_{j}^{\beta }{T}^{\alpha }\right)}^{{p}^{\prime }}\le {\left({m}_{1}^{-\beta }+{T}^{\alpha }\right)}^{{p}^{\prime }}{m}_{j}^{\beta {p}^{\prime }}$, by using (2.11), we see that

$$\begin{align}\hfill & {{\Vert}\left({\mathcal{O}}_{3}F\right)\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}}\hfill \\ \hfill & \quad \quad \le \;{\int }_{0}^{T}{\Vert}{\mathcal{L}}^{\beta \left(p-{p}^{\prime }\right)},\sum _{j=1}^{\infty },{F}_{j},\left(\tau \right),{E}_{\alpha ,\alpha },\left(-{m}_{j}^{\beta }{\left(T-\tau \right)}^{\alpha }\right),{\left(T-\tau \right)}^{\alpha -1},\frac{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)},{e}_{j}{\Vert}\mathrm{d}\tau \hfill \\ \hfill & \quad \quad ={\int }_{0}^{T}{\left\{\sum _{j=1}^{\infty }{F}_{j}^{2}\left(\tau \right){E}_{\alpha ,\alpha }^{2}\left(-{m}_{j}^{\beta }{\left(T-\tau \right)}^{\alpha }\right){\left(T-\tau \right)}^{2\alpha -2}\frac{{E}_{\alpha ,1}^{2}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}^{2}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}{m}_{j}^{2\beta \left(p-{p}^{\prime }\right)}\right\}}^{1/2}\mathrm{d}\tau \hfill \\ \hfill & \quad \quad \le {\hat{c}}_{\alpha }{M}_{5}{\int }_{0}^{T}{\left\{\sum _{j=1}^{\infty }{F}_{j}^{2}\left(\tau \right){m}_{j}^{-2\beta p}{\left(T-\tau \right)}^{-2\alpha p}{\left(T-\tau \right)}^{2\alpha -2}{t}^{-2\alpha {q}^{\prime }}{m}_{j}^{2\beta {p}^{\prime }}{m}_{j}^{2\beta \left(p-{p}^{\prime }\right)}\right\}}^{1/2}\mathrm{d}\tau .\hfill \end{align}$$

Thus, by some simple computations, one can get

$$\begin{equation}{{\Vert}\left({\mathcal{O}}_{3}F\right)\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}}\le {\hat{c}}_{\alpha }{M}_{5}{t}^{-\alpha {q}^{\prime }}{\int }_{0}^{T}{\Vert}F\left(t,\cdot \right){\Vert}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \le {M}_{6}{t}^{-\alpha {q}^{\prime }}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}},\end{equation} \tag{ 3.13 }$$

with ${M}_{6}={\hat{c}}_{\alpha }{M}_{5}{T}^{s}$, where the inequality ${\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q}}\le {T}^{s}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}}$ has been used. Bring (3.11)–(3.13) and (3.2) together, we have

$$\begin{equation*}{{\Vert}u\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}}\le {M}_{7}\left({{\Vert}\varphi {\Vert}}_{{V}_{\beta p}}+{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}}\right){t}^{-\alpha {q}^{\prime }},\end{equation*}$$

where M₇ = ∑_4≤n≤6M_n. Since the function t → t^−αq' is clearly contained in the space ${L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;\mathbb{R}\right)$, we can take the ${L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;\mathbb{R}\right)$-norm on both sides of the above inequality, namely

$$\begin{equation}{{\Vert}u{\Vert}}_{{L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}\right)}\le {M}_{8}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}+{M}_{8}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}},\end{equation} \tag{ 3.14 }$$

where ${M}_{8}={M}_{7}{{\Vert}{t}^{-\alpha {q}^{\prime }}{\Vert}}_{{L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;\mathbb{R}\right)}$.

Step 2: We prove $u\in {C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)$. Let us consider 0 < t₁ < t₂ ≤ T. By (3.2), the difference u(t₂, x) − u(t₁, x) can be calculated as

$$\begin{align}\hfill & u\left({t}_{2},x\right)-u\left({t}_{1},x\right)\hfill \\ \hfill & \quad \quad =\sum _{j=1}^{\infty }{\left.{F}_{j}\left(t\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)\right\vert }_{t={t}_{1}}^{t={t}_{2}}{e}_{j}\left(x\right)+\sum _{j=1}^{\infty }{\varphi }_{j}{\left.\frac{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\right\vert }_{t={t}_{1}}^{t={t}_{2}}{e}_{j}\left(x\right)\hfill \\ \hfill & \quad \quad \quad \quad -\;\sum _{j=1}^{\infty }{\left.\left({F}_{j}\left(r\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{r}^{\alpha }\right)\right)\right\vert }_{r=T}{\left.\frac{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\right\vert }_{t={t}_{1}}^{t={t}_{2}}{e}_{j}\left(x\right).\hfill \end{align}$$

By using the differentiation identities

$$\begin{equation*}{D}_{\omega }{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)=-{m}_{j}^{\beta }{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)\;\;\mathrm{a}\mathrm{n}\mathrm{d}\;\;{D}_{\omega }\left({{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)\right)={\omega }^{\alpha -2}{E}_{\alpha ,\alpha -1}\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right),\end{equation*}$$

see, for example [58, 59, 60], we have

$$\begin{align}\hfill & {\left.{F}_{j}\left(t\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)\right\vert }_{t={t}_{1}}^{t={t}_{2}}\hfill \\ \hfill & ={\int }_{0}^{{t}_{1}}{F}_{j}\left(\tau \right){\left.{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)\right\vert }_{\omega ={t}_{1}-\tau }^{\omega ={t}_{2}-\tau }\mathrm{d}\tau +{\int }_{{t}_{1}}^{{t}_{2}}{F}_{j}\left(\tau \right){{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\left({t}_{2}-\tau \right)}^{\alpha }\right)\mathrm{d}\tau \hfill \\ \hfill & ={\int }_{0}^{{t}_{1}}{\int }_{{t}_{1}-\tau }^{{t}_{2}-\tau }{F}_{j}\left(\tau \right){\omega }^{\alpha -2}{E}_{\alpha ,\alpha -1}\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)\mathrm{d}\omega \mathrm{d}\tau +{\int }_{{t}_{1}}^{{t}_{2}}{F}_{j}\left(\tau \right){{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\left({t}_{2}-\tau \right)}^{\alpha }\right)\mathrm{d}\tau ,\hfill \end{align}$$

and

$$\begin{equation*}{\left.{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)\right\vert }_{t={t}_{1}}^{t={t}_{2}}=-{m}_{j}^{\beta }{\int }_{{t}_{1}}^{{t}_{2}}{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)\mathrm{d}\omega .\end{equation*}$$

Combining the above arguments gives

$$\begin{align}\hfill & u\left({t}_{2},x\right)-u\left({t}_{1},x\right)\hfill \\ \hfill & =\sum _{j=1}^{\infty }{\int }_{0}^{{t}_{1}}{\int }_{{t}_{1}-\tau }^{{t}_{2}-\tau }{F}_{j}\left(\tau \right){\omega }^{\alpha -2}{E}_{\alpha ,\alpha -1}\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)\mathrm{d}\omega \mathrm{d}\tau {e}_{j}\left(x\right)\hfill \\ \hfill & \quad \quad \quad \quad +\sum _{j=1}^{\infty }{\int }_{{t}_{1}}^{{t}_{2}}{F}_{j}\left(\tau \right){{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\left({t}_{2}-\tau \right)}^{\alpha }\right)\mathrm{d}\tau {e}_{j}\left(x\right)-{\mathcal{L}}^{\beta }\sum _{j=1}^{\infty }{\varphi }_{j}{\int }_{{t}_{1}}^{{t}_{2}}\frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\mathrm{d}\omega {e}_{j}\left(x\right)\hfill \\ \hfill & \quad \quad \quad \quad +\;{\mathcal{L}}^{\beta }\sum _{j=1}^{\infty }{\int }_{0}^{T}{\int }_{{t}_{1}}^{{t}_{2}}{F}_{j}\left(\tau \right){{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\left(T-\tau \right)}^{\alpha }\right)\frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\mathrm{d}\omega \mathrm{d}\tau {e}_{j}\left(x\right)\hfill \\ \hfill & {:=}{\mathcal{I}}_{1}+{\mathcal{I}}_{2}+{\mathcal{I}}_{3}+{\mathcal{I}}_{4}.\hfill \end{align} \tag{ 3.15 }$$

Now, we will establish estimates for ${\mathcal{I}}_{j}$, j = 1, 2, 3, 4, and show that ${\mathcal{I}}_{j}$ tends to 0 as t₂ − t₁ → 0. Firstly, by the inequality (2.11), we see that the absolute value of ${E}_{\alpha ,\alpha -1}\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)$ is bounded by ${\hat{c}}_{\alpha }{m}_{j}^{-\beta p}{\omega }^{-\alpha p}$. This implies

$$\begin{equation*}{\omega }^{\alpha -2}\left\vert {E}_{\alpha ,\alpha -1}\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)\right\vert \le {\hat{c}}_{\alpha }{\omega }^{\alpha q-2}{m}_{j}^{-\beta p}.\end{equation*}$$

Moreover, for 0 < τ < t₁, we have

$$\begin{equation*}{\int }_{{t}_{1}-\tau }^{{t}_{2}-\tau }{\omega }^{\alpha q-2}\mathrm{d}\omega =\frac{1}{1-\alpha q}\frac{{\left({t}_{2}-\tau \right)}^{1-\alpha q}-{\left({t}_{1}-\tau \right)}^{1-\alpha q}}{{\left({t}_{1}-\tau \right)}^{1-\alpha q}{\left({t}_{2}-\tau \right)}^{1-\alpha q}},\end{equation*}$$

where we note that the estimates ${\left({t}_{2}-\tau \right)}^{1-\alpha q}-{\left({t}_{1}-\tau \right)}^{1-\alpha q}\le {\left({t}_{2}-{t}_{1}\right)}^{1-\alpha q}$ and ${\left({t}_{2}-\tau \right)}^{1-\alpha q}\ge {\left({t}_{1}-\tau \right)}^{s}{\left({t}_{2}-{t}_{1}\right)}^{1-\alpha q-s}$ can be showed easily from 0 < αq < 1, and 1 − αq − s > 0. Hence, we deduce

$$\begin{align}\hfill {\Vert}{\mathcal{I}}_{1}{\Vert}& \le {\int }_{0}^{{t}_{1}}{\Vert}\sum _{j=1}^{\infty },{\int }_{{t}_{1}-\tau }^{{t}_{2}-\tau },{F}_{j},\left(\tau \right),{\omega }^{\alpha -2},{E}_{\alpha ,\alpha -1},\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right),\mathrm{d},\omega ,{e}_{j}{\Vert}\text{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }{m}_{1}^{-\beta p}{\int }_{0}^{{t}_{1}}{\int }_{{t}_{1}-\tau }^{{t}_{2}-\tau }{\omega }^{\alpha q-2}\mathrm{d}\omega {\Vert}F\left(\tau ,\cdot \right){\Vert}\mathrm{d}\tau \hfill \\ \hfill & \le \frac{{\hat{c}}_{\alpha }{m}_{1}^{-\beta p}}{1-\alpha q}{\int }_{0}^{{t}_{1}}{\Vert}F\left(\tau ,\cdot \right){\Vert}{\left({t}_{1}-\tau \right)}^{\alpha q-s-1}\mathrm{d}\tau {\left({t}_{2}-{t}_{1}\right)}^{s}.\hfill \end{align}$$

This leads to

$$\begin{equation}\;{\Vert}{\mathcal{I}}_{1}{\Vert}\le {M}_{9}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}}{\left({t}_{2}-{t}_{1}\right)}^{s},\end{equation} \tag{ 3.16 }$$

where we let ${M}_{9}=\frac{{\hat{c}}_{\alpha }{m}_{1}^{-\beta p}}{1-\alpha q}$. Secondly, an estimate for the term ${\mathcal{I}}_{2}$ can be shown by using (2.11) as follows.

$$\begin{align}\hfill {\Vert}{\mathcal{I}}_{2}{\Vert}& \le {\int }_{{t}_{1}}^{{t}_{2}}{\Vert}\sum _{j=1}^{\infty },{F}_{j},\left(\tau \right),{E}_{\alpha ,\alpha },\left(-{m}_{j}^{\beta }{\left({t}_{2}-\tau \right)}^{\alpha }\right),{e}_{j}{\Vert}{\left({t}_{2}-\tau \right)}^{\alpha -1}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }{\int }_{{t}_{1}}^{{t}_{2}}{\Vert}F\left(\tau ,\cdot \right){\Vert}{\left({t}_{2}-\tau \right)}^{\alpha q-s-1}{\left({t}_{2}-\tau \right)}^{\alpha p+s}\mathrm{d}\tau \hfill \\ \hfill & \le {M}_{10}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}}{\left({t}_{2}-{t}_{1}\right)}^{s},\hfill \end{align} \tag{ 3.17 }$$

where we let ${M}_{10}={\hat{c}}_{\alpha }{T}^{\alpha p}$. Thirdly, we will estimate the term ${\mathcal{I}}_{3}$. We have

$$\begin{equation*}{\Vert}{\mathcal{I}}_{3}{\Vert}={\Vert}{\mathcal{L}}^{\beta },\sum _{j=1}^{\infty },{\varphi }_{j},{\int }_{{t}_{1}}^{{t}_{2}},\frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)},\mathrm{d},\omega ,{e}_{j}{\Vert}.\end{equation*}$$

Here, the fraction can be estimated as follows

$$\begin{equation}\left\vert \frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\right\vert \le {\hat{c}}_{\alpha }{c}_{\alpha }^{-1}{\left[\frac{1+{m}_{j}^{\beta }{T}^{\alpha }}{1+{m}_{j}^{\beta }{\omega }^{\alpha }}\right]}^{p}{\left[\frac{1+{m}_{j}^{\beta }{T}^{\alpha }}{1+{m}_{j}^{2\beta }{\omega }^{2\alpha }}\right]}^{q}{\omega }^{\alpha -1},\end{equation} \tag{ 3.18 }$$

by using (2.11) and ${{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)={E}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right){\omega }^{\alpha -1}$. Moreover, we can see that

$$\begin{equation*}\frac{1+{m}_{j}^{\beta }{T}^{\alpha }}{1+{m}_{j}^{2\beta }{\omega }^{2\alpha }}\le \left({m}_{1}^{-\beta }+{T}^{\alpha }\right){m}_{j}^{\beta }{m}_{j}^{-2\beta }{\omega }^{-2\alpha }\le \left({m}_{1}^{-\beta }+{T}^{\alpha }\right){m}_{j}^{-\beta }{\omega }^{-2\alpha }.\end{equation*}$$

Taking these estimates together, we thus obtain the following chain of the inequalities

$$\begin{align}\hfill {\Vert}{\mathcal{I}}_{3}{\Vert}& \le {\left\{\sum _{j=1}^{\infty }{\varphi }_{j}^{2}{m}_{j}^{2\beta }{\left[{\int }_{{t}_{1}}^{{t}_{2}}\left\vert \frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\right\vert \mathrm{d}\omega \right]}^{2}\right\}}^{1/2}\hfill \\ \hfill & \le {M}_{11}{\left\{\sum _{j=1}^{\infty }{\varphi }_{j}^{2}{m}_{j}^{2\beta }{\left[{\int }_{{t}_{1}}^{{t}_{2}}{\omega }^{-\alpha p}{m}_{j}^{-\beta q}{\omega }^{-2\alpha q}{\omega }^{\alpha -1}\mathrm{d}\omega \right]}^{2}\right\}}^{1/2}\hfill \\ \hfill & \le {M}_{11}{\left\{\sum _{j=1}^{\infty }{\varphi }_{j}^{2}{m}_{j}^{2\beta p}{\left[{\int }_{{t}_{1}}^{{t}_{2}}{\omega }^{-\alpha q-1}\mathrm{d}\omega \right]}^{2}\right\}}^{1/2},\hfill \end{align}$$

which implies that

$$\begin{equation}{\Vert}{\mathcal{I}}_{3}{\Vert}\le {M}_{12}{t}_{1}^{-2\alpha q}{\left({t}_{2}-{t}_{1}\right)}^{\alpha q}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}},\end{equation} \tag{ 3.19 }$$

where ${M}_{11}={\hat{c}}_{\alpha }{c}_{\alpha }^{-1}{T}^{\alpha p}{\left({m}_{1}^{-\beta }+{T}^{\alpha }\right)}^{q}$, and M₁₂ = M₁₁[αq]⁻¹. Fourthly, we proceed to estimate ${\mathcal{I}}_{4}$. According to (3.18), we have $\left\vert \frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\right\vert \le {M}_{11}{m}_{j}^{-\beta q}{\omega }^{-\alpha q-1}$. Moreover, ${{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\left(T-\tau \right)}^{\alpha }\right)\le {\hat{c}}_{\alpha }{m}_{j}^{-\beta p}{\left(T-\tau \right)}^{\alpha q-1}$ can be established by using the inequalities (2.11). Hence, we obtain

$$\begin{align}\hfill {\Vert}{\mathcal{I}}_{4}{\Vert}& \le {\int }_{0}^{T}{\Vert}{\mathcal{L}}^{\beta },\sum _{j=1}^{\infty },{\int }_{{t}_{1}}^{{t}_{2}},{F}_{j},\left(\tau \right),{{\sim}{E}}_{\alpha ,\alpha },\left(-{m}_{j}^{\beta }{\left(T-\tau \right)}^{\alpha }\right),\frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)},\mathrm{d},\omega ,{e}_{j}{\Vert}\mathrm{d}\tau \hfill \\ \hfill & \le {\int }_{0}^{T}{\left\{\sum _{j=1}^{\infty }{m}_{j}^{2\beta }{F}_{j}^{2}\left(\tau \right){\left[{\int }_{{t}_{1}}^{{t}_{2}}\left\vert {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\left(T-\tau \right)}^{\alpha }\right)\frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\right\vert \mathrm{d}\omega \right]}^{2}\right\}}^{1/2}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }{M}_{11}{\int }_{0}^{T}{\left\{\sum _{j=1}^{\infty }{m}_{j}^{2\beta }{F}_{j}^{2}\left(\tau \right){\left[{\int }_{{t}_{1}}^{{t}_{2}}{m}_{j}^{-\beta p}{\left(T-\tau \right)}^{\alpha q-1}{m}_{j}^{-\beta q}{\omega }^{-\alpha q-1}\mathrm{d}\omega \right]}^{2}\right\}}^{1/2}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }{M}_{11}\frac{{t}_{2}^{\alpha q}-{t}_{1}^{\alpha q}}{{t}_{1}^{2\alpha q}}{\int }_{0}^{T}{\Vert}F\left(\tau ,\cdot \right){\Vert}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau ,\hfill \end{align}$$

and we arrive at

$$\begin{equation}{\Vert}{\mathcal{I}}_{4}{\Vert}\le {\hat{c}}_{\alpha }{M}_{13}{t}_{1}^{-2\alpha q}{\left({t}_{2}-{t}_{1}\right)}^{\alpha q}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}},\end{equation} \tag{ 3.20 }$$

where M₁₃ = M₁₁T^s. We deduce from (3.16), (3.17), (3.19), (3.20) that ${\Vert}{\sum }_{1\le \hspace{0.17em}j\le 4},{\mathcal{I}}_{j}{\Vert}$ tends to 0 as t₂ − t₁ tends 0 for 0 < t₁ < t₂ ≤ T. Thus, u belongs to the set C((0, T ]; L²(D)). On the other hand, by assumption (R3), we have 0 < αq − s < αq and ${\mathcal{X}}_{2,\alpha q-s}\left(J{\times}D\right)\subset {\mathcal{X}}_{2,\alpha q}\left(J{\times}D\right)$. Therefore, the assumptions on φ and F in this theorem also fulfill lemma 3.2. Hence, the inequality (3.3) holds, i.e.,

$$\begin{equation}{t}^{\alpha q}{\Vert}u\left(t,\cdot \right){\Vert}\le {M}_{4}\left({{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}+{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q}}\right),\quad \quad t{ >}0.\end{equation} \tag{ 3.21 }$$

This implies u belongs to ${C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)$. Moreover, by taking the supremum on both sides of (3.21) on (0, T ], we obtain

$$\begin{equation}{{\Vert}u{\Vert}}_{{C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)}\le \;{M}_{4}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}+{T}^{s}{M}_{4}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}}.\end{equation} \tag{ 3.22 }$$

Step 3: We prove u ∈ C^s([0, T ]; V_−βq'). In this step, we establish the continuity of the solution on the closed interval [0, T ]. Now, we consider 0 ≤ t₁ < t₂ ≤ T. If t₁ = 0, then ${\mathcal{I}}_{1}=0$. If t₁ > 0, then combining (2.2) in the same way as in (3.16) gives

$$\begin{align}\hfill {{\Vert}{\mathcal{I}}_{1}{\Vert}}_{{\mathbf{V}}_{-\beta {q}^{\prime }}}\le & {\int }_{0}^{{t}_{1}}{{\Vert}\sum _{j=1}^{\infty },{\int }_{{t}_{1}-\tau }^{{t}_{2}-\tau },{F}_{j},\left(\tau \right),{\omega }^{\alpha -2},{E}_{\alpha ,\alpha -1},\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right),\mathrm{d},\omega ,{e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta {q}^{\prime }}}\mathrm{d}\tau \hfill \\ \hfill \le & {\int }_{0}^{{t}_{1}}\left\{\sum _{j=1}^{\infty }{m}_{j}^{-2\beta {q}^{\prime }}\left({\int }_{{t}_{1}-\tau }^{{t}_{2}-\tau }{F}_{j}\left(\tau \right){\omega }^{\alpha -2}{E}_{\alpha ,\alpha -1}\right.\right.{\left.{\left.\;\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)\mathrm{d}\omega {e}_{j},{e}_{j}\right)}_{-\beta {q}^{\prime },\beta {q}^{\prime }}^{2}\right\}}^{1/2}\mathrm{d}\tau \hfill \\ \hfill \le & {\int }_{0}^{{t}_{1}}\left\{\sum _{j=1}^{\infty }{m}_{j}^{-2\beta {q}^{\prime }}{F}_{j}^{2}\left(\tau \right)\left\vert {\int }_{{t}_{1}-\tau }^{{t}_{2}-\tau }{\omega }^{\alpha -2}\left\vert {E}_{\alpha ,\alpha -1}\right.\right.\right.{\left.{\left.\left.\;\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)\right\vert \mathrm{d}\omega \right\vert }^{2}\right\}}^{1/2}\mathrm{d}\tau ,\hfill \end{align}$$

and so

$$\begin{equation}{{\Vert}{\mathcal{I}}_{1}{\Vert}}_{{\mathbf{V}}_{-\beta {q}^{\prime }}}\le \;\frac{{\hat{c}}_{\alpha }{m}_{1}^{-\beta {q}^{\prime }-\beta p}}{1-\alpha q}{\left({t}_{2}-{t}_{1}\right)}^{s}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}}.\end{equation} \tag{ 3.23 }$$

On the other hand, the inequality (3.17) also holds for all 0 ≤ t₁ < t₂ ≤ T. Hence, the same way as in the proof (3.17) shows that

$$\begin{align}\hfill {{\Vert}{\mathcal{I}}_{2}{\Vert}}_{{\mathbf{V}}_{-\beta {q}^{\prime }}}& \le {\int }_{{t}_{1}}^{{t}_{2}}{{\Vert}\sum _{j=1}^{\infty },{F}_{j},\left(\tau \right),{E}_{\alpha ,\alpha },\left(-{m}_{j}^{\beta }{\left({t}_{2}-\tau \right)}^{\alpha }\right),{e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta {q}^{\prime }}}{\left({t}_{2}-\tau \right)}^{\alpha -1}\mathrm{d}\tau \hfill \\ \hfill & \le {m}_{1}^{-\beta {q}^{\prime }}{\hat{c}}_{\alpha }{\left({t}_{2}-{t}_{1}\right)}^{\alpha p+s}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}}\hfill \\ \hfill & \le {m}_{1}^{-\beta {q}^{\prime }}{\hat{c}}_{\alpha }{T}^{\alpha p}{\left({t}_{2}-{t}_{1}\right)}^{s}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}}.\hfill \end{align}$$

Now, we will establish estimates for ${{\Vert}{I}_{3}{\Vert}}_{{\mathbf{V}}_{-\beta {q}^{\prime }}}$ and ${{\Vert}{I}_{4}{\Vert}}_{{\mathbf{V}}_{-\beta {q}^{\prime }}}$. Indeed, we have

$$\begin{align}\hfill \left\vert \frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\right\vert & \le {\hat{c}}_{\alpha }{c}_{\alpha }^{-1}{\left[\frac{1+{m}_{j}^{\beta }{T}^{\alpha }}{1+{m}_{j}^{\beta }{\omega }^{\alpha }}\right]}^{p-{p}^{\prime }}{\left[\frac{1+{m}_{j}^{\beta }{T}^{\alpha }}{1+{m}_{j}^{\beta }{\omega }^{\alpha }}\right]}^{q+{p}^{\prime }}{\omega }^{\alpha -1},\hfill \\ \hfill & \le {\hat{c}}_{\alpha }{c}_{\alpha }^{-1}{\left({m}_{1}^{-\beta }+{T}^{\alpha }\right)}^{p-{p}^{\prime }}{m}_{j}^{\beta \left(p-{p}^{\prime }\right)}{T}^{\alpha \left(q+{p}^{\prime }\right)}{\omega }^{-\alpha \left(q+{p}^{\prime }\right)}{\omega }^{\alpha -1}.\hfill \end{align}$$

Thus, we can derive

$$\begin{equation}\left\vert \frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\right\vert \le {M}_{14}{m}_{j}^{\beta \left(p-{p}^{\prime }\right)}{\omega }^{\alpha \left(p-{p}^{\prime }\right)-1},\end{equation} \tag{ 3.24 }$$

where we let

$$\begin{equation*}{M}_{14}={\hat{c}}_{\alpha }{c}_{\alpha }^{-1}{\left({m}_{1}^{-\beta }+{T}^{\alpha }\right)}^{p-{p}^{\prime }}{T}^{\alpha \left(q+{p}^{\prime }\right)}.\end{equation*}$$

This leads to

$$\begin{align}\hfill {{\Vert}{I}_{3}{\Vert}}_{{\mathbf{V}}_{-\beta {q}^{\prime }}}& \le {\left\{\sum _{j=1}^{\infty }{\varphi }_{j}^{2}{m}_{j}^{2\beta }{m}_{j}^{-2\beta {q}^{\prime }}{\left[{\int }_{{t}_{1}}^{{t}_{2}}\left\vert \frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\right\vert \mathrm{d}\omega \right]}^{2}\right\}}^{1/2}\hfill \\ \hfill & \le {M}_{14}{\left\{\sum _{j=1}^{\infty }{\varphi }_{j}^{2}{m}_{j}^{2\beta }{m}_{j}^{-2\beta {q}^{\prime }}{\left[{\int }_{{t}_{1}}^{{t}_{2}}{m}_{j}^{\beta \left(p-{p}^{\prime }\right)}{\omega }^{\alpha \left(p-{p}^{\prime }\right)-1}\mathrm{d}\omega \right]}^{2}\right\}}^{1/2}\hfill \\ \hfill & \le {M}_{14}{\left\{\sum _{j=1}^{\infty }{\varphi }_{j}^{2}{m}_{j}^{2\beta p}{\left[{\int }_{{t}_{1}}^{{t}_{2}}{\omega }^{\alpha \left(p-{p}^{\prime }\right)-1}\mathrm{d}\omega \right]}^{2}\right\}}^{1/2}.\hfill \end{align}$$

Thus, by letting ${M}_{15}=\frac{{M}_{14}}{\alpha \left(p-{p}^{\prime }\right)}{T}^{\alpha \left(p-{p}^{\prime }\right)-s}$, we obtain the estimate

$$\begin{align}\hfill {{\Vert}{I}_{3}{\Vert}}_{{\mathbf{V}}_{-\beta {q}^{\prime }}}& \le \frac{{M}_{14}}{\alpha \left(p-{p}^{\prime }\right)}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}\left({t}_{2}^{\alpha \left(p-{p}^{\prime }\right)}-{t}_{1}^{\alpha \left(p-{p}^{\prime }\right)}\right)\hfill \\ \hfill & \le {M}_{15}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}{\left({t}_{2}-{t}_{1}\right)}^{s},\hfill \end{align} \tag{ 3.25 }$$

where we have used that

$$\begin{equation*}{t}_{2}^{\alpha \left(p-{p}^{\prime }\right)}-{t}_{1}^{\alpha \left(p-{p}^{\prime }\right)}\le {\left({t}_{2}-{t}_{1}\right)}^{\alpha \left(p-{p}^{\prime }\right)}\le {T}^{\alpha \left(p-{p}^{\prime }\right)-s}{\left({t}_{2}-{t}_{1}\right)}^{s}\end{equation*}$$

for ${p}^{\prime }\le p-\frac{s}{\alpha }$. By employing (3.24) and following the same way as in (3.25), we have

$$\begin{align}\hfill {{\Vert}{\mathcal{I}}_{4}{\Vert}}_{{\mathbf{V}}_{-\beta {q}^{\prime }}}& \le \;{\int }_{0}^{T}{{\Vert}{\mathcal{L}}^{\beta },\sum _{j=1}^{\infty },{\int }_{{t}_{1}}^{{t}_{2}},{F}_{j},\left(\tau \right),{{\sim}{E}}_{\alpha ,\alpha },\left(-{m}_{j}^{\beta }{\left(T-\tau \right)}^{\alpha }\right),\frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)},\mathrm{d},\omega ,{e}_{j}{\Vert}}_{\mathbf{V}-\beta {q}^{\prime }}\mathrm{d}\tau \hfill \\ \hfill & \le {\int }_{0}^{T}\left\{\sum _{j=1}^{\infty }{m}_{j}^{2\beta {p}^{\prime }}{F}_{j}^{2}\left(\tau \right)\left[{\int }_{{t}_{1}}^{{t}_{2}}\left\vert {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\left(T-\tau \right)}^{\alpha }\right)\right.\right.\right.{\left.{\left.\left.\frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\right\vert \mathrm{d}\omega \right]}^{2}\right\}}^{1/2}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }{M}_{14}{\int }_{0}^{T}\left\{\sum _{j=1}^{\infty }{m}_{j}^{2\beta {p}^{\prime }}{F}_{j}^{2}\left(\tau \right)\left[{\int }_{{t}_{1}}^{{t}_{2}}{\left(T-\tau \right)}^{\alpha q-1}{m}_{j}^{-\beta {p}^{\prime }}\right.\right.{\left.{\left.{\omega }^{\alpha \left(p-{p}^{\prime }\right)-1}\mathrm{d}\omega \right]}^{2}\right\}}^{1/2}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }\frac{{M}_{14}}{\alpha \left(p-{p}^{\prime }\right)}\left({t}_{2}^{\alpha \left(p-{p}^{\prime }\right)}-{t}_{1}^{\alpha \left(p-{p}^{\prime }\right)}\right){\int }_{0}^{T}{\Vert}F\left(\tau ,\cdot \right){\Vert}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau ,\hfill \end{align}$$

where

$$\begin{equation*}\vert {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\left(T-\tau \right)}^{\alpha }\right)\vert \le {m}_{j}^{-\beta p}{\left(T-\tau \right)}^{\alpha q-1}.\end{equation*}$$

This implies that

$$\begin{equation}{{\Vert}{\mathcal{I}}_{4}{\Vert}}_{{\mathbf{V}}_{-\beta {q}^{\prime }}}\le {M}_{16}{\left({t}_{2}-{t}_{1}\right)}^{s}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}},\end{equation} \tag{ 3.26 }$$

where ${M}_{16}={\hat{c}}_{\alpha }{T}^{s}{M}_{15}$. Combining the above arguments guarantees that u belongs to C^s([0, T ]; V_−βq'). Moreover, there also exists a positive constant M₁₇ such that

$$\begin{equation}{\left\vert \left\vert \left\vert u\right\vert \right\vert \right\vert }_{{C}^{s}\left(\left[0,T\right];{\mathbf{V}}_{-\beta {q}^{\prime }}\right)}\le {M}_{17}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}+{M}_{17}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}}.\end{equation} \tag{ 3.27 }$$

Finally, the inequality (3.10) is obtained by taking the inequality (3.14), (3.22) and (3.27) together. The proof is complete. □

In the next theorem, we will investigate the time-space fractional derivative of the mild solution u. More specifically, we investigate ${}^{\mathrm{c}}D_{t}^{\alpha }{\mathcal{L}}^{-\beta \left(q-\hat{q}\right)}u$, for a suitably chosen number $\hat{q}\le q$. We also establish the continuity of ${}^{\mathrm{c}}D_{t}^{\alpha }{\mathcal{L}}^{-\beta q}u$ on the interval (0, T ] without establishing it at t = 0 since this requires a strong assumption of F, for example, F must be continuous on whole interval [0, T ].

Theorem 3.4. Let $p,q,s,{p}^{\prime },{q}^{\prime },\hat{p},\hat{q},r,\hat{r}$ be defined by (R1), (R3), (R4), (R5).

• (a)  
  If $\varphi \in {\mathbf{V}}_{\beta \left(p+\hat{q}\right)}$, and $F\in {L}^{\frac{1}{\alpha q-s}+\hat{r}}\left(0,T;{L}^{2}\left(D\right)\right)$, then FVP (3.1) has a unique solution u such that

  $$\begin{align}\hfill & u\in {L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}\right)\cap {C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)\cap {C}^{s}\left(\left[0,T\right];{\mathbf{V}}_{-\beta {q}^{\prime }}\right),\hfill \\ \hfill & {}^{\mathrm{c}}D_{t}^{\alpha }u\in {L}^{\frac{1}{\alpha }-\hat{r}}\left(0,T;{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}\right).\hfill \end{align}$$

  Moreover, there exists a positive constant C₃ such that

  $$\begin{equation}{{\Vert}{}^{\mathrm{c}}D_{t}^{\alpha }u{\Vert}}_{{L}^{\frac{1}{\alpha }-\hat{r}}\left(0,T;{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}\right)}\le \;{C}_{3}{{\Vert}F{\Vert}}_{{L}^{\frac{1}{\alpha q-s}+\hat{r}}\left(0,T;{L}^{2}\left(D\right)\right)}+{C}_{3}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}.\end{equation} \tag{ 3.28 }$$
• (b)  
  If $\varphi \in {\mathbf{V}}_{\beta \left(p+\hat{q}\right)}$, and $F\in {L}^{\frac{1}{\alpha q-s}+\hat{r}}\left(0,T;{L}^{2}\left(D\right)\right)\cap {C}_{w}^{\alpha }\left(\left(0,T\right];{V}_{-\beta q}\right)$, then FVP (3.1) has a unique solution u such that

  $$\begin{align}\hfill & u\in {L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}\right)\cap {C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)\cap {C}^{s}\left(\left[0,T\right];{\mathbf{V}}_{-\beta {q}^{\prime }}\right),\hfill \\ \hfill & {}^{\mathrm{c}}D_{t}^{\alpha }u\in {L}^{\frac{1}{\alpha }-\hat{r}}\left(0,T;{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}\right)\cap {C}_{w}^{\alpha }\left(\left(0,T\right];{V}_{-\beta q}\right).\hfill \end{align}$$

  Moreover, there exists a positive constant C₄ such that

  $$\begin{align}\hfill & {{\Vert}{}^{\mathrm{c}}D_{t}^{\alpha }u{\Vert}}_{{L}^{\frac{1}{\alpha }-\hat{r}}\left(0,T;{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}\right)}+{{\Vert}{}^{\mathrm{c}}D_{t}^{\alpha }u{\Vert}}_{{C}_{w}^{\alpha }\left(\left(0,T\right];{\mathbf{V}}_{-\beta q}\right)}\hfill \\ \hfill & \quad \quad \le {C}_{4}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}+{C}_{4}{{\Vert}F{\Vert}}_{{L}^{\frac{1}{\alpha q-s}+\hat{r}}\left(0,T;{L}^{2}\left(D\right)\right)}+{C}_{4}{{\Vert}F{\Vert}}_{{C}_{w}^{\alpha }\left(\left(0,T\right];{\mathbf{V}}_{-\beta q}\right)}.\hfill \end{align} \tag{ 3.29 }$$

Proof. (a) By (R5), we have 0 < αq − s < 1, and $\frac{1}{\alpha q-s}+\hat{r}{ >}\frac{1}{\alpha q-s}{ >}1$. Thus, one can deduce from (2.5) that

$$\begin{equation}{L}^{\frac{1}{\alpha q-s}+\hat{r}}\left(0,T;{L}^{2}\left(D\right)\right)\subset {\mathcal{X}}_{2,\alpha q-s}\left(J{\times}D\right),\end{equation} \tag{ 3.30 }$$

where we have used the inclusion (2.4). Moreover, the Sobolev embedding ${\mathbf{V}}_{\beta \left(p+\hat{q}\right)}\hookrightarrow {\mathbf{V}}_{\beta p},$ holds. Therefore, the assumptions of this theorem also fulfills part (b) of theorem 3.3. Hence, FVP (3.1) has a unique solution

$$\begin{equation*}\;u\in {L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}\right)\cap {C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)\cap {C}^{s}\left(\left[0,T\right];{\mathbf{V}}_{-\beta {q}^{\prime }}\right).\end{equation*}$$

Now, we prove ${}^{\mathrm{c}}D_{t}^{\alpha }u$ exists and belongs to ${L}^{\frac{1}{\alpha }-\hat{r}}\left(0,T;{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}\right)\cap {C}_{w}^{\alpha }\left(\left(0,T\right];{\mathbf{V}}_{-\beta q}\right)$. It follows from the identities

$$\begin{align}\hfill {}^{\mathrm{c}}D_{t}^{\alpha }{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)& =-{m}_{j}^{\beta }{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right),\hfill \\ \hfill {}^{\mathrm{c}}D_{t}^{\alpha }{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)& =-{m}_{j}^{\beta }{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{t}^{\alpha }\right),\hfill \end{align}$$

see, for example [58, 59, 60], and equation (2.9) that

$$\begin{align}\hfill {}^{\mathrm{c}}D_{t}^{\alpha }{u}_{j}\left(t\right)& ={}^{\mathrm{c}}D_{t}^{\alpha }\left[{F}_{j}\left(t\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)\right]+\left[{\varphi }_{j}-{\left.\left({F}_{j}\left(r\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{r}^{\alpha }\right)\right)\right\vert }_{r=T}\right]\frac{{}^{\mathrm{c}}D_{t}^{\alpha }{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\hfill \\ \hfill & ={F}_{j}\left(t\right)-{m}_{j}^{\beta }{F}_{j}\left(t\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)-{\varphi }_{j}\frac{{m}_{j}^{\beta }{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\hfill \\ \hfill & \quad {\left.+\left({F}_{j}\left(r\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{r}^{\alpha }\right)\right)\right\vert }_{r=T}\frac{{m}_{j}^{\beta }{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}{:=}{F}_{j}\left(t\right)+{\psi }_{j}^{\left(1\right)}\left(t\right)+{\psi }_{j}^{\left(2\right)}\left(t\right)+{\psi }_{j}^{\left(3\right)}\left(t\right),\hfill \end{align}$$

for all $j\in \mathbb{N}$. Firstly, let us consider the sum ${\sum }_{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{\psi }_{j}^{\left(1\right)}\left(t\right){e}_{j}$, for ${n}_{1},{n}_{2}\in \mathbb{N}$, 1 ≤ n₁ < n₂. By the definition of the dual space ${\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}$ of ${\mathbf{V}}_{\beta \left(q-\hat{q}\right)}$, and the identity (2.2) of their dual inner product, we have

$$\begin{align}\hfill & {{\Vert}\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}},{\psi }_{j}^{\left(1\right)},\left(t\right),{e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}\hfill \\ \hfill \quad \quad \quad \quad & \quad \le {\int }_{0}^{t}{{\Vert}\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}},{m}_{j}^{\beta },{F}_{j},\left(\tau \right),{{\sim}{E}}_{\alpha ,\alpha },\left(-{m}_{j}^{\beta }{\left(t-\tau \right)}^{\alpha }\right),{e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}\mathrm{d}\tau \hfill \\ \hfill & \quad \le {\int }_{0}^{t}{\left\{\sum _{i=1}^{\infty }{m}_{i}^{2\beta \left(p+\hat{q}\right)}{\left(\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{F}_{j}\left(\tau \right){{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\left(t-\tau \right)}^{\alpha }\right){e}_{j},{e}_{i}\right)}_{-\beta \left(q-\hat{q}\right),\beta \left(q-\hat{q}\right)}^{2}\right\}}^{1/2}\mathrm{d}\tau \hfill \\ \hfill & \quad \le {\int }_{0}^{t}{\left\{\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{m}_{j}^{2\beta \left(p+\hat{q}\right)}{F}_{j}^{2}\left(\tau \right){{\sim}{E}}_{\alpha ,\alpha }^{2}\left(-{m}_{j}^{\beta }{\left(t-\tau \right)}^{\alpha }\right)\right\}}^{1/2}\mathrm{d}\tau .\hfill \end{align}$$

Assumption (R5) shows that $0{< }p+\hat{q}{< }1$. Hence, by using the inequalities (2.11), we have $\vert {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\left(t-\tau \right)}^{\alpha }\right)\vert \le {\hat{c}}_{\alpha }{m}_{j}^{-\beta \left(p+\hat{q}\right)}{\left(t-\tau \right)}^{-\alpha \left(p+\hat{q}\right)}{\left(t-\tau \right)}^{\alpha -1}$. This together with the above argument gives

$$\begin{equation}{{\Vert}\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}},{\psi }_{j}^{\left(1\right)},\left(t\right),{e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}\le {M}_{18}{t}^{-\alpha }{\int }_{0}^{t}{\left(t-\tau \right)}^{\alpha \left(q-\hat{q}\right)-1}{\left\{\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{F}_{j}^{2}\left(\tau \right)\right\}}^{1/2}\mathrm{d}\tau ,\end{equation} \tag{ 3.31 }$$

where ${M}_{18}={\hat{c}}_{\alpha }{T}^{\alpha }$. Secondy, we proceed to establish an estimate for the sum ${\sum }_{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{\psi }_{j}^{\left(2\right)}\left(t\right){e}_{j}$. Using the inequality (2.11), the absolute value of $\frac{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}$ is bounded by ${\hat{c}}_{\alpha }{c}_{\alpha }^{-1}{T}^{\alpha }{t}^{-\alpha }$. Therefore, we derive

$$\begin{align}\hfill {{\Vert}\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}},{\psi }_{j}^{\left(2\right)},\left(t\right),{e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}=& {\left\{\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{m}_{j}^{2\beta \left(p+\hat{q}\right)}{\varphi }_{j}^{2}\frac{{E}_{\alpha ,1}^{2}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}^{2}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\right\}}^{1/2}\hfill \\ \hfill \le & {\hat{c}}_{\alpha }{c}_{\alpha }^{-1}{T}^{\alpha }{\left\{\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{m}_{j}^{2\beta \left(p+\hat{q}\right)}{\varphi }_{j}^{2}{t}^{-2\alpha }\right\}}^{1/2},\hfill \end{align}$$

which shows that

$$\begin{equation}{{\Vert}\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}},{\psi }_{j}^{\left(2\right)},\left(t\right),{e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}\le \;{M}_{19}{t}^{-\alpha }{\left\{\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{m}_{j}^{2\beta \left(p+\hat{q}\right)}{\varphi }_{j}^{2}\right\}}^{1/2},\end{equation} \tag{ 3.32 }$$

where ${M}_{19}{:=}{\hat{c}}_{\alpha }{c}_{\alpha }^{-1}{T}^{\alpha }$. Thirdly, we proceed to establish an estimate for the sum ${\sum }_{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{\psi }_{j}^{\left(3\right)}\left(t\right){e}_{j}$. By a similar argument as in (3.32), we have

$$\begin{align}\hfill & {{\Vert}\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}},{\psi }_{j}^{\left(3\right)},\left(t\right),{e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}\hfill \\ \hfill & \quad \quad \le {\int }_{0}^{T}{{\Vert}\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}},{F}_{j},\left(\tau \right),{{\sim}{E}}_{\alpha ,\alpha },\left(-{m}_{j}^{\beta }{\left(T-\tau \right)}^{\alpha }\right),\frac{{m}_{j}^{\beta }{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)},{e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}\mathrm{d}\tau \hfill \\ \hfill & \quad \quad \le {\int }_{0}^{T}{\left\{\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{m}_{j}^{2\beta \left(p+\hat{q}\right)}{F}_{j}^{2}\left(\tau \right){{\sim}{E}}_{\alpha ,\alpha }^{2}\left(-{m}_{j}^{\beta }{\left(T-\tau \right)}^{\alpha }\right)\frac{{E}_{\alpha ,1}^{2}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}^{2}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\right\}}^{1/2}\mathrm{d}\tau \hfill \\ \hfill & \quad \quad \le {\hat{c}}_{\alpha }{M}_{19}{t}^{-\alpha }{\int }_{0}^{T}{\left\{\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{m}_{j}^{2\beta \left(p+\hat{q}\right)}{F}_{j}^{2}\left(\tau \right){m}_{j}^{-2\beta \left(p+\hat{q}\right)}{\left(T-\tau \right)}^{2\alpha \left(q-\hat{q}\right)-2}\right\}}^{1/2}\mathrm{d}\tau .\hfill \end{align}$$

Therefore, we obtain the following estimate

$$\begin{equation} {{\Vert}\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}},{\psi }_{j}^{\left(3\right)},\left(t\right),{e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}\le \;{\hat{c}}_{\alpha }{M}_{19}{t}^{-\alpha }{\int }_{0}^{T}{\left(T-\tau \right)}^{\alpha \left(q-\hat{q}\right)-1}{\left\{\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{F}_{j}^{2}\left(\tau \right)\right\}}^{1/2}\mathrm{d}\tau .\end{equation} \tag{ 3.33 }$$

For almost every τ in the interval (0, T ), by (3.30), we have that F(τ, ⋅) belongs to L²(D). This implies ∑_1≤j≤nF_j(τ)e_j is a Cauchy sequence in L²(D). This together with the embedding

$$\begin{equation*}{L}^{2}\left(D\right)\hookrightarrow {\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}\end{equation*}$$

implies that ∑_1≤j≤nF_j(τ)e_j is also a Cauchy sequence in ${\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}$. On the other hand, it follows from $\varphi \in {\mathbf{V}}_{\beta \left(p+\hat{q}\right)}$ that

$$\begin{equation*}\underset{{n}_{1},{n}_{2}\to \infty }{\mathrm{lim}}\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{\varphi }_{j}^{2}{m}_{j}^{2\beta \left(p+\hat{q}\right)}=0.\end{equation*}$$

By (R5), $0\le \hat{q}\le \frac{s}{\alpha }$, and we obtain the inclusion ${\mathcal{X}}_{2,\alpha q-s}\left(J{\times}D\right)\subset {\mathcal{X}}_{2,\alpha \left(q-\hat{q}\right)}\left(J{\times}D\right)$. We deduce that $F\in {\mathcal{X}}_{2,\alpha \left(q-\hat{q}\right)}\left(J{\times}D\right)$, and

$$\begin{equation}\underset{{n}_{1},{n}_{2}\to \infty }{\mathrm{lim}}{\int }_{0}^{T}{\left(T-\tau \right)}^{\alpha \left(q-\hat{q}\right)-1}{\left\{\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{F}_{j}^{2}\left(\tau \right)\right\}}^{1/2}\mathrm{d}\tau =0,\end{equation} \tag{ 3.34 }$$

by the dominated convergence theorem. Combining these with the estimates (3.31)–(3.33), we have that

$$\begin{equation*}\underset{{n}_{1},{n}_{2}\to \infty }{\mathrm{lim}}{{\Vert}\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}},{}^{\mathrm{c}}D_{t}^{\alpha },{u}_{j},\left(t\right),{e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}=0.\end{equation*}$$

Hence ${\sum }_{j=1}^{n}{}^{\mathrm{c}}D_{t}^{\alpha }{u}_{j}\left(t\right){e}_{j}$ is a Cauchy sequence and a convergent sequence in ${\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}$. We then conclude that ${}^{\mathrm{c}}D_{t}^{\alpha }u\left(t,\cdot \right)={\sum }_{j=1}^{\infty }{}^{\mathrm{c}}D_{t}^{\alpha }{u}_{j}\left(t\right){e}_{j}$ finitely exists in the space ${\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}$. Moreover, by taking the inequalities (3.31)–(3.33), there exists a constant M₂₀ > 0 such that

$$\begin{align}\hfill {{\Vert}{}^{\mathrm{c}}D_{t}^{\alpha }u\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}& \le {M}_{20}{{\Vert}F\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}+{M}_{20}\left({{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}+{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha \left(q-\hat{q}\right)}}\right){t}^{-\alpha }\hfill \\ \hfill & \le {M}_{20}{\Vert}F\left(t,\cdot \right){\Vert}+{M}_{20}\left({{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}+{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha \left(q-\hat{q}\right)}}\right){t}^{-\alpha }.\hfill \end{align} \tag{ 3.35 }$$

Now, it follows from $0{< }\hat{r}\le \frac{1-\alpha }{\alpha }$ and 0 < αq − s < α that $1\le \frac{1}{\alpha }-\hat{r}{< }\frac{1}{\alpha q-s}+\hat{r}$. This implies the following Sobolev embedding

$$\begin{equation*}{L}^{\frac{1}{\alpha q-s}+\hat{r}}\left(0,T;{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}\right)\hookrightarrow {L}^{\frac{1}{\alpha }-\hat{r}}\left(0,T;{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}\right).\end{equation*}$$

Moreover, by the assumption (R5), $\hat{q}{< }\frac{s}{\alpha }$, we have $\frac{1}{\alpha q-s}+\hat{r}{ >}\frac{1}{\alpha \left(q-\hat{q}\right)}$. This implies that there exists a constant C_* > 0 such that

$$\begin{equation}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha \left(q-\hat{q}\right)}}\le {C}_{{\ast}}{{\Vert}F{\Vert}}_{{L}^{\frac{1}{\alpha q-s}+\hat{r}}\left(0,T;{L}^{2}\left(D\right)\right)}.\end{equation} \tag{ 3.36 }$$

Hence, we deduce from (3.35) that there exists a constant M₂₁ > 0 satisfying

$$\begin{equation}{{\Vert}{}^{\mathrm{c}}D_{t}^{\alpha }u{\Vert}}_{{L}^{\frac{1}{\alpha }-\hat{r}}\left(0,T;{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}\right)}\le \;{M}_{21}{{\Vert}F{\Vert}}_{{L}^{\frac{1}{\alpha q-s}+\hat{r}}\left(0,T;{L}^{2}\left(D\right)\right)}+{M}_{21}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}},\end{equation} \tag{ 3.37 }$$

where we note that ${{\Vert}{t}^{-\alpha }{\Vert}}_{{L}^{\frac{1}{\alpha }-\hat{r}}\left(0,T;\mathbb{R}\right)}{< }\infty$. The inequality (3.28) is proved by letting C₃ = M₂₁.

(b) It is clear that the assumptions of this part also satisfy part (a). Therefore, by part (a), it is necessary to prove $u\in {C}_{w}^{\alpha }\left(\left(0,T\right];{\mathbf{V}}_{-\beta q}\right)$, i.e.,

$$\begin{equation}\underset{{t}_{2}-{t}_{1}\to 0}{\mathrm{lim}}{{\Vert}{}^{\mathrm{c}}D_{t}^{\alpha }u\left({t}_{2},\cdot \right)-{}^{\mathrm{c}}D_{t}^{\alpha }u\left({t}_{1},\cdot \right){\Vert}}_{{\mathbf{V}}_{-\beta q}}=0,\end{equation} \tag{ 3.38 }$$

where we note 0 < t₁ < t₂ ≤ T. After some simple computations we find that

$$\begin{equation}{}^{\mathrm{c}}D_{t}^{\alpha }u\left({t}_{2},x\right)-{}^{\mathrm{c}}D_{t}^{\alpha }u\left({t}_{1},x\right)=F\left({t}_{2},x\right)-F\left({t}_{1},x\right)+\sum _{1\le n\le 4}{\mathcal{J}}_{n},\end{equation} \tag{ 3.39 }$$

where ${\mathcal{J}}_{n}=-{\mathcal{L}}^{\beta }{\mathcal{I}}_{n}$, and ${\mathcal{I}}_{n}$ is defined by (3.15). Since F is in ${C}_{w}^{\alpha }\left(\left(0,T\right];{\mathbf{V}}_{-\beta q}\right)$, we have just to prove ${{\Vert}{\mathcal{J}}_{n}{\Vert}}_{{\mathbf{V}}_{-\beta q}}$ approaches 0 as t₂ − t₁ approaches 0. Let us first consider ${{\Vert}{\mathcal{J}}_{1}{\Vert}}_{{\mathbf{V}}_{-\beta q}}$. The inequalities (2.11) yields that

$$\begin{align}\hfill {{\Vert}{\mathcal{J}}_{1}{\Vert}}_{{\mathbf{V}}_{-\beta q}}\le & {\int }_{0}^{{t}_{1}}{{\Vert}{\mathcal{L}}^{\beta },\sum _{j=1}^{\infty },{\int }_{{t}_{1}-\tau }^{{t}_{2}-\tau },{F}_{j},\left(\tau \right),{\omega }^{\alpha -2},{E}_{\alpha ,\alpha -1},\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right),\mathrm{d},\omega ,{e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta q}}\mathrm{d}\tau \hfill \\ \hfill \le & {\int }_{0}^{{t}_{1}}{\left\{\sum _{j=1}^{\infty }{m}_{j}^{2\beta }{m}_{j}^{-2\beta q}{F}_{j}^{2}\left(\tau \right){\left\vert {\int }_{{t}_{1}-\tau }^{{t}_{2}-\tau }{\omega }^{\alpha -2}\left\vert {E}_{\alpha ,\alpha -1}\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)\right\vert \mathrm{d}\omega \right\vert }^{2}\right\}}^{1/2}\mathrm{d}\tau \hfill \\ \hfill \le & {\hat{c}}_{\alpha }{\int }_{0}^{{t}_{1}}{\left\{\sum _{j=1}^{\infty }{m}_{j}^{2\beta }{m}_{j}^{-2\beta q}{F}_{j}^{2}\left(\tau \right){\left\vert {\int }_{{t}_{1}-\tau }^{{t}_{2}-\tau }{\omega }^{\alpha -2}{m}_{j}^{-\beta p}{\omega }^{-\alpha p}\mathrm{d}\omega \right\vert }^{2}\right\}}^{1/2}\mathrm{d}\tau .\hfill \end{align}$$

We recall that, by (3.30), F belongs to ${\mathcal{X}}_{2,\alpha q-s}\left(J{\times}D\right)$. Thus, we can deduce from the above inequality that

$$\begin{equation}{{\Vert}{\mathcal{J}}_{1}{\Vert}}_{{\mathbf{V}}_{-\beta q}}\le \;{\hat{c}}_{\alpha }{\int }_{0}^{{t}_{1}}{\Vert}F\left(\tau ,\cdot \right){\Vert}\left\vert {\int }_{{t}_{1}-\tau }^{{t}_{2}-\tau }{\omega }^{\alpha q-2}\mathrm{d}\omega \right\vert \mathrm{d}\tau \le \frac{{\hat{c}}_{\alpha }}{1-\alpha q}{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}}{\left({t}_{2}-{t}_{1}\right)}^{s},\end{equation} \tag{ 3.40 }$$

where we have used the same argument as in the estimate (3.16). Let us secondly consider ${{\Vert}{\mathcal{J}}_{2}{\Vert}}_{{\mathbf{V}}_{-\beta q}}$. We have

$$\begin{align}\hfill {{\Vert}{\mathcal{J}}_{2}{\Vert}}_{{\mathbf{V}}_{-\beta q}}& \le {\int }_{{t}_{1}}^{{t}_{2}}{{\Vert}{\mathcal{L}}^{\beta },\sum _{j=1}^{\infty },{F}_{j},\left(\tau \right),{E}_{\alpha ,\alpha },\left(-{m}_{j}^{\beta }{\left({t}_{2}-\tau \right)}^{\alpha }\right),{e}_{j}{\Vert}}_{{V}_{-\beta q}}{\left({t}_{2}-\tau \right)}^{\alpha -1}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }{\int }_{{t}_{1}}^{{t}_{2}}{\left\{\sum _{j=1}^{\infty }{m}_{j}^{2\beta }{m}_{j}^{-2\beta q}{F}_{j}^{2}\left(\tau \right){m}_{j}^{-2\beta p}{\left({t}_{2}-\tau \right)}^{-2\alpha p}\right\}}^{1/2}{\left({t}_{2}-\tau \right)}^{\alpha -1}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }{\int }_{{t}_{1}}^{{t}_{2}}{\Vert}F\left(\tau ,\cdot \right){\Vert}{\left({t}_{2}-\tau \right)}^{\alpha q-s-1}{\left({t}_{2}-\tau \right)}^{s}\mathrm{d}\tau \le {\hat{c}}_{\alpha }{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q-s}}{\left({t}_{2}-{t}_{1}\right)}^{s},\hfill \end{align} \tag{ 3.41 }$$

where (2.11) has been used. Thirdly, we consider the norm ${{\Vert}{\mathcal{J}}_{3}{\Vert}}_{{V}_{-\beta q}}$. It is clear that

$$\begin{equation*}{\mathcal{J}}_{3}={\mathcal{L}}^{\beta }{\mathcal{I}}_{3}={\mathcal{L}}^{2\beta }\sum _{j=1}^{\infty }{\varphi }_{j}{\int }_{{t}_{1}}^{{t}_{2}}\frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\mathrm{d}\omega {e}_{j}.\end{equation*}$$

Hence, we deduce

$$\begin{equation*}{{\Vert}{\mathcal{J}}_{3}{\Vert}}_{{\mathbf{V}}_{-\beta q}}={\left\{\sum _{j=1}^{\infty }{m}_{j}^{4\beta }{m}_{j}^{-2\beta q}{\varphi }_{j}^{2}{\left\vert {\int }_{{t}_{1}}^{{t}_{2}}\frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\mathrm{d}\omega \right\vert }^{2}\right\}}^{1/2}.\end{equation*}$$

By applying (2.11), the absolute value of $\frac{{E}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}$ is bounded by ${\hat{c}}_{\alpha }{c}_{\alpha }^{-1}\frac{1+{m}_{j}^{\beta }{T}^{\alpha }}{1+{\left({m}_{j}^{\beta }{\omega }^{\alpha }\right)}^{2}}$. This is associated with $1+{m}_{j}^{\beta }{T}^{\alpha }\le \left({m}_{1}^{-\beta }+{T}^{\alpha }\right){m}_{j}^{\beta }$ that $\frac{1+{m}_{j}^{\beta }{T}^{\alpha }}{1+{\left({m}_{j}^{\beta }{\omega }^{\alpha }\right)}^{2}}\le \left({m}_{1}^{-\beta }+{T}^{\alpha }\right){m}_{j}^{-\beta }{\omega }^{-2\alpha }$. Thus, we obtain

$$\begin{align}\hfill {{\Vert}{\mathcal{J}}_{3}{\Vert}}_{{\mathbf{V}}_{-\beta q}}& \le {\hat{c}}_{\alpha }{c}_{\alpha }^{-1}\left({m}_{1}^{-\beta }+{T}^{\alpha }\right){\left\{\sum _{j=1}^{\infty }{m}_{j}^{4\beta }{m}_{j}^{-2\beta q}{\varphi }_{j}^{2}{\left\vert {\int }_{{t}_{1}}^{{t}_{2}}{m}_{j}^{-\beta }{\omega }^{-2\alpha }{\omega }^{\alpha -1}\mathrm{d}\omega \right\vert }^{2}\right\}}^{1/2}\hfill \\ \hfill & \le \frac{{M}_{23}}{\alpha }{t}_{1}^{-2\alpha }{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}\left({t}_{2}^{\alpha }-{t}_{1}^{\alpha }\right),\hfill \end{align} \tag{ 3.42 }$$

where ${M}_{23}={\hat{c}}_{\alpha }{c}_{\alpha }^{-1}\left({m}_{1}^{-\beta }+{T}^{\alpha }\right)$. Finally, we can look at ${{\Vert}{\mathcal{J}}_{4}{\Vert}}_{{\mathbf{V}}_{-\beta q}}$ as follows:

$$\begin{align}\hfill {{\Vert}{\mathcal{J}}_{4}{\Vert}}_{{\mathbf{V}}_{-\beta q}}\le & {\int }_{0}^{T}{{\Vert}{\mathcal{L}}^{2\beta },\sum _{j=1}^{\infty },{\int }_{{t}_{1}}^{{t}_{2}},{F}_{j},\left(\tau \right),{{\sim}{E}}_{\alpha ,\alpha },\left(-{m}_{j}^{\beta }{\left(T-\tau \right)}^{\alpha }\right),\frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)},\mathrm{d},\omega ,{e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta q}}\mathrm{d}\tau \hfill \\ \hfill \le & {\int }_{0}^{T}\left\{\sum _{j=1}^{\infty }{m}_{j}^{4\beta }{m}_{j}^{-2\beta q}{F}_{j}^{2}\left(\tau \right)\left\vert {\int }_{{t}_{1}}^{{t}_{2}}{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\left(T-\tau \right)}^{\alpha }\right)\right.\right.{\left.{\left.\frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\mathrm{d}\omega \right\vert }^{2}\right\}}^{1/2}\mathrm{d}\tau \hfill \\ \hfill \le & {M}_{24}{\int }_{0}^{T}\left\{\sum _{j=1}^{\infty }{m}_{j}^{4\beta }{m}_{j}^{-2\beta q}{F}_{j}^{2}\left(\tau \right)\left\vert {\int }_{{t}_{1}}^{{t}_{2}}{m}_{j}^{-\beta p}{\left(T-\tau \right)}^{\alpha q-1}\right.\right.{\left.{\left.{m}_{j}^{-\beta }{\omega }^{-\alpha -1}\mathrm{d}\omega \right\vert }^{2}\right\}}^{1/2}\mathrm{d}\tau \hfill \end{align}$$

where the fraction $\frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}$ can be estimated in the same way as in the proof of (3.42), and ${M}_{24}={\hat{c}}_{\alpha }{M}_{23}$. This leads to

$$\begin{equation*}{{\Vert}{\mathcal{J}}_{4}{\Vert}}_{{\mathbf{V}}_{-\beta q}}\le \;\frac{{M}_{24}}{\alpha }{t}_{1}^{-2\alpha }\left({t}_{2}^{\alpha }-{t}_{1}^{\alpha }\right){\int }_{0}^{T}{\Vert}F\left(\tau ,\cdot \right){\Vert}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau ,\end{equation*}$$

which shows that

$$\begin{equation}{{\Vert}{\mathcal{J}}_{4}{\Vert}}_{{\mathbf{V}}_{-\beta q}}\le \;\frac{{M}_{24}}{\alpha }{t}_{1}^{-2\alpha }{\left\vert \left\vert \left\vert F\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q}}\left({t}_{2}^{\alpha }-{t}_{1}^{\alpha }\right).\end{equation} \tag{ 3.43 }$$

This implies (3.38) by taking (3.39)–(3.43) together. Thus, ${}^{\mathrm{c}}D_{t}^{\alpha }u$ is contained in C((0, T ]; V_−βq).

On the other hand, it is easy to see that the estimates (3.31)–(3.33) also hold for $\hat{q}=0$. Hence, we deduce from (3.35) and (3.36) that

$$\begin{align}\hfill & {t}^{\alpha }{{\Vert}{}^{\mathrm{c}}D_{t}^{\alpha }u\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{-\beta q}}\hfill \\ \hfill & \quad \quad \le {M}_{20}{t}^{\alpha }{{\Vert}F\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{-\beta q}}+{M}_{20}\left({{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}+{C}_{{\ast}}{{\Vert}F{\Vert}}_{{L}^{\frac{1}{\alpha q-s}+\hat{r}}\left(0,T;{L}^{2}\left(D\right)\right)}\right).\hfill \end{align} \tag{ 3.44 }$$

Now ${}^{\mathrm{c}}D_{t}^{\alpha }u\in {C}_{w}^{\alpha }\left(\left(0,T\right];{\mathbf{V}}_{-\beta q}\right)$. In addition, there exists a positive constant C^' > 0 such that

$$\begin{align}\hfill & \quad \quad {{\Vert}{}^{\mathrm{c}}D_{t}^{\alpha }u{\Vert}}_{{C}_{w}^{\alpha }\left(\left(0,T\right];{\mathbf{V}}_{-\beta q}\right)}\hfill \\ \hfill & \quad \quad \le {M}_{20}{{\Vert}F{\Vert}}_{{C}_{w}^{\alpha }\left(\left(0,T\right];{\mathbf{V}}_{-\beta q}\right)}+{C}^{\prime }{M}_{20}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}+{C}_{{\ast}}{M}_{20}{{\Vert}F{\Vert}}_{{L}^{\frac{1}{\alpha q-s}+\hat{r}}\left(0,T;{L}^{2}\left(D\right)\right)}.\hfill \end{align}$$

We can complete the proof by taking (3.28) and the above inequality together.□

In this section, we study the existence, uniqueness, and regularity of mild solutions of FVP (1.1)–(1.3) corresponding to the nonlinear source function F(t, x, u(t, x)). It is suitable considering assumptions that u(t, ⋅) and F(t, ⋅, u(t, ⋅)) belong to the same spatial space H. In view of most considerations of PDEs, we let H = L²(D).

We introduce the following assumptions on the numbers $p,q,{p}^{\prime },{q}^{\prime },\hat{p},\hat{q},r,\hat{r}$.

• (R1b) 0 < q < p < 1 such that p + q = 1;
• (R4b) $0{< }{p}^{\prime }{< }p, \quad {q}^{\prime }=1-{p}^{\prime },\quad \quad 0{< }r\le \frac{1-\alpha {q}^{\prime }}{\alpha {q}^{\prime }}$;
• (R4c) $0{< }{p}^{\prime }\le p-q,\quad \quad {q}^{\prime }=1-{p}^{\prime },\quad \quad 0{< }r\le \frac{1-\alpha {q}^{\prime }}{\alpha {q}^{\prime }}$;
• (R5b) $0\le \hat{q}{< }q, \quad \hat{p}=1-\hat{q},\quad 0{< }\hat{r}\le \frac{1-\alpha }{\alpha }$.

In our work, we will assume on F(t, ⋅, u(t, ⋅)) the following assumptions

• (A1) F(t, ⋅, 0) = 0, and there exists a constant K > 0 such that, for all v₁, v₂ ∈ L²(D) and t ∈ J,
  $$\begin{equation*}{\Vert}F\left(t,\cdot ,{v}_{1}\right)-F\left(t,\cdot ,{v}_{2}\right){\Vert}\le K{\Vert}{v}_{1}-{v}_{2}{\Vert}.\end{equation*}$$
• (A2) F(t, ⋅, 0) = 0, and there exists a constant K_* > 0 such that, for all v₁, v₂ ∈ L²(D) and t₁, t₂ ∈ J,
  $$\begin{equation*}{\Vert}F\left({t}_{1},\cdot ,{v}_{1}\right)-F\left({t}_{2},\cdot ,{v}_{2}\right){\Vert}\le {K}_{{\ast}}\left(\vert {t}_{1}-{t}_{2}\vert +{\Vert}{v}_{1}-{v}_{2}{\Vert}\right).\end{equation*}$$

Note that the assumption (A1), (A2) imply that, for v ∈ L²(D),

$$\begin{equation}{\Vert}F\left(t,\cdot ,v\right){\Vert}\le K{\Vert}v{\Vert}.\end{equation} \tag{ 4.1 }$$

We try to develop the ideas of the linear FVP (3.1) to deal with the nonlinear FVP (1.1)–(1.3). In section 3, for the linear function F(t, x) we assume that

$$\begin{equation}F\in {\mathcal{X}}_{2,\alpha q}\left(J{\times}D\right),\quad \mathrm{o}\mathrm{r}\quad F\in {\mathcal{X}}_{2,\alpha q-s}\left(J{\times}D\right),\quad \mathrm{o}\mathrm{r}\quad F\in {L}^{\frac{1}{\alpha q-s}+\hat{r}}\left(0,T;{L}^{2}\left(D\right)\right),\end{equation} \tag{ 4.2 }$$

where $p,q,s,\hat{r}$ are defined by (R1), (R3), (R5). However, we cannot suppose that the nonlinear source function F(t, x, u(t, x)) satisfies the same assumptions as in (4.2), and then find the solution u. A natural idea might be to combine the idea of lemma 3.2 with the inequality (4.1), i.e., we predict the solution u may be contained in the set

$$\begin{equation*}{\mathbf{W}}_{\gamma ,\eta }^{\rho }\left(J{\times}D\right){:=}\left\{w\in {\mathcal{X}}_{2,\eta }\left(J{\times}D\right):\quad {\Vert}w\left(t,\cdot \right){\Vert}\le \rho {t}^{-\gamma },\quad \mathrm{f}\mathrm{o}\mathrm{r}\;0{< }t\le T\right\},\end{equation*}$$

for ρ > 0, 0 < γ ≤ η < 1.

The prediction will be proved in the next lemma. However, it is necessary to give some useful notes on ${\mathbf{W}}_{\gamma ,\eta }^{\rho }\left(J{\times}D\right)$ as follows. For $w\in {\mathbf{W}}_{\gamma ,\eta }^{\rho }\left(J{\times}D\right)$, we see

$$\begin{equation*}\underset{0\le t\le T} {\text{ess}}\hspace{0.17em}{\text{sup}}{\int }_{0}^{t}{\Vert}w\left(\tau ,\cdot \right){\Vert}{\left(t-\tau \right)}^{\eta -1}\mathrm{d}\tau \le \rho \hspace{0.17em}\underset{0\le t\le T}{\text{ess}}\hspace{0.17em}{\text{sup}}{\int }_{0}^{t}{\tau }^{-\gamma }{\left(t-\tau \right)}^{\eta -1}\mathrm{d}\tau .\end{equation*}$$

The function τ → τ^−γ(t − τ)^η−1 is integrable on (0, t) since both numbers −γ and η − 1 are greater than −1. In addition, we have ${\int }_{0}^{t}{\tau }^{-\gamma }{\left(t-\tau \right)}^{\eta -1}\mathrm{d}\tau ={t}^{\eta -\gamma }B\left(\eta ,1-\gamma \right)$, where B(⋅, ⋅) is the beta function see, for example, [58, 59, 60]. Hence, we have

$$\begin{equation}{\left\vert \left\vert \left\vert w\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\eta }}\le \rho {T}^{\eta -\gamma }B\left(\eta ,1-\gamma \right).\end{equation} \tag{ 4.3 }$$

Moreover, if γ < η, then there always exists a real number p such that $1{< }\frac{1}{\eta }{< }p{< }\frac{1}{\gamma }$. This implies that the function t^−γ belongs to L^p(0, T; L²(D)). Therefore, we can obtain the following inclusions

$$\begin{equation*}{\mathbf{W}}_{\gamma ,\eta }^{\rho }\left(J{\times}D\right)\subset {L}^{p}\left(0,T;{L}^{2}\left(D\right)\right)\subset {\mathcal{X}}_{2,\eta }\left(J{\times}D\right).\end{equation*}$$

In the following lemma, we will consider the case γ = η, which we will denote by ${\mathbf{W}}_{\gamma }^{\rho }\left(J{\times}D\right){:=}{\mathbf{W}}_{\gamma ,\eta }^{\rho }\left(J{\times}D\right)$.

Now, the Sobolev embedding V_βp ↪ L²(D) shows that there exists a positive constant C_D depending on D, β, q such that ${\Vert}v{\Vert}\le {C}_{D}{{\Vert}v{\Vert}}_{{\mathbf{V}}_{\beta p}}$ for all v ∈ V_βp. In this section, we let

$$\begin{equation*}{k}_{0}\left(T\right)=KB\left(\alpha q,1-\alpha q\right){T}^{\alpha q}\left[{\hat{c}}_{\alpha }{m}_{1}^{-\beta p}+{\hat{c}}_{\alpha }^{2}{c}_{\alpha }^{-1}{\left({m}_{1}^{-\beta }+{T}^{\alpha }\right)}^{p}\right],\end{equation*}$$

and

$$\begin{equation*}{M}_{0}={C}_{D}{T}^{\alpha q}+{\hat{c}}_{\alpha }{c}_{\alpha }^{-1}{T}^{\alpha q}{\left({m}_{1}^{-\beta }+{T}^{\alpha }\right)}^{p}.\end{equation*}$$

Lemma 4.1. Let p, q be defined by (R1 ) and ϕ be a function on D. Let $\left\{{w}_{\left(n\right)}\right\}$ be defined by w₍₀₎ = ϕ,

$$\begin{align}\hfill {w}_{\left(n\right)}\left(t,x\right)& =\left(\mathcal{O}{w}_{\left(n-1\right)}\right)\left(t,x\right)\hfill \\ \hfill & =\left({\mathcal{O}}_{1}F\left({w}_{\left(n-1\right)}\right)\right)\left(t,x\right)+{\mathcal{O}}_{2}\left(t\right)\phi \left(x\right)+\left({\mathcal{O}}_{3}F\left({w}_{\left(n-1\right)}\right)\right)\left(t,x\right),\quad n\in \mathbb{N},n\ge 1.\hfill \end{align} \tag{ 4.4 }$$

If ϕ belongs to V_βp, F satisfies (A1), and k₀(T ) < 1, then

$$\begin{equation}{\left\{{w}_{\left(n\right)}\right\}}_{n\ge 0}\subset {\mathbf{W}}_{\alpha q}^{{\hat{C}}_{0}}\left(J{\times}D\right),\end{equation} \tag{ 4.5 }$$

where ${\hat{C}}_{0}{:=}{{\sim}{C}}_{0}{{\Vert}\phi {\Vert}}_{{\mathbf{V}}_{\beta p}}$ and ${{\sim}{C}}_{0}=\frac{{M}_{0}}{1-{k}_{0}\left(T\right)}$.

Proof. First, we have

$$\begin{equation*}{\Vert}{w}_{\left(0\right)}{\Vert}={\Vert}\phi {\Vert}\le {C}_{D}{{\Vert}\phi {\Vert}}_{{\mathbf{V}}_{\beta p}}\le {M}_{0}{{\Vert}\phi {\Vert}}_{{\mathbf{V}}_{\beta p}}{t}^{-\alpha q}\le {\hat{C}}_{0}{t}^{-\alpha q}.\end{equation*}$$

Hence, inequality (4.3) and αq < 1, imply ${w}_{\left(0\right)}\in {\mathbf{W}}_{\alpha q}^{{\hat{C}}_{0}}\left(J{\times}D\right)$. Now, we assume that w_(n−1) belongs to ${\mathbf{W}}_{\alpha q}^{{\hat{C}}_{0}}\left(J{\times}D\right)$ for some n ≥ 1. Then, by using (4.3), we have

$$\begin{equation} {\left\vert \left\vert \left\vert {w}_{\left(n-1\right)}\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q}}\le {\hat{C}}_{0}B\left(\alpha q,1-\alpha q\right).\end{equation} \tag{ 4.6 }$$

By induction, the inclusion (4.5) will be proved by showing that w_(n) belongs to ${\mathbf{W}}_{\alpha q}^{{\hat{C}}_{0}}\left(J{\times}D\right)$. Indeed, by using the same arguments as in the proof of (3.6), we have

$$\begin{align}\hfill {\Vert}\left({\mathcal{O}}_{1}F\left({w}_{\left(n-1\right)}\right)\right)\left(t,\cdot \right){\Vert}& \le {\hat{c}}_{\alpha }{m}_{1}^{-\beta p}{\int }_{0}^{t}{\Vert}F\left(\tau ,\cdot ,{w}_{\left(n-1\right)}\left(\tau ,\cdot \right)\right){\Vert}{\left(t-\tau \right)}^{\alpha q-1}\mathrm{d}\tau ,\hfill \\ \hfill & \le K{\hat{c}}_{\alpha }{m}_{1}^{-\beta p}{\left\vert \left\vert \left\vert {w}_{\left(n-1\right)}\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q}}\le {M}_{26}{\hat{C}}_{0}{t}^{-\alpha q},\hfill \end{align} \tag{ 4.7 }$$

where we have used (4.1), (4.6) and let ${M}_{26}=K{\hat{c}}_{\alpha }{m}_{1}^{-\beta p}B\left(\alpha q,1-\alpha q\right){T}^{\alpha q}$. On the other hand, the norm ${\Vert}{\mathcal{O}}_{2}\left(t\right)\phi {\Vert}$ is estimated by (3.7), i.e.,

$$\begin{equation*}{\Vert}{\mathcal{O}}_{2}\left(t\right)\phi {\Vert}\le {M}_{0}{{\Vert}\phi {\Vert}}_{{\mathbf{V}}_{\beta p}}{t}^{-\alpha q},\end{equation*}$$

where we note that

$$\begin{equation*}{M}_{2}={\hat{c}}_{\alpha }{c}_{\alpha }^{-1}{T}^{\alpha q}{\left({m}_{1}^{-\beta }+{T}^{\alpha }\right)}^{p}\le {M}_{0}.\end{equation*}$$

The norm ${\Vert}\left({\mathcal{O}}_{3}F\left({w}_{\left(n-1\right)}\right)\right)\left(t,\cdot \right){\Vert}$ can be estimated in the same way as in the proof of (3.8). That is,

$$\begin{align}\hfill {\Vert}\left({\mathcal{O}}_{3}F\left({w}_{\left(n-1\right)}\right)\right)\left(t,\cdot \right){\Vert}& \le {M}_{3}{t}^{-\alpha q}{\int }_{0}^{T}{\Vert}F\left(\tau ,\cdot ,{w}_{\left(n-1\right)}\left(\tau ,\cdot \right)\right){\Vert}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \\ \hfill & \le K{M}_{3}{t}^{-\alpha q}{\left\vert \left\vert \left\vert {w}_{\left(n-1\right)}\right\vert \right\vert \right\vert }_{{\mathcal{X}}_{2,\alpha q}}\le {M}_{27}{\hat{C}}_{0}{t}^{-\alpha q},\hfill \end{align} \tag{ 4.8 }$$

where (4.1), (4.6) have been used. Here, M₂₇ = KM₃B(αq, 1 − αq), ${M}_{3}={\hat{c}}_{\alpha }^{2}{c}_{\alpha }^{-1}{T}^{\alpha q}{\left({m}_{1}^{-\beta }+{T}^{\alpha }\right)}^{p}$.

We deduce from the above arguments and ${w}_{\left(n\right)}\left(t,\cdot \right)=\mathcal{O}\left(t,\cdot \right){w}_{\left(n-1\right)}$ that

$$\begin{align}\hfill {\Vert}{w}_{\left(n\right)}\left(t,\cdot \right){\Vert}& \le {\Vert}\left({\mathcal{O}}_{1}F\left({w}_{\left(n-1\right)}\right)\right)\left(t,\cdot \right){\Vert}+{\Vert}{\mathcal{O}}_{2}\left(t\right)\phi {\Vert}+{\Vert}\left({\mathcal{O}}_{3}F\left({w}_{\left(n-1\right)}\right)\right)\left(t,\cdot \right){\Vert}\hfill \\ \hfill & \le {k}_{0}\left(T\right){\hat{C}}_{0}{t}^{-\alpha q}+{M}_{0}{{\Vert}\phi {\Vert}}_{{\mathbf{V}}_{\beta p}}{t}^{-\alpha q},\hfill \end{align} \tag{ 4.9 }$$

by noting that k₀(T ) = M₂₆ + M₂₇. Since ${\hat{C}}_{0}=\frac{{M}_{0}}{1-{k}_{0}\left(T\right)}{{\Vert}\phi {\Vert}}_{{\mathbf{V}}_{\beta p}}$, the above inequality implies that ${\Vert}{w}_{\left(n\right)}\left(t,\cdot \right){\Vert}\le {\hat{C}}_{0}{t}^{-\alpha q}$. Therefore, from αq < 1, we obtain the inclusion (4.5).□

Next, it is necessary to give a definition of mild solutions of FVP (1.1)–(1.3).

Definition 4.2. Let F be defined by (A1) or (A2). If a function u belongs to ${L}^{{p}_{2}}\left(0,T;{L}^{{p}_{1}}\left(D\right)\right)$, for some p₁, p₂ ≥ 1, and satisfies equation (2.10) where F stands for the nonlinear source function F(t, x, u(t, x)), then u is said to be a mild solution of FVP (1.1)–(1.3).

The following theorems presents existence, uniqueness, and regularity of a mild solution of FVP (1.1)–(1.3).

Theorem 4.3. 

• (a)  
  Let p, q, r, p', q' be defined by (R1), (R4b). If φ belongs to V_βp, F satisfies (A1), and k₀(T ) < 1, then FVP (1.1)–(1.3) has a unique solution

  $$\begin{equation*}u\in {L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}\right)\cap {C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right),\end{equation*}$$

  and there exists a positive constant C₅ such that

  $$\begin{equation*}{{\Vert}u{\Vert}}_{{L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}\right)}+{{\Vert}u{\Vert}}_{{C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)}\le \;{C}_{5}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}.\end{equation*}$$
• (b)  
  Let p, q, r, p', q' be defined by (R1b), (R4c). If φ belongs to V_βp, F satisfies (A1), and k₀(T ) < 1, then FVP (1.1)–(1.3) has a unique solution

  $$\begin{equation*}u\in {L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}\right)\cap {C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)\cap {C}^{\alpha q}\left(\left[0,T\right];{\mathbf{V}}_{-\beta {q}^{\prime }}\right),\end{equation*}$$

  and there exists a positive constant C₆ such that

  $$\begin{equation*}{{\Vert}u{\Vert}}_{{L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}\right)}+{{\Vert}u{\Vert}}_{{C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)}+{\left\vert \left\vert \left\vert u\right\vert \right\vert \right\vert }_{{C}_{w}^{\alpha q}\left(\left[0,T\right];{\mathbf{V}}_{-\beta {q}^{\prime }}\right)}\le \;{C}_{6}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}.\end{equation*}$$

Proof. (a) We divide the proof of this part into the following steps.

Step 1: We prove the existence and uniqueness of a mild solution. In order to prove the existence of a mild solution of FVP (1.1)–(1.3), we will construct a convergent sequence in ${L}^{\frac{1}{\alpha q}-r}\left(0,T;{L}^{2}\left(D\right)\right)$ whose limit will be a mild solution of the problem. Here, r is defined by (R2). Let ${\left\{{w}_{\left(n\right)}\right\}}_{n\ge 0}$ be a sequence defined by lemma 4.1 with respect to ϕ = φ ∈ V_βp, then ${\left\{{w}_{\left(n\right)}\right\}}_{n\ge 0}\subset {\mathbf{W}}_{\alpha q}^{{\hat{C}}_{0}}\left(J{\times}D\right)$ where ${\hat{C}}_{0}{:=}\frac{{M}_{2}}{1-{k}_{0}\left(T\right)}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}$. Therefore,

$$\begin{equation*}{\Vert}{w}_{\left(n\right)}\left(t,\cdot \right){\Vert}\le {\hat{C}}_{0}{t}^{-\alpha q},\quad 0{< }t\le T,\end{equation*}$$

for all n ≥ 1. This together with t^−αq belonging to ${L}^{\frac{1}{\alpha q}-r}\left(0,T;\mathbb{R}\right)$ implies that ${\left\{{w}_{\left(n\right)}\right\}}_{n\ge 0}$ is a bounded sequence in ${L}^{\frac{1}{\alpha q}-r}\left(0,T;{L}^{2}\left(D\right)\right)$. Now, we will show that ${\left\{{w}_{\left(n\right)}\right\}}_{n\ge 0}$ is convergent by proving that it is also a Cauchy sequence. For fixed n ≥ 1 and k ≥ 1, the definition (4.4) of ${\left\{{w}_{\left(n\right)}\right\}}_{n\ge 0}$ yields that

$$\begin{align}\hfill {w}_{\left(n+k\right)}\left(t,x\right)-{w}_{\left(n\right)}& ={\mathcal{O}}_{1}\left[F\left({w}_{\left(n-1+k\right)}\right)-F\left({w}_{\left(n-1\right)}\right)\right]\hfill \\ \hfill & \quad +{\mathcal{O}}_{3}\left[F\left({w}_{\left(n-1+k\right)}\right)-F\left({w}_{\left(n-1\right)}\right)\right].\hfill \end{align}$$

Since F satisfies lemma 4.1, the latter equation shows that we can apply the same arguments as in lemma 4.1 with ϕ = 0. Hence, one can deduce

$$\begin{align}\hfill & {\Vert}{w}_{\left(1+k\right)}\left(t,\cdot \right)-{w}_{\left(1\right)}\left(t,\cdot \right){\Vert}\hfill \\ \hfill & \quad \quad \le {\Vert}\left({\mathcal{O}}_{1}\left[F\left({w}_{\left(0+k\right)}\right)-F\left({w}_{\left(0\right)}\right)\right]\right)\left(t,\cdot \right){\Vert}+{\Vert}\left({\mathcal{O}}_{3}\left[F\left({w}_{\left(0+k\right)}\right)-F\left({w}_{\left(0\right)}\right)\right]\right)\left(t,\cdot \right){\Vert}\hfill \\ \hfill & \quad \quad \le {\hat{c}}_{\alpha }{m}_{1}^{-\beta p}{\int }_{0}^{t}{\Vert}F\left(\tau ,\cdot ,{w}_{\left(0+k\right)}\left(\tau ,\cdot \right)\right)-F\left(\tau ,\cdot ,{w}_{\left(0\right)}\left(\tau ,\cdot \right)\right){\Vert}{\left(t-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \\ \hfill & \quad \quad \quad +{M}_{3}{t}^{-\alpha q}{\int }_{0}^{T}{\Vert}F\left(\tau ,\cdot ,{w}_{\left(0+k\right)}\left(\tau ,\cdot \right)\right)-F\left(\tau ,\cdot ,{w}_{\left(0\right)}\left(\tau ,\cdot \right)\right){\Vert}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \end{align}$$

where we combined the estimates (4.7), (4.8). From ${\left\{{w}_{\left(n\right)}\right\}}_{n\ge 0}\subset {\mathbf{W}}_{\alpha q}^{{\hat{C}}_{0}}\left(J{\times}D\right)$, we have

$$\begin{equation*} {\Vert}{w}_{\left(0+k\right)}\left(\tau ,\cdot \right)-{w}_{\left(0\right)}\left(\tau ,\cdot \right){\Vert}\le 2{\hat{C}}_{0}{t}^{-\alpha q}.\end{equation*}$$

Thus, by noting the identity

$$\begin{equation*}{\int }_{0}^{t}{\left(t-\tau \right)}^{a-1}{\tau }^{b-1}\mathrm{d}\tau ={t}^{a+b-1}B\left(a,b\right),\end{equation*}$$

where a, b > 0 and B(⋅, ⋅) is the beta function, we find that

$$\begin{align}\hfill & {\Vert}{w}_{\left(1+k\right)}\left(t,\cdot \right)-{w}_{\left(1\right)}\left(t,\cdot \right){\Vert}\hfill \\ \hfill & \quad \quad \le {\hat{c}}_{\alpha }{m}_{1}^{-\beta p}K{\int }_{0}^{t}{\Vert}{w}_{\left(0+k\right)}\left(\tau ,\cdot \right)-{w}_{\left(0\right)}\left(\tau ,\cdot \right){\Vert}{\left(t-\tau \right)}^{\alpha q-1}\mathrm{d}\tau ,\hfill \\ \hfill & \quad \quad \quad +{M}_{3}K{t}^{-\alpha q}{\int }_{0}^{T}{\Vert}{w}_{\left(0+k\right)}\left(\tau ,\cdot \right)-{w}_{\left(0\right)}\left(\tau ,\cdot \right){\Vert}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \\ \hfill & \quad \quad \le {\hat{c}}_{\alpha }{m}_{1}^{-\beta p}K\left(2{\hat{C}}_{0}\right){\int }_{0}^{t}{\tau }^{-\alpha q}{\left(t-\tau \right)}^{\alpha q-1}\mathrm{d}\tau +{M}_{3}K\left(2{\hat{C}}_{0}\right){t}^{-\alpha q}{\int }_{0}^{T}{\tau }^{-\alpha q}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \\ \hfill & \quad \quad \le \left({\hat{c}}_{\alpha }{m}_{1}^{-\beta p}K{T}^{\alpha q}B\left(\alpha q,1-\alpha q\right)+{M}_{3}KB\left(\alpha q,1-\alpha q\right)\right)\left(2{\hat{C}}_{0}\right){t}^{-\alpha q}. \hfill \end{align}$$

From the definition of k₀(T ), we conclude that

$$\begin{equation*}{\Vert}{w}_{\left(1+k\right)}\left(t,\cdot \right)-{w}_{\left(1\right)}\left(t,\cdot \right){\Vert}\le {k}_{0}\left(T\right)\left(2{\hat{C}}_{0}\right){t}^{-\alpha q}.\end{equation*}$$

Iterating this method n-times shows

$$\begin{equation*}\;{\Vert}{w}_{\left(n+k\right)}\left(t,\cdot \right)-{w}_{\left(n\right)}\left(t,\cdot \right){\Vert}\le {k}_{0}^{n}\left(T\right)\left(2{\hat{C}}_{0}\right){t}^{-\alpha q}.\end{equation*}$$

Taking the ${L}^{\frac{1}{\alpha q}-r}\left(0,T;\mathbb{R}\right)$-norm of both sides of the above inequality directly implies

$$\begin{equation}\;{{\Vert}{w}_{\left(n+k\right)}-{w}_{\left(n\right)}{\Vert}}_{{L}^{\frac{1}{\alpha q}-r}\left(0,T;{L}^{2}\left(D\right)\right)}\le {k}_{0}^{n}\left(T\right)\left(2{\hat{C}}_{0}\right){{\Vert}{t}^{-\alpha q}{\Vert}}_{{L}^{\frac{1}{\alpha q}-r}\left(0,T;\mathbb{R}\right)}.\end{equation} \tag{ 4.10 }$$

Here, we emphasise that the constants in (4.10) also do not depend on (n, k). Therefore, by letting n go to infinity, we obtain

$$\begin{equation*}\underset{n,k\to \infty }{\mathrm{lim}}{{\Vert}{w}_{\left(n+k\right)}-{w}_{\left(n\right)}{\Vert}}_{{L}^{\frac{1}{\alpha q}-r}\left(0,T;{L}^{2}\left(D\right)\right)}=0,\end{equation*}$$

i.e., ${\left\{{w}_{\left(n\right)}\right\}}_{n\ge 0}$ is a bounded Cauchy sequence in ${L}^{\frac{1}{\alpha q}-r}\left(0,T;{L}^{2}\left(D\right)\right)$. Hence, there exists a function u in ${L}^{\frac{1}{\alpha q}-r}\left(0,T;{L}^{2}\left(D\right)\right)$ such that

$$\begin{equation*}u=\underset{n\to \infty }{\mathrm{lim}}{w}_{\left(n\right)},\quad \mathrm{i}\mathrm{n}\hspace{0.17em}{L}^{\frac{1}{\alpha q}-r}\left(0,T;{L}^{2}\left(D\right)\right),\end{equation*}$$

and u satisfies equation (2.10), i.e., u is a mild solution of FVP problem (1.1)–(1.3). Moreover, the boundedness (4.9) of ${\left\{{w}_{\left(n\right)}\right\}}_{n\ge 0}$ gives

$$\begin{equation}{\Vert}u\left(t,\cdot \right){\Vert}\le {{\sim}{C}}_{0}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}{t}^{-\alpha q},\end{equation} \tag{ 4.11 }$$

and so that

$$\begin{equation*}{{\Vert}u{\Vert}}_{{L}^{\frac{1}{\alpha q}-r}\left(0,T;{L}^{2}\left(D\right)\right)}\le {M}_{28}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}\end{equation*}$$

where ${M}_{28}={{\sim}{C}}_{0}{{\Vert}{t}^{-\alpha q}{\Vert}}_{{L}^{\frac{1}{\alpha q}-r}\left(0,T;\mathbb{R}\right)}$.

Now, we show the uniqueness of the solution u. Assume that ${\sim}{u}$ is another solution of FVP (1.1)–(1.3). Then, by applying the same argument as in (4.10), we also have

$$\begin{equation*}\;{{\Vert}u-{\sim}{u}{\Vert}}_{{L}^{\frac{1}{\alpha q}-r}\left(0,T;{L}^{2}\left(D\right)\right)}\le {k}_{0}^{n}\left(T\right)\left(2{\hat{C}}_{0}\right){{\Vert}{t}^{-\alpha q}{\Vert}}_{{L}^{\frac{1}{\alpha q}-r}\left(0,T;\mathbb{R}\right)},\end{equation*}$$

for all $n\in \mathbb{N}$, n ≥ 1. Thus ${{\Vert}u-{\sim}{u}{\Vert}}_{{L}^{\frac{1}{\alpha q}-r}\left(0,T;{L}^{2}\left(D\right)\right)}=0$ by letting n go to infinity. Hence, $u={\sim}{u}$ in ${L}^{\frac{1}{\alpha q}-r}\left(0,T;{L}^{2}\left(D\right)\right)$.

Step 2: We prove that $u\in {L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}\right)$. This will be proved by using the inequality (4.11). We now apply the same arguments as in the proofs of (3.11), and (3.13) to estimate ${{\Vert}u\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}}$ as follows. First, we have

$$\begin{align}\hfill & {{\Vert}\left({\mathcal{O}}_{1}F\left(u\right)\right)\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}}\hfill \\ \hfill & \quad \quad \le {\hat{c}}_{\alpha }{\int }_{0}^{t}{\left\{\sum _{j=1}^{\infty }{F}_{j}^{2}\left(\tau ,u\left(\tau \right)\right){m}_{j}^{2\beta \left(p-{p}^{\prime }\right)}{m}_{j}^{-2\beta p}{\left(t-\tau \right)}^{-2\alpha p}{\left(t-\tau \right)}^{2\alpha -2}\right\}}^{1/2}\mathrm{d}\tau \hfill \\ \hfill & \quad \quad \le {\hat{c}}_{\alpha }{m}_{1}^{-\beta {p}^{\prime }}{\int }_{0}^{t}{\Vert}F\left(\tau ,\cdot ,u\left(\tau ,\cdot \right)\right){\Vert}{\left(t-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \\ \hfill & \quad \quad \le {\hat{c}}_{\alpha }{m}_{1}^{-\beta {p}^{\prime }}K{\int }_{0}^{t}{\Vert}u\left(\tau ,\cdot \right){\Vert}{\left(t-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \\ \hfill & \quad \quad \le {\hat{c}}_{\alpha }{m}_{1}^{-\beta {p}^{\prime }}K{\hat{C}}_{0}{\int }_{0}^{t}{\tau }^{-\alpha q}{\left(t-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \le {M}_{29}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}{t}^{-\alpha {q}^{\prime }},\hfill \end{align} \tag{ 4.12 }$$

where we let

$$\begin{equation*}{M}_{29}={\hat{c}}_{\alpha }{m}_{1}^{-\beta {p}^{\prime }}K{{\sim}{C}}_{0}B\left(\alpha q,1-\alpha q\right){T}^{\alpha {q}^{\prime }}.\end{equation*}$$

Secondly,

$$\begin{align}\hfill {{\Vert}\left({\mathcal{O}}_{3}F\left(u\right)\right)\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}}& \le {M}_{6}{t}^{-\alpha {q}^{\prime }}{\int }_{0}^{T}{\Vert}F\left(\tau ,\cdot ,u\left(\tau ,\cdot \right)\right){\Vert}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \\ \hfill & \le {M}_{6}{t}^{-\alpha {q}^{\prime }}K{\int }_{0}^{T}{\Vert}u\left(\tau ,\cdot \right){\Vert}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \\ \hfill & \le {M}_{6}{t}^{-\alpha {q}^{\prime }}K{\hat{C}}_{0}{\int }_{0}^{t}{\tau }^{-\alpha q}{\left(t-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \\ \hfill & \le {M}_{30}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}{t}^{-\alpha {q}^{\prime }}, \hfill \end{align} \tag{ 4.13 }$$

where we let ${M}_{30}={M}_{6}K{{\sim}{C}}_{0}B\left(\alpha q,1-\alpha q\right)$. We recall that ${{\Vert}{\mathcal{O}}_{2}\left(t\right)\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}}$ have been estimated by (3.12). According to the above arguments, we arrive at the estimate

$$\begin{align}\hfill {{\Vert}u\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}}\le & {{\Vert}\left({\mathcal{O}}_{1}F\left(u\right)\right)\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}}+{{\Vert}{\mathcal{O}}_{2}\left(t\right)\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}}+{{\Vert}\left({\mathcal{O}}_{3}F\left(u\right)\right)\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}} \hfill \\ \hfill \le & {M}_{31}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}{t}^{-\alpha {q}^{\prime }},\hfill \end{align}$$

for M₃₁ = M₂₉ + M₅ + M₃₀. By taking the ${L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;\mathbb{R}\right)$-norm, then the latter inequalities can be transformed into the following estimate

$$\begin{equation}{{\Vert}u{\Vert}}_{{L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}\right)}\le \;{M}_{32}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}},\end{equation} \tag{ 4.14 }$$

where ${M}_{32}={M}_{31}{{\Vert}{t}^{-\alpha {q}^{\prime }}{\Vert}}_{{L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;\mathbb{R}\right)}$.

Step 3: We prove that $u\in {C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)$. Let us consider 0 < t₁ < t₂ ≤ T. By the same arguments as in (3.15), we have

$$\begin{align}\hfill & u\left({t}_{2},x\right)-u\left({t}_{1},x\right)\hfill \\ \hfill & \quad \quad =\sum _{j=1}^{\infty }{\int }_{0}^{{t}_{1}}{\int }_{{t}_{1}-\tau }^{{t}_{2}-\tau }{F}_{j}\left(\tau ,u\left(\tau \right)\right){\omega }^{\alpha -2}{E}_{\alpha ,\alpha -1}\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)\mathrm{d}\omega \mathrm{d}\tau {e}_{j}\left(x\right)\hfill \\ \hfill & \quad \quad \quad \quad +\sum _{j=1}^{\infty }{\int }_{{t}_{1}}^{{t}_{2}}{F}_{j}\left(\tau ,u\left(\tau \right)\right){{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\left({t}_{2}-\tau \right)}^{\alpha }\right)\mathrm{d}\tau {e}_{j}\left(x\right)\hfill \\ \hfill & \quad \quad \quad \quad -{\mathcal{L}}^{\beta }\sum _{j=1}^{\infty }{\varphi }_{j}{\int }_{{t}_{1}}^{{t}_{2}}\frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\mathrm{d}\omega {e}_{j}\left(x\right)\hfill \\ \hfill & \quad \quad \quad \quad +{\mathcal{L}}^{\beta }\sum _{j=1}^{\infty }{\int }_{0}^{T}{\int }_{{t}_{1}}^{{t}_{2}}{F}_{j}\left(\tau ,u\left(\tau \right)\right){{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\left(T-\tau \right)}^{\alpha }\right)\frac{{{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\mathrm{d}\omega \mathrm{d}\tau {e}_{j}\left(x\right)\hfill \\ \hfill & \quad \quad {:=}{\mathcal{I}}_{1}^{N}+{\mathcal{I}}_{2}^{N}+{\mathcal{I}}_{3}+{\mathcal{I}}_{4}^{N}.\hfill \end{align} \tag{ 4.15 }$$

Here, by (3.19), ${\Vert}{\mathcal{I}}_{3}{\Vert}$ tends to 0 as t₂ − t₁ tends to 0. In what follows, we will establish the convergence for ${\Vert}{\mathcal{I}}_{n}^{N}{\Vert}$, n = 1, 2, 4 which can be treated similarly as in (3.16), (3.17), (3.20) based on the Lipschitzian assumption (A1). We first see that

$$\begin{align}\hfill {\Vert}{\mathcal{I}}_{1}^{N}{\Vert}& \le {\hat{c}}_{\alpha }{m}_{1}^{-\beta p}{\int }_{0}^{{t}_{1}}{\left\{\sum _{j=1}^{\infty }{F}_{j}^{2}\left(\tau ,u\left(\tau \right)\right){\left\vert {\int }_{{t}_{1}-\tau }^{{t}_{2}-\tau }{\omega }^{\alpha -2}{\omega }^{-\alpha p}\mathrm{d}\omega \right\vert }^{2}\right\}}^{1/2}\mathrm{d}\tau \hfill \\ \hfill & \le \frac{{\hat{c}}_{\alpha }{m}_{1}^{-\beta p}}{1-\alpha q}{\int }_{0}^{{t}_{1}}{\Vert}F\left(\tau ,\cdot ,u\left(\tau ,\cdot \right)\right){\Vert}\left[{\left({t}_{1}-\tau \right)}^{\alpha q-1}-{\left({t}_{2}-\tau \right)}^{\alpha q-1}\right]\mathrm{d}\tau \hfill \\ \hfill & \le \frac{{\hat{c}}_{\alpha }{m}_{1}^{-\beta p}}{1-\alpha q}K{\int }_{0}^{{t}_{1}}{\Vert}u\left(\tau ,\cdot \right){\Vert}\left[{\left({t}_{1}-\tau \right)}^{\alpha q-1}-{\left({t}_{2}-\tau \right)}^{\alpha q-1}\right]\mathrm{d}\tau \hfill \\ \hfill & \le \frac{{\hat{c}}_{\alpha }{m}_{1}^{-\beta p}}{1-\alpha q}K{{\sim}{C}}_{0}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}{\int }_{0}^{{t}_{1}}{\tau }^{-\alpha q}\left[{\left({t}_{1}-\tau \right)}^{\alpha q-1}-{\left({t}_{2}-\tau \right)}^{\alpha q-1}\right]\mathrm{d}\tau .\hfill \end{align} \tag{ 4.16 }$$

Note that

$$\begin{equation*}{\int }_{0}^{{t}_{1}}{\tau }^{-\alpha q}{\left({t}_{1}-\tau \right)}^{\alpha q-1}\mathrm{d}\tau =B\left(\alpha q,1-\alpha q\right).\end{equation*}$$

Thus, due to the substitution τ = t₂μ, we have

$$\begin{align}\hfill & {\int }_{0}^{{t}_{1}}{\tau }^{-\alpha q}\left[{\left({t}_{1}-\tau \right)}^{\alpha q-1}-{\left({t}_{2}-\tau \right)}^{\alpha q-1}\right]\mathrm{d}\tau \hfill \\ \hfill & \quad \quad \quad \quad =B\left(\alpha q,1-\alpha q\right)-{\int }_{0}^{{t}_{1}/{t}_{2}}{\mu }^{-\alpha q}{\left(1-\mu \right)}^{\alpha q-1}\mathrm{d}\mu \hfill \\ \hfill & \quad \quad \quad \quad =B\left(\alpha q,1-\alpha q\right)-\left[B\left(\alpha q,1-\alpha q\right)-{\int }_{{t}_{1}/{t}_{2}}^{1}{\mu }^{-\alpha q}{\left(1-\mu \right)}^{\alpha q-1}\mathrm{d}\mu \right] \hfill \\ \hfill & \quad \quad \quad \quad \le {\left(\frac{{t}_{2}}{{t}_{1}}\right)}^{\alpha q}{\int }_{{t}_{1}/{t}_{2}}^{1}{\left(1-\mu \right)}^{\alpha q-1}\mathrm{d}\mu =\frac{1}{\alpha q}{\left(\frac{{t}_{2}}{{t}_{1}}-1\right)}^{\alpha q}.\hfill \end{align} \tag{ 4.17 }$$

As a consequence, $\underset{{t}_{2}-{t}_{1}\to 0}{\mathrm{lim}}{\int }_{0}^{{t}_{1}}{\tau }^{-\alpha q}\left[{\left({t}_{1}-\tau \right)}^{\alpha q-1}-{\left({t}_{2}-\tau \right)}^{\alpha q-1}\right]\mathrm{d}\tau =0$, and so $\underset{{t}_{2}-{t}_{1}\to 0}{\mathrm{lim}}{\Vert}{\mathcal{I}}_{1}^{N}{\Vert}=0$. Secondly we proceed to deal with ${\mathcal{I}}_{2}^{N}$. Now 0 < p₀ < p, by (2.11), we have

$$\begin{equation}\vert {E}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{\left({t}_{2}-\tau \right)}^{\alpha }\right)\vert \le {\hat{c}}_{\alpha }{m}_{1}^{-\beta \left(p-{p}_{0}\right)}{\left({t}_{2}-\tau \right)}^{-\alpha \left(p-{p}_{0}\right)}.\end{equation} \tag{ 4.18 }$$

We deduce the following chain of estimates

$$\begin{align}\hfill {\Vert}{\mathcal{I}}_{2}^{N}{\Vert}\le & {\int }_{{t}_{1}}^{{t}_{2}}{\Vert}\sum _{j=1}^{\infty },{F}_{j},\left(\tau ,u\left(\tau \right)\right),{E}_{\alpha ,\alpha },\left(-{m}_{j}^{\beta }{\left({t}_{2}-\tau \right)}^{\alpha }\right),{e}_{j}{\Vert}{\left({t}_{2}-\tau \right)}^{\alpha -1}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }{m}_{1}^{-\beta \left(p-{p}_{0}\right)}{\int }_{{t}_{1}}^{{t}_{2}}{\Vert}F\left(\tau ,\cdot ,u\left(\tau ,\cdot \right)\right){\Vert}{\left({t}_{2}-\tau \right)}^{\alpha q-1+\alpha {p}_{0}}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }{m}_{1}^{-\beta \left(p-{p}_{0}\right)}K{\int }_{{t}_{1}}^{{t}_{2}}{\Vert}u\left(\tau ,\cdot \right){\Vert}{\left({t}_{2}-\tau \right)}^{\alpha q-1}\mathrm{d}\tau {\left({t}_{2}-{t}_{1}\right)}^{\alpha {p}_{0}}\hfill \\ \hfill & \le {\hat{c}}_{\alpha }{m}_{1}^{-\beta \left(p-{p}_{0}\right)}K{{\sim}{C}}_{0}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}{\int }_{0}^{{t}_{2}}{\tau }^{-\alpha q}{\left({t}_{2}-\tau \right)}^{\alpha q-1}\mathrm{d}\tau {\left({t}_{2}-{t}_{1}\right)}^{\alpha {p}_{0}}\hfill \\ \hfill & \le {\hat{c}}_{\alpha }{m}_{1}^{-\beta \left(p-{p}_{0}\right)}K{{\sim}{C}}_{0}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}B\left(\alpha q,1-\alpha q\right){\left({t}_{2}-{t}_{1}\right)}^{\alpha {p}_{0}}.\hfill \end{align} \tag{ 4.19 }$$

This implies $\underset{{t}_{2}-{t}_{1}\to 0}{\mathrm{lim}}{\Vert}{\mathcal{I}}_{2}^{N}{\Vert}=0$. Next, we thirdly proceed to consider ${\mathcal{I}}_{4}^{N}$. The same argument as in (3.20) gives

$$\begin{align}\hfill {\Vert}{\mathcal{I}}_{4}^{N}{\Vert}& \le {\hat{c}}_{\alpha }{M}_{11}\frac{{t}_{2}^{\alpha q}-{t}_{1}^{\alpha q}}{{t}_{1}^{2\alpha q}}{\int }_{0}^{T}{\Vert}F\left(\tau ,\cdot ,u\left(\tau ,\cdot \right)\right){\Vert}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }{M}_{11}K{{\sim}{C}}_{0}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}\frac{{t}_{2}^{\alpha q}-{t}_{1}^{\alpha q}}{{t}_{1}^{2\alpha q}}{\int }_{0}^{T}{\tau }^{-\alpha q}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }{M}_{11}K{{\sim}{C}}_{0}B\left(\alpha q,1-\alpha q\right){{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}\frac{{t}_{2}^{\alpha q}-{t}_{1}^{\alpha q}}{{t}_{1}^{2\alpha q}},\hfill \end{align} \tag{ 4.20 }$$

and we arrive at $\underset{{t}_{2}-{t}_{1}\to 0}{\mathrm{lim}}{\Vert}{\mathcal{I}}_{4}^{N}{\Vert}=0$. The above arguments prove u ∈ C((0, T ]; L²(D)). This combines with (4.11) so that $u\in {C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)$, and

$$\begin{equation}{{\Vert}u{\Vert}}_{{C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)}\le {{\sim}{C}}_{0}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}.\end{equation} \tag{ 4.21 }$$

We complete step 1 by combining the inequalities (4.14) and (4.21).

(b) According to part (a), we have just to prove that u ∈ C^αq([0, T ]; V_−βq'). In this part, we consider 0 ≤ t₁ < t₂ ≤ T. Let us first show

$$\begin{equation}{{\Vert}{\mathcal{I}}_{1}^{N}{\Vert}}_{{\mathbf{V}}_{-\beta {q}^{\prime }}}\le {M}_{33}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}{\left({t}_{2}-{t}_{1}\right)}^{\alpha q},\end{equation} \tag{ 4.22 }$$

for some positive constant M₃₃, where the case t₁ = 0 is trivial. It is necessary to prove (4.22) for t₁ > 0. From the proof of the estimate (3.23), we have

$$\begin{align}\hfill {{\Vert}{\mathcal{I}}_{1}^{N}{\Vert}}_{{\mathbf{V}}_{-\beta {q}^{\prime }}}& \le \;{\int }_{0}^{{t}_{1}}\left\{\sum _{j=1}^{\infty }{m}_{j}^{-2\beta {q}^{\prime }}{F}_{j}^{2}\left(\tau ,u\left(\tau \right)\right)\left\vert {\int }_{{t}_{1}-\tau }^{{t}_{2}-\tau }\right.\right.{\left.{\left.{\omega }^{\alpha -2}\left\vert {E}_{\alpha ,\alpha -1}\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)\right\vert \mathrm{d}\omega \right\vert }^{2}\right\}}^{1/2}\mathrm{d}\tau .\hfill \end{align}$$

In addition, the inequalities (2.11) yield that, $\vert {E}_{\alpha ,\alpha -1}\left(-{m}_{j}^{\beta }{\omega }^{\alpha }\right)\vert \le {\hat{c}}_{\alpha }{m}_{j}^{-\beta {p}^{\prime }}{\omega }^{-\alpha {p}^{\prime }}$. This associates with αq' − 2 = αq − 2 + α(p − p') so that

$$\begin{align}\hfill {{\Vert}{\mathcal{I}}_{1}^{N}{\Vert}}_{{\mathbf{V}}_{-\beta {q}^{\prime }}}& \le {\hat{c}}_{\alpha }{m}_{1}^{-\beta }{\int }_{0}^{{t}_{1}}{\Vert}F\left(\tau ,\cdot ,u\left(\tau ,\cdot \right)\right){\Vert}\left\vert {\int }_{{t}_{1}-\tau }^{{t}_{2}-\tau }{\omega }^{\alpha q-2+\alpha \left(p-{p}^{\prime }\right)}\mathrm{d}\omega \right\vert \mathrm{d}\tau \hfill \\ \hfill & \le \frac{{\hat{c}}_{\alpha }{m}_{1}^{-\beta }}{1-\alpha {q}^{\prime }}K{\int }_{0}^{{t}_{1}}{\Vert}u\left(\tau ,\cdot \right){\Vert}\left[{\left({t}_{1}-\tau \right)}^{\alpha q-1+\alpha \left(p-{p}^{\prime }\right)}-{\left({t}_{2}-\tau \right)}^{\alpha q-1+\alpha \left(p-{p}^{\prime }\right)}\right]\mathrm{d}\tau \hfill \\ \hfill & \le \frac{{\hat{c}}_{\alpha }{m}_{1}^{-\beta }}{1-\alpha {q}^{\prime }}K{\hat{C}}_{0}{\int }_{0}^{{t}_{1}}{\tau }^{-\alpha q}{\left({t}_{1}-\tau \right)}^{\alpha \left(p-{p}^{\prime }\right)}\left[{\left({t}_{1}-\tau \right)}^{\alpha q-1}-{\left({t}_{2}-\tau \right)}^{\alpha q-1}\right]\mathrm{d}\tau \hfill \\ \hfill & \le \frac{{\hat{c}}_{\alpha }{m}_{1}^{-\beta }}{1-\alpha {q}^{\prime }}K{\hat{C}}_{0}{t}_{1}^{\alpha \left(p-{p}^{\prime }\right)}\frac{1}{\alpha q}{\left(\frac{{t}_{2}}{{t}_{1}}-1\right)}^{\alpha q}\le {M}_{33}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}{\left({t}_{2}-{t}_{1}\right)}^{\alpha q},\hfill \end{align}$$

where we let

$$\begin{equation*}{M}_{33}=\frac{{\hat{c}}_{\alpha }{m}_{1}^{-\beta }}{\left(1-\alpha {q}^{\prime }\right)\alpha q}K{{\sim}{C}}_{0}{T}^{\alpha \left(p-{p}^{\prime }\right)-\alpha q}.\end{equation*}$$

Here, we have used (3.15), (4.17), and α(p − p') ≥ αq by (R4b). Secondly, we are going to consider ${\mathcal{I}}_{2}^{N}$. The Sobolev embedding L²(D) ↪ V_−βq' yields that there exists a positive constant M₃₄ such that

$$\begin{align}\hfill {{\Vert}{\mathcal{I}}_{2}^{N}{\Vert}}_{{\mathbf{V}}_{-\beta {q}^{\prime }}}& \le {M}_{34}{\Vert}{\mathcal{I}}_{2}^{N}{\Vert}\hfill \\ \hfill & \le {M}_{34}{\hat{c}}_{\alpha }{m}_{1}^{-\beta {p}^{\prime }}K{{\sim}{C}}_{0}B\left(\alpha q,1-\alpha q\right){{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}{\left({t}_{2}-{t}_{1}\right)}^{\alpha \left(p-{p}^{\prime }\right)}\hfill \\ \hfill & \le {M}_{34}{\hat{c}}_{\alpha }{m}_{1}^{-\beta {p}^{\prime }}{T}^{\alpha \left(p-{p}^{\prime }\right)-\alpha q}K{{\sim}{C}}_{0}B\left(\alpha q,1-\alpha q\right){{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}{\left({t}_{2}-{t}_{1}\right)}^{\alpha q},\hfill \end{align}$$

where we applied (4.19) with respect to 0 < p₀ = p − p' < p. Thirdly, we now consider ${\mathcal{I}}_{3}^{N}$. By applying the same arguments as in (3.25), one can get

$$\begin{align}\hfill {{\Vert}{\mathcal{I}}_{3}^{N}{\Vert}}_{{\mathbf{V}}_{-\beta {q}^{\prime }}}& \le \frac{{M}_{14}}{\alpha \left(p-{p}^{\prime }\right)}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}\left({t}_{2}^{\alpha \left(p-{p}^{\prime }\right)}-{t}_{1}^{\alpha \left(p-{p}^{\prime }\right)}\right)\hfill \\ \hfill & \le \frac{{M}_{14}}{\alpha \left(p-{p}^{\prime }\right)}{T}^{\alpha \left(p-{p}^{\prime }\right)-\alpha q}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}{\left({t}_{2}-{t}_{1}\right)}^{\alpha q}. \hfill \end{align} \tag{ 4.23 }$$

Finally, the arguments proving (3.26) give

$$\begin{align}\hfill {{\Vert}{\mathcal{I}}_{4}{\Vert}}_{{\mathbf{V}}_{-\beta {q}^{\prime }}}& \le {\hat{c}}_{\alpha }\frac{{M}_{14}}{\alpha \left(p-{p}^{\prime }\right)}\left({t}_{2}^{\alpha \left(p-{p}^{\prime }\right)}-{t}_{1}^{\alpha \left(p-{p}^{\prime }\right)}\right){\int }_{0}^{T}{\Vert}F\left(\tau ,\cdot ,u\left(\tau ,\cdot \right)\right){\Vert}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }\frac{{M}_{14}}{\alpha \left(p-{p}^{\prime }\right)}{\left({t}_{2}-{t}_{1}\right)}^{\alpha \left(p-{p}^{\prime }\right)}K{\int }_{0}^{T}{\Vert}u\left(\tau ,\cdot \right){\Vert}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }\frac{{M}_{14}}{\alpha \left(p-{p}^{\prime }\right)}{T}^{\alpha \left(p-{p}^{\prime }\right)-\alpha q}K{{\sim}{C}}_{0}B\left(\alpha q,1-\alpha q\right){{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}{\left({t}_{2}-{t}_{1}\right)}^{\alpha q}.\hfill \end{align}$$

Taking the above estimates for ${\mathcal{I}}_{n}^{N}$, 1 ≤ n ≤ 4 together, we conclude that u belongs to C^αq([0, T ]; V_−βq'). Moreover, there exists a positive constant M₃₅ such that

$$\begin{equation*}{\left\vert \left\vert \left\vert u\right\vert \right\vert \right\vert }_{{C}^{\alpha q}\left(\left[0,T\right];{\mathbf{V}}_{-\beta {q}^{\prime }}\right)}\le {M}_{35}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}.\end{equation*}$$

By combining this inequality with (4.14), (4.21), we complete the proof.□

Theorem 4.4. Let $p,q,{p}^{\prime },{q}^{\prime },\hat{p},\hat{q},r,\hat{r}$ be defined by (R1), (R4b), (R5b). If φ belongs to ${\mathbf{V}}_{\beta \left(p+\hat{q}\right)}$, F satisfies the assumptions (A2), and k₀(T ) < 1, then FVP (1.1)–(1.3) has a unique solution u satisfying that

$$\begin{align}\hfill & u\in {L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}\right)\cap {C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right),\hfill \\ \hfill & {}^{\mathrm{c}}D_{t}^{\alpha }u\in {L}^{\frac{1}{\alpha \hat{q}}-\hat{r}}\left(0,T;{\mathbf{V}}_{-\beta \left(q+\hat{p}\right)}\right)\cap {C}_{w}^{\alpha }\left(\left(0,T\right];{\mathbf{V}}_{-\beta q}\right).\hfill \end{align}$$

Moreover, there exists a constant C₇ > 0 such that

$$\begin{equation}{{\Vert}{}^{\mathrm{c}}D_{t}^{\alpha }u{\Vert}}_{{L}^{\frac{1}{\alpha }-\hat{r}}\left(0,T;{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}\right)}+{{\Vert}{}^{\mathrm{c}}D_{t}^{\alpha }u\left(t,\cdot \right){\Vert}}_{{C}_{w}^{\alpha }\left(\left(0,T\right];{\mathbf{V}}_{-\beta q}\right)}\le {C}_{7}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}.\end{equation} \tag{ 4.24 }$$

Proof. Since F satisfies (A2), F also satisfies (A1) with respect to the Lipschitz constant K_*. In addition, the Sobolev imbedding ${\mathbf{V}}_{\beta \left(p+\hat{q}\right)}\hookrightarrow {\mathbf{V}}_{\beta p}$ shows that φ belongs to V_βp. Hence, by theorem 4.3, FVP (1.1)–(1.3) has a unique solution

$$\begin{equation*}u\in {L}^{\frac{1}{\alpha {q}^{\prime }}-r}\left(0,T;{\mathbf{V}}_{\beta \left(p-{p}^{\prime }\right)}\right)\cap {C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right).\end{equation*}$$

Moreover, the inequality (4.11) also holds. We deduce that, for 0 < t ≤ T,

$$\begin{equation}{\Vert}F\left(t,\cdot ,u\left(t,\cdot \right)\right){\Vert}\le {K}_{{\ast}}{{\sim}{C}}_{0}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}}{t}^{-\alpha q}\le {M}_{36}{K}_{{\ast}}{{\sim}{C}}_{0}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}{t}^{-\alpha q}.\end{equation} \tag{ 4.25 }$$

The remainder of this proof falls naturally into two steps as follows.

Step 1: We prove ${}^{\mathrm{c}}D_{t}^{\alpha }u$ finitely exists and belongs to ${L}^{\frac{1}{\alpha }-\hat{r}}\left(0,T;{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}\right)$. By the same way as in part (a) of theorem 3.4, we have

$$\begin{align}\hfill {}^{\mathrm{c}}D_{t}^{\alpha }{u}_{j}\left(t\right)& ={F}_{j}\left(t,u\left(t\right)\right)-{m}_{j}^{\beta }{F}_{j}\left(t,u\left(t\right)\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)\hfill \\ \hfill & \quad \quad -{\varphi }_{j}\frac{{m}_{j}^{\beta }{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}+{\left.\left({F}_{j}\left(r,u\left(r\right)\right)\star {{\sim}{E}}_{\alpha ,\alpha }\left(-{m}_{j}^{\beta }{r}^{\alpha }\right)\right)\right\vert }_{r=T}\frac{{m}_{j}^{\beta }{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)}\hfill \\ \hfill & {:=}{F}_{j}\left(t,u\left(t\right)\right)+{\psi }_{j}^{N,1}\left(t\right)+{\psi }_{j}^{N,2}\left(t\right)+{\psi }_{j}^{N,3}\left(t\right),\hfill \end{align}$$

for all $j\in \mathbb{N}$, j ≥ 1. In view of (4.25), F(t, ⋅, u(t, ⋅)) is contained in L²(D) for 0 < t ≤ T. This associates with the Sobolev embedding ${L}^{2}\left(D\right)\hookrightarrow {\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}$ that F(t, ⋅, u(t, ⋅)) is contained in ${\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}$, namely ${\sum }_{j=1}^{\infty }{F}_{j}\left(t,u\left(t\right)\right){e}_{j}$ is contained in ${\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}$. On the other hand, ${\psi }_{j}^{N,2}={\psi }_{j}^{\left(2\right)}$, and the norm ${{\Vert}{\sum }_{j=1}^{\infty }{\psi }_{j}^{N,2}\left(t\right){e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}$ exists finitely by (3.32). Now, we consider ${{\Vert}{\sum }_{j=1}^{\infty }{\psi }_{j}^{N,n}\left(t\right){e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}$, n = 1, 3. According to the estimates (3.31) and (3.33), the following ones hold:

$$\begin{align}\hfill {{\Vert}\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}},{\psi }_{j}^{N,1},\left(t\right),{e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}& \le {\hat{c}}_{\alpha }{T}^{\alpha }{t}^{-\alpha }{\int }_{0}^{t}{\left(t-\tau \right)}^{\alpha \left(q-\hat{q}\right)-1}\;{\left\{\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{F}_{j}^{2}\left(\tau ,u\left(\tau \right)\right)\right\}}^{1/2}\mathrm{d}\tau ,\hfill \\ \hfill {{\Vert}\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}},{\psi }_{j}^{N,3},\left(t\right),{e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}& \le {\hat{c}}_{\alpha }{M}_{19}{t}^{-\alpha }{\int }_{0}^{T}{\left(T-\tau \right)}^{\alpha \left(q-\hat{q}\right)-1}\;{\left\{\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{F}_{j}^{2}\left(\tau ,u\left(\tau \right)\right)\right\}}^{1/2}\mathrm{d}\tau .\hfill \end{align}$$

For 0 < τ < T, we have F(τ, ⋅, u(τ, ⋅)) belonging to L²(D). This follows that the sequence $\left\{{G}_{n}\left(\tau \right)\right\}$, which is defined by ${G}_{n}\left(\tau \right)={\left\{{\sum }_{j\ge n}{F}_{j}^{2}\left(\tau ,u\left(\tau \right)\right)\right\}}^{1/2}$, converges pointwise to 0 as n goes to infinity. Moreover, by (4.25), we have

$$\begin{equation*}\left\vert {\left(t-\tau \right)}^{\alpha \left(q-\hat{q}\right)-1}{G}_{n}\left(\tau \right)\right\vert \le \;{M}_{36}{K}_{{\ast}}{{\sim}{C}}_{0}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}{\left(t-\tau \right)}^{\alpha \left(q-\hat{q}\right)-1}{\tau }^{-\alpha q}.\end{equation*}$$

The function $\tau \to {\left(t-\tau \right)}^{\alpha \left(q-\hat{q}\right)-1}{\tau }^{-\alpha q}$ is integrable on the open interval (0, t), t > 0, since

$$\begin{equation*}{\int }_{0}^{t}{\left(t-\tau \right)}^{\alpha \left(q-\hat{q}\right)-1}{\tau }^{-\alpha q}\mathrm{d}\tau ={t}^{-\alpha \hat{q}}B\left(\alpha \left(q-\hat{q}\right),1-\alpha q\right).\end{equation*}$$

Therefore, the dominated convergence theorem yields that

$$\begin{equation*}\underset{n\to \infty }{\mathrm{lim}}{\int }_{0}^{t}{\left(t-\tau \right)}^{\alpha \left(q-\hat{q}\right)-1}{G}_{n}\left(\tau \right)\mathrm{d}\tau =0.\end{equation*}$$

This together with ${\left\{{\sum }_{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{F}_{j}^{2}\left(\tau ,u\left(\tau \right)\right)\right\}}^{1/2}\le {G}_{{n}_{1}}\left(\tau \right)$ gives

$$\begin{equation*}\underset{{n}_{1},{n}_{2}\to \infty }{\mathrm{lim}}{\int }_{0}^{t}{\left(t-\tau \right)}^{\alpha \left(q-\hat{q}\right)-1}{\left\{\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{F}_{j}^{2}\left(\tau ,u\left(\tau \right)\right)\right\}}^{1/2}\mathrm{d}\tau =0.\end{equation*}$$

Similarly, we also have

$$\begin{equation*}\underset{{n}_{1},{n}_{2}\to \infty }{\mathrm{lim}}{\int }_{0}^{T}{\left(T-\tau \right)}^{\alpha \left(q-\hat{q}\right)-1}{\left\{\sum _{{n}_{1}\le \hspace{0.17em}j\le {n}_{2}}{F}_{j}^{2}\left(\tau ,u\left(\tau \right)\right)\right\}}^{1/2}\mathrm{d}\tau =0.\end{equation*}$$

We deduce ${{\Vert}{\sum }_{j=1}^{\infty }{\psi }_{j}^{N,n}\left(t\right){e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}$, n = 1, 3 exist finitely. Taking all the above arguments together, we conclude that ${{\Vert}{\sum }_{j=1}^{\infty }{}^{\mathrm{c}}D_{t}^{\alpha }{u}_{j}\left(t\right){e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}$ finitely exists. In addition, the Sobolev embedding ${L}^{2}\left(D\right)\hookrightarrow {\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}$ yields that there exists a positive constant M₃₇ such that

$$\begin{equation*}{{\Vert}F\left(t,\cdot ,u\left(t,\cdot \right)\right){\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}\le {M}_{37}{\Vert}F\left(t,\cdot ,u\left(t,\cdot \right)\right){\Vert}.\end{equation*}$$

Hence,

$$\begin{align}\hfill {{\Vert}{}^{\mathrm{c}}D_{t}^{\alpha }u\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}& \le {{\Vert}F\left(t,\cdot ,u\left(t,\cdot \right)\right){\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}+\sum _{1\le n\le 3}{{\Vert}\sum _{j=1}^{\infty },{\psi }_{j}^{N,n},\left(t\right),{e}_{j}{\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}\hfill \\ \hfill & \le {M}_{37}{\Vert}F\left(t,\cdot ,u\left(t,\cdot \right)\right){\Vert}+{\hat{c}}_{\alpha }{\int }_{0}^{t}{\left(t-\tau \right)}^{\alpha \left(q-\hat{q}\right)-1}{\Vert}F\left(\tau ,\cdot ,u\left(\tau ,\cdot \right)\right){\Vert}\mathrm{d}\tau \hfill \\ \hfill & \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{M}_{19}{t}^{-\alpha }{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}+{\hat{c}}_{\alpha }{M}_{19}{t}^{-\alpha }\;{\int }_{0}^{T}{\left(T-\tau \right)}^{\alpha \left(q-\hat{q}\right)-1}{\Vert}F\left(\tau ,\cdot ,u\left(\tau ,\cdot \right)\right){\Vert}\mathrm{d}\tau .\hfill \end{align}$$

We now note that

$$\begin{equation*}{\int }_{0}^{t}{\left(t-\tau \right)}^{\alpha \left(q-\hat{q}\right)-1}{\tau }^{-\alpha q}\mathrm{d}\tau \le {T}^{\alpha -\alpha \hat{q}}B\left(\alpha \left(q-\hat{q}\right),1-\alpha q\right){t}^{-\alpha },\end{equation*}$$

and

$$\begin{equation*}{\int }_{0}^{T}{\left(T-\tau \right)}^{\alpha \left(q-\hat{q}\right)-1}{\tau }^{-\alpha q}\mathrm{d}\tau ={T}^{-\alpha \hat{q}}B\left(\alpha \left(q-\hat{q}\right),1-\alpha q\right).\end{equation*}$$

This combines with (4.25) and there exists a constant M₃₈ > 0 such that

$$\begin{equation}{{\Vert}{}^{\mathrm{c}}D_{t}^{\alpha }u\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}}\le {M}_{38}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}{t}^{-\alpha },\end{equation} \tag{ 4.26 }$$

which leads to

$$\begin{equation}{{\Vert}{}^{\mathrm{c}}D_{t}^{\alpha }u{\Vert}}_{{L}^{\frac{1}{\alpha }-\hat{r}}\left(0,T;{\mathbf{V}}_{-\beta \left(q-\hat{q}\right)}\right)}\le {M}_{38}{{\Vert}{t}^{-\alpha }{\Vert}}_{{L}^{\frac{1}{\alpha }-\hat{r}}\left(0,T;\mathbb{R}\right)}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}.\end{equation} \tag{ 4.27 }$$

Step 2: We prove ${}^{\mathrm{c}}D_{t}^{\alpha }u\in {C}_{w}^{\alpha }\left(\left(0,T\right];{\mathbf{V}}_{-\beta q}\right)$. We consider 0 < t₁ < t₂ ≤ T. A similar argument as in (3.39) yields

$$\begin{equation*}{}^{\mathrm{c}}D_{t}^{\alpha }u\left({t}_{2},x\right)-{}^{\mathrm{c}}D_{t}^{\alpha }u\left({t}_{1},x\right)=F\left({t}_{2},x,u\left({t}_{2},x\right)\right)-F\left({t}_{1},x,u\left({t}_{1},x\right)\right)+\sum _{1\le n\le 4}{\mathcal{J}}_{n}^{N},\end{equation*}$$

where ${\mathcal{J}}_{n}^{N}=-{\mathcal{L}}^{\beta }{\mathcal{I}}_{n}^{N}$ and ${\mathcal{I}}_{n}^{N}$ is defined by (4.15). By applying the Sobolev embedding L²(D) ↪ V_−βq, there exists a positive constant M₃₉ such that

$$\begin{align}\hfill & \underset{{t}_{2}-{t}_{1}\to 0}{\mathrm{lim}}{{\Vert}F\left({t}_{2},\cdot ,u\left({t}_{2},\cdot \right)\right)-F\left({t}_{1},\cdot ,u\left({t}_{1},\cdot \right)\right){\Vert}}_{{\mathbf{V}}_{-\beta q}}\hfill \\ \hfill & \quad \quad \quad \quad \le \underset{{t}_{2}-{t}_{1}\to 0}{\mathrm{lim}}{M}_{39}{\Vert}F\left({t}_{2},\cdot ,u\left({t}_{2},\cdot \right)\right)-F\left({t}_{1},\cdot ,u\left({t}_{1},\cdot \right)\right){\Vert}\hfill \\ \hfill & \quad \quad \quad \quad \le \underset{{t}_{2}-{t}_{1}\to 0}{\mathrm{lim}}{M}_{39}{K}_{{\ast}}\left(\vert {t}_{2}-{t}_{1}\vert +{\Vert}u\left({t}_{2},\cdot \right)-u\left({t}_{1},\cdot \right){\Vert}\right)=0, \hfill \end{align}$$

where we note that $u\in {C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)$. From (3.40) and (4.17), we have

$$\begin{align}\hfill {{\Vert}{\mathcal{J}}_{1}^{N}{\Vert}}_{{\mathbf{V}}_{-\beta q}}& \le {\hat{c}}_{\alpha }{\int }_{0}^{{t}_{1}}{\Vert}F\left(\tau ,\cdot ,u\left(\tau ,\cdot \right)\right){\Vert}\left\vert {\int }_{{t}_{1}-\tau }^{{t}_{2}-\tau }{\omega }^{\alpha q-2}\mathrm{d}\omega \right\vert \mathrm{d}\tau \hfill \\ \hfill & \le \frac{{\hat{c}}_{\alpha }{M}_{36}{K}_{{\ast}}{{\sim}{C}}_{0}}{1-\alpha q}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}{\int }_{0}^{{t}_{1}}{\tau }^{-\alpha q}\left[{\left({t}_{1}-\tau \right)}^{\alpha q-1}-{\left({t}_{2}-\tau \right)}^{\alpha q-1}\right]\mathrm{d}\tau \hfill \\ \hfill & \le \frac{{\hat{c}}_{\alpha }{M}_{36}{K}_{{\ast}}{{\sim}{C}}_{0}}{\left(1-\alpha q\right)\alpha q}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}{\left(\frac{{t}_{2}}{{t}_{1}}-1\right)}^{\alpha q}.\hfill \end{align}$$

In addition, by

$$\begin{equation*}{\int }_{{t}_{1}}^{{t}_{2}}{\tau }^{-\alpha q}{\left({t}_{2}-\tau \right)}^{\alpha q-1}\mathrm{d}\tau ={\int }_{{t}_{1}/{t}_{2}}^{1}{\mu }^{-\alpha q}{\left(1-\mu \right)}^{\alpha q-1}\mathrm{d}\mu \le \frac{1}{\alpha q}{\left(\frac{{t}_{2}}{{t}_{1}}-1\right)}^{\alpha q}\end{equation*}$$

and (4.25), we can obtain the following chain of the inequalities

$$\begin{align}\hfill {{\Vert}{\mathcal{J}}_{2}^{N}{\Vert}}_{{\mathbf{V}}_{-\beta q}}& \le {\hat{c}}_{\alpha }{\int }_{{t}_{1}}^{{t}_{2}}{\Vert}F\left(\tau ,\cdot ,u\left(\tau ,\cdot \right)\right){\Vert}{\left({t}_{2}-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }{M}_{36}{K}_{{\ast}}{{\sim}{C}}_{0}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}{\int }_{{t}_{1}}^{{t}_{2}}{\tau }^{-\alpha q}{\left({t}_{2}-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \\ \hfill & \le {\hat{c}}_{\alpha }{M}_{36}{K}_{{\ast}}{{\sim}{C}}_{0}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}\frac{1}{\alpha q}{t}_{1}^{-\alpha q}{\left({t}_{2}-{t}_{1}\right)}^{\alpha q}.\hfill \end{align} \tag{ 4.28 }$$

Finally, the norm ${{\Vert}{\mathcal{J}}_{3}^{N}{\Vert}}_{{V}_{-\beta q}}$ has been estimated by (3.42), and the norm ${{\Vert}{\mathcal{J}}_{4}^{N}{\Vert}}_{{V}_{-\beta q}}$ can be estimated as follows:

$$\begin{align}\hfill {{\Vert}{\mathcal{J}}_{4}^{N}{\Vert}}_{{\mathbf{V}}_{-\beta q}}& \le \frac{{M}_{24}}{\alpha }{t}_{1}^{-2\alpha }\left({t}_{2}^{\alpha }-{t}_{1}^{\alpha }\right){\int }_{0}^{T}{\Vert}F\left(\tau ,\cdot ,u\left(\tau ,\cdot \right)\right){\Vert}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \\ \hfill & \le \frac{{M}_{24}}{\alpha }{t}_{1}^{-2\alpha }\left({t}_{2}^{\alpha }-{t}_{1}^{\alpha }\right){M}_{36}{K}_{{\ast}}{{\sim}{C}}_{0}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}{\int }_{0}^{T}{\tau }^{-\alpha q}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \hfill \\ \hfill & \le \frac{{M}_{24}}{\alpha }{t}_{1}^{-2\alpha }\left({t}_{2}^{\alpha }-{t}_{1}^{\alpha }\right){M}_{36}{K}_{{\ast}}{{\sim}{C}}_{0}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}B\left(\alpha q,1-\alpha q\right).\hfill \end{align}$$

It follows from the above arguments that ${}^{\mathrm{c}}D_{t}^{\alpha }u$ belongs to C((0, T]; V_−βq). On the other hand, the estimate (4.26) also holds for $\hat{p}=0$ and $\hat{q}=1$, i.e., we have

$$\begin{equation*}{t}^{\alpha }{{\Vert}{}^{\mathrm{c}}D_{t}^{\alpha }u\left(t,\cdot \right){\Vert}}_{{\mathbf{V}}_{-\beta q}}\le {M}_{38}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta p}},\end{equation*}$$

for 0 < t ≤ T. Therefore, ${}^{\mathrm{c}}D_{t}^{\alpha }u\in {C}_{w}^{\alpha }\left(\left(0,T\right];{\mathbf{V}}_{-\beta q}\right)$ there exists a constant M₄₀ > 0 such that

$$\begin{equation}{{\Vert}{}^{\mathrm{c}}D_{t}^{\alpha }u\left(t,\cdot \right){\Vert}}_{{C}_{w}^{\alpha }\left(\left(0,T\right];{\mathbf{V}}_{-\beta q}\right)}\le {M}_{40}{{\Vert}\varphi {\Vert}}_{{\mathbf{V}}_{\beta \left(p+\hat{q}\right)}}.\end{equation} \tag{ 4.29 }$$

by the Sobolev embedding ${\mathbf{V}}_{\beta \left(p+\hat{q}\right)}\hookrightarrow {\mathbf{V}}_{\beta p}$. The inequality (4.24) is derived by taking the inequality (4.27) and (4.29) together. We finally complete the proof. □

Remark 4.1. At the beginning part of step 1 of the above proof, we recall that ${\psi }_{j}^{N,2}={\psi }_{j}^{\left(2\right)}$. This means that ${\psi }_{j}^{N,2}$ (with respect to the nonlinear case) can be similarly estimated in the same way as in the linear case. More precisely, this term can be estimated as (3.31). Here, the formula of ${\psi }_{j}^{\left(2\right)}$ is given by

$$\begin{equation}{\psi }_{j}^{\left(2\right)}=-{\varphi }_{j}\frac{{m}_{j}^{\beta }{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{t}^{\alpha }\right)}{{E}_{\alpha ,1}\left(-{m}_{j}^{\beta }{T}^{\alpha }\right)},\end{equation} \tag{ 4.30 }$$

see the proof of theorem 3.4. The appearance of the factor ${m}_{j}^{\beta }$ in (4.30) tells that we need $\varphi \in {\mathbf{V}}_{\beta \left(p+\hat{q}\right)}$ to obtain (3.31). In summary, in order to bound the Caputo fractional derivative ${}^{\mathrm{c}}D_{t}^{\alpha }u$, we need the stronger assumption $\varphi \in {\mathbf{V}}_{\beta \left(p+\hat{q}\right)}$ rather than φ ∈ V_βp.

In the previous section, we found a solution u of FVP (1.1)–(1.3) in the set ${\mathbf{W}}_{\alpha q}^{{\hat{C}}_{0}}\left(J{\times}D\right)$. This allows that ${\Vert}u\left(t,\cdot \right){\Vert}\le {\hat{C}}_{0}{t}^{-\alpha q}$ for all 0 < t ≤ T. Then, we obtain $u\in {C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)$ by establishing the time continuity of u, which corresponds to the boundedness

$$\begin{equation}\underset{0{< }t\le T}{\mathrm{sup}}{t}^{\alpha q}{\Vert}u\left(t,\cdot \right){\Vert}{< }+\infty .\end{equation} \tag{ 5.1 }$$

However, the existence given in theorem 4.3 requires the assumption k₀(T ) < 1, which is equivalent to KT^αq < M₄₁, where M₄₁ is a constant. This can occur if K or T is small enough.

'Under what conditions is the contractivity condition k₀(T ) < 1 satisfied?' This motivates the result in this section.

The purpose of this section is to discuss global existence of solutions, namely, existence of solutions without any assumptions on K and T. To overcome the difficulties of finding solutions in ${C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)$, we shall seek solutions in a wider/weaker space than ${C}_{w}^{\alpha q}\left(\left(0,T\right];{L}^{2}\left(D\right)\right)$. The alternative solution space we are going to find is to take inspiration from replacing the supremum (5.1) by the following integral

$$\begin{equation}{\int }_{0}^{T}{\left({t}^{b}{\mathrm{e}}^{-\rho t}{\Vert}u\left(t,\cdot \right){\Vert}\right)}^{\mu }\;\mathrm{d}t{< }+\infty ,\end{equation} \tag{ 5.2 }$$

with suitable parameter b, ρ, and μ. We expect that the mapping $\mathcal{O}$ (formulated by lemma 4.1) on the alternative solution space is contracted as ρ tends to positive infinity.

The above arguments motivate us to denote by ${L}_{\rho ,b}^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)$, μ ≥ 1, ρ > 0, b > 0, the weighted Lebesgue space of all functions v: (0, T ) → L²(D) such that

$$\begin{equation*}{{\Vert}v{\Vert}}_{{L}_{\rho ,b}^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)}{:=}{\left({\int }_{0}^{T}{\left({t}^{b}{\mathrm{e}}^{-\rho t}{\Vert}v\left(t,\cdot \right){\Vert}\right)}^{\mu }\;\mathrm{d}t\right)}^{\frac{1}{\mu }}{< }\infty .\end{equation*}$$

In the next theorem, we present global existence for FVP (1.1)–(1.3) in ${L}_{\rho ,b}^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)$. It is helpful to introduce the following special function

$$\begin{equation*}{}_{1}\mathcal{F}_{1}\left(a,b,z\right){:=}\frac{{\Gamma}\left(b\right)}{{\Gamma}\left(b-a\right){\Gamma}\left(a\right)}{\int }_{0}^{1}{\left(1-\tau \right)}^{b-a-1}{\tau }^{a-1}{\mathrm{e}}^{z\tau }\mathrm{d}\tau ,\quad b{ >}a{ >}0,z\in \mathbb{C},\end{equation*}$$

which is called the Kummer function or hypergeometric function. We recall the following asymptotic behavior of this function ${}_{1}\mathcal{F}_{1}\left(a,b,z\right){:=}{\Gamma}\left(b\right){\left({\Gamma}\left(a\right)\right)}^{-1}{\mathrm{e}}^{z}{z}^{-\left(b-a\right)}\left(1+O{\vert z\vert }^{-1}\right)$, see [62] (lemma 8) or [63] (chapter 13). Due to a simple integration by substitution, for all t ∈ (0, T ) we observe that

$$\begin{align}\hfill {\int }_{0}^{t}{\left(t-\tau \right)}^{a-1}{\tau }^{b-1}{\mathrm{e}}^{z\tau }\mathrm{d}\tau & ={t}^{a+b-1}{\int }_{0}^{t}{\left(1-\tau \right)}^{a-1}{\tau }^{b-1}{\mathrm{e}}^{zt\tau }\mathrm{d}\tau \hfill \\ \hfill & ={t}^{a+b-1}\frac{{\Gamma}\left(a\right){\Gamma}\left(b\right)}{{\Gamma}\left(a+b\right)}{}_{1}\mathcal{F}_{1}\left(b,a+b,zt\right)\hfill \\ \hfill & ={t}^{a+b-1}\frac{{\mathrm{e}}^{zt}{\Gamma}\left(a\right)}{{\left(zt\right)}^{a}}\left(1+O\left({\vert zt\vert }^{-1}\right)\right).\hfill \end{align} \tag{ 5.3 }$$

Theorem 5.1. Assume that $\frac{1}{2}{< }\alpha {< }1$. Let p, q be defined by (R1) such that $\frac{1}{2\alpha }{< }q{< }1$. Let μ and b be such that $\frac{1}{\alpha q}{< }\mu {< }2$, $\alpha q-\frac{1}{\mu }{< }b{< }1-\frac{1}{\mu }$. If φ belongs to V_βp, F satisfies (A1), then there exists $\hat{\rho }{ >}0$ such that FVP (1.1)–(1.3) has a unique solution ${u}_{G}\in {L}_{\rho ,b}^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)$ with $\rho \ge \hat{\rho }$, and furthermore

$$\begin{equation*}{\int }_{0}^{T}{\left({t}^{b}{\mathrm{e}}^{-\rho t}{\Vert}{u}_{G}\left(t,\cdot \right){\Vert}\right)}^{\mu }\;\mathrm{d}t{< }+\infty .\end{equation*}$$

Proof. For ${v}_{1},{v}_{2}\in {L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)$, we first estimate the ${L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)$-norm of ${\mathcal{O}}_{1}{v}_{1}-{\mathcal{O}}_{1}{v}_{2}$. The main idea is splitting the quantity ∥F(τ, ⋅, v₁(τ, ⋅)) − F(τ, ⋅, v₂(τ, ⋅))∥(t − τ)^αq−1 into the product of (t − τ)^αq−1τ^−be^ρτ and τ^be^−ρτ∥F(τ, ⋅, v₁(τ, ⋅)) − F(τ, ⋅, v₂(τ, ⋅))∥. Then the ${L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)$-norm of ∥v₁(τ, ⋅) − v₂(τ, ⋅)∥ can be obtained by applying the Hölder inequality and the Lipschitz assumption (A1). Indeed, one can see that

$$\begin{align*}\hfill & {\int }_{0}^{T}{\left({t}^{b}{\mathrm{e}}^{-\rho t}{\int }_{0}^{t}{\Vert}{v}_{1}\left(\tau ,\cdot \right)-{v}_{2}\left(\tau ,\cdot \right){\Vert}{\left(t-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \right)}^{\mu }\;\mathrm{d}t\hfill \\ \hfill & \le {\int }_{0}^{T}{\left({t}^{b}{\mathrm{e}}^{-\rho t}\right)}^{\mu }{\left({\int }_{0}^{t}{\left(t-\tau \right)}^{\frac{\left(\alpha q-1\right)\mu }{\mu -1}}{\tau }^{-\frac{b\mu }{\mu -1}}{\mathrm{e}}^{\frac{\rho \mu \tau }{\mu -1}}\mathrm{d}\tau \right)}^{\mu -1}\;{\int }_{0}^{T}{\left({\tau }^{b}{\mathrm{e}}^{-\rho \tau }{\Vert}{v}_{1}\left(\tau ,\cdot \right)-{v}_{2}\left(\tau ,\cdot \right){\Vert}\right)}^{\mu }\mathrm{d}\tau \;\mathrm{d}t\hfill \\ \hfill & \le {C}_{1}^{\mu -1}{{\Vert}{v}_{1}-{v}_{2}{\Vert}}_{{L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)}^{\mu }{\int }_{0}^{T}{\left({t}^{b}{\mathrm{e}}^{-\rho t}\right)}^{\mu }\;{\left({t}^{\frac{\left(\alpha q-b\right)\mu -1}{\mu -1}}{\mathrm{e}}^{\frac{\rho \mu t}{\mu -1}}{\left(\rho t\right)}^{\frac{1-\alpha q\mu }{\mu -1}}\left(1+{t}^{-1}O\left({\rho }^{-1}\right)\right)\right)}^{\mu -1}\;\mathrm{d}t\hfill \\ \hfill & \le {C}_{1}^{\mu -1}{{\Vert}{v}_{1}-{v}_{2}{\Vert}}_{{L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)}^{\mu }{\left(T+O\left({\rho }^{-1}\right)\right)}^{\mu -1}{\rho }^{1-\alpha q\mu }{\int }_{0}^{T}{t}^{1-\mu }\;\mathrm{d}t,\hfill \end{align*}$$

where we denote ${C}_{1}{:=}{\Gamma}\left(\left(\alpha q\mu -1\right)/\left(\mu -1\right)\right){\left(\left(\mu -1\right)/\mu \right)}^{\left(\alpha q\mu -1\right)/\left(\mu -1\right)}$. Here, the asymptotic behavior (5.3) had been used in the second estimate, where we note that

$$\begin{equation*}\frac{\left(\alpha q-1\right)\mu }{\mu -1}+1=\frac{\alpha q\mu -1}{\mu -1}{ >}\frac{\alpha q\left(1/\left(\alpha q\right)\right)-1}{\mu -1}=0,\end{equation*}$$

as μ > 1/(αq), and 1 − bμ/(μ − 1) > 0 as b < (μ − 1)/μ. Moreover, by taking ρ large enough we can bound ${\left(T+O\left({\rho }^{-1}\right)\right)}^{\mu -1}$ by a constant independently of x, t. Since μ > 1/(αq), the factor ρ^1−αqμ obviously tends to zero as ρ tends to infinity. Furthermore, the latter improper integral is convergent as μ < 2. Summarizing, we can find a constant ρ₁ > 0 such that

$$\begin{align*}\hfill & {{\Vert}{\mathcal{O}}_{1}{v}_{1}-{\mathcal{O}}_{1}{v}_{2}{\Vert}}_{{L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)}^{\mu }\hfill \\ \hfill & \quad \quad \le {\left({\hat{c}}_{\alpha }{m}_{1}^{-\beta p}\right)}^{\mu }{\int }_{0}^{T}{\left({t}^{b}{\mathrm{e}}^{-\rho t}{\int }_{0}^{t}{\Vert}F\left(\tau ,\cdot ,{v}_{1}\left(\tau ,\cdot \right)\right)-F\left(\tau ,\cdot ,{v}_{2}\left(\tau ,\cdot \right)\right){\Vert}{\left(t-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \right)}^{\mu }\;\mathrm{d}t\hfill \\ \hfill & \quad \quad \le {\left(K{\hat{c}}_{\alpha }{m}_{1}^{-\beta p}\right)}^{\mu }{\int }_{0}^{T}{\left({t}^{b}{\mathrm{e}}^{-\rho t}{\int }_{0}^{t}{\Vert}{v}_{1}\left(\tau ,\cdot \right)-{v}_{2}\left(\tau ,\cdot \right){\Vert}{\left(t-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \right)}^{\mu }\;\mathrm{d}t\hfill \end{align*}$$

so we arrive at the following estimate

$$\begin{equation}{{\Vert}{\mathcal{O}}_{1}{v}_{1}-{\mathcal{O}}_{1}{v}_{2}{\Vert}}_{{L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)}\le \frac{1}{4}{{\Vert}{v}_{1}-{v}_{2}{\Vert}}_{{L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)},\end{equation} \tag{ 5.4 }$$

for all ρ ≥ ρ₁, where we have used (4.7) in the first estimate and the Lipschitz assumption (A1) in the second estimate.

Second, we will estimate the ${L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)$-norm of the difference ${\mathcal{O}}_{3}{v}_{1}-{\mathcal{O}}_{3}{v}_{2}$. Based on estimating the ${L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)$-norm as above, this can be treated by combining the Hölder inequality, the Lipschitz assumption (A1), and the inequality (4.8). Indeed, we see that

$$\begin{align*}\hfill & {{\Vert}{\mathcal{O}}_{3}\left(t,\cdot \right){v}_{1}-{\mathcal{O}}_{3}\left(t,\cdot \right){v}_{2}{\Vert}}_{{L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)}^{\mu }\hfill \\ \hfill & \quad \quad \le {M}_{3}^{\mu }{\int }_{0}^{T}{\left({t}^{b}{\mathrm{e}}^{-\rho t}{t}^{-\alpha q}{\int }_{0}^{T}{\Vert}F\left(\tau ,\cdot ,{v}_{1}\left(\tau ,\cdot \right)\right)-F\left(\tau ,\cdot ,{v}_{2}\left(\tau ,\cdot \right)\right){\Vert}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \right)}^{\mu }\;\mathrm{d}t\hfill \\ \hfill & \quad \quad \le {\left(K{M}_{3}\right)}^{\mu }{\int }_{0}^{T}{\left({t}^{b}{\mathrm{e}}^{-\rho t}{t}^{-\alpha q}{\int }_{0}^{T}{\Vert}{v}_{1}\left(\tau ,\cdot \right)-{v}_{2}\left(\tau ,\cdot \right){\Vert}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau \right)}^{\mu }\;\mathrm{d}t\hfill \\ \hfill & \quad \quad \le {\left(K{M}_{3}\right)}^{\mu }{C}_{1}^{\mu -1}{{\Vert}{v}_{1}-{v}_{2}{\Vert}}_{{L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)}^{\mu }\hfill \\ \hfill & \quad \quad \quad \quad {\int }_{0}^{T}{\left({t}^{b-\alpha q}{\mathrm{e}}^{-\rho t}\right)}^{\mu }\underset{{:=}{\sigma }_{b,\rho }\left(T\right)}{\underbrace{{\left({T}^{\frac{\left(\alpha q-b\right)\mu -1}{\mu -1}}{\mathrm{e}}^{\frac{\rho \mu T}{\mu -1}}{\left(\rho T\right)}^{\frac{1-\alpha q\mu }{\mu -1}}\left(1+{T}^{-1}O\left({\rho }^{-1}\right)\right)\right)}^{\mu -1}}}\;\mathrm{d}t,\hfill \end{align*}$$

where the asymptotic behavior (5.3) was employed. Let us denote by ${\mathcal{I}}_{b,\rho }$ the latter integral, then

$$\begin{equation*}{\mathcal{I}}_{b,\rho }={\sigma }_{b,\rho }\left(T\right){\int }_{0}^{T}{t}^{\left(b-\alpha q\right)\mu }{\mathrm{e}}^{-\rho \mu t}\;\mathrm{d}t.\end{equation*}$$

Note that (b − αq)μ > ((αq − 1/μ) − αq)μ = −1 as b > αq − 1/μ. Hence, using the asymptotic behavior (5.3), we have

$$\begin{align*}\hfill {\mathcal{I}}_{b,\rho }& \le {\sigma }_{b,\rho }\left(T\right){T}^{\left(b-\alpha q\right)\mu +1}{\mathrm{e}}^{-\rho \mu T}{\left(\rho \mu T\right)}^{-1}\left(1+O\left({\left(\rho \mu T\right)}^{-1}\right)\right)\hfill \\ \hfill & ={\left(1+{T}^{-1}O\left({\rho }^{-1}\right)\right)}^{\mu -1}\left(1+O\left({\left(\rho \mu T\right)}^{-1}\right)\right)\frac{{\mu }^{-1}{T}^{-\alpha q\mu }}{{\rho }^{\alpha q\mu }},\hfill \end{align*}$$

where the latter right-hand side tends to zero as ρ tends to infinity. Thus, there exists a ρ₂ > 0 such that

$$\begin{equation}{{\Vert}{\mathcal{O}}_{3}{v}_{1}-{\mathcal{O}}_{3}{v}_{2}{\Vert}}_{{L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)}\le \frac{1}{4}{{\Vert}{v}_{1}-{v}_{2}{\Vert}}_{{L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)},\end{equation} \tag{ 5.5 }$$

for all ρ ≥ ρ₂. Taking the estimates (5.4) and (5.5) together gives

$$\begin{align*}\hfill {{\Vert}\mathcal{O}{v}_{1}-\mathcal{O}{v}_{2}{\Vert}}_{{L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)}& \le \sum _{j\in \left\{1;3\right\}}{{\Vert}{\mathcal{O}}_{j}{v}_{1}-{\mathcal{O}}_{j}{v}_{2}{\Vert}}_{{L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)}\hfill \\ \hfill & \le \frac{1}{2}{{\Vert}{v}_{1}-{v}_{2}{\Vert}}_{{L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)},\hfill \end{align*}$$

for all ρ ≥ max{ρ₁; ρ₂}, namely, $\mathcal{O}:{L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)\to {L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)$ is a contraction mapping. This means that $\mathcal{O}$ has only one fixed point in ${L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)$, and so FVP (1.1)–(1.3) has a unique solution u_G in the weighted Lebesgue space ${L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)$. The desired inequality is obvious. □

Remark 5.1. In fact, we tried to find solutions in the space

$$\begin{equation*}{C}_{w}^{b,\rho }\left(\left(0,T\right];{L}^{2}\left(D\right)\right){:=}\left\{v\in C\left(\left(0,T\right];{L}^{2}\left(D\right)\right)\left\vert {{\Vert}v{\Vert}}_{{C}_{w}^{b,\rho }\left(\left(0,T\right];{L}^{2}\left(D\right)\right)}\right.{:=}\underset{0{< }t\le T}{\mathrm{sup}}{t}^{b}{\mathrm{e}}^{-\rho t}{\Vert}v\left(t,\cdot \right){\Vert}{< }+\infty \right\}.\end{equation*}$$

Note that the following inclusion holds

$$\begin{equation*}{C}_{w}^{b,\rho }\left(\left(0,T\right];{L}^{2}\left(D\right)\right)\subset {L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right).\end{equation*}$$

Indeed, if $v\in {C}_{w}^{b,\rho }\left(\left(0,T\right];{L}^{2}\left(D\right)\right)$ then

$$\begin{equation*}{{\Vert}v{\Vert}}_{{L}_{\rho ,b}^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)}\le {\left({\int }_{0}^{T}\mathrm{d}t\right)}^{\frac{1}{\mu }}\left(\underset{0{< }t\le T}{\mathrm{sup}}{t}^{b}{\mathrm{e}}^{-\rho t}{\Vert}v\left(t,\cdot \right){\Vert}\right)={T}^{\frac{1}{\mu }}{{\Vert}v{\Vert}}_{{C}_{w}^{b,\rho }\left(\left(0,T\right];{L}^{2}\left(D\right)\right)}{< }+\infty .\end{equation*}$$

In order to find solutions in this space, for all ${v}_{1},{v}_{2}\in {C}_{w}^{b,\rho }\left(\left(0,T\right];{L}^{2}\left(D\right)\right)$, it requires to bound the following quantities

$$\begin{align*}\hfill & {Q}_{1}\left(b,\rho \right){:=}\underset{0{< }t\le T}{\mathrm{sup}}{t}^{b}{\mathrm{e}}^{-\rho t}{\int }_{0}^{t}{\Vert}{v}_{1}\left(\tau ,\cdot \right)-{v}_{2}\left(\tau ,\cdot \right){\Vert}{\left(t-\tau \right)}^{\alpha q-1}\mathrm{d}\tau ,\hfill \\ \hfill & {Q}_{2}\left(b,\rho \right){:=}\underset{0{< }t\le T}{\mathrm{sup}}{t}^{b}{\mathrm{e}}^{-\rho t}{t}^{-\alpha q}{\int }_{0}^{T}{\Vert}{v}_{1}\left(\tau ,\cdot \right)-{v}_{2}\left(\tau ,\cdot \right){\Vert}{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau ,\hfill \end{align*}$$

by ${k}_{j}\left(\rho \right){{\Vert}{v}_{1}-{v}_{2}{\Vert}}_{{C}_{w}^{b,\rho }\left(\left(0,T\right];{L}^{2}\left(D\right)\right)}$ with k_j(ρ), j = 1, 2, tend to zero as ρ tends to infinity. Unfortunately, it does not occur with the term Q₂(b, ρ). Indeed, since ${v}_{1},{v}_{2}\in {C}_{w}^{b,\rho }\left(\left(0,T\right];{L}^{2}\left(D\right)\right)$ we have

$$\begin{equation*}{\Vert}{v}_{1}\left(t,\cdot \right)-{v}_{2}\left(t,\cdot \right){\Vert}\le {t}^{-b}{\mathrm{e}}^{\rho t}{{\Vert}{v}_{1}-{v}_{2}{\Vert}}_{{C}_{w}^{b,\rho }\left(\left(0,T\right];{L}^{2}\left(D\right)\right)},\end{equation*}$$

which gives

$$\begin{equation*}{Q}_{2}\left(b,\rho \right)\le \left(\underset{0{< }t\le T}{\mathrm{sup}}\underset{{:=}{\hat{\sigma }}_{b,\rho }\left(t,T\right)}{\underbrace{{t}^{b}{\mathrm{e}}^{-\rho t}{t}^{-\alpha q}{\int }_{0}^{T}{\tau }^{-b}{\mathrm{e}}^{\rho \tau }{\left(T-\tau \right)}^{\alpha q-1}\mathrm{d}\tau }}\right){{\Vert}{v}_{1}-{v}_{2}{\Vert}}_{{C}_{w}^{b,\rho }\left(\left(0,T\right];{L}^{2}\left(D\right)\right)}.\end{equation*}$$

The following conclusions are obvious:

• Due to the asymptotic behavior (5.3), the supremum of ${\hat{\sigma }}_{b,\rho }\left(t,T\right)$ on (0, T ] tends to infinity as ρ tends to infinity. Hence, $\mathcal{O}:{L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)\to {L}_{b,\rho }^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)$ cannot be a contraction mapping for arbitrary T. This is the main reason why we did not find solutions in ${C}_{w}^{b,\rho }\left(\left(0,T\right];{L}^{2}\left(D\right)\right)$.
• The idea of using the space ${L}_{\rho ,b}^{\mu }\left(0,T;{L}^{2}\left(D\right)\right)$ fortuitously came when we realized that
  $$\begin{equation*}{\int }_{0}^{T}{\left({\hat{\sigma }}_{b,\rho }\left(t,T\right)\right)}^{\mu }\;\mathrm{d}t\rho \to +\infty {\to }0\end{equation*}$$
  with a suitable number μ ≥ 1. Here, we replaced the supremum (5.1) by the integral (5.2).
• One can show that Q₁(b, ρ) is bounded by ${k}_{1}\left(\rho \right){{\Vert}{v}_{1}-{v}_{2}{\Vert}}_{{C}_{w}^{b,\rho }\left(\left(0,T\right];{L}^{2}\left(D\right)\right)}$, where k₁(ρ) tends to zero as ρ tends to infinity. This means that we can establish the existence of a mild solution to the initial value problem (1.1), (1.2), (1.4) in ${C}_{w}^{b,\rho }\left(\left(0,T\right];{L}^{2}\left(D\right)\right)$ without any assumptions on K and T.

We would like to thank the editor and anonymous referees for their valuable comments and suggestions. This research was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.02-2019.09.