Approximation of solutions to integro-differential time fractional order parabolic equations in $L^{p}$-spaces

Abstract

In this paper we study the initial boundary value problem for a class of integro-differential time fractional order parabolic equations with a small positive parameter ε. Using the Laplace transform, Mittag-Leffler operator family, $C_{0}$-semigroup, resolvent operator, and weighted function space, we get the existence of a mild solution. For suitable indices $p\in [1,+\infty )$ and $s\in (1,+\infty )$, we first prove that the mild solution of the approximating problem converges to that of the corresponding limit problem in $L^{p}((0,T), L^{s}(\Omega ))$ as $\varepsilon \rightarrow 0^{+}$. Then for the linear approximating problem with ε and the corresponding limit problem, we give the continuous dependence of the solutions. Finally, for a class of semilinear approximating problems and the corresponding limit problems with initial data in $L^{s}(\Omega )$, we prove the local existence and uniqueness of the mild solution and then give the continuous dependence on the initial data.

1 Introduction

In this paper, we study the following initial boundary value problem (IBVP) for integro-differential time fractional order parabolic equations with a small positive parameter ε in a suitable $L^{p}$-space:

$$\begin{aligned} \textstyle\begin{cases} D_{t}^{\alpha}u_{\varepsilon}(t)+\chi (\varepsilon )(-A)u_{ \varepsilon}(t)+(-A)(k^{\varepsilon}\ast u_{\varepsilon})(t)=f_{ \varepsilon}(t), \\ u_{\varepsilon}(0)=u_{0,\varepsilon}, \end{cases}\displaystyle \end{aligned}$$

(1.1)

where the kernel $k(t): \mathbf{R}\rightarrow \mathbf{R}$ is a continuous function such that

$$\begin{aligned} k^{\varepsilon}(t)=\frac{1}{\varepsilon}k \biggl(\frac{t}{\varepsilon} \biggr), \qquad \bigl(k^{\varepsilon}\ast u_{\varepsilon} \bigr) (t)= \int _{0}^{t}k^{ \varepsilon}(t-s)u_{\varepsilon}(s) \,ds, \end{aligned}$$

(1.2)

$\alpha \in (0,1)$, and $D_{t}^{\alpha}u(t)$ represents the Caputo fractional derivative of u defined by

$$\begin{aligned} D_{t}^{\alpha}u(t)= \int _{0}^{t}g_{1-\alpha}(t-s)u'(s) \,ds,\qquad g_{ \beta}(t)=\frac{t^{\beta -1}}{\Gamma (\beta )},\quad t>0,\beta >0, \end{aligned}$$

(1.3)

where the Gelfand–Shilov function $g_{\beta}(t)$ satisfies the semigroup property $g_{\alpha}\ast g_{\beta}=g_{\alpha +\beta}$, while χ is a positive scalar function defined in $(0,\varepsilon _{0}]$ for a given real number $\varepsilon _{0}>0$ such that

$$\begin{aligned} \chi (\varepsilon )\rightarrow \chi _{0} \quad \text{as } \varepsilon \rightarrow 0^{+}. \end{aligned}$$

(1.4)

In general, A is a linear unbounded operator defined on a bounded domain $\Omega \subset \mathbf{R}^{n}$ with $C^{2}$-boundary, here we consider that A is a second-order differential operator in divergence form,

$$\begin{aligned} A:=A(x,D_{x})=\sum_{i,j=1}^{n}D_{x_{i}} \bigl(a_{ij}(x)D_{x_{j}} \bigr)+a_{0}(x), \end{aligned}$$

(1.5)

endowed with vanishing Dirichlet boundary condition on ∂Ω, and there are positive constants $c_{0}$, $c_{1}$, γ such that $a_{0}(x)\geq \gamma$, $a_{0}\in C(\overline{\Omega})$, $a_{ij}=a_{ji}\in C^{1}(\overline{\Omega})$, and

$$\begin{aligned} c_{0} \vert \xi \vert ^{2}\leq \sum _{i,j=1}^{n}a_{ij}(x)\xi _{i}\xi _{j}\leq c_{1} \vert \xi \vert ^{2}, \quad\forall x\in \overline{\Omega}, \forall \xi \in \mathbf{R}^{n}. \end{aligned}$$

(1.6)

The time-fractional integro-differential equations are very popular due to their applications in real-world problems in physics, mechanics, civil and mechanical engineering. In the recent years, the research on time-fractional differential equations has attracted many authors. For fractional calculus and Mittag-Leffler function, we refer to references [15, 20]. For abstract integro-differential equations and abstract evolution equations, we refer to the monographs [19, 22] and papers [13, 14, 24]. For the fractional integro-differential equations, we refer to the references [8–10, 12]. For fractional evolution equations in Banach spaces, we refer to the PhD thesis [2]. Zhang and Li [25] considered the global well-posedness and blow-up of solutions of the Cauchy problem for a time-fractional superdiffusion equation. Li [16] discussed the regularity of mild solutions for a fractional abstract Cauchy problem with order $\alpha \in (1, 2)$. Chen and Tang [5, 6] considered the homogenization of nonlinear nonlocal p-Laplacian operators with convolution kernels.

For $\alpha =1$, Lorenzi and Messina [17] considered the approximation of solutions to linear integro-differential parabolic equation (1.1) in $L^{p}$-spaces and extended the results to the corresponding nonlinear equation [18]. For $\alpha \in (1, 2)$, Zhao and Tang [26] discussed the approximation of solutions to nonlinear integro-differential time fractional wave equation (1.1) in the $L^{p}$-space.

In this paper, we first consider the linear integro-differential parabolic equation (1.1) with $\alpha \in (0,1]$, the time-fractional order evolution equation in a Banach space. Our main task is to show that, under suitable assumptions on the functions $f_{\varepsilon}$, f and the initial data $u_{0,\varepsilon}$, $u_{0}$, for suitable indices $p\in [1,+\infty )$ and $s\in (1,+\infty )$, in $L^{p}((0,T), L^{s}(\Omega ))$-norm, the mild solution $u_{\varepsilon}(t)$ of problem (1.1) converges to the mild solution $u(t)$ of the limit problem

$$\begin{aligned} \textstyle\begin{cases} D_{t}^{\alpha}u(t)+(1+\chi _{0})(-A)u(t)=f(t), \quad t\in (0,T], \\ u(0)=u_{0}, \end{cases}\displaystyle \end{aligned}$$

(1.7)

as $\varepsilon \rightarrow 0^{+}$.

In general, any order Caputo fractional derivative of $u(t)$ for $\alpha >0$ is defined by [21]

$$\begin{aligned} D_{t}^{\alpha}u(t)= \int _{0}^{t}g_{m-\alpha}(t-s) \frac{d^{m}}{ds^{m}}u(s)\,ds, \end{aligned}$$

(1.8)

where m is the smallest integer greater than or equal to α, and $g_{m-\alpha}(t)$ is defined by (1.3) with $m-\alpha$ instead of β.

Similarly, the Riemann–Liouville fractional integral is defined by

$$\begin{aligned} J_{t}^{\alpha}f(t)=(g_{\alpha}\ast f) (t)= \int _{0}^{t}g_{\alpha}(t-s)f(s)\,ds, \end{aligned}$$

(1.9)

thus we have

$$\begin{aligned} D_{t}^{\alpha}J_{t}^{\alpha}f(t)=f(t),\qquad J_{t}^{\alpha}D_{t}^{ \alpha}f(t)=f(t)-\sum _{k=0}^{m-1}f^{(k)}(0) \frac{t^{k}}{k!}. \end{aligned}$$

(1.10)

Applying the properties of Laplace transform and taking into account that $\widehat{{g_{\alpha}}}(\lambda )=\lambda ^{-\alpha}$ and $\widehat{(J_{t}^{\alpha}f)}(\lambda )=\widehat{(g_{\alpha}*f)}( \lambda )=\widehat{{g_{\alpha}}}(\lambda )\hat{f}(\lambda )=\lambda ^{- \alpha}\hat{f}(\lambda )$, the second equation in (1.10) implies that

$$\begin{aligned} \widehat{D_{t}^{\alpha}f}(\lambda )=\lambda ^{\alpha}\widehat{f}( \lambda )-\sum_{k=0}^{m-1}f^{(k)}(0) \lambda ^{\alpha -1-k}. \end{aligned}$$

(1.11)

For the linear time fractional order parabolic equation (1.1) with a small positive parameter ε, we now consider a semilinear approximating problem

$$\begin{aligned} \textstyle\begin{cases} D_{t}^{\alpha}u_{\varepsilon}(t)+\chi (\varepsilon )(-A)u_{ \varepsilon}(t)+(-A)(k^{\varepsilon}\ast u_{\varepsilon})(t)=f_{ \varepsilon}(t)+N[u_{\varepsilon}](t), \\ u_{\varepsilon}(0,x)=u_{0,\varepsilon}(x), \quad x\in \Omega , \\ u_{\varepsilon}(t,x)=0, \quad (t,x)\in (0,T)\times \partial \Omega , \end{cases}\displaystyle \end{aligned}$$

(1.12)

and the corresponding semilinear limit problem as $\varepsilon \rightarrow 0^{+}$, namely

$$\begin{aligned} \textstyle\begin{cases} D_{t}^{\alpha}u(t,x)+(1+\chi _{0})(-A)u(t,x)=f(t,x)+N[u](t,x), \\ u(0,x)=u_{0}(x), \quad x\in \Omega , \\ u(t,x)=0, \quad (t,x)\in (0,T)\times \partial \Omega , \end{cases}\displaystyle \end{aligned}$$

(1.13)

where N is a nonlinear operator admitting the following representation:

$$\begin{aligned} N(u) (t,x)=\psi \bigl(t,u(t,x) \bigr), \end{aligned}$$

(1.14)

and ψ is given later.

For Banach spaces $(X,\|\cdot \|_{X})$ and $(Y,\|\cdot \|_{Y})$, we consider the closed linear operator $A: D(A)\subset X\rightarrow Y$, the notation $[D(A)]$ represents the domain of A endowed with graph norm $\|x\|_{1}=\|x\|_{X}+\|Ax\|_{Y}$, $x\in D(A)$. Recalling the classical theory of semigroups [11], the Mittag-Leffler function $E_{\alpha}$ and the nonnegative Wright function $\Psi _{\alpha}$ have the relation and property (see [2])

$$\begin{aligned}& E_{\alpha}(-z) = \int _{0}^{\infty}\Psi _{\alpha}(t)e^{-zt} \,dt,\quad z \in \mathbb{C}, \end{aligned}$$

(1.15)

$$\begin{aligned}& \int _{0}^{\infty}\Psi _{\alpha}(s)s^{r} \,ds = \frac{\Gamma (r+1)}{\Gamma (\alpha r+1)},\quad r>-1, \end{aligned}$$

(1.16)

where the Gamma function is defined by $\Gamma (\alpha )={\int _{0}^{+\infty}t^{\alpha -1}e^{-t}\,dt}$. In this paper we will use also the Beta function $\mathrm{B}:(0,\infty )\times (0,\infty )\rightarrow (0,\infty )$ defined by

$$ \mathrm{B}(a,b)= \int _{0}^{1}(1-s)^{a-1}s^{b-1} \,ds. $$

(1.17)

Using the subordination principle, we can write the Mittag-Leffler family associated to the operator A in the form

$$\begin{aligned} E_{\alpha}(tA)= \int _{0}^{\infty}\Psi _{\alpha}(s)e^{st^{\alpha}A} \,ds, \end{aligned}$$

(1.18)

where $\{e^{tA}\}_{t\geq 0}$ is an analytic semigroup associated with the operator A defined in (1.5).

Regarding the convergence of approximating IBVP to the α-order time fractional evolutional equation $D_{t}^{\alpha}u_{\varepsilon}+\chi (\varepsilon )(-A)u_{\varepsilon}+(-A)(k^{ \varepsilon}\ast u_{\varepsilon})=N[u_{\varepsilon}]$ to the corresponding limit IBVP of the equation $D_{t}^{\alpha}u+(1+\chi _{0})(-A)u=N[u]$ as scale parameter $\varepsilon \rightarrow 0^{+}$, for a general class of sectorial operators $A=\sum_{i,j=1}^{n}D_{x_{i}}(a_{ij}(x)D_{x_{j}})+a_{0}(x)$ defined in equation (1.5), the novelty of this paper is to extend the results in [17, 18] from $\alpha =1$ (classical integro-differential parabolic equation) to $\alpha \in (0,1]$ (time fractional order integro-differential parabolic equation). Also we consider a more general kernel $k(t)$ in equation (1.1) instead of the kernel included in the series representation in [17].

This paper is divided into five sections. Section 2 is devoted to showing some basic properties of the resolvent operators $F^{\varepsilon}_{\alpha}$, $G^{\varepsilon}_{\alpha}$ defined in equations (2.2)–(2.5) and the nonlinear operator N. Our main results are stated in Sect. 3, and their proofs are given in Sect. 4. A discussion is given in Sect. 5.

2 Properties of resolvent operators

We first state some assumptions on the operator A and the initial data $u_{0,\varepsilon}(x)$ to our abstract Cauchy problems of the linear equation (1.1) and the semilinear equation (1.12), respectively. We assume that X is a given general Banach space with a norm denoted by $\|\cdot \|_{X}$. And approximation results we want to study in $L^{p}((0,T); X)$ involve the norm denoted by $\|f\|_{L^{p}((0,T); X)}=\int _{0}^{T}\|f(t)\|_{X}^{p}\,dt$. Furthermore, we state the following assumptions:

• (H1) The operator $A: D(A)\subset X\rightarrow X$ is a closed linear operator, with $D(A)$ dense in X, and for some $\phi \in (\frac{\pi}{2},\pi )$ there is a positive constant $C_{0}=C_{0}(\phi )$ such that

  $$\begin{aligned} \Sigma _{0,\phi}= \bigl\{ \lambda \in \mathbb{C}: \bigl\vert \arg ( \lambda ) \bigr\vert < \phi \bigr\} \subset \rho (A), \end{aligned}$$

  and the resolvent operator $R(\lambda ,A)=(\lambda I-A)^{-1}$ satisfies

  $$\begin{aligned} \bigl\Vert R(\lambda ,A) \bigr\Vert \leq \frac{C_{0}}{ \vert \lambda \vert },\quad \forall \lambda \in \Sigma _{0,\phi}. \end{aligned}$$

• (H2) The kernel $k(t)\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{+})$ is such that $\widehat{k}(\lambda )$ exists for $\operatorname{Re} (\lambda )>0$, $\widehat{k}(\lambda )$ can be extended to $\sum_{0,\phi}$, and $\| \widehat{k}(\lambda )\|=O (\frac{1}{|\lambda |} )$ as $|\lambda |\rightarrow +\infty$. For $p\in [1,+\infty )$, $\alpha \in (0,1]$, there exist constants $\theta _{0}>\frac{1}{p}+\alpha$, $k_{0},r>0$, and $r_{0}>r$, $C>0$ such that

  $$\begin{aligned} \bigl\vert \widehat{k}(\lambda )-k_{0} \bigr\vert \leq C \vert \lambda \vert ^{\theta _{0}}, \quad \forall \lambda \in \Sigma _{0,\phi}\cap B(0,r_{0}). \end{aligned}$$

• (H3) There exist constants M, $C>0$ such that

  $$\begin{aligned} \bigl\Vert \bigl(\lambda I- \bigl(1+\chi (\varepsilon ) \bigr)A \bigr)^{-1} \bigr\Vert _{\mathcal{L}(X)}\leq \frac{M}{ \vert \lambda \vert }, \quad\forall \lambda \in \Sigma _{0,\phi}, \forall \varepsilon \in (0, \varepsilon _{0}). \end{aligned}$$

• (H4) The sequence $\{f_{\varepsilon}\}_{\varepsilon >0}\subset C([0,T],X)$ converges to f in $C([0,T],X)$ as $\varepsilon \rightarrow 0^{+}$.

• (H5) Functions $u_{0,\varepsilon},u_{0}\in X$ and $\{u_{0,\varepsilon}\}_{\varepsilon >0}$ converges to $u_{0}$ in X as $\varepsilon \rightarrow 0^{+}$.

Remark 2.1

Assumptions (H1), (H2), and (H3) have been considered in [7–10, 12], with the operator $B(t)=K(t)A$ for a given function $K(t)$ and $\alpha =1$.

In the sequel, for $r>0$ and $\theta \in (\frac{\pi}{2},\phi )$, we denote a sector by

$$\begin{aligned} \Sigma _{r,\theta}= \bigl\{ \lambda \in \mathbb{C}: \vert \lambda \vert \geq r, \bigl\vert \arg (\lambda ) \bigr\vert < \theta \bigr\} . \end{aligned}$$

(2.1)

In addition, for a given Banach space, $\rho (F_{\alpha}^{\varepsilon})$ and $\rho (G_{\alpha}^{\varepsilon})$ are the sets

$$\begin{aligned} &\rho \bigl(F_{\alpha}^{\varepsilon} \bigr)= \bigl\{ \lambda \in \mathbb{C}: F_{ \alpha}^{\varepsilon}(\lambda )= \bigl(\lambda ^{\alpha}I-\chi ( \varepsilon )A-\widehat{k}(\varepsilon \lambda )A \bigr)^{-1}\in \mathcal{L}(X) \bigr\} , \end{aligned}$$

(2.2)

$$\begin{aligned} &\rho \bigl(G_{\alpha}^{\varepsilon} \bigr)= \bigl\{ \lambda \in \mathbb{C}: G_{ \alpha}^{\varepsilon}(\lambda ) =\lambda ^{\alpha -1} \bigl(\lambda ^{ \alpha}I-\chi (\varepsilon )A-\widehat{k}(\varepsilon \lambda )A \bigr)^{-1} \in \mathcal{L}(X) \bigr\} , \end{aligned}$$

(2.3)

and $\rho (F_{\alpha})$ and $\rho (G_{\alpha})$ are the sets

$$\begin{aligned} &\rho (F_{\alpha})= \bigl\{ \lambda \in \mathbb{C}:F_{\alpha}(\lambda )= \bigl(\lambda ^{\alpha}I-\chi _{0}A-k_{0}A \bigr)^{-1}\in \mathcal{L}(X) \bigr\} , \end{aligned}$$

(2.4)

$$\begin{aligned} &\rho (G_{\alpha})= \bigl\{ \lambda \in \mathbb{C}:G_{\alpha}(\lambda )= \lambda ^{\alpha -1} \bigl(\lambda ^{\alpha}I-\chi _{0}A-k_{0}A \bigr)^{-1} \in \mathcal{L}(X) \bigr\} . \end{aligned}$$

(2.5)

Next we collect some properties established in [7, 23].

Lemma 2.1

([7])

Under Assumptions (H1), (H2), and (H3), there exists an $r>0$ such that $\sum_{r,\phi}\subset \rho (G^{\varepsilon}_{\alpha})$, $\sum_{r,\phi}\subset \rho (F^{\varepsilon}_{\alpha})$, and the operator-valued functions $F^{\varepsilon}_{\alpha}, G^{\varepsilon}_{\alpha}:\sum_{r,\phi} \rightarrow \mathcal{L}(X)$ are analytic. Moreover, there exists a constant M such that

$$\begin{aligned} \bigl\Vert F^{\varepsilon}_{\alpha}(\lambda ) \bigr\Vert \leq \frac{M}{ \vert \lambda \vert ^{\alpha}},\qquad \bigl\Vert G^{\varepsilon}_{\alpha}( \lambda ) \bigr\Vert \leq \frac{M}{ \vert \lambda \vert },\quad \forall \lambda \in \Sigma _{r,\phi}. \end{aligned}$$

Definition 2.1

([7, 23])

The operator families $\{S_{\alpha}^{\varepsilon}(t)\}_{t\geq 0}$ and $\{S_{\alpha}(t)\}_{t\geq 0}$ are defined by

$$\begin{aligned} S_{\alpha}^{\varepsilon}(t)&=\frac{1}{2\pi i} \int _{\Gamma _{r,\theta}}e^{ \lambda t}\lambda ^{\alpha -1} \bigl( \lambda ^{\alpha}I-\chi ( \varepsilon )A-\widehat{k}(\varepsilon \lambda )A \bigr)^{-1}\,d\lambda = \frac{1}{2\pi i} \int _{\Gamma _{r,\theta}}e^{\lambda t}G_{\alpha}^{ \varepsilon}( \lambda )\,d\lambda , \\ S_{\alpha}(t)&=\frac{1}{2\pi i} \int _{\Gamma _{r,\theta}}e^{\lambda t} \lambda ^{\alpha -1} \bigl( \lambda ^{\alpha}I-\chi _{0}A-k_{0}A \bigr)^{-1}\,d\lambda =\frac{1}{2\pi i} \int _{\Gamma _{r,\theta}}e^{\lambda t}G_{ \alpha}(\lambda )\,d\lambda , \end{aligned}$$

and the operator families $\{T_{\alpha}^{\varepsilon}(t)\}_{t\geq 0}$ and $\{T_{\alpha}(t)\}_{t\geq 0}$ are defined by

$$\begin{aligned}& T_{\alpha}^{\varepsilon}(t) =\frac{1}{2\pi i} \int _{\Gamma _{r,\theta}}e^{ \lambda t} \bigl(\lambda ^{\alpha}I-\chi (\varepsilon )A-\widehat{k}( \varepsilon \lambda )A \bigr)^{-1}\,d \lambda =\frac{1}{2\pi i} \int _{ \Gamma _{r,\theta}}e^{\lambda t}F_{\alpha}^{\varepsilon}( \lambda )\,d\lambda , \\& T_{\alpha}(t) =\frac{1}{2\pi i} \int _{\Gamma _{r,\theta}}e^{\lambda t} \bigl(\lambda ^{\alpha}I-\chi _{0}A-k_{0}A \bigr)^{-1}\,d\lambda = \frac{1}{2\pi i} \int _{\Gamma _{r,\theta}}e^{\lambda t}F_{\alpha}( \lambda )\,d\lambda , \end{aligned}$$

where $\Gamma _{1}(r,\theta )=\{te^{i\theta}: t\geq r\}$, $\Gamma _{2}(r,\theta )=\{re^{i\xi}:|\xi |\leq \theta \}$, $\Gamma _{3}(r,\theta )=\{te^{-i\theta}:t\geq r\}$, $\theta \in (\pi /2,\phi )$, and $\Gamma _{r,\theta}=\bigcup_{i=1}^{3}\Gamma _{i}(r,\theta )$, oriented counterclockwise.

Using the Laplace transform, we give the definitions of $L^{q}$-mild solutions to the equations (1.1), (1.7), (1.12), and (1.13), respectively.

Definition 2.2

Let $T>0$, $\varepsilon \in (0,\varepsilon _{0}]$, $u_{0,\varepsilon}, u_{0}\in L^{q}(\Omega )$. Functions $u_{\varepsilon}\in \mathrm{C}([0,T],L^{q}(\Omega ))$ and $u\in \mathrm{C}([0,T],L^{q}(\Omega ))$ are called $L^{q}$-mild solutions of equations (1.1) and (1.7) in $[0,T]$ if $u_{\varepsilon}$ and u respectively satisfy the following equations:

$$\begin{aligned}& u_{\varepsilon}(t) =\mathrm{S}_{\alpha}^{\varepsilon}(t)u_{0, \varepsilon}+ \int _{0}^{t}\mathrm{T}_{\alpha}^{\varepsilon}(t-s)f_{ \varepsilon}(s) \,ds, \end{aligned}$$

(2.6)

$$\begin{aligned}& u(t) =\mathrm{S}_{\alpha}(t)u_{0}+ \int _{0}^{t}\mathrm{T}_{\alpha}(t-s)f(s)\,ds. \end{aligned}$$

(2.7)

Definition 2.3

Let $T>0$, $\varepsilon \in (0,\varepsilon _{0}]$, $u_{0,\varepsilon}, u_{0}\in L^{q}(\Omega )$. Functions $u_{\varepsilon}\in \mathrm{C}([0,T],L^{q}(\Omega ))$ and $u\in \mathrm{C}([0,T],L^{q}(\Omega ))$ are called $L^{q}$-mild solutions of equations (1.12) and (1.13) in $[0,T]$ if $u_{\varepsilon}$ and u respectively satisfy the following equations:

$$\begin{aligned}& u_{\varepsilon}(t) =\mathrm{S}_{\alpha}^{\varepsilon}(t)u_{0, \varepsilon}+ \int _{0}^{t}\mathrm{T}_{\alpha}^{\varepsilon}(t-s)f_{ \varepsilon}(s) \,ds+ \int _{0}^{t}\mathrm{T}_{\alpha}^{\varepsilon}(t-s)Nu_{ \varepsilon}(s) \,ds, \end{aligned}$$

(2.8)

$$\begin{aligned}& u(t) =\mathrm{S}_{\alpha}(t)u_{0}+ \int _{0}^{t}\mathrm{T}_{\alpha}(t-s)f(s)\,ds+ \int _{0}^{t}\mathrm{T}_{\alpha}(t-s)Nu(s) \,ds. \end{aligned}$$

(2.9)

Lemma 2.2

([7])

Under Assumptions (H1), (H2), and (H3), the operator-valued function $S_{\alpha}^{\varepsilon}(t)$ defined in Definition 2.1is (i) exponentially bounded in $\mathcal{L}(X)$; (ii) exponentially bounded in $\mathcal{L}(D[A])$; (iii) strongly continuous on $[0,\infty )$ and uniformly continuous on $(0,\infty )$ in $\mathcal{L}(X)$; (iv) strongly continuous on $[0,\infty )$ in $\mathcal{L}(D[A])$. The operator function $T_{\alpha}^{\varepsilon}(t)$ defined in Definition 2.1is (v) exponentially bounded in $\mathcal{L}(X)$; (vi) uniformly continuous on $(0,\infty )$ in $\mathcal{L}(X)$.

Lemma 2.3

Let $1\leq q\leq r<\infty$ be such that $\frac{n}{2}(\frac{1}{q}-\frac{1}{r})<1$. If the operator A is defined in equation (1.5) and satisfies Assumption (H1), then for every $\varphi \in L^{q}(\Omega )$ we have

$$\begin{aligned}& \bigl\Vert S_{\alpha}(t)\varphi \bigr\Vert _{L^{r}(\Omega )} \leq Ct^{- \frac{\alpha n}{2}(\frac{1}{q}-\frac{1}{r})} \Vert \varphi \Vert _{L^{q}( \Omega )}, \end{aligned}$$

(2.10)

$$\begin{aligned}& \bigl\Vert T_{\alpha}(t)\varphi \bigr\Vert _{L^{r}(\Omega )} \leq Ct^{- \frac{\alpha n}{2}(\frac{1}{q}-\frac{1}{r})+\alpha -1} \Vert \varphi \Vert _{L^{q}( \Omega )}, \end{aligned}$$

(2.11)

where C is a positive constant. Moreover, the operator family $\{S_{\alpha}(t)\}_{t\geq 0}$ is strongly continuously in $L^{r}(\Omega )$.

Proof

The proof of estimate (2.10) can be found in [18]. In order to compare the operator families $\{S_{\alpha}(t)\}_{t\geq 0}$ and $\{T_{\alpha}(t)\}_{t\geq 0}$, here we give the proof of estimate (2.10) again. From Theorem 3.1 [23], we get

$$\begin{aligned}& S_{\alpha}(t)\varphi = \int _{0}^{\infty}\Psi _{\alpha}(s)e^{(k_{0}+ \chi _{0})st^{\alpha}A} \varphi \,ds, \end{aligned}$$

(2.12)

$$\begin{aligned}& T_{\alpha}(t)\varphi =t^{\alpha -1} \int _{0}^{\infty}\alpha s\Psi _{ \alpha}(s)e^{(k_{0}+\chi _{0})st^{\alpha}A} \varphi \,ds. \end{aligned}$$

(2.13)

From the proof of Lemma 2.1 [1] and using equation (1.16) of the Wright function $\Psi _{\alpha}$, we have

$$ \begin{aligned} \bigl\Vert S_{\alpha}(t)\varphi \bigr\Vert _{L^{r}(\Omega )}&\leq \int _{0}^{ \infty}\Psi _{\alpha}(s) \bigl\Vert e^{(k_{0}+\chi _{0})st^{\alpha}A}\varphi \bigr\Vert _{L^{r}( \Omega )}\,ds \\ &\leq Ct^{-\frac{\alpha n}{2}(\frac{1}{q}-\frac{1}{r})} \int _{0}^{ \infty}\Psi _{\alpha}(s)s^{-\frac{n}{2}(\frac{1}{q}-\frac{1}{r})} \,ds \Vert \varphi \Vert _{L^{q}(\Omega )} \\ &=Ct^{-\frac{\alpha n}{2}(\frac{1}{q}-\frac{1}{r})} \Vert \varphi \Vert _{L^{q}( \Omega )} \frac{\Gamma (-\frac{n}{2}(\frac{1}{q}-\frac{1}{r})+1)}{\Gamma (-\frac{\alpha n}{2}(\frac{1}{q}-\frac{1}{r})+1)} \\ &=Ct^{-\frac{\alpha n}{2}(\frac{1}{q}-\frac{1}{r})} \Vert \varphi \Vert _{L^{q}( \Omega )}. \end{aligned} $$

Similarly, we have

$$ \begin{aligned} \bigl\Vert T_{\alpha}(t)\varphi \bigr\Vert _{L^{r}(\Omega )}&\leq t^{ \alpha -1} \int _{0}^{\infty}\alpha s\Psi _{\alpha}(s) \bigl\Vert e^{(k_{0}+ \chi _{0})st^{\alpha}A}\varphi \bigr\Vert _{L^{r}(\Omega )}\,ds \\ &\leq Ct^{-\frac{\alpha n}{2}(\frac{1}{q}-\frac{1}{r})+\alpha -1} \int _{0}^{\infty}\Psi _{\alpha}(s)s^{-\frac{n}{2}(\frac{1}{q}- \frac{1}{r})+1} \,ds \Vert \varphi \Vert _{L^{q}(\Omega )} \\ &=Ct^{-\frac{\alpha n}{2}(\frac{1}{q}-\frac{1}{r})+\alpha -1} \Vert \varphi \Vert _{L^{q}(\Omega )} \frac{\Gamma (-\frac{n}{2}(\frac{1}{q}-\frac{1}{r})+2)}{\Gamma (-\frac{\alpha n}{2}(\frac{1}{q}-\frac{1}{r})+\alpha +1)} \\ &=Ct^{-\frac{\alpha n}{2}(\frac{1}{q}-\frac{1}{r})+\alpha -1} \Vert \varphi \Vert _{L^{q}(\Omega )}. \end{aligned} $$

Now, for $0\leq t_{1}\leq t_{2}$, we have

$$\begin{aligned} \bigl\Vert S_{\alpha}(t_{2})\varphi -S_{\alpha}(t_{1}) \varphi \bigr\Vert _{L^{r}}\leq \int _{0}^{\infty}\Psi _{\alpha}(s) \bigl\Vert e^{(k_{0}+\chi _{0})st_{1}^{ \alpha}A} \bigl[e^{(k_{0}+\chi _{0})s(t_{2}^{\alpha}-t_{1}^{\alpha})A} \varphi -\varphi \bigr] \bigr\Vert _{L^{r}}\,ds, \end{aligned}$$

thus, $\forall s>0$, as $t_{2}\rightarrow t_{1}^{+}$, one gets

$$\begin{aligned} \bigl\Vert e^{(k_{0}+\chi _{0})st_{1}^{\alpha}A} \bigl[e^{(k_{0}+\chi _{0})s(t_{2}^{ \alpha}-t_{1}^{\alpha})A}\varphi -\varphi \bigr] \bigr\Vert _{L^{r}}\leq C \bigl\Vert e^{(k_{0}+ \chi _{0})s(t_{2}^{\alpha}-t_{1}^{\alpha})A}\varphi -\varphi \bigr\Vert _{L^{r}} \rightarrow 0. \end{aligned}$$

Now Lebesgue’s dominated convergence theorem and the fact $\int _{0}^{\infty}\Psi _{\alpha}(s)\,ds=1$ imply that

$$\begin{aligned} \lim_{t_{2}\rightarrow t_{1}^{+}} \int _{0}^{\infty}\Psi _{\alpha}(s) \bigl\Vert e^{(k_{0}+\chi _{0})Ast_{1}^{\alpha}} \bigl[e^{(k_{0}+\chi _{0})As(t_{2}^{ \alpha}-t_{1}^{\alpha})}\varphi -\varphi \bigr] \bigr\Vert _{L^{r}}\,ds=0, \end{aligned}$$

therefore,

$$ \lim_{t_{2}\rightarrow t_{1}^{+}} \bigl\Vert S_{\alpha}(t_{2}) \varphi -S_{ \alpha}(t_{1})\varphi \bigr\Vert _{L^{r}(\Omega )}=0, $$

so the operator family $\{S_{\alpha}(t)\}_{t\geq 0}$ is strongly continuously in $L^{r}(\Omega )$. This completes the proof of Lemma 2.3. □

Lemma 2.4

There exists a positive constant C depending only on α, T, M, θ, r such that

$$\begin{aligned}& \bigl\Vert S_{\alpha}^{\varepsilon}(t) \bigr\Vert _{\mathcal{L}(X)} \leq C,\quad \forall t\in [0,T], \end{aligned}$$

(2.14)

$$\begin{aligned}& \bigl\Vert T_{\alpha}^{\varepsilon}(t) \bigr\Vert _{\mathcal{L}(X)} \leq C, \quad \forall t\in [0,T]. \end{aligned}$$

(2.15)

Proof

From (i) and (v) in Lemma 2.2, for any $\lambda \in \Sigma _{r,\varphi}$, $t\in [0,T]$, we have

$$\begin{aligned}& \bigl\Vert S_{\alpha}^{\varepsilon}(t) \bigr\Vert _{\mathcal{L}(X)} \leq Ce^{rT}\leq C, \\& \bigl\Vert T_{\alpha}^{\varepsilon}(t) \bigr\Vert _{\mathcal{L}(X)} 1\leq Ce^{rT}\leq C. \end{aligned}$$

This completes the proof of Lemma 2.4. □

Lemma 2.5

Under Assumptions (H1)–(H3), for $p\in [1, +\infty )$, $\alpha +\frac{1}{p}>1$, $\theta _{0}\in (\alpha +\frac{1}{p}, +\infty )$, and $\varepsilon _{1}\in (0, \min \{\varepsilon _{0},\frac{r_{0}}{r}\})$, there exists a positive constant C depending only on α, T, M, θ, r, $\varepsilon _{0}$, $\theta _{0}$, p, $r_{0}$ such that

$$\begin{aligned}& \bigl\Vert S_{\alpha}^{\varepsilon}-S_{\alpha} \bigr\Vert _{L^{p}((0,T),\mathcal{L}(X))} \leq C \bigl(\varepsilon ^{\frac{1}{p}}+\varepsilon ^{\theta _{0}}+ \bigl\vert \chi ( \varepsilon )-\chi _{0} \bigr\vert \bigr), \quad \forall \varepsilon \in (0, \varepsilon _{1}], \end{aligned}$$

(2.16)

$$\begin{aligned}& \bigl\Vert T_{\alpha}^{\varepsilon}-T_{\alpha} \bigr\Vert _{L^{p}((0,T),\mathcal{L}(X))} \leq C \bigl(\varepsilon ^{\frac{1}{p}+\alpha -1}+\varepsilon ^{\theta _{0}}+ \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert \bigr), \quad \forall \varepsilon \in (0, \varepsilon _{1}]. \end{aligned}$$

(2.17)

Proof

From Lemma 2.1, we have

$$ \begin{aligned} & \bigl\Vert \bigl(\lambda ^{\alpha}I- \bigl( \chi (\varepsilon )+ \widehat{k}(\varepsilon \lambda ) \bigr)A \bigr)^{-1}- \bigl(\lambda ^{\alpha}I-(k_{0}+ \chi _{0})A \bigr)^{-1} \bigr\Vert _{\mathcal{L}(X)} \\ &\quad \leq \bigl\Vert \bigl\{ \lambda ^{\alpha}I- \bigl(\chi (\varepsilon )+ \widehat{k}( \varepsilon \lambda ) \bigr)A \bigr\} ^{-1} \bigl\{ \bigl( \chi (\varepsilon )-\chi _{0} \bigr)+ \bigl( \widehat{k}(\varepsilon \lambda )-k_{0} \bigr) \bigr\} \bigr\Vert _{\mathcal{L}(X)} \\ &\qquad {}\times \bigl\Vert A \bigl(\lambda ^{\alpha}I-(k_{0}+\chi _{0})A \bigr)^{-1} \bigr\Vert _{\mathcal{L}(X)} \\ &\quad \leq \frac{M}{ \vert \lambda \vert ^{\alpha}} \bigl( \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert + \bigl\vert \widehat{k}(\varepsilon \lambda )-k_{0} \bigr\vert \bigr) \bigl\Vert A \bigl(\lambda ^{\alpha}-(k_{0}+ \chi _{0})A \bigr)^{-1} \bigr\Vert _{\mathcal{L}(X)} \\ &\quad \leq \frac{M}{ \vert \lambda \vert ^{\alpha}} \bigl( \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert + \bigl\vert \widehat{k}(\varepsilon \lambda )-k_{0} \bigr\vert \bigr) \bigl\Vert I-\lambda ^{\alpha} \bigl( \lambda ^{\alpha}-(k_{0}+\chi _{0})A \bigr)^{-1} \bigr\Vert _{\mathcal{L}(X)} \\ &\quad \leq \frac{M(1+C_{0})}{ \vert \lambda \vert ^{\alpha}(k_{0}+\chi _{0})} \bigl( \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert + \bigl\vert \widehat{k}(\varepsilon \lambda )-k_{0} \bigr\vert \bigr), \end{aligned} $$

then

$$ \begin{aligned} & \bigl\Vert S_{\alpha}^{\varepsilon}(t)-S_{\alpha}(t) \bigr\Vert _{ \mathcal{L}(X)} \\ &\quad \leq \frac{1}{2\pi} \biggl\Vert \int _{\Gamma _{r,\theta}}e^{\lambda t} \lambda ^{\alpha -1} \bigl\{ \bigl(\lambda ^{\alpha} I- \bigl(\chi (\varepsilon )+ \widehat{k}(\varepsilon \lambda ) \bigr)A \bigr)^{-1}- \bigl(\lambda ^{\alpha}-(k_{0}+ \chi _{0})A \bigr)^{-1} \bigr\} \,d\lambda \biggr\Vert _{\mathcal{L}(X)} \\ &\quad \leq \frac{M(1+C_{0})}{2\pi (k_{0}+\chi _{0})} \int _{\Gamma _{r, \theta}}\frac{e^{t\operatorname{Re}\lambda}}{ \vert \lambda \vert } \bigl( \bigl\vert \chi ( \varepsilon )- \chi _{0} \bigr\vert + \bigl\vert \widehat{k}( \varepsilon \lambda )-k_{0} \bigr\vert \bigr)\,d\lambda , \end{aligned} $$

and

$$ \begin{aligned} & \bigl\Vert T_{\alpha}^{\varepsilon}(t)-T_{\alpha}(t) \bigr\Vert _{ \mathcal{L}(X)} \\ &\quad \leq \frac{1}{2\pi} \biggl\Vert \int _{\Gamma _{r,\theta}}e^{\lambda t} \bigl\{ \bigl[\lambda ^{\alpha} I- \bigl(\chi (\varepsilon )+\widehat{k}( \varepsilon \lambda ) \bigr)A \bigr]^{-1}- \bigl[\lambda ^{\alpha}-(k_{0}+ \chi _{0})A \bigr]^{-1} \bigr\} \,d\lambda \biggr\Vert _{\mathcal{L}(X)} \\ &\quad \leq \frac{M(1+C_{0})}{2\pi (k_{0}+\chi _{0})} \int _{\Gamma _{r, \theta}}\frac{e^{t\operatorname{Re}\lambda}}{ \vert \lambda \vert ^{\alpha}} \bigl( \bigl\vert \chi ( \varepsilon )-\chi _{0} \bigr\vert + \bigl\vert \widehat{k}( \varepsilon \lambda )-k_{0} \bigr\vert \bigr)\,d\lambda . \end{aligned} $$

Hence, we have

$$ \begin{aligned} & \bigl\Vert S_{\alpha}^{\varepsilon}-S_{\alpha} \bigr\Vert _{L^{p}((0,T),X)} \\ &\quad \leq \frac{M(1+C_{0})}{2\pi (k_{0}+\chi _{0})} \int _{\Gamma _{r, \theta}} \frac{ \vert \chi (\varepsilon )-\chi _{0} \vert + \vert \widehat{k}(\varepsilon \lambda )-k_{0} \vert }{ \vert \lambda \vert } \biggl( \int _{0}^{+\infty}e^{tp\operatorname{Re}(\lambda )}\,dt \biggr)^{\frac{1}{p}}\,d\lambda \\ &\quad =p^{-\frac{1}{p}}\frac{M(1+C_{0})}{2\pi (k_{0}+\chi _{0})} \sum_{j=1}^{3} \int _{\Gamma _{j}(r,\theta )} \frac{ \vert \chi (\varepsilon )-\chi _{0} \vert + \vert \widehat{k}(\varepsilon \lambda )-k_{0} \vert }{ \vert \operatorname{Re}(\lambda ) \vert ^{\frac{1}{p}} \vert \lambda \vert }\,d\lambda \end{aligned} $$

(2.18)

and

$$ \begin{aligned} & \bigl\Vert T_{\alpha}^{\varepsilon}-T_{\alpha} \bigr\Vert _{L^{p}((0,T),X)} \\ &\quad \leq \frac{M(1+C_{0})}{2\pi (k_{0}+\chi _{0})} \int _{\Gamma _{r, \theta}} \frac{ \vert \chi (\varepsilon )-\chi _{0} \vert + \vert \widehat{k}(\varepsilon \lambda )-k_{0} \vert }{ \vert \lambda \vert ^{\alpha}} \biggl( \int _{0}^{+\infty}e^{\operatorname{Re}(\lambda )}\,dt \biggr)^{\frac{1}{p}}\,d\lambda \\ &\quad =p^{-\frac{1}{p}}\frac{M(1+C_{0})}{2\pi (k_{0}+\chi _{0})}\sum_{j=1}^{3} \int _{\Gamma _{j}(r,\theta )} \frac{ \vert \chi (\varepsilon )-\chi _{0} \vert + \vert \widehat{k}(\varepsilon \lambda )-k_{0} \vert }{ \vert \operatorname{Re}(\lambda ) \vert ^{\frac{1}{p}} \vert \lambda \vert ^{\alpha}}\,d\lambda . \end{aligned} $$

(2.19)

We first estimate the integrals over $\Gamma _{j}(r,\theta )$, $j=1,3$. From Assumption (H2), there exists a positive constant C such that for $\Gamma _{1}(r,\theta )=\{te^{i\theta}: t\geq r\}$ and $\Gamma _{3}(r,\theta )=\{te^{-i\theta}:t\geq r\}$, we have

$$ \begin{aligned} \int _{\Gamma _{j}(r,\theta )} \frac{ \vert \widehat{k}(\varepsilon \lambda )-k_{0} \vert }{ \vert \operatorname{Re}\lambda \vert ^{\frac{1}{p}} \vert \lambda \vert }\,d\lambda &\leq \cos ^{-\frac{1}{p}}(\theta )\varepsilon ^{\frac{1}{p}} \int _{\varepsilon r}^{+\infty} \frac{ \vert \widehat{k}(se^{i\theta})-k_{0} \vert }{s^{1+\frac{1}{p}}}\,ds \\ &\leq C\cos ^{-\frac{1}{p}}(\theta )\varepsilon ^{\frac{1}{p}} \biggl( \int _{0}^{r_{0}}s^{\theta _{0}-1-\frac{1}{p}}\,ds+ \int _{r_{0}}^{+ \infty}s^{-1-\frac{1}{p}}\,ds \biggr) \\ &=C\varepsilon ^{\frac{1}{p}}, \end{aligned} $$

(2.20)

where $I_{1}={\int _{0}^{r_{0}}s^{\theta _{0}-1-\frac{1}{p}}\,ds}$ converges for $\theta _{0}>\frac{1}{p}$, and $I_{2}={\int _{r_{0}}^{+\infty}s^{-1-\frac{1}{p}}\,ds}$ converges for $p\geq 1$.

Similarly, there exists a positive constant C such that

$$\begin{aligned} & \int _{\Gamma _{j}(r,\theta )} \frac{ \vert \widehat{k}(\varepsilon \lambda )-k_{0} \vert }{ \vert \operatorname{Re}\lambda \vert ^{\frac{1}{p}} \vert \lambda \vert ^{\alpha}}\,d\lambda \\ \begin{aligned} &\quad \leq \cos ^{-\frac{1}{p}}(\theta )\varepsilon ^{\frac{1}{p}+\alpha -1} \int _{\varepsilon r}^{+\infty} \frac{ \vert \widehat{k}(se^{i\theta})-k_{0} \vert }{s^{\alpha +\frac{1}{p}}}\,ds \\ &\quad \leq C\cos ^{-\frac{1}{p}}(\theta )\varepsilon ^{\frac{1}{p}+\alpha -1} \biggl( \int _{0}^{r_{0}}s^{\theta _{0}-1-\frac{1}{p}-\alpha}\,ds+ \int _{r_{0}}^{+ \infty}s^{-1-\frac{1}{p}-\alpha}\,ds \biggr)\end{aligned} \\ &\quad =C\varepsilon ^{\frac{1}{p}+\alpha -1}, \end{aligned}$$

(2.21)

where $I_{3}={\int _{0}^{r_{0}}s^{\theta _{0}-1-\frac{1}{p}- \alpha}\,ds}$ converges for $\theta _{0}>\frac{1}{p}+\alpha$ and $I_{4}={\int _{r_{0}}^{+\infty}s^{-1-\frac{1}{p}-\alpha}\,ds}$ converges for $p\geq 1$ and $\alpha \in (0,1)$.

For the function $\chi (\varepsilon )$, we can also get

$$ \int _{\Gamma _{j}(r,\theta )} \frac{ \vert \chi (\varepsilon )-\chi _{0} \vert }{ \vert \operatorname{Re}\lambda \vert ^{\frac{1}{p}} \vert \lambda \vert }\,d\lambda = \int _{r}^{+\infty} \frac{ \vert \chi (\varepsilon )-\chi \vert }{s^{1+\frac{1}{p}}\cos ^{\frac{1}{p}}(\theta )}\,ds \leq C \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert r^{-\frac{1}{p}} $$

(2.22)

and

$$ \int _{\Gamma _{j}(r,\theta )} \frac{ \vert \chi (\varepsilon )-\chi _{0} \vert }{ \vert \operatorname{Re}\lambda \vert ^{\frac{1}{p}} \vert \lambda \vert ^{\alpha}}\,d\lambda = \int _{r}^{+\infty} \frac{ \vert \chi (\varepsilon )-\chi \vert }{s^{\alpha +\frac{1}{p}}\cos ^{\frac{1}{p}}(\theta )}\,ds \leq C \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert r^{1-\alpha -\frac{1}{p}}, $$

(2.23)

where C is a positive constant.

From (2.20)–(2.23), we get the estimates of the integrals over $\Gamma _{j}(r,\theta )$ ($j=1,3$):

$$\begin{aligned}& \int _{\Gamma _{j}(r,\theta )} \frac{ \vert \chi (\varepsilon )-\chi _{0} \vert + \vert \widehat{k}(\varepsilon \lambda )-k_{0} \vert }{ \vert \operatorname{Re}\lambda \vert ^{\frac{1}{p}} \vert \lambda \vert }\,d\lambda \leq C\varepsilon ^{\frac{1}{p}}+C \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert r^{- \frac{1}{p}}, \end{aligned}$$

(2.24)

$$\begin{aligned}& \int _{\Gamma _{j}(r,\theta )} \frac{ \vert \chi (\varepsilon )-\chi _{0} \vert + \vert \widehat{k}(\varepsilon \lambda )-k_{0} \vert }{ \vert \operatorname{Re}\lambda \vert ^{\frac{1}{p}} \vert \lambda \vert ^{\alpha}}\,d\lambda \leq C\varepsilon ^{\frac{1}{p}+\alpha -1}+C \bigl\vert \chi ( \varepsilon )-\chi _{0} \bigr\vert r^{1-\alpha -\frac{1}{p}}. \end{aligned}$$

(2.25)

We now estimate the integrals over $\Gamma _{2}(r,\theta )$ in (2.18) and (2.19). Choosing $\varepsilon \in (0,\varepsilon _{1}]$ with $\varepsilon _{1}\leq \min \{\varepsilon _{0},\frac{r_{0}}{r}\}$ and making use of assumptions (iv) and (v) in Lemma 2.2, we have

$$ \begin{aligned} & \int _{\Gamma _{2}(r,\theta )} \frac{ \vert \chi (\varepsilon )-\chi _{0} \vert + \vert \widehat{k}(\varepsilon \lambda )-k_{0} \vert }{ \vert \operatorname{Re}\lambda \vert ^{\frac{1}{p}} \vert \lambda \vert }\,d\lambda \\ &\quad = \int _{\Gamma _{2}(r,\theta )} \frac{ \vert \widehat{k}(\varepsilon \lambda )-k_{0} \vert }{ \vert \operatorname{Re}\lambda \vert ^{\frac{1}{p}} \vert \lambda \vert }\,d\lambda + \int _{\Gamma _{2}(r,\theta )} \frac{ \vert \chi (\varepsilon )-\chi _{0} \vert }{ \vert \operatorname{Re}\lambda \vert ^{\frac{1}{p}} \vert \lambda \vert }\,d\lambda \\ &\quad \leq r^{-\frac{1}{p}} \biggl[ \int _{-\theta}^{\theta} \frac{ \vert \widehat{k}(\varepsilon re^{i\xi})-k_{0} \vert }{\cos ^{\frac{1}{p}}(\xi )}\,d\xi + \int _{-\theta}^{\theta} \frac{ \vert \chi (\varepsilon )-\chi _{0} \vert }{\cos ^{\frac{1}{p}}(\xi )}\,d\xi \biggr] \\ &\quad \leq Cr^{-\frac{1}{p}} \biggl[(\varepsilon r)^{\theta _{0}} \int _{- \theta}^{\theta}\cos ^{-\frac{1}{p}}(\xi )\,d\xi + \bigl\vert \chi (\varepsilon )- \chi _{0} \bigr\vert \int _{-\theta}^{\theta}\cos ^{-\frac{1}{p}}(\xi )\,d\xi \biggr] \\ &\quad \leq C \bigl(\varepsilon ^{\theta _{0}}+ \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert \bigr), \end{aligned} $$

(2.26)

and

$$\begin{aligned}& \int _{\Gamma _{2}(r,\theta )} \frac{ \vert \chi (\varepsilon )-\chi _{0} \vert + \vert \widehat{k}(\varepsilon \lambda )-k_{0} \vert }{ \vert \operatorname{Re}\lambda \vert ^{\frac{1}{p}} \vert \lambda \vert ^{\alpha}}\,d\lambda \\& \quad = \int _{\Gamma _{2}(r,\theta )} \frac{ \vert \widehat{k}(\varepsilon \lambda )-k_{0} \vert }{ \vert \operatorname{Re}\lambda \vert ^{\frac{1}{p}} \vert \lambda \vert ^{\alpha}}\,d\lambda + \int _{\Gamma _{2}(r,\theta )} \frac{ \vert \chi (\varepsilon )-\chi _{0} \vert }{ \vert \operatorname{Re}\lambda \vert ^{\frac{1}{p}} \vert \lambda \vert ^{\alpha}}\,d\lambda \\& \quad \leq r^{1-\alpha +\frac{1}{p}} \biggl[ \int _{-\theta}^{\theta} \frac{ \vert \widehat{k}(\varepsilon re^{i\xi})-k_{0} \vert }{\cos ^{\frac{1}{p}}(\xi )}\,d\xi + \int _{-\theta}^{\theta} \frac{ \vert \chi (\varepsilon )-\chi _{0} \vert }{\cos ^{\frac{1}{p}}(\xi )}\,d\xi \biggr] \\& \quad \leq Cr^{1-\alpha +\frac{1}{p}}(\varepsilon r)^{\theta _{0}} \int _{- \theta}^{\theta}\cos ^{-\frac{1}{p}}(\xi )\,d\xi + \bigl\vert \chi (\varepsilon )- \chi _{0} \bigr\vert \int _{-\theta}^{\theta}\cos ^{-\frac{1}{p}}(\xi )\,d\xi \\& \quad \leq C \bigl(\varepsilon ^{\theta _{0}}+ \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert \bigr). \end{aligned}$$

(2.27)

From (2.18), (2.24), and (2.26), we get the estimate (2.16); and from (2.19), (2.25), and (2.27), we get the estimate (2.17). This completes the proof of Lemma 2.5. □

Lemma 2.6

If $f_{\varepsilon}$ and f satisfy Assumption (H4), and $\frac{1}{p}+\alpha >1$ ($p\geq 1$), then

$$ \begin{aligned} & \bigl\Vert T_{\alpha}^{\varepsilon} \ast f_{\varepsilon}-T_{\alpha}\ast f \bigr\Vert _{L^{r}((0,T),X)} \\ &\quad \leq C \Vert f_{\varepsilon}-f \Vert _{C([0,T],X)}+C \bigl(\varepsilon ^{\frac{1}{p}+ \alpha -1}+\varepsilon ^{\theta _{0}}+ \bigl\vert \chi (\varepsilon )- \chi _{0} \bigr\vert \bigr) \Vert f \Vert _{C([0,T],X)}, \end{aligned} $$

where C is a positive constant depending only on α, T, M, θ, r, $\varepsilon _{0}$, $\theta _{0}$, p, $r_{0}$, and

$$ \frac{1}{r}= \textstyle\begin{cases} \frac{1}{p}+\frac{1}{q}-1, &\textit{if } \frac{1}{p}+\frac{1}{q}\geq 1, p \geq 1,q\geq 1, \\ \frac{1}{p}, &\textit{if } \frac{1}{p}+\frac{1}{q}< 1, p>1,q>1, \end{cases} $$

the convolution is defined by ${(T\ast f)(t)=\int _{0}^{t}T(t-s)f(s)\,ds,}$ and we have

$$\begin{aligned} T_{\alpha}^{\varepsilon}\ast f_{\varepsilon}\rightarrow T_{\alpha} \ast f \quad \textit{in } L^{r} \bigl((0,T);X \bigr) \textit{ as } \varepsilon \rightarrow 0^{+}. \end{aligned}$$

(2.28)

Proof

From the identity

$$\begin{aligned} T_{\alpha}^{\varepsilon}\ast f_{\varepsilon}-T_{\alpha}\ast f=T_{ \alpha}^{\varepsilon}\ast (f_{\varepsilon}-f)+ \bigl(T_{\alpha}^{ \varepsilon}-T_{\alpha} \bigr)\ast f, \end{aligned}$$

we have

$$ \begin{aligned} & \bigl\Vert T_{\alpha}^{\varepsilon} \ast f_{\varepsilon}-T_{\alpha}\ast f \bigr\Vert _{L^{r}((0,T),X)} \\ &\quad = \bigl\Vert T_{\alpha}^{\varepsilon}\ast (f_{\varepsilon}-f)+ \bigl(T_{\alpha}^{ \varepsilon}-T_{\alpha} \bigr)\ast f \bigr\Vert _{L^{r}((0,T),X)} \\ &\quad \leq \bigl\Vert T_{\alpha}^{\varepsilon}\ast (f_{\varepsilon}-f) \bigr\Vert _{L^{r}((0,T),X)}+ \bigl\Vert \bigl(T_{\alpha}^{\varepsilon}-T_{\alpha} \bigr)\ast f \bigr\Vert _{L^{r}((0,T),X)}. \end{aligned} $$

(2.29)

For $\frac{1}{p}+\frac{1}{q}\geq 1$ ($p\geq 1$, $q\geq 1$), we can use Young’s inequality for convolution, together with Minkowski’s inequality and Lemmas 2.4 and 2.5, to obtain

$$ \begin{aligned} & \bigl\Vert T_{\alpha}^{\varepsilon}\ast f_{\varepsilon}-T_{ \alpha}\ast f \bigr\Vert _{L^{r}((0,T),X)} \\ &\quad \leq \bigl\Vert T_{\alpha}^{\varepsilon} \bigr\Vert _{L^{p}((0,T),X)} \Vert f_{\varepsilon}-f \Vert _{L^{q}((0,T),X)}+ \bigl\Vert T_{\alpha}^{\varepsilon}-T_{\alpha} \bigr\Vert _{L^{p}((0,T),X)} \Vert f \Vert _{L^{q}((0,T),X)} \\ &\quad \leq C \bigl\{ \Vert f_{\varepsilon}-f \Vert _{C([0,T],X)}+ \bigl( \varepsilon ^{ \frac{1}{p}+\alpha -1}+\varepsilon ^{\theta _{0}}+ \bigl\vert \chi ( \varepsilon )- \chi _{0} \bigr\vert \bigr) \Vert f \Vert _{C([0,T],X)} \bigr\} . \end{aligned} $$

For $\frac{1}{p}+\frac{1}{q}<1$ ($p>1$, $q>1$), we first choose $r=p$ in (2.29) and then use Young’s inequality for convolution, together with Lemmas 2.4 and 2.5, to get

$$ \begin{aligned} & \bigl\Vert T_{\alpha}^{\varepsilon}\ast f_{\varepsilon}-T_{ \alpha}\ast f \bigr\Vert _{L^{p}((0,T),X)} \\ &\quad \leq \bigl\Vert T_{\alpha}^{\varepsilon} \bigr\Vert _{L^{p}((0,T),X)} \Vert f_{\varepsilon}-f \Vert _{L^{1}((0,T),X)}+ \bigl\Vert T_{\alpha}^{\varepsilon}-T_{\alpha} \bigr\Vert _{L^{p}((0,T),X)} \Vert f \Vert _{L^{1}((0,T),X)} \\ &\quad \leq C \bigl\{ \Vert f_{\varepsilon}-f \Vert _{C([0,T],X)}+ \bigl( \varepsilon ^{ \frac{1}{p}+\alpha -1}+\varepsilon ^{\theta _{0}}+ \bigl\vert \chi ( \varepsilon )- \chi _{0} \bigr\vert \bigr) \Vert f \Vert _{C([0,T],X)} \bigr\} . \end{aligned} $$

Thus the estimate (2.28) holds true for $\frac{1}{p}+\frac{1}{q}<1$ ($p>1$, $q>1$).

Since $L^{q}((0,T),X)\hookrightarrow L^{1}((0,T),X)$, we easily get the convergence (2.28). This completes the proof of Lemma 2.6. □

Lemma 2.7

If $u_{0,\varepsilon}$ and $u_{0}$ satisfy Assumption (H5), then for $p\geq 1$ and for any $\varepsilon \in (0,\varepsilon _{1}]$, we have

$$ \bigl\Vert S_{\alpha}^{\varepsilon}u_{0,\varepsilon}-S_{\alpha}u_{0} \bigr\Vert _{L^{p}((0,T),X)} \leq C \bigl\{ \Vert u_{0,\varepsilon}-u_{0} \Vert + \bigl(\varepsilon ^{ \frac{1}{p}}+\varepsilon ^{\theta _{0}}+ \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert \bigr) \Vert u_{0} \Vert \bigr\} , $$

(2.30)

for some positive constant C depending only on α, T, M, θ, r, $\varepsilon _{0}$, $\theta _{0}$, p, $r_{0}$. Moreover,

$$\begin{aligned} S_{\alpha}^{\varepsilon}u_{0,\varepsilon}\rightarrow S_{\alpha}u_{0} \quad \textit{in } L^{p} \bigl((0,T),X \bigr) \textit{ as } \varepsilon \rightarrow 0^{+}. \end{aligned}$$

(2.31)

Proof

Due to the identity $S_{\alpha}^{\varepsilon}u_{0,\varepsilon}-S_{\alpha}u_{0}=S_{\alpha}^{ \varepsilon}(u_{0,\varepsilon}-u_{0})+(S_{\alpha}^{\varepsilon}-S_{ \alpha})u_{0}$, from Minkowski’s inequality and Lemmas 2.5 and 2.6, we obtain that

$$ \begin{aligned} & \bigl\Vert S_{\alpha}^{\varepsilon}u_{0,\varepsilon}-S_{\alpha}u_{0} \bigr\Vert _{L^{p}((0,T),X)} \\ &\quad = \bigl\Vert S_{\alpha}^{\varepsilon}(u_{0,\varepsilon}-u_{0})+ \bigl(S_{\alpha}^{ \varepsilon}-S_{\alpha} \bigr)u_{0} \bigr\Vert _{L^{p}((0,T),X)} \\ &\quad \leq \bigl\Vert S_{\alpha}^{\varepsilon}(u_{0,\varepsilon}-u_{0}) \bigr\Vert _{L^{p}((0,T),X)}+ \bigl\Vert \bigl(S_{\alpha}^{\varepsilon}-S_{\alpha} \bigr)u_{0} \bigr\Vert _{L^{p}((0,T),X)} \\ &\quad \leq \bigl\Vert S_{\alpha}^{\varepsilon} \bigr\Vert _{L^{p}((0,T),X)} \Vert u_{0,\varepsilon}-u_{0} \Vert + \bigl\Vert S_{\alpha}^{\varepsilon}-S_{\alpha} \bigr\Vert _{L^{p}((0,T),X)} \Vert u_{0} \Vert \\ &\quad \leq C \bigl\{ \Vert u_{0,\varepsilon}-u_{0} \Vert + \bigl( \varepsilon ^{ \frac{1}{p}}+\varepsilon ^{\theta _{0}}+ \bigl\vert \chi _{0}-\chi (\varepsilon ) \bigr\vert \bigr) \Vert u_{0} \Vert \bigr\} . \end{aligned} $$

(2.32)

This completes the proof of Lemma 2.7. □

3 Statement of the problems and main results

Now we consider the existence of solutions $u_{\varepsilon}$ and u to problem (1.1) with a small parameter ε and the limit problem (1.7) as $\varepsilon \rightarrow 0^{+}$, respectively, and then we focus on the convergence of $u_{\varepsilon}$ to u in the space $X_{\rho}$, where

$$\begin{aligned} X_{\rho}= \Bigl\{ u\in C( \bigl(0,T], L^{\rho q}( \Omega ) \bigr): \sup_{t\in (0,T]}t^{ \frac{\alpha n}{2\rho q}(\rho -1)} \bigl\Vert u(t) \bigr\Vert _{L^{\rho q}(\Omega )}< \infty \Bigr\} \end{aligned}$$

(3.1)

is endowed with the norm

$$\begin{aligned} \Vert u \Vert _{X_{\rho}}=\sup_{t\in (0,T]}t^{\frac{\alpha n}{2\rho q}(\rho -1)} \bigl\Vert u(t) \bigr\Vert _{L^{\rho q}(\Omega )}. \end{aligned}$$

(3.2)

Define the space

$$\begin{aligned} Y_{\rho}= \bigl\{ u\in L^{p} \bigl((0,T), L^{\rho q}(\Omega ) \bigr): \bigl\Vert t^{ \frac{\alpha n}{2\rho q}(\rho -1)}u(t,x) \bigr\Vert _{L^{p}((0,T),L^{\rho q}( \Omega ))}< \infty \bigr\} \end{aligned}$$

(3.3)

endowed with the norm

$$\begin{aligned} \Vert u \Vert _{Y_{\rho}}= \bigl\Vert t^{\frac{\alpha n}{2\rho q}(\rho -1)}u(t,x) \bigr\Vert _{L^{p}((0,T), L^{\rho q}(\Omega ))}. \end{aligned}$$

(3.4)

We now state our main results.

Theorem 3.1

Suppose the operator A defined in equation (1.5) satisfies Assumptions (H1), (H3), the coefficient χ satisfies the convergence condition (1.4), the convolution kernel k satisfies Assumption (H2), and $\rho \geq 1$, $q\geq 1$, $q>\frac{\alpha n}{2}(\rho -1)$, $\alpha \in (\frac{1}{\rho}, 1]$. Then

• (i) There exist constants $T>0$ and $R>0$ such that the problem (1.1) admits an $L^{q}$-mild solution $u_{\varepsilon}: [0,T]\rightarrow L^{q}(\Omega )$ in the ball $\mathbf{B}_{R}$ which is unique in $C([0,T], L^{q}(\Omega ))\cap X_{\rho}$, for any $(f_{\varepsilon},u_{0,\varepsilon})\in \mathrm{C}([0,T], L^{\rho q}( \Omega ))\times L^{\rho q}(\Omega )$, $\varepsilon \in [0,\varepsilon _{0}]$.

• (ii) For constants $T>0$ and $R>0$, in the ball $\mathbf{B}_{R}$ the problem (1.7) admits an $L^{q}$-mild solution $u: [0,T]\rightarrow L^{q}(\Omega )$ which is unique in $C([0,T], L^{q}(\Omega ))\cap X_{\rho}$, for any $(f,u_{0})\in \mathrm{C}([0,T], L^{q}(\Omega ))\times L^{q}(\Omega )$.

• (iii) The solutions $u_{\varepsilon}$, u depend continuously on the initial data, that is, if $u_{\varepsilon}$, $v_{\varepsilon}$ and u, v are solutions of problems (1.1) and (1.7) starting from $u_{0,\varepsilon}$, $v_{0,\varepsilon}$ and $u_{0}$, $v_{0}$, respectively, then

  $$\begin{aligned}& \sup_{t\in (0,T]}t^{\frac{\alpha n}{2\rho q}(\rho -1)} \bigl\Vert u_{ \varepsilon}(t)-v_{\varepsilon}(t) \bigr\Vert _{L^{\rho q}(\Omega )} \leq C \Vert u_{0, \varepsilon}-v_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}, \end{aligned}$$

  (3.5)
  $$\begin{aligned}& \sup_{t\in (0,T]}t^{\frac{\alpha n}{2\rho q}(\rho -1)} \bigl\Vert u(t)-v(t) \bigr\Vert _{L^{ \rho q}(\Omega )} \leq C \Vert u_{0}-v_{0} \Vert _{L^{q}(\Omega )}. \end{aligned}$$

  (3.6)

Theorem 3.2

Suppose the operator A defined in equation (1.5) satisfies Assumptions (H1), (H3), the nonlinear operator $N: [0,T]\times X\rightarrow X$ defined by equation (1.14) is continuous with respect to t, and there exists a constant $M>0$ such that

$$\begin{aligned}& \bigl\Vert N(t,x) \bigr\Vert \leq M \bigl(1+ \Vert x \Vert ^{\rho -1} \bigr), \quad \forall x,y\in X, \end{aligned}$$

(3.7)

$$\begin{aligned}& \bigl\Vert N(t,x)-N(t,y) \bigr\Vert \leq M \bigl(1+ \Vert x \Vert ^{\rho -1}+ \Vert y \Vert ^{\rho -1} \bigr) \Vert x-y \Vert ,\quad \forall x,y\in X. \end{aligned}$$

(3.8)

Assume that the coefficient χ satisfies the convergence condition (1.4), the convolution kernel k satisfies Assumption (H2), and $\rho \geq 1$, $q\geq 1$, $\alpha >\rho \beta$, $\alpha \in (\frac{1}{\rho}, 1]$, $q>\frac{\alpha n(\rho -1)}{2}$. Then

• (i) There exist constants $T>0$ and $R>0$ such that the problem (1.12) admits an $L^{q}$-mild solution $u_{\varepsilon}:[0,T]\rightarrow L^{q}(\Omega )$ in the ball $\mathbf{B}_{R}$ which is unique in $\mathrm{C}([0,T], L^{q}(\Omega ))\cap X_{\rho}$ for any $(f_{\varepsilon}, u_{0,\varepsilon})\in \mathrm{C}([0,T], L^{\rho q}( \Omega ))\times L^{\rho q}(\Omega )$, $\varepsilon \in [0,\varepsilon _{0}]$.

• (ii) For constants $T>0$ and $R>0$, in the ball $\mathbf{B}_{R}$ the problem (1.13) admits an $L^{q}$-mild solution $u:[0,T]\rightarrow L^{q}(\Omega )$ which is unique in $\mathrm{C}([0,T], L^{q}(\Omega ))\cap X_{\rho}$ for any $(f, u_{0})\in \mathrm{C}([0,T], L^{q}(\Omega ))\times L^{q}(\Omega )$.

• (iii) The solutions $u_{\varepsilon}$ and u depend continuously on the initial data, that is, if $u_{\varepsilon}$, $v_{\varepsilon}$ and u, v are solutions of the problem (1.12) and the problem (1.13) starting from $u_{0,\varepsilon}$, $v_{0,\varepsilon}$ and $u_{0}$, $v_{0}$, respectively, then we have

  $$\begin{aligned}& \sup_{t\in (0,T]}t^{\frac{\alpha n}{2\rho q}(\rho -1)} \bigl\Vert u_{ \varepsilon}(t)-v_{\varepsilon}(t) \bigr\Vert _{L^{\rho q}(\Omega )} \leq C \Vert u_{0, \varepsilon}-v_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}, \end{aligned}$$

  (3.9)
  $$\begin{aligned}& \sup_{t\in (0,T]}t^{\frac{\alpha n}{2\rho q}(\rho -1)} \bigl\Vert u(t)-v(t) \bigr\Vert _{L^{ \rho q}(\Omega )} \leq C \Vert u_{0}-v_{0} \Vert _{L^{q}(\Omega )}. \end{aligned}$$

  (3.10)

In the following, we denote $\beta =\frac{\alpha n}{2\rho q}(\rho -1)$.

Theorem 3.3

Suppose the operator A defined in equation (1.5) satisfies Assumptions (H1), (H3), the coefficient χ satisfies the convergence condition (1.4), the convolution kernel k satisfies Assumption (H2), and $\rho \geq 1$, $p,q\geq 1$, $\alpha \in (\frac{1}{\rho}, 1]$, $\frac{1}{p}+\alpha \geq 1$. If $f_{\varepsilon}$, $f\in C([0,T];L^{\rho q}(\Omega ))$ and $u_{0,\varepsilon}$, $u_{0}\in L^{\rho q}(\Omega )$ are such that

$$\begin{aligned} \Vert f_{\varepsilon}-f \Vert _{C([0,T], L^{\rho q}(\Omega ))}\rightarrow 0 \quad \textit{and}\quad \Vert u_{0,\varepsilon}-u_{0} \Vert _{L^{\rho q}(\Omega )} \rightarrow 0 \quad \textit{as } \varepsilon \rightarrow 0^{+}, \end{aligned}$$

(3.11)

then the mild solution $u_{\varepsilon}$ of the approximating problem (1.1) converges in the space $Y_{\rho}$ to the mild solution u of the limit problem (1.7) as $\varepsilon \rightarrow 0^{+}$. More precisely, there exists a positive constant C such that

$$ \begin{aligned} \Vert u_{\varepsilon}-u \Vert _{Y_{\rho}}\leq{}& C \bigl\{ \Vert u_{0,\varepsilon}-u_{0} \Vert _{L^{\rho q}(\Omega )}+ \Vert f_{\varepsilon}-f \Vert _{C([0,T], L^{\rho q}( \Omega ))} \\ & {} + \bigl(\varepsilon ^{\frac{1}{p}}+\varepsilon ^{\theta _{0}}+ \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert \bigr) \Vert u_{0} \Vert _{L^{\rho q}(\Omega )} \\ & {} + \bigl(\varepsilon ^{\frac{1}{p}+\alpha -1}+\varepsilon ^{ \theta _{0}}+ \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert \bigr) \Vert f \Vert _{C([0,T], L^{ \rho q}(\Omega ))} \bigr\} . \end{aligned} $$

(3.12)

Theorem 3.4

Suppose the operator A defined in equation (1.5) satisfies Assumptions (H1), (H3), the nonlinear operator $N: [0,T]\times X\rightarrow X$ defined by equation (1.14) is continuous with respect to t and satisfies (3.7) and (3.8). Assume that the coefficient χ satisfies the convergence condition (1.4), the convolution kernel k satisfies Assumption (H2), and $\rho \geq 1$, $p, q\geq 1$, $q>\frac{\alpha n(\rho -1)}{2}$, $\alpha \in ( \frac{1}{\rho}, 1]$, $\alpha >\rho \beta$, $\frac{1}{p}+\alpha \geq 1$. If $f_{\varepsilon}$, $f\in C([0,T], L^{\rho q}(\Omega ))$ and $u_{0,\varepsilon}$, $u_{0}\in L^{\rho q}(\Omega )$ are such that

$$\begin{aligned} \Vert f_{\varepsilon}-f \Vert _{C([0,T], L^{\rho q}(\Omega ))}\rightarrow 0 \quad \textit{and} \quad \Vert u_{0,\varepsilon}-u_{0} \Vert _{L^{\rho q}(\Omega )} \rightarrow 0 \quad \textit{as } \varepsilon \rightarrow 0^{+}, \end{aligned}$$

(3.13)

then the mild solution $u_{\varepsilon}$ of the approximating problem (1.12) converges in the space $Y_{\rho}$ to the mild solution u of the limit problem (1.13) as $\varepsilon \rightarrow 0^{+}$. More precisely, there exists a positive constant C such that

$$ \begin{aligned} \Vert u_{\varepsilon}-u \Vert _{Y_{\rho}}\leq{}& C \bigl\{ \Vert u_{0,\varepsilon}-u_{0} \Vert _{L^{\rho q}(\Omega )}+ \Vert f_{\varepsilon}-f \Vert _{C([0,T],L^{\rho q}( \Omega ))} \\ & {} + \bigl(\varepsilon ^{\frac{1}{p}}+\varepsilon ^{\theta _{0}}+ \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert \bigr) \Vert u_{0} \Vert _{L^{\rho q}(\Omega )} \\ & {} + \bigl(\varepsilon ^{\frac{1}{p}+\alpha -1}+\varepsilon ^{ \theta _{0}}+ \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert \bigr) \bigl( \Vert u_{0} \Vert _{L^{ \rho q}(\Omega )}+ \Vert u_{0} \Vert _{L^{\rho q}(\Omega )}^{\rho} \\ & {} + \Vert f \Vert _{C([0,T],L^{\rho q}(\Omega ))}+ \bigl\Vert f(t) \bigr\Vert _{C([0,T], L^{ \rho q}(\Omega ))}^{\rho} \bigr) \bigr\} . \end{aligned} $$

(3.14)

4 Proofs of Theorems 3.1–3.4

Proof of Theorem 3.1.

For the approximating problem (1.1) with a small positive parameter ε and the limit problem (1.7) as $\varepsilon \rightarrow 0^{+}$, due to Definition 2.3, we can transform them into the integral equations (2.6) and (2.7) by the operator families $\{S_{\alpha}(t)\}_{t\geq 0}$, $\{T_{\alpha}(t)\}_{t\geq 0}$ and the Laplace transform. Direct computation then implies that

$$\begin{aligned}& \rho \beta =\rho \frac{\alpha n}{2\rho q}(\rho -1)= \frac{\alpha n}{2q}(\rho -1)< 1, \\& \frac{n}{2} \biggl(\frac{1}{q}-\frac{1}{\rho q} \biggr) = \frac{n(\rho -1)}{2\rho q}=\frac{\beta}{\alpha}< \frac{1}{\rho \alpha}< 1, \end{aligned}$$

from which, together with $u_{0,\varepsilon}\in L^{\rho q}(\Omega )$ and Lemma 2.2 (i), we obtain

$$ \begin{aligned} \Vert u_{\varepsilon} \Vert _{X_{\rho}}&\leq \bigl\Vert S_{\alpha}^{ \varepsilon}(t)u_{0,\varepsilon} \bigr\Vert _{X_{\rho}}+ \biggl\Vert \int _{0}^{t} \mathrm{T}_{\alpha}^{\varepsilon}(t-s)f_{\varepsilon}(s) \,ds \biggr\Vert _{X_{ \rho}} \\ &=\sup_{t\in (0,T]}t^{\beta} \bigl\Vert S_{\alpha}^{\varepsilon}(t)u_{0, \varepsilon} \bigr\Vert _{L^{\rho q}(\Omega )}+ \sup_{t\in (0,T]}t^{\beta} \biggl\Vert \int _{0}^{t}\mathrm{T}_{\alpha}^{\varepsilon}(t-s)f_{ \varepsilon}(s) \,ds \biggr\Vert _{L^{\rho q}(\Omega )} \\ &\leq \sup_{t\in (0,T]}Ce^{rt}t^{\beta} \Vert u_{0,\varepsilon} \Vert _{L^{ \rho q}(\Omega )}+\sup_{t\in (0,T]}t^{\beta} \int _{0}^{t} \bigl\Vert \mathrm{T}_{\alpha}^{\varepsilon}(t-s)f_{\varepsilon}(s) \bigr\Vert _{L^{\rho q}( \Omega )}\,ds \\ &\leq Ce^{rT}T^{\beta} \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+C \sup_{t\in (0,T]}t^{\beta} \int _{0}^{t}e^{r(t-s)} \bigl\Vert f_{\varepsilon}(s) \bigr\Vert _{L^{\rho q}(\Omega )}\,ds \\ &\leq Ce^{rT}T^{\beta} \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+C \Vert f_{ \varepsilon} \Vert _{C([0,T],L^{\rho q}(\Omega ))}\sup _{t\in (0,T]}e^{rt}t^{ \beta} \int _{0}^{t}e^{-rs}\,ds \\ &\leq Ce^{rT}T^{\beta} \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+ \frac{C}{r}T^{\beta} \bigl(e^{rT}-1 \bigr) \Vert f_{\varepsilon} \Vert _{C([0,T],L^{\rho q}( \Omega ))}\leq R. \end{aligned} $$

Next, we will prove that $u_{\varepsilon}$ is an $L^{q}$-mild solution. Indeed, as $t\rightarrow 0^{+}$, we have

$$ \begin{aligned} & \bigl\Vert u_{\varepsilon}(t)-u_{0,\varepsilon} \bigr\Vert _{L^{q}(\Omega )} \\ &\quad \leq \bigl\Vert S_{\alpha}^{\varepsilon}(t)u_{0,\varepsilon}-u_{0,\varepsilon} \bigr\Vert _{L^{q}(\Omega )}+ \biggl\Vert \int _{0}^{t}\mathrm{T}_{\alpha}^{ \varepsilon}(t-s)f_{\varepsilon}(s) \biggr\Vert _{L^{q}(\Omega )} \\ &\quad \leq C \bigl\Vert S_{\alpha}^{\varepsilon}(t)u_{0,\varepsilon}-u_{0, \varepsilon} \bigr\Vert _{L^{\rho q}(\Omega )}+ \int _{0}^{t} \bigl\Vert \mathrm{T}_{ \alpha}^{\varepsilon}(t-s)f_{\varepsilon}(s) \bigr\Vert _{L^{ q}(\Omega )}\,ds \\ &\quad \leq C \bigl\Vert S_{\alpha}^{\varepsilon}(t)u_{0,\varepsilon}-u_{0, \varepsilon} \bigr\Vert _{L^{\rho q}(\Omega )}+C \int _{0}^{t}e^{r(t-s)} \bigl\Vert f_{ \varepsilon}(s) \bigr\Vert _{L^{ q}(\Omega )}\,ds \\ &\quad \leq C \bigl\Vert S_{\alpha}^{\varepsilon}(t)u_{0,\varepsilon}-u_{0, \varepsilon} \bigr\Vert _{L^{\rho q}(\Omega )}+C \Vert f_{\varepsilon} \Vert _{C([0,T],L^{ \rho q}(\Omega ))} e^{rt} \int _{0}^{t}e^{-rs}\,ds \\ &\quad = C \bigl\Vert S_{\alpha}^{\varepsilon}(t)u_{0,\varepsilon}-u_{0,\varepsilon} \bigr\Vert _{L^{\rho q}(\Omega )}+\frac{C}{r} \bigl(e^{rt}-1 \bigr) \Vert f_{\varepsilon} \Vert _{C([0,T],L^{ \rho q}(\Omega ))}\rightarrow 0. \end{aligned} $$

Therefore, $u_{\varepsilon}\in C([0,T], L^{q}(\Omega ))\cap C((0,T], L^{\rho q}( \Omega ))$ is an $L^{q}$-mild solution of the approximating problem (1.1). In order to obtain the continuous dependence on the initial data of the solutions, we assume that $u_{\varepsilon}$ and $v_{\varepsilon}$ are solutions of the problem (1.1) starting from $u_{0,\varepsilon}, v_{0,\varepsilon}\in L^{\rho q}(\Omega )$, respectively. For $T>0$, we have

$$ \begin{aligned} t^{\beta} \bigl\Vert u_{\varepsilon}(t)-v_{\varepsilon}(t) \bigr\Vert _{L^{ \rho q}(\Omega )}&\leq t^{\beta} \bigl\Vert S_{\alpha}^{\varepsilon}(t)u_{0, \varepsilon}-S_{\alpha}^{\varepsilon}(t)v_{0,\varepsilon} \bigr\Vert _{L^{\rho q}( \Omega )} \\ &\leq Ce^{rT}T^{\beta} \Vert u_{0,\varepsilon}-v_{0,\varepsilon} \Vert _{L^{ \rho q}(\Omega )}\leq C \Vert u_{0,\varepsilon}-v_{0,\varepsilon} \Vert _{L^{ \rho q}(\Omega )}. \end{aligned} $$

Next, we prove the existence of a solution to the linear limit problem (1.7). Using Lemma 2.3 and the integral equation (2.9), we have

$$\begin{aligned} \Vert u \Vert _{X_{\rho}}&\leq \bigl\Vert S_{\alpha}(t)u_{0} \bigr\Vert _{X_{\rho}}+ \biggl\Vert \int _{0}^{t}\mathrm{T}_{\alpha}(t-s)f(s)\,ds \biggr\Vert _{X_{\rho}} \\ &=\sup_{t\in (0,T]}t^{\beta} \bigl\Vert S_{\alpha}(t)u_{0} \bigr\Vert _{L^{\rho q}( \Omega )}+\sup _{t\in (0,T]}t^{\beta} \biggl\Vert \int _{0}^{t}\mathrm{T}_{ \alpha}(t-s)f(s)\,ds \biggr\Vert _{L^{\rho q}(\Omega )} \\ &\leq \sup_{t\in (0,T]}t^{\beta}t^{-\beta} \Vert u_{0} \Vert _{L^{q}(\Omega )}+ \sup_{t\in (0,T]}t^{\beta} \int _{0}^{t} \bigl\Vert \mathrm{T}_{\alpha}(t-s)f(s) \bigr\Vert _{L^{\rho q}(\Omega )}\,ds \\ &\leq C \Vert u_{0} \Vert _{L^{q}(\Omega )}+C\sup _{t\in (0,T]}t^{\beta} \int _{0}^{t}(t-s)^{- \beta +\alpha -1} \bigl\Vert f(s) \bigr\Vert _{L^{ q}(\Omega )}\,ds \\ &\leq C \Vert u_{0} \Vert _{L^{q}(\Omega )}+C \Vert f \Vert _{C([0,T],L^{q}(\Omega ))} \sup_{t\in (0,T]}t^{\beta} \int _{0}^{t}(t-s)^{\alpha -1-\beta}\,ds \\ &\leq C \Vert u_{0} \Vert _{L^{q}(\Omega )}+CT^{\alpha} \Vert f \Vert _{C([0,T];L^{q}( \Omega ))}\leq R. \end{aligned}$$

The contraction principle implies the existence of the mild solution u in the ball $\mathbf{B}_{R}$. We will prove that u is an $L^{q}$-mild solution and is unique in $X_{\rho}$. Indeed, as $t\rightarrow 0^{+}$, we have

$$ \begin{aligned} \bigl\Vert u(t)-u_{0} \bigr\Vert _{L^{q}(\Omega )}&\leq \bigl\Vert S_{\alpha}(t)u_{0}-u_{0} \bigr\Vert _{L^{q}(\Omega )}+ \int _{0}^{t} \bigl\Vert \mathrm{T}_{\alpha}(t-s)f(s) \bigr\Vert _{L^{q}( \Omega )}\,ds \\ &\leq \bigl\Vert S_{\alpha}(t)u_{0}-u_{0} \bigr\Vert _{L^{q}(\Omega )}+ \int _{0}^{t}(t-s)^{ \alpha -1} \bigl\Vert f(s) \bigr\Vert _{L^{q}(\Omega )}\,ds \\ &\leq \bigl\Vert S_{\alpha}(t)u_{0}-u_{0} \bigr\Vert _{L^{q}(\Omega )}+C \Vert f \Vert _{C([0,T],L^{q}( \Omega ))} \int _{0}^{t}(t-s)^{\alpha -1}\,ds \\ &= \bigl\Vert S_{\alpha}(t)u_{0}-u_{0} \bigr\Vert _{L^{q}(\Omega )}+\frac{C}{\alpha}t^{ \alpha} \Vert f \Vert _{C([0,T],L^{q}(\Omega ))}\rightarrow 0. \end{aligned} $$

Therefore, $u\in \mathrm{C}([0,T], L^{q}(\Omega ))\cap \mathrm{C}((0,T], L^{ \rho q}(\Omega ))$ is an $L^{q}$-mild solution of the linear limit problem (1.7). In order to obtain the continuous dependence on the initial data of the solutions, we assume that u and v are solutions of the problem (1.7) starting from $u_{0}, v_{0}\in L^{q}(\Omega )$, respectively. For $T>0$, we have

$$ \begin{aligned} t^{\beta} \bigl\Vert u(t)-v(t) \bigr\Vert _{L^{\rho q}}&\leq t^{\beta} \bigl\Vert S_{ \alpha}(t)u_{0}-S_{\alpha}(t)v_{0} \bigr\Vert _{L^{\rho q}(\Omega )} \\ &\leq Ct^{\beta}t^{-\frac{\alpha n}{2}(\frac{1}{q}-\frac{1}{\rho q})} \Vert u_{0}-v_{0} \Vert _{L^{q}(\Omega )}=C \Vert u_{0}-v_{0} \Vert _{L^{q}(\Omega )}, \end{aligned} $$

finishing the proof of Theorem 3.1. □

Proof of Theorem 3.2.

For $\rho \beta <1$, $\frac{n}{2}(\frac{1}{q}-\frac{1}{\rho q})<1$, we can choose T such that

$$\begin{aligned}& Ce^{rT}T^{\beta} \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )} + \frac{CT^{\beta}(e^{rT}-1)}{r} \bigl[ \Vert f_{\varepsilon} \Vert _{C([0,T],L^{\rho q}( \Omega ))}+M \bigr] \\& \quad {} +\frac{CMR^{\rho}e^{rT}T^{1-\rho \beta +\beta}}{1-\rho \beta}\leq R, \\& C \Vert u_{0} \Vert _{L^{q}(\Omega )} +\frac{CT^{\alpha}}{\alpha -\beta} \Vert f \Vert _{C([0,T];L^{q}( \Omega ))} \\& \quad {} +CMT^{\alpha +\beta} \biggl[\frac{1}{\alpha}+R^{\rho}T^{-\rho \beta} \mathrm{B}(\alpha ,1-\rho \beta ) \biggr]\leq R, \\& \frac{CMe^{rT}T^{\beta}\Gamma (1- \beta )}{r^{1-\beta}} + \frac{2CMR^{\rho -1}e^{rT}T^{\beta}\Gamma (1-\rho \beta )}{r^{1-\rho \beta}} \leq \frac{1}{2}, \\& MCT^{\alpha}\mathrm{B}(\alpha ,1-\beta ) +2R^{\rho -1}T^{\alpha + \beta -\rho \beta} \mathrm{B}(\alpha ,1-\rho \beta )\leq \frac{1}{2}, \end{aligned}$$

where $\Gamma (\alpha )$ and $\mathrm{B}(\alpha ,\beta )$ are the Gamma and Beta functions, respectively. Let $\mathbf{B}_{R}$ be the closed ball in $X_{\rho}$ centered at the origin with radius R. Then, for the nonlinear approximating problem (1.12), define an operator $\Lambda :\mathbf{B}_{R}\rightarrow \mathbf{B}_{R}$ by

$$ \Lambda u_{\varepsilon}(t)=\mathrm{S}_{\alpha}^{\varepsilon}(t)u_{0, \varepsilon}+ \int _{0}^{t}\mathrm{T}_{\alpha}^{\varepsilon}(t-s)f_{ \varepsilon}(s) \,ds+ \int _{0}^{t}\mathrm{T}_{\alpha}^{\varepsilon}(t-s)Nu_{ \varepsilon}(s) \,ds. $$

(4.1)

By the assumptions on the nonlinear operator N, due to Lemma 2.2 and for $u_{\varepsilon}\in X_{\rho}$, we have

$$ \begin{aligned} & \Vert \Lambda u_{\varepsilon} \Vert _{X_{\rho}} \\ &\quad \leq \bigl\Vert \mathrm{S}_{\alpha}^{\varepsilon}(t)u_{0,\varepsilon} \bigr\Vert _{X_{ \rho}}+ \biggl\Vert \int _{0}^{t}\mathrm{T}_{\alpha}^{\varepsilon}(t-s)f_{ \varepsilon}(s) \,ds \biggr\Vert _{X_{\rho}} + \biggl\Vert \int _{0}^{t} \mathrm{T}_{\alpha}^{\varepsilon}(t-s)Nu_{\varepsilon}(s) \,ds \biggr\Vert _{X_{ \rho}} \\ &\quad \leq \sup_{t\in (0,T]}t^{\beta} \bigl\Vert \mathrm{S}_{\alpha}^{\varepsilon}(t)u_{0, \varepsilon} \bigr\Vert _{L^{\rho q}(\Omega )}+\sup_{t\in (0,T]}t^{\beta} \int _{0}^{t} \bigl\Vert \mathrm{T}_{\alpha}^{\varepsilon}(t-s)f_{\varepsilon}(s) \bigr\Vert _{L^{\rho q}(\Omega )}\,ds \\ &\qquad {} +\sup_{t\in [0,T]}t^{\beta} \int _{0}^{t} \bigl\Vert \mathrm{T}_{\alpha}^{ \varepsilon}(t-s)Nu_{\varepsilon}(s) \bigr\Vert _{L^{\rho q}(\Omega )}\,ds \\ &\quad \leq \sup_{t\in (0,T]}t^{\beta}Ce^{rt} \Vert u_{0,\varepsilon} \Vert _{L^{ \rho q}(\Omega )} +\sup_{t\in (0,T]} \int _{0}^{t}t^{\beta} \bigl\Vert \mathrm{T}_{\alpha}^{\varepsilon}(t-s)f_{\varepsilon}(s) \bigr\Vert _{L^{\rho q}( \Omega )}\,ds \\ & \qquad {} +CM\sup_{t\in (0,T]}t^{\beta} \int _{0}^{t}e^{r(t-s)} \bigl(1+ \bigl\Vert u_{ \varepsilon}(s) \bigr\Vert _{L^{\rho q}(\Omega )}^{\rho} \bigr)\,ds, \end{aligned} $$

thus,

$$ \begin{aligned} & \Vert \Lambda u_{\varepsilon} \Vert _{X_{\rho}} \\ &\quad \leq Ce^{rT}T^{\beta} \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+CT^{ \beta} \Vert f_{\varepsilon} \Vert _{C([0,T],L^{\rho q}(\Omega ))}\sup_{t\in (0,T]}e^{rt} \int _{0}^{t}e^{-rs}\,ds \\ & \qquad {} +CM\sup_{t\in (0,T]}e^{rt}t^{\beta} \int _{0}^{t}e^{-rs}\,ds \\ &\qquad {} +CM\sup_{t\in (0,T]}e^{rt}t^{\beta} \int _{0}^{t}e^{-rs}s^{- \rho \beta}s^{\rho \beta} \bigl\Vert u_{\varepsilon}(s) \bigr\Vert _{L^{\rho q}(\Omega )}^{ \rho}\,ds \\ &\quad \leq Ce^{rT}T^{\beta} \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+ \frac{C}{r}T^{\beta} \bigl(e^{rT}-1 \bigr) \Vert f_{\varepsilon} \Vert _{C([0,T],L^{\rho q}( \Omega ))} \\ & \qquad {} +\frac{C}{r}T^{\beta}M \bigl(e^{rT}-1 \bigr)+ \frac{C}{1-\rho \beta}R^{\rho}Me^{rT}T^{1- \rho \beta +\beta}\leq R. \end{aligned} $$

If $u_{\varepsilon}, v_{\varepsilon}\in \mathbf{B}_{R}$, we have

$$\begin{aligned} & \Vert \Lambda u_{\varepsilon}-\Lambda v_{\varepsilon} \Vert _{X_{ \rho}} \\ &\quad \leq \sup_{t\in (0,T]}t^{\beta} \int _{0}^{t} \bigl\Vert \mathrm{T}_{\alpha}^{ \varepsilon}(t-s) \bigl(Nu_{\varepsilon}(s)-Nv_{\varepsilon}(s) \bigr) \bigr\Vert _{L^{ \rho q}(\Omega )} \,ds \\ &\quad \leq MC\sup_{t\in (0,T]}t^{\beta} \int _{0}^{t}e^{r(t-s)} \bigl(1+ \bigl\Vert u_{ \varepsilon}(s) \bigr\Vert _{L^{\rho q}(\Omega )}^{\rho -1}+ \bigl\Vert v_{\varepsilon}(s) \bigr\Vert _{L^{\rho q}(\Omega )}^{\rho -1} \bigr) \\ &\qquad {} \times \bigl\Vert u_{\varepsilon}(s)-v_{\varepsilon}(s) \bigr\Vert _{L^{\rho q}( \Omega )}\,ds \\ &\quad \leq MC \Vert u_{\varepsilon}-v_{\varepsilon} \Vert _{X_{\rho}}\sup _{t\in (0,T]}e^{rt}t^{ \beta} \int _{0}^{t}e^{-rs}s^{-\beta}\,ds \\ &\qquad {} +MC \Vert u_{\varepsilon}-v_{\varepsilon} \Vert _{X_{\rho}}\sup _{t\in (0,T]}e^{rt}t^{ \beta} \int _{0}^{t}e^{-rs} s^{-\beta}s^{-\beta (\rho -1)} \Vert u_{ \varepsilon} \Vert _{X_{\rho}}^{\rho -1}\,ds \\ &\qquad {} +MC \Vert u_{\varepsilon}-v_{\varepsilon} \Vert _{X_{\rho}}\sup _{t\in (0,T]}e^{rt}t^{ \beta} \int _{0}^{t}e^{-rs} s^{-\beta}s^{-\beta (\rho -1)} \Vert v_{ \varepsilon} \Vert _{X_{\rho}}^{\rho -1}\,ds, \end{aligned}$$

thus,

$$ \begin{aligned} & \Vert \Lambda u_{\varepsilon}-\Lambda v_{\varepsilon} \Vert _{X_{ \rho}} \\ &\quad \leq CMe^{rT}T^{\beta}\Gamma (1- \beta )r^{\beta -1} \Vert u_{\varepsilon}-v_{ \varepsilon} \Vert _{X_{\rho}} \\ &\qquad {} + CMe^{rT}T^{\beta}\Gamma (1-\rho \beta )r^{\rho \beta -1} \Vert u_{ \varepsilon}-v_{\varepsilon} \Vert _{X_{\rho}} \Vert u_{\varepsilon} \Vert _{X_{\rho}}^{ \rho -1} \\ &\qquad {} +CMe^{rT}T^{\beta}\Gamma (1-\rho \beta )r^{\rho \beta -1} \Vert u_{ \varepsilon}-v_{\varepsilon} \Vert _{X_{\rho}} \Vert v_{\varepsilon} \Vert _{X_{\rho}}^{ \rho -1} \\ &\quad \leq CMe^{rT}T^{\beta}\Gamma (1-\beta )r^{\beta -1} \Vert u_{\varepsilon}-v_{ \varepsilon} \Vert _{X_{\rho}} \\ & \qquad {} +CMR^{\rho -1}e^{rT}T^{\beta}\Gamma (1-\rho \beta )r^{\rho \beta -1} \Vert u_{\varepsilon}-v_{\varepsilon} \Vert _{X_{\rho}} \\ &\qquad {} +CMR^{\rho -1}e^{rT}T^{\beta}\Gamma (1-\rho \beta )r^{\rho \beta -1} \Vert u_{\varepsilon}-v_{\varepsilon} \Vert _{X_{\rho}} \\ &\quad \leq CMe^{rT}T^{\beta} \Vert u_{\varepsilon}-v_{\varepsilon} \Vert _{X_{\rho}} \bigl(r^{\beta -1}\Gamma (1-\beta )+2R^{\rho -1}r^{\rho \beta -1} \Gamma (1-\rho \beta ) \bigr) \\ &\quad \leq \frac{1}{2} \Vert u_{\varepsilon}-v_{\varepsilon} \Vert _{X_{\rho}}. \end{aligned} $$

This yields that Λ is a contraction operator on $X_{\rho}$. We shall prove that $u_{\varepsilon}$ is an $L^{q}$-mild solution and is unique in $X_{\rho}$. Indeed, we have

$$ \begin{aligned} & \bigl\Vert u_{\varepsilon}(t)-u_{0,\varepsilon} \bigr\Vert _{L^{q}(\Omega )} \\ &\quad \leq \bigl\Vert S_{\alpha}^{\varepsilon}(t)u_{0,\varepsilon}-u_{0,\varepsilon} \bigr\Vert _{L^{q}(\Omega )} \\ &\qquad {}+ \biggl\Vert \int _{0}^{t}\mathrm{T}_{\alpha}^{\varepsilon}(t-s)f_{ \varepsilon}(s) \,ds \biggr\Vert _{L^{q}(\Omega )}+ \biggl\Vert \int _{0}^{t} \mathrm{T}_{\alpha}^{\varepsilon}(t-s)Nu_{\varepsilon}(s) \,ds \biggr\Vert _{L^{q}( \Omega )} \\ &\quad \leq C \bigl\Vert S_{\alpha}^{\varepsilon}(t)u_{0,\varepsilon}-u_{0, \varepsilon} \bigr\Vert _{L^{\rho q}(\Omega )}+C \int _{0}^{t}e^{r(t-s)} \bigl\Vert f_{ \varepsilon}(s) \bigr\Vert _{L^{ q}(\Omega )}\,ds \\ &\qquad {}+ MC \int _{0}^{t}e^{r(t-s)} \bigl(1+ \bigl\Vert u_{\varepsilon}(s) \bigr\Vert _{L^{q}( \Omega )}^{\rho} \bigr)\,ds \\ &\quad \leq C \bigl\Vert S_{\alpha}^{\varepsilon}(t)u_{0,\varepsilon}-u_{0, \varepsilon} \bigr\Vert _{L^{\rho q}(\Omega )}+C \Vert f_{\varepsilon} \Vert _{C([0,T],L^{ \rho q}(\Omega ))}e^{rt} \int _{0}^{t}e^{-rs}\,ds \\ &\qquad {}+ MC \int _{0}^{t}e^{r(t-s)}\,ds+MC \Vert u_{\varepsilon} \Vert _{X_{\rho}}^{ \rho}e^{rt} \int _{0}^{t}e^{-rs}s^{-\rho \beta}\,ds, \end{aligned} $$

thus, as $t\rightarrow 0^{+}$, we obtain

$$ \begin{aligned} \bigl\Vert u_{\varepsilon}(t)-u_{0,\varepsilon} \bigr\Vert _{L^{q}(\Omega )}\leq{}& C \bigl\Vert S_{\alpha}^{\varepsilon}(t)u_{0,\varepsilon}-u_{0, \varepsilon} \bigr\Vert _{L^{\rho q}(\Omega )}+\frac{C}{r} \bigl(e^{rt}-1 \bigr) \Vert f_{ \varepsilon} \Vert _{C([0,T],L^{\rho q}(\Omega ))} \\ & {} + \frac{MC}{r} \bigl(e^{rt}-1 \bigr)+\frac{MC}{1-\rho \beta}e^{rt}t^{1- \rho \beta} \Vert u_{\varepsilon} \Vert _{X_{\rho}}\rightarrow 0. \end{aligned} $$

Therefore, $u_{\varepsilon}\in C([0,T], L^{q}(\Omega ))\cap C((0,T], L^{\rho q}( \Omega ))$ is an $L^{q}$-mild solution for the nonlinear approximating problem (1.12).

If $v_{\varepsilon}\in X_{\rho}$ is a solution of problem (1.12), taking $0< T'\leq T$ such that $\|v_{\varepsilon}\|_{X_{\rho}}^{T'}\leq R$, the uniqueness in $\mathbf{B}_{R}$ implies that $u_{\varepsilon}(t,x)=v_{\varepsilon}(t,x)$ for all $t\in [0,T']$. Now, we set

$$\begin{aligned} R'=\max \Bigl\{ \sup_{t\in (0,T]}t^{\beta} \bigl\Vert u_{\varepsilon}(t) \bigr\Vert _{L^{ \rho q}(\Omega )},\sup_{t\in (0,T]}t^{\beta} \bigl\Vert v_{\varepsilon}(t) \bigr\Vert _{L^{ \rho q}(\Omega )} \Bigr\} . \end{aligned}$$

For $t\in [T',T]$, we have

$$ \begin{aligned} &t^{\beta} \bigl\Vert u_{\varepsilon}(t)-v_{\varepsilon}(t) \bigr\Vert _{L^{ \rho q}(\Omega )} \\ &\quad \leq MCt^{\beta} \int _{0}^{t}e^{r(t-s)} \bigl(1+ \bigl\Vert u_{\varepsilon}(s) \bigr\Vert _{L^{ \rho q}(\Omega )}^{\rho -1}+ \bigl\Vert v_{\varepsilon}(s) \bigr\Vert _{L^{\rho q}( \Omega )}^{\rho -1} \bigr) \bigl\Vert u_{\varepsilon}(s)-v_{\varepsilon}(s) \bigr\Vert _{L^{ \rho q}(\Omega )}\,ds \\ &\quad \leq MCt^{\beta}e^{rt} \int _{0}^{t}e^{-rs}s^{-\beta} \bigl(1+s^{-( \rho -1)\beta} \Vert u_{\varepsilon} \Vert _{X_{\rho}}^{\rho -1} +s^{-(\rho -1) \beta} \Vert v_{\varepsilon} \Vert _{X_{\rho}}^{\rho -1} \bigr) \\ &\qquad {}\times s^{\beta} \bigl\Vert u_{\varepsilon}(s)-v_{\varepsilon}(s) \bigr\Vert _{L^{ \rho q}(\Omega )}\,ds \\ &\quad \leq MC \bigl(1+2T^{\prime -(\rho -1)\beta}R^{\prime \rho -1} \bigr)e^{rt} \int _{T'}^{t}e^{-rs}s^{- \beta}s^{\beta} \bigl\Vert u_{\varepsilon}(s)-v_{\varepsilon}(s) \bigr\Vert _{L^{\rho q}( \Omega )} \,ds. \end{aligned} $$

Defining a function $\eta :[0,T]\rightarrow [0,+\infty )$ by $\eta _{\varepsilon}(t)=t^{\beta}\|u_{\varepsilon}(t)-v_{\varepsilon}(t) \|_{L^{\rho q}(\Omega )}$, we have

$$\begin{aligned} \eta _{\varepsilon}(t)\leq MC \bigl(1+2T^{\prime -(\rho -1)\beta}R^{\prime \rho -1} \bigr)e^{rt} \int _{0}^{t}e^{-rs}s^{-\beta}\eta _{\varepsilon}(s)\,ds, \end{aligned}$$

so, due to the singular Gronwall’s lemma [4, Theorem 4], we obtain the uniqueness.

In order to obtain the continuous dependence on the initial data of the solutions, for $T>0$, we assume that $u_{\varepsilon}$ and $v_{\varepsilon}$ are solutions of equation (1.12) starting from $u_{0,\varepsilon}, v_{0,\varepsilon}\in L^{\rho q}(\Omega )$, respectively. From the choice of T, it follows that

$$ \begin{aligned} &t^{\beta} \bigl\Vert u_{\varepsilon}(t)-v_{\varepsilon}(t) \bigr\Vert _{L^{ \rho q}(\Omega )} \\ &\quad \leq Ce^{rt}t^{\beta} \Vert u_{0,\varepsilon}-v_{0,\varepsilon} \Vert _{L^{ \rho q}(\Omega )} \\ &\qquad {}+MC \int _{0}^{t}t^{\beta}e^{r(t-s)} \bigl(1+ \bigl\Vert u_{\varepsilon}(s) \bigr\Vert _{L^{\rho q}(\Omega )}^{\rho -1}+ \bigl\Vert v_{\varepsilon}(s) \bigr\Vert _{L^{\rho q}( \Omega )}^{\rho -1} \bigr) \bigl\Vert u_{\varepsilon}(s)-v_{\varepsilon}(s) \bigr\Vert _{L^{ \rho q}(\Omega )}\,ds \\ &\quad \leq Ce^{rT}T^{\beta} \Vert u_{0,\varepsilon}-v_{0,\varepsilon} \Vert _{L^{ \rho q}(\Omega )} \\ &\qquad {}+CMe^{rT}T^{\beta} \bigl\Vert u_{\varepsilon}(s)-v_{\varepsilon}(s) \bigr\Vert _{X_{ \rho}} \biggl(\frac{1}{r^{1-\beta}}\Gamma (1-\beta )+ \frac{2}{r^{1-\rho \beta}}R^{\rho -1}\Gamma (1-\rho \beta ) \biggr) \\ &\quad \leq C \Vert u_{0,\varepsilon}-v_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+ \frac{1}{2} \Vert u_{\varepsilon}-v_{\varepsilon} \Vert _{X_{\rho}}. \end{aligned} $$

For the nonlinear limit problem (1.13), let $\mathbf{B}_{R}$ be a closed ball in $X_{\rho}$, centered at the origin and with radius R, and define a map $\Gamma :\mathbf{B}_{R}\rightarrow \mathbf{B}_{R}$ by

$$ \Gamma u(t)=\mathrm{S}_{\alpha}(t)u_{0}+ \int _{0}^{t}\mathrm{T}_{ \alpha}(t-s)f(s) \,ds+ \int _{0}^{t}\mathrm{T}_{\alpha}(t-s)Nu(s) \,ds. $$

(4.2)

For $u\in X_{\rho}$, we have

$$ \begin{aligned} \Vert \Gamma u \Vert _{X_{\rho}}\leq{}& \bigl\Vert \mathrm{S}_{\alpha}(t)u_{0} \bigr\Vert _{X_{\rho}}+ \biggl\Vert \int _{0}^{t}\mathrm{T}_{\alpha}(t-s)f(s)\,ds \biggr\Vert _{X_{\rho}} \\ & {} + \biggl\Vert \int _{0}^{t}\mathrm{T}_{\alpha}(t-s)Nu(s)\,ds \biggr\Vert _{X_{\rho}} \\ \leq{}& \sup_{t\in (0,T]}t^{\beta} \bigl\Vert \mathrm{S}_{\alpha}(t)u_{0} \bigr\Vert _{L^{ \rho q}(\Omega )}+\sup _{t\in (0,T]}t^{\beta} \int _{0}^{t} \bigl\Vert \mathrm{T}_{\alpha}(t-s)f(s) \bigr\Vert _{L^{\rho q}(\Omega )}\,ds \\ & {} +\sup_{t\in (0,T]}t^{\beta} \int _{0}^{t} \bigl\Vert \mathrm{T}_{\alpha}(t-s)Nu(s) \bigr\Vert _{L^{\rho q}(\Omega )}\,ds. \end{aligned} $$

Using Lemmas 2.3–2.4, one gets

$$ \begin{aligned} \Vert \Gamma u \Vert _{X_{\rho}}\leq{}& C \Vert u_{0} \Vert _{L^{q}(\Omega )}+C \sup_{t\in (0,T]}t^{\beta} \int _{0}^{t}(t-s)^{\alpha -1-\beta} \bigl\Vert f(s) \bigr\Vert _{L^{q}(\Omega )}\,ds \\ & {} + CM\sup_{t\in (0,T]}t^{\beta} \int _{0}^{t}(t-s)^{\alpha -1} \bigl(1+ \bigl\Vert u(s) \bigr\Vert _{L^{\rho q}(\Omega )}^{\rho} \bigr)\,ds \\ \leq{}& C \Vert u_{0} \Vert _{L^{q}(\Omega )}+\frac{C}{\alpha -\beta} \bigl\Vert f(s) \bigr\Vert _{C([0,T];L^{q}(\Omega ))}T^{\alpha}+ \frac{CM}{\alpha}T^{\alpha + \beta} \\ & {} + CM\sup_{t\in (0,T]}t^{\beta} \int _{0}^{t}(t-s)^{\alpha -1} \bigl\Vert u(s) \bigr\Vert _{L^{\rho q}(\Omega )}^{\rho}\,ds \\ \leq{}& C \Vert u_{0} \Vert _{L^{q}(\Omega )}+\frac{C}{\alpha -\beta} \bigl\Vert f(s) \bigr\Vert _{C([0,T];L^{q}( \Omega ))}T^{\alpha}+ \frac{CM}{\alpha}T^{\alpha +\beta} \\ & {} + CMR^{\rho}\sup_{t\in (0,T]}t^{\beta} \int _{0}^{t}(t-s)^{ \alpha -1}s^{-\rho \beta} \,ds \\ \leq{}& C \Vert u_{0} \Vert _{L^{q}(\Omega )}+\frac{C}{\alpha -\beta} \bigl\Vert f(s) \bigr\Vert _{C([0,T],L^{q}( \Omega ))}T^{\alpha}+ \frac{CM}{\alpha}T^{\alpha +\beta} \\ & {} + CMR^{\rho}T^{\alpha -\rho \beta +\beta}\mathrm{B}(\alpha ,1- \rho \beta )\leq R. \end{aligned} $$

For $u,v\in \mathbf{B}_{R}$, we also get

$$\begin{aligned} & \Vert \Gamma u-\Gamma v \Vert _{X_{\rho}} \\ &\quad \leq \sup_{t\in (0,T]}t^{\beta} \int _{0}^{t} \bigl\Vert \mathrm{T}_{\alpha}(t-s) \bigl(Nu(s)-Nv(s) \bigr) \bigr\Vert _{L^{\rho q}(\Omega )}\,ds \\ &\quad \leq MC\sup_{t\in (0,T]}t^{\beta} \int _{0}^{t}(t-s)^{\alpha -1} \bigl(1+ \bigl\Vert u(s) \bigr\Vert _{L^{\rho q}(\Omega )}^{\rho -1} \\ &\qquad {}+ \bigl\Vert v(s) \bigr\Vert _{L^{\rho q}(\Omega )}^{\rho -1} \bigr) \bigl\Vert u(s)-v(s) \bigr\Vert _{L^{ \rho q}(\Omega )}\,ds \\ &\quad \leq MC \Vert u-v \Vert _{X_{\rho}}\sup_{t\in (0,T]}t^{\beta} \int _{0}^{t}(t-s)^{ \alpha -1}s^{-\beta} \,ds \\ &\qquad {}+MC \Vert u-v \Vert _{X_{\rho}}\sup_{t\in [0,T]}t^{\beta} \int _{0}^{t}(t-s)^{ \alpha -1}s^{-\beta}s^{-(\rho -1)\beta} \bigl( \Vert u \Vert _{X_{\rho}}^{\rho -1}+ \|v \bigr) \|_{X_{\rho}}^{\rho -1})\,ds \\ &\quad \leq MC \Vert u-v \Vert _{X_{\rho}}T^{\alpha}\mathrm{B}(\alpha ,1-\beta ) \\ &\qquad {}+2MC \Vert u-v \Vert _{X_{\rho}}R^{\rho -1}T^{\alpha +\beta -\rho \beta} \mathrm{B}(\alpha ,1-\rho \beta ) \\ &\quad \leq MC \Vert u-v \Vert _{X_{\rho}} \bigl(T^{\alpha}\mathrm{B}( \alpha ,1-\beta )+2R^{ \rho -1}T^{\alpha +\beta -\rho \beta}\mathrm{B}(\alpha ,1-\rho \beta ) \bigr) \\ &\quad \leq \frac{1}{2} \Vert u-v \Vert _{X_{\rho}}. \end{aligned}$$

The contraction principle implies the existence of a mild solution u in $\mathbf{B}_{R}$. We will prove that u is an $L^{q}$-mild solution and is unique in $X_{\rho}$. Indeed, we have

$$ \begin{aligned} & \bigl\Vert u(t)-u_{0} \bigr\Vert _{L^{q}(\Omega )} \\ &\quad \leq \bigl\Vert S_{\alpha}(t)u_{0}-u_{0} \bigr\Vert _{L^{q}(\Omega )} \\ &\qquad {}+ \int _{0}^{t} \bigl\Vert \mathrm{T}_{\alpha}(t-s)f(s) \bigr\Vert _{L^{q}(\Omega )}\,ds+ \int _{0}^{t} \bigl\Vert \mathrm{T}_{\alpha}(t-s)Nu(s) \bigr\Vert _{L^{q}(\Omega )}\,ds \\ &\quad \leq \bigl\Vert S_{\alpha}(t)u_{0}-u_{0} \bigr\Vert _{L^{q}(\Omega )}+C \int _{0}^{t}(t-s)^{ \alpha -1} \bigl\Vert f(s) \bigr\Vert _{L^{q}(\Omega )}\,ds \\ &\qquad {}+CM \int _{0}^{t}(t-s)^{\alpha -1} \bigl(1+ \bigl\Vert u(s) \bigr\Vert _{L^{q}(\Omega )}^{ \rho} \bigr)\,ds \\ &\quad \leq \bigl\Vert S_{\alpha}(t)u_{0}-u_{0} \bigr\Vert _{L^{q}(\Omega )}+C \Vert f \Vert _{C([0,T],L^{q}( \Omega ))} \int _{0}^{t}(t-s)^{\alpha -1}\,ds \\ &\qquad {}+CM \int _{0}^{t}(t-s)^{\alpha -1}\,ds+CM \int _{0}^{t}(t-s)^{ \alpha -1} \bigl\Vert u(s) \bigr\Vert _{L^{q}(\Omega )}^{\rho}\,ds, \end{aligned} $$

thus, as $t\rightarrow 0^{+}$, we obtain

$$ \begin{aligned} \bigl\Vert u(t)-u_{0} \bigr\Vert _{L^{q}(\Omega )}\leq{}& \bigl\Vert S_{\alpha}(t)u_{0}-u_{0} \bigr\Vert _{L^{q}(\Omega )}+C\alpha ^{-1}t^{\alpha} \Vert f \Vert _{C([0,T],L^{q}( \Omega ))}+CM\alpha ^{-1}t^{\alpha} \\ & {} +CMt^{\alpha -\rho \beta}\mathrm{B}(\alpha ,1-\rho \beta ) \Vert u \Vert _{X_{\rho}}^{\rho}\rightarrow 0. \end{aligned} $$

Therefore, $u\in \mathrm{C}([0,T],L^{q}(\Omega ))\cap \mathrm{C}((0,T],L^{\rho q}( \Omega ))$ is an $L^{q}$-mild solution for the limit problem (1.13). If $v\in X_{\rho}$ is a solution of problem (1.13), we may take $0< T'\leq T$ such that $\|v\|_{X_{\rho}}^{T'}\leq R$. Then, by the uniqueness in $\mathbf{B}_{R}$, we have $u(t,x)=v(t,x)$ for all $t\in [0,T']$. Set

$$\begin{aligned} R'=\max \Bigl\{ \sup_{t\in (0,T]}t^{\beta} \bigl\Vert u(t) \bigr\Vert _{L^{\rho q}(\Omega )}, \sup_{t\in (0,T]}t^{\beta} \bigl\Vert v(t) \bigr\Vert _{L^{\rho q}(\Omega )} \Bigr\} . \end{aligned}$$

For $t\in [T',T]$, we have

$$ \begin{aligned} &t^{\beta} \bigl\Vert u(t)-v(t) \bigr\Vert _{L^{\rho q}(\Omega )} \\ &\quad \leq MCt^{\beta} \int _{0}^{t}(t-s)^{\alpha -1} \bigl(1+ \bigl\Vert u(s) \bigr\Vert _{L^{ \rho q}(\Omega )}^{\rho -1}+ \bigl\Vert v(s) \bigr\Vert _{L^{\rho q}(\Omega )}^{\rho -1} \bigr) \bigl\Vert u(s)-v(s) \bigr\Vert _{L^{\rho q}(\Omega )}\,ds \\ &\quad \leq MCt^{\beta} \int _{0}^{t}(t-s)^{\alpha -1} \bigl(1+s^{-(\rho -1) \beta} \Vert u \Vert _{X_{\rho}}^{\rho -1}+s^{-(\rho -1)\beta} \Vert v \Vert _{X_{\rho}}^{ \rho -1} \bigr) \\ &\qquad {} \times \bigl\Vert u(s)-v(s) \bigr\Vert _{L^{\rho q}(\Omega )}\,ds \\ &\quad \leq MC \bigl(1+2 \bigl(T' \bigr)^{-(\rho -1)\beta}R^{\prime \rho -1} \bigr) \bigl(T' \bigr)^{-\beta}t^{\beta} \int _{0}^{t}(t-s)^{\alpha -1}s^{\beta} \bigl\Vert u(s)-v(s) \bigr\Vert _{L^{\rho q}( \Omega )}\,ds. \end{aligned} $$

Now, let $\xi :[0,T]\rightarrow [0,+\infty )$ be defined by $\xi (t)=t^{\beta}\|u(t)-v(t)\|_{L^{\rho q}(\Omega )}$, thus we have

$$\begin{aligned} \xi (t)\leq MC \bigl(1+2 \bigl(T' \bigr)^{-(\rho -1)\beta}R^{\prime \rho -1} \bigr) \bigl(T' \bigr)^{- \beta}t^{\beta} \int _{0}^{t}(t-s)^{\alpha -1}\xi (s)\,ds. \end{aligned}$$

Apply the singular Gronwall’s Lemma [4, Theorem 4], we obtain the uniqueness of the solution.

In order to obtain the continuous dependence on the initial data of the solution, we assume that u and v are solutions of problem (1.13) starting from $u_{0}, v_{0}\in L^{\rho q}(\Omega )$. From the choice of T, it follows that

$$ \begin{aligned} &t^{\beta} \bigl\Vert u(t)-v(t) \bigr\Vert _{L^{\rho q}(\Omega )} \\ &\quad \leq C \Vert u_{0}-v_{0} \Vert _{L^{q}(\Omega )} \\ &\qquad {}+MC\sup_{t\in (0,T]}t^{\beta} \int _{0}^{t}(t-s)^{\alpha -1} \bigl(1+ \bigl\Vert u(s) \bigr\Vert _{L^{\rho q}(\Omega )}^{\rho -1}+ \bigl\Vert v(s) \bigr\Vert _{L^{\rho q}( \Omega )}^{\rho -1} \bigr) \\ &\qquad {}\times \bigl\Vert u(s)-v(s) \bigr\Vert _{L^{\rho q}( \Omega )}\,ds \\ &\quad \leq C \Vert u_{0}-v_{0} \Vert _{L^{q}(\Omega )} \\ &\qquad {}+MC \Vert u-v \Vert _{X_{\rho}} \bigl(T^{\alpha}\mathrm{B}( \alpha ,1- \beta )+2R^{\rho -1}T^{\alpha +\beta -\rho \beta}\mathrm{B}(\alpha ,1- \rho \beta ) \bigr) \\ &\quad \leq C \Vert u_{0}-v_{0} \Vert _{L^{q}(\Omega )}+ \frac{1}{2} \Vert u-v \Vert _{X_{\rho}}, \end{aligned} $$

which finishes the proof of Theorem 3.2. □

To give the proof of Theorems 3.3 and 3.4, we need the fact that the solutions $u,u_{\varepsilon}\in L^{\infty}_{\rho}((0,T), L^{\rho q}(\Omega ))$, where the space

$$ \begin{aligned} &L^{\infty}_{\rho} \bigl((0,T),L^{\rho q}(\Omega ) \bigr) \\ &\quad = \Bigl\{ u\in L^{\infty} \bigl((0,T),L^{\rho q}(\Omega ) \bigr): \sup_{t\in (0,T)}t^{ \frac{\alpha n(\rho -1)}{2\rho q}} \bigl\Vert u(t) \bigr\Vert _{L^{\rho q}(\Omega )}< \infty \Bigr\} \end{aligned} $$

(4.3)

is endowed with the norm

$$\begin{aligned} \Vert u \Vert _{L^{\infty}_{\rho}((0,T),L^{\rho q}(\Omega ))}=\sup_{t\in (0,T)}t^{ \frac{\alpha n(\rho -1)}{2\rho q}} \bigl\Vert u(t) \bigr\Vert _{L^{\rho q}(\Omega )}. \end{aligned}$$

(4.4)

Lemma 4.1

If $u_{\varepsilon}$, u are defined by equations (2.6) and (2.7), respectively, then we have

$$\begin{aligned} &\Vert u_{\varepsilon} \Vert _{L_{\rho}^{\infty}((0,T),L^{\rho q}(\Omega ))} \leq C \bigl( \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+ \Vert f_{\varepsilon} \Vert _{L^{1}((0,T),L^{\rho q}(\Omega ))} \bigr), \end{aligned}$$

(4.5)

$$\begin{aligned} &\Vert u \Vert _{L_{\rho}^{\infty}((0,T),L^{\rho q}(\Omega ))}\leq C \Vert u_{0} \Vert _{L^{q}( \Omega )}+\frac{C}{\alpha -\beta}T^{\alpha} \Vert f \Vert _{C([0,T],L^{q}( \Omega ))}. \end{aligned}$$

(4.6)

Proof

From equations (2.6) and (2.7), and estimates (2.14) and (2.15), we get

$$ \begin{aligned} \bigl\Vert u_{\varepsilon}(t) \bigr\Vert _{L^{\rho q}(\Omega )}&\leq \bigl\Vert \mathrm{S}_{\alpha}^{\varepsilon}(t) \bigr\Vert _{\mathcal{L}(L^{\rho q}( \Omega ))} \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+ \bigl\Vert \mathrm{T}_{ \alpha}^{\varepsilon}(t) \bigr\Vert _{\mathcal{L}(L^{\rho q}(\Omega ))}\ast \bigl\Vert f_{ \varepsilon}(t) \bigr\Vert _{L^{\rho q}(\Omega )} \\ &\leq C \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+C \Vert f_{\varepsilon} \Vert _{L^{1}((0,T),L^{\rho q}(\Omega ))} \\ &\leq C \bigl( \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+ \Vert f_{\varepsilon} \Vert _{L^{1}((0,T),L^{\rho q}(\Omega ))} \bigr), \end{aligned} $$

and

$$\begin{aligned}& \Vert u_{\varepsilon} \Vert _{L_{\rho}^{\infty}((0,T),L^{\rho q})} \leq CT^{\beta} \bigl( \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+ \Vert f_{ \varepsilon} \Vert _{L^{1}((0,T),L^{\rho q}(\Omega ))} \bigr) \\& \hphantom{\Vert u_{\varepsilon} \Vert _{L_{\rho}^{\infty}((0,T),L^{\rho q})}}\leq C \bigl( \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+ \Vert f_{\varepsilon} \Vert _{L^{1}((0,T),L^{\rho q}(\Omega ))} \bigr), \\& \bigl\Vert u(t) \bigr\Vert _{L^{\rho q}(\Omega )} \leq \bigl\Vert \mathrm{S}_{\alpha}(t)u_{0} \bigr\Vert _{L^{ \rho q}(\Omega )}+ \bigl\Vert \mathrm{T}_{\alpha}(t)\ast f(t) \bigr\Vert _{L^{\rho q}( \Omega )} \\& \hphantom{\bigl\Vert u(t) \bigr\Vert _{L^{\rho q}(\Omega )}} \leq Ct^{-\beta} \Vert u_{0} \Vert _{L^{ q}(\Omega )}+Ct^{-\beta +\alpha -1} \ast \bigl\Vert f(t) \bigr\Vert _{L^{q}(\Omega )}, \end{aligned}$$

therefore

$$ \begin{aligned} \Vert u \Vert _{L_{\rho}^{\infty}((0,T);L^{\rho q})}&\leq C\sup _{t \in (0,T)}t^{-\beta}t^{\beta} \Vert u_{0} \Vert _{L^{ q}(\Omega )}+C\sup_{t \in (0,T)}t^{\beta} \bigl(t^{-\beta +\alpha -1}\ast \bigl\Vert f(t) \bigr\Vert _{L^{q}(\Omega )} \bigr) \\ &\leq C \Vert u_{0} \Vert _{L^{ q}(\Omega )}+\frac{C}{\alpha -\beta}T^{\alpha} \Vert f \Vert _{C([0,T],L^{q}(\Omega ))}, \end{aligned} $$

ending the proof of Lemma 4.1. □

Lemma 4.2

If $u_{\varepsilon}$ and u are defined by equations (2.8) and (2.9), respectively, then

$$\begin{aligned}& \Vert u_{\varepsilon} \Vert _{L_{\rho}^{\infty}((0,T), L^{\rho q}(\Omega ))} \leq MC \bigl( \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+ \Vert f_{ \varepsilon} \Vert _{L^{1}((0,T), L^{\rho q}(\Omega ))} \bigr), \end{aligned}$$

(4.7)

$$\begin{aligned}& \Vert u \Vert _{L_{\rho}^{\infty}((0,T), L^{\rho q})} \leq C \bigl( \Vert u_{0} \Vert _{L^{q}( \Omega )}+ \Vert u_{0} \Vert _{L^{q}(\Omega )}^{\rho} \\& \hphantom{\Vert u \Vert _{L_{\rho}^{\infty}((0,T), L^{\rho q})} \leq} {} + \Vert f \Vert _{C([0,T], L^{q}(\Omega ))}+ \Vert f \Vert _{C([0,T], L^{q}( \Omega )}^{\rho} \bigr), \end{aligned}$$

(4.8)

where $C=C(\alpha ,r,\theta ,T,\rho )>0$ is a constant.

Proof

From equations (2.8), (2.9) and Lemmas 2.3–2.5, we get

$$ \begin{aligned} & \bigl\Vert u_{\varepsilon}(t) \bigr\Vert _{L^{\rho q}(\Omega )} \\ &\quad \leq \bigl\Vert \mathrm{S}_{\alpha}^{\varepsilon}(t) \bigr\Vert _{\mathcal{L}(L^{\rho q}( \Omega ))} \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )} \\ & \qquad {} + \bigl\Vert \mathrm{T}_{\alpha}^{\varepsilon}(t) \bigr\Vert _{\mathcal{L}(L^{ \rho q}(\Omega ))}\ast \bigl\Vert f_{\varepsilon}(t) \bigr\Vert _{L^{\rho q}(\Omega )}+ \bigl\Vert \mathrm{T}_{\alpha}^{\varepsilon}(t) \bigr\Vert _{\mathcal{L}(L^{\rho q}( \Omega ))}\ast \bigl\Vert Nu_{\varepsilon}(t) \bigr\Vert _{L^{\rho q}(\Omega )} \\ &\quad \leq C \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+C \Vert f_{\varepsilon} \Vert _{L^{1}((0,T),L^{\rho q}(\Omega ))}+ MC \int _{0}^{t} \bigl(1+ \Vert u_{ \varepsilon} \Vert _{L^{\rho q}(\Omega )}^{\rho} \bigr)\,ds \\ &\quad \leq C \bigl( \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+ \Vert f_{ \varepsilon} \Vert _{L^{1}((0,T),L^{\rho q}(\Omega ))}+MT \bigr)+MC \int _{0}^{t} \bigl\Vert u_{\varepsilon}(s) \bigr\Vert _{L^{\rho q}(\Omega )}^{\rho}\,ds. \end{aligned} $$

Applying a generalization of Gronwall’s inequality proved in [3], we get

$$ \begin{aligned} & \bigl\Vert u_{\varepsilon}(t) \bigr\Vert _{L^{\rho q}(\Omega )} \\ &\quad \leq \bigl[C \bigl( \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+ \Vert f_{ \varepsilon} \Vert _{L^{1}((0,T),L^{\rho q}(\Omega ))}+MT \bigr)^{1-\rho}+( \rho -1)tMC \bigr]^{\rho -1} \\ &\quad \leq C \bigl( \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+ \Vert f_{ \varepsilon} \Vert _{L^{1}((0,T),L^{\rho q}(\Omega ))}+MT \bigr)+C \bigl[( \rho -1)TMC \bigr]^{\rho -1}, \end{aligned} $$

thus

$$ \begin{aligned} \Vert u_{\varepsilon} \Vert _{L_{\rho}^{\infty}((0,T), L^{\rho q})} \leq{}& CT^{\beta} \bigl( \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+ \Vert f_{ \varepsilon} \Vert _{L^{1}((0,T), L^{\rho q}(\Omega ))}+MT \bigr) \\ & {} +CT^{\beta} \bigl[(\rho -1)TMC \bigr]^{\rho -1} \\ \leq{}& CM \bigl( \Vert u_{0,\varepsilon} \Vert _{L^{\rho q}(\Omega )}+ \Vert f_{ \varepsilon} \Vert _{L^{1}((0,T), L^{\rho q}(\Omega ))} \bigr) \end{aligned} $$

and

$$ \begin{aligned} \bigl\Vert t^{\beta}u(t) \bigr\Vert _{L^{\rho q}(\Omega )}\leq{}& t^{\beta} \bigl\Vert \mathrm{S}_{\alpha}(t)u_{0} \bigr\Vert _{L^{\rho q}(\Omega )}+t^{\beta} \bigl\Vert \mathrm{T}_{\alpha}(t) \ast f(t) \bigr\Vert _{L^{\rho q}(\Omega )} \\ & {} +t^{\beta} \bigl\Vert \mathrm{T}_{\alpha}(t)\ast Nu(t) \bigr\Vert _{L^{\rho q}( \Omega )} \\ \leq{}& Ct^{\beta}t^{-\beta} \Vert u_{0} \Vert _{L^{ q}(\Omega )}+t^{\beta}Ct^{- \beta +\alpha -1}\ast \bigl\Vert f(t) \bigr\Vert _{L^{q}(\Omega )} \\ & {} +t^{\beta}MCt^{\alpha -1}\ast \bigl(1+ \Vert u \Vert _{L^{\rho q}(\Omega )}^{ \rho} \bigr) \\ \leq{}& C \biggl( \Vert u_{0} \Vert _{L^{ q}(\Omega )}+ \frac{t^{\alpha}}{\alpha -\beta} \Vert f \Vert _{C([0,T],L^{q}(\Omega ))}+ \frac{t^{\alpha +\beta}}{\alpha} \biggr) \\ & {} +Ct^{\beta} \int _{0}^{t}(t-s)^{\alpha -1}s^{-\rho \beta}s^{ \rho \beta} \bigl\Vert u(s) \bigr\Vert _{L^{\rho q}(\Omega )}^{\rho}\,ds \\ \leq{}& C \biggl( \Vert u_{0} \Vert _{L^{ q}(\Omega )}+ \frac{T^{\alpha}}{\alpha -\beta} \Vert f \Vert _{C([0,T],L^{q}(\Omega ))}+ \frac{T^{\alpha +\beta}}{\alpha} \biggr) \\ & {} +CT^{\beta}T^{-\rho \beta} \int _{0}^{t}(t-s)^{\alpha -1}s^{ \rho \beta} \bigl\Vert u(s) \bigr\Vert _{L^{\rho q}(\Omega )}^{\rho}\,ds. \end{aligned} $$

Applying the singular Gronwall’s inequality [4, Theorem 4], we get

$$ \begin{aligned} & \bigl\Vert t^{\beta}u(t) \bigr\Vert _{L^{\rho q}(\Omega )} \\ &\quad \leq C \biggl( \Vert u_{0} \Vert _{L^{q}(\Omega )}+ \frac{T^{\alpha}}{\alpha -\beta} \Vert f \Vert _{C([0,T],L^{q}(\Omega ))}+ \frac{T^{\alpha +\beta}}{\alpha} \biggr) \\ &\qquad {} +C\alpha ^{-1}t^{\alpha}T^{\beta -\rho \beta} \biggl( \Vert u_{0} \Vert _{L^{q}( \Omega )}+\frac{T^{\alpha}}{\alpha -\beta} \Vert f \Vert _{C([0,T],L^{q}( \Omega ))}+\frac{T^{\alpha +\beta}}{\alpha} \biggr)^{\rho}, \end{aligned} $$

therefore,

$$ \begin{aligned} & \Vert u \Vert _{L_{\rho}^{\infty}((0,T),L^{\rho q})} \\ &\quad \leq C \biggl( \Vert u_{0} \Vert _{L^{q}(\Omega )}+ \frac{T^{\alpha}}{\alpha -\beta} \Vert f \Vert _{C([0,T],L^{q}(\Omega ))}+ \frac{T^{\alpha +\beta}}{\alpha} \biggr) \\ &\qquad {} +C\frac{T^{\beta -\rho \beta +\alpha}}{\alpha} \biggl( \Vert u_{0} \Vert _{L^{ q}(\Omega )}+\frac{T^{\alpha}}{\alpha -\beta} \Vert f \Vert _{C([0,T],L^{q}( \Omega ))}+ \frac{T^{\alpha +\beta}}{\alpha} \biggr)^{\rho} \\ &\quad \leq C \bigl( \Vert u_{0} \Vert _{L^{ q}(\Omega )}+ \Vert u_{0} \Vert _{L^{ q}(\Omega )}^{ \rho}+ \Vert f \Vert _{C([0,T],L^{q}(\Omega ))}+ \bigl\Vert f(t) \bigr\Vert _{C([0,T],L^{q}(\Omega ))}^{ \rho} \bigr), \end{aligned} $$

ending the proof of Lemma 4.2. □

Proof of Theorem 3.3.

The space $Y_{\rho}=L^{p}((0,T),L^{\rho q}(\Omega ))$ is defined in (3.3). Using equations (2.8) and (2.9), as well as Lemmas 2.4–2.7, we have

$$\begin{aligned} & \Vert u_{\varepsilon}-u \Vert _{Y_{\rho}} \\ &\quad \leq \bigl\Vert \mathrm{S}_{\alpha}^{\varepsilon}(t) (u_{0,\varepsilon}-u_{0}) \bigr\Vert _{Y_{\rho}}+ \bigl\Vert \bigl(\mathrm{S}_{\alpha}^{\varepsilon}(t)-\mathrm{S}_{ \alpha}(t) \bigr)u_{0} \bigr\Vert _{Y_{\rho}} \\ &\qquad {}+ \biggl\Vert \int _{0}^{t} \bigl(\mathrm{T}_{\alpha}^{\varepsilon}(t-s)f_{ \varepsilon}(s)- \mathrm{T}_{\alpha}(t-s)f(s) \bigr)\,ds \biggr\Vert _{Y_{\rho}} \\ &\quad \leq C \bigl\{ \Vert u_{0,\varepsilon}-u_{0} \Vert _{L^{\rho q}(\Omega )}+ \bigl( \varepsilon ^{\frac{1}{p}}+\varepsilon ^{\theta _{0}}+ \bigl\vert \chi ( \varepsilon )-\chi _{0} \bigr\vert \bigr) \Vert u_{0} \Vert _{L^{\rho q}(\Omega )} \bigr\} \\ &\qquad {}+C \bigl\{ \Vert f_{\varepsilon}-f \Vert _{L^{p}((0,T),L^{1}(\Omega ))}+ \bigl[ \varepsilon ^{\frac{1}{p}+\alpha -1}+\varepsilon ^{\theta _{0}} \\ &\qquad {}+ \bigl\vert \chi _{0}-\chi (\varepsilon ) \bigr\vert \bigr] \Vert f \Vert _{L^{p}((0,T),L^{1}( \Omega ))} \bigr\} \\ &\quad \leq C \bigl\{ \Vert u_{0,\varepsilon}-u_{0} \Vert _{L^{\rho q}(\Omega )}+ \Vert f_{ \varepsilon}-f \Vert _{C([0,T],L^{\rho q}(\Omega ))} \\ &\qquad {}+ \bigl(\varepsilon ^{\frac{1}{p}}+\varepsilon ^{\theta _{0}}+ \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert \bigr) \Vert u_{0} \Vert _{L^{\rho q}(\Omega )} \\ &\qquad {}+ \bigl(\varepsilon ^{\frac{1}{p}+\alpha -1}+\varepsilon ^{ \theta _{0}}+ \bigl\vert \chi _{0}-\chi (\varepsilon ) \bigr\vert \bigr) \Vert f \Vert _{C([0,T],L^{ \rho q}(\Omega ))} \bigr\} , \end{aligned}$$

ending the proof of Theorem 3.3. □

Proof of Theorem 3.4.

From Lemma 4.2, we have

$$\begin{aligned}& \Vert u_{\varepsilon} \Vert _{L_{\rho}^{\infty}((0,T),L^{\rho q})} \leq CM \bigl( \Vert u_{0, \varepsilon} \Vert _{L^{\rho q}(\Omega )}+ \Vert f_{\varepsilon} \Vert _{L^{1}((0,T),L^{ \rho q}(\Omega ))} \bigr)\leq C, \\& \Vert u \Vert _{L_{\rho}^{\infty}((0,T),L^{\rho q})} \leq C \bigl( \Vert u_{0} \Vert _{L^{ q}( \Omega )}+ \Vert u_{0} \Vert _{L^{ q}(\Omega )}^{\rho} \\& \hphantom{\Vert u \Vert _{L_{\rho}^{\infty}((0,T),L^{\rho q})} \leq} {} + \Vert f \Vert _{C([0,T],L^{q}(\Omega ))}+ \Vert f \Vert _{C([0,T],L^{q}( \Omega ))}^{\rho} \bigr)\leq C. \end{aligned}$$

Thus,

$$\begin{aligned} &t^{\beta} \bigl\Vert u_{\varepsilon}(t)-u(t) \bigr\Vert _{L^{\rho q}( \Omega )} \\ &\quad = t^{\beta} \biggl\Vert \mathrm{S}_{\alpha}^{\varepsilon}(t)u_{0, \varepsilon}+ \int _{0}^{t}\mathrm{T}_{\alpha}^{\varepsilon}(t-s)f_{ \varepsilon}(s) \,ds + \int _{0}^{t}\mathrm{T}_{\alpha}^{\varepsilon}(t-s)Nu_{ \varepsilon}(s) \,ds \\ &\qquad {}-\mathrm{S}_{\alpha}(t)u_{0}- \int _{0}^{t}\mathrm{T}_{\alpha}(t-s)f(s)- \int _{0}^{t}\mathrm{T}_{\alpha}(t-s)Nu(s)\,ds \biggr\Vert _{L^{\rho q}( \Omega )} \\ &\quad \leq t^{\beta} \bigl\Vert \mathrm{S}_{\alpha}^{\varepsilon}(t) (u_{0, \varepsilon}-u_{0}) \bigr\Vert _{L^{\rho q}(\Omega )} +t^{\beta} \bigl\Vert \bigl(\mathrm{S}_{ \alpha}^{\varepsilon}(t)- \mathrm{S}_{\alpha}(t) \bigr)u_{0} \bigr\Vert _{L^{\rho q}( \Omega )} \\ &\qquad {}+t^{\beta} \int _{0}^{t} \bigl\Vert \mathrm{T}_{\alpha}^{\varepsilon}(t-s) \bigl(f_{ \varepsilon}(s)-f(s) \bigr) \bigr\Vert _{L^{\rho q}(\Omega )}\,ds \\ &\qquad {}+t^{\beta} \int _{0}^{t} \bigl\Vert \bigl( \mathrm{T}_{\alpha}^{\varepsilon}(t-s)- \mathrm{T}_{\alpha}(t-s) \bigr)f(s) \bigr\Vert _{L^{\rho q}(\Omega )}\,ds \\ &\qquad {}+t^{\beta} \int _{0}^{t} \bigl\Vert \mathrm{T}_{\alpha}^{\varepsilon}(t-s) \bigl(Nu_{ \varepsilon}(s)-Nu(s) \bigr) \bigr\Vert _{L^{\rho q}(\Omega )}\,ds \\ &\qquad {}+t^{\beta} \int _{0}^{t} \bigl\Vert \bigl( \mathrm{T}_{\alpha}^{\varepsilon}(t-s)- \mathrm{T}_{\alpha}(t-s) \bigr)Nu(s) \bigr\Vert _{L^{\rho q}(\Omega )}\,ds \\ &\quad \leq D_{\varepsilon}^{1}(t)+Mt^{\beta} \bigl\Vert \mathrm{T}_{\alpha}^{ \varepsilon}(t)-\mathrm{T}_{\alpha}(t) \bigr\Vert _{\mathcal{L}(L^{\rho q}( \Omega ))}\ast \bigl(1+ \bigl\Vert u(t) \bigr\Vert _{L^{\rho q}(\Omega )}^{\rho} \bigr) \\ &\qquad {}+Mt^{\beta}e^{rt} \int _{0}^{t}e^{-rs} \bigl(1+ \bigl\Vert u_{\varepsilon}(s) \bigr\Vert _{L^{ \rho q}(\Omega )}^{\rho -1} \\ &\qquad {}+ \bigl\Vert u(s) \bigr\Vert _{L^{\rho q}(\Omega )}^{\rho -1} \bigr) \bigl\Vert u_{\varepsilon}(s)-u(s) \bigr\Vert _{L^{\rho q}(\Omega )}\,ds \\ &\quad \leq D_{\varepsilon}^{1}(t)+D_{\varepsilon}^{2}(t) \\ &\qquad {}+ Me^{rT}T^{\beta} \int _{0}^{t} \bigl(1+2Cs^{\beta -\rho \beta} \bigr)s^{- \beta}s^{\beta} \bigl\Vert u_{\varepsilon}(s)-u(s) \bigr\Vert _{L^{\rho q}(\Omega )}\,ds, \end{aligned}$$

where

$$\begin{aligned}& D_{\varepsilon}^{1}(t) =t^{\beta} \bigl\Vert \mathrm{S}_{\alpha}^{ \varepsilon}(t) (u_{0,\varepsilon}-u_{0}) \bigr\Vert _{L^{\rho q}(\Omega )} +t^{ \beta} \bigl\Vert \bigl( \mathrm{S}_{\alpha}^{\varepsilon}(t)-\mathrm{S}_{\alpha}(t) \bigr)u_{0} \bigr\Vert _{L^{\rho q}(\Omega )} \\& \hphantom{D_{\varepsilon}^{1}(t) =} {} +t^{\beta} \int _{0}^{t} \bigl\Vert \mathrm{T}_{\alpha}^{\varepsilon}(t-s) \bigl(f_{ \varepsilon}(s)-f(s) \bigr) \bigr\Vert _{L^{\rho q}(\Omega )}\,ds \\& \hphantom{D_{\varepsilon}^{1}(t) =} {} +t^{\beta} \int _{0}^{t} \bigl\Vert \bigl( \mathrm{T}_{\alpha}^{\varepsilon}(t-s)- \mathrm{T}_{\alpha}(t-s) \bigr)f(s) \bigr\Vert _{L^{\rho q}(\Omega )}\,ds, \\& D_{\varepsilon}^{2}(t) =Mt^{\beta} \bigl\Vert \mathrm{T}_{\alpha}^{\varepsilon}(t)- \mathrm{T}_{\alpha}(t) \bigr\Vert _{\mathcal{L}(L^{\rho q}(\Omega ))}\ast \bigl(1+ \bigl\Vert u(t) \bigr\Vert _{L^{\rho q}(\Omega )}^{\rho} \bigr). \end{aligned}$$

Applying Gronwall’s inequality, we obtain

$$ \begin{aligned} &t^{\beta} \bigl\Vert u_{\varepsilon}(t)-u(t) \bigr\Vert _{L^{\rho q}( \Omega )} \\ &\quad \leq D_{\varepsilon}^{1}(t)+D_{\varepsilon}^{2}(t)+Me^{rT}T^{\beta} \int _{0}^{t}\exp \biggl\{ \frac{(1-\rho \beta )(s^{1-\beta})+2C(1-\beta )(s^{1-\rho \beta})}{(1-\beta )(1-\rho \beta )} \biggr\} \\ & \qquad {} \times \bigl[D_{\varepsilon}^{1}(s)+D_{\varepsilon}^{2}(s) \bigr] \bigl(s^{- \beta}+2Cs^{-\rho \beta} \bigr)\,ds. \end{aligned} $$

Taking the $Y_{\rho}$-norm and using Young’s inequality, we get

$$ \begin{aligned} \Vert u_{\varepsilon}-u \Vert _{Y_{\rho}}\leq{}& \bigl( \bigl\Vert D_{ \varepsilon}^{1}(t) \bigr\Vert _{L^{p}(0,T)}+ \bigl\Vert D_{\varepsilon}^{2}(t) \bigr\Vert _{L^{p}(0,T)} \bigr) \\ & {} +Me^{rT}T^{\beta +1}\exp \biggl(\frac{T^{1-\beta}}{1-\beta}+ \frac{2CT^{1-\rho \beta}}{1-\rho \beta} \biggr) \\ & {} + \bigl( \bigl\Vert D_{\varepsilon}^{1}(t) \bigr\Vert _{L^{p}(0,T)}+ \bigl\Vert D_{\varepsilon}^{2}(t) \bigr\Vert _{L^{p}(0,T)} \bigr) \biggl(\frac{T^{1-\beta}}{1-\beta}+ \frac{2CT^{1-\rho \beta}}{1-\rho \beta} \biggr) \\ \leq{}& C \bigl( \bigl\Vert D_{\varepsilon}^{1}(t) \bigr\Vert _{L^{p}(0,T)}+ \bigl\Vert D_{\varepsilon}^{2}(t) \bigr\Vert _{L^{p}(0,T)} \bigr). \end{aligned} $$

Applying Young’s inequality again, from Lemmas 2.4–2.7, we deduce that

$$\begin{aligned}& \bigl\Vert D_{\varepsilon}^{1}(t) \bigr\Vert _{L^{p}(0,T)} \leq C \bigl\{ \Vert u_{0, \varepsilon}-u_{0} \Vert _{L^{\rho q}(\Omega )}+ \Vert f_{\varepsilon}-f \Vert _{C([0,T],L^{ \rho q}(\Omega ))} \\& \hphantom{\bigl\Vert D_{\varepsilon}^{1}(t) \bigr\Vert _{L^{p}(0,T)} \leq} {} + \bigl(\varepsilon ^{\frac{1}{p}}+\varepsilon ^{\theta _{0}}+ \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert \bigr) \Vert u_{0} \Vert _{L^{\rho q}(\Omega )} \\& \hphantom{\bigl\Vert D_{\varepsilon}^{1}(t) \bigr\Vert _{L^{p}(0,T)} \leq} {} + \bigl(\varepsilon ^{\frac{1}{p}+\alpha -1}+\varepsilon ^{ \theta _{0}}+ \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert \bigr) \Vert f \Vert _{C([0,T],L^{ \rho q}(\Omega ))} \bigr\} , \\& \bigl\Vert D_{\varepsilon}^{2}(t) \bigr\Vert _{L^{p}(0,T)} \leq C \bigl\Vert t^{\beta} \bigl(\mathrm{T}_{ \alpha}^{\varepsilon}(t)- \mathrm{T}_{\alpha}(t) \bigr) \bigr\Vert _{L^{p}((0,T), \mathcal{L}(L^{\rho q}(\Omega )))} \\& \hphantom{\bigl\Vert D_{\varepsilon}^{2}(t) \bigr\Vert _{L^{p}(0,T)} \leq} {} \times \bigl(T+ \bigl\Vert t^{-\beta}t^{\beta}u(t) \bigr\Vert _{L^{1}((0,T),L^{ \rho q}(\Omega ))}^{\rho} \bigr) \\& \hphantom{\bigl\Vert D_{\varepsilon}^{2}(t) \bigr\Vert _{L^{p}(0,T)} } \leq C \bigl\Vert t^{\beta} \bigl(\mathrm{T}_{\alpha}^{\varepsilon}(t)- \mathrm{T}_{ \alpha}(t) \bigr) \bigr\Vert _{L^{p}((0,T),\mathcal{L}(L^{\rho q}(\Omega )))} \\& \hphantom{\bigl\Vert D_{\varepsilon}^{2}(t) \bigr\Vert _{L^{p}(0,T)} \leq} {} \times \biggl(T+\frac{T^{1-\rho \beta}}{1-\rho \beta} \bigl\Vert t^{\beta}u(t) \bigr\Vert _{L^{\infty}((0,T),L^{\rho q}(\Omega ))}^{\rho} \biggr) \\& \hphantom{\bigl\Vert D_{\varepsilon}^{2}(t) \bigr\Vert _{L^{p}(0,T)} }\leq C \bigl\Vert t^{\beta} \bigl(\mathrm{T}_{\alpha}^{\varepsilon}(t)- \mathrm{T}_{ \alpha}(t) \bigr) \bigr\Vert _{L^{p}((0,T),\mathcal{L}(L^{\rho q}(\Omega )))} \\& \hphantom{\bigl\Vert D_{\varepsilon}^{2}(t) \bigr\Vert _{L^{p}(0,T)} \leq} {} \times \biggl\{ T+\frac{T^{1-\rho \beta}}{1-\rho \beta}C \bigl[ \Vert u_{0} \Vert _{L^{\rho q}(\Omega )}+ \Vert u_{0} \Vert _{L^{\rho q}(\Omega )}^{\rho} \\& \hphantom{\bigl\Vert D_{\varepsilon}^{2}(t) \bigr\Vert _{L^{p}(0,T)} \leq} {} + \Vert f \Vert _{C([0,T]),L^{\rho q}(\Omega ))}+ \Vert f \Vert _{C([0,T],L^{\rho q}( \Omega ))}^{\rho} \bigr] \biggr\} , \end{aligned}$$

therefore,

$$ \begin{aligned} \Vert u_{\varepsilon}-u \Vert _{Y_{\rho}}\leq{}& C \bigl\{ \Vert u_{0, \varepsilon}-u_{0} \Vert _{L^{\rho q}(\Omega )}+ \Vert f_{\varepsilon}-f \Vert _{C([0,T],L^{ \rho q}(\Omega ))} \\ & {} + \bigl(\varepsilon ^{\frac{1}{p}}+\varepsilon ^{\theta _{0}}+ \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert \bigr) \Vert u_{0} \Vert _{L^{\rho q}(\Omega )} \\ & {} + \bigl(\varepsilon ^{\frac{1}{p}+\alpha -1}+\varepsilon ^{ \theta _{0}}+ \bigl\vert \chi (\varepsilon )-\chi _{0} \bigr\vert \bigr) \bigl( \Vert u_{0} \Vert _{L^{ \rho q}(\Omega )}+ \Vert u_{0} \Vert _{L^{\rho q}(\Omega )}^{\rho} \\ & {} + \Vert f \Vert _{C([0,T],L^{\rho q}(\Omega ))}+ \Vert f \Vert _{C([0,T],L^{\rho q}( \Omega ))}^{\rho} \bigr) \bigr\} , \end{aligned} $$

completing the proof of Theorem 3.4. □

5 Discussion

This paper mainly discusses two problems. One is the well-posedness of mild solutions of integro-differential time-fractional order ordinary differential equations (1.1), (1.7), (1.12), and (1.13). The other is the convergence of mild solutions $u_{\varepsilon}(t,x)$ to the initial boundary value problems (1.1) and (1.12) which contain the small positive scale parameter ε, and the asymptotic behavior of the approximating mild solution $u_{\varepsilon}(t,x)$ near the limit mild solution $u(t,x)$ as $\varepsilon \rightarrow 0^{+}$, where $u(t,x)$ is the mild solution of the limit problems (1.7) and (1.13), respectively.

An interesting problem is that the second-order differential operator A in equation (1.1) and equation (1.12) has the following divergence form:

$$\begin{aligned} A:=A(x,t,D_{x})=\sum_{i,j=1}^{n}D_{x_{i}} \bigl(a_{ij}(x,t)D_{x_{j}} \bigr)+a_{0}(x). \end{aligned}$$

(5.1)

In contrast to the differential operator $A=\sum_{i,j=1}^{n}D_{x_{i}}(a_{ij}(x)D_{x_{j}})+a_{0}(x)$ defined in equation (1.5), the difference is that $a_{ij}(t,x)$ depends on the time variable t. For the special case $a_{ij}(t,x)=K(t)\tilde{a}_{ij}(x)$, one can see Remark 2.1. For the problems (1.1) and (1.12) with A defined in equation (5.1), our method is invalid. So the question is, for example, how to characterize the analytic semigroup of the operator A? And how to define a mild solution by the Laplace transform?