1 Introduction

In recent decades, fractional calculus has had many applications in various fields such as mechanic, biological, physical science, and applied science. This topic has increasingly asserted its role in the field of applied mathematics, especially the subjects are investigating the properties of the concept of derivative. Specifically, the situations with integer order in PDEs are not working well. Therefore, PDEs with fractional derivatives are a generalization equation with integer-order partial derivatives and a strong theoretical and practical interest. There have been many authors researching this field, for example, [115]. According to our search results, the extended results for a coupled nonlinear fractional pseudo-parabolic equation are still limited. This is the great impetus that motivated us to study the following model. In this paper, we extend the coupled system and consider the following coupled nonlinear fractional pseudo-parabolic equation:

$$ \textstyle\begin{cases} \partial ^{\alpha }_{t} (u(x,t)-a\Delta u(x,t) ) - \Delta u(x,t)= \mathcal{F}(u,v),& (x,t) \in \Omega \times (0,T), \\ \partial ^{\alpha }_{t} (v(x,t)-a\Delta v(x,t) ) - \Delta v(x,t)= \mathcal{G}(u,v),& (x,t) \in \Omega \times (0,T), \\ u(x,t)=v(x,t)=0,&x\in \partial \Omega , t\in (0,T], \\ u_{t}(x,0)=v_{t}(x,0)=0,& { x\in \partial \Omega ,} \\ u(x,0)=f(x),& x\in \Omega , \\ v(x,0)=g(x),& x\in \Omega , \end{cases} $$

where T, a are positive numbers and \(\Omega \in \mathbb{R}^{n}\), \(n \ge 1\), is an open bounded domain with a smooth boundary Ω. Note that \(\mathbb{L}^{2}(\Omega )\), \(\mathbb{H}_{0}^{1}(\Omega )\), \(\mathbb{H}^{2}( \Omega )\) denote the usual Sobolev spaces. The symmetric uniform elliptic operator \(\Delta : \mathbb{L}^{2}(\Omega ) \to \mathbb{L}^{2}(\Omega ) \) is defined by

$$\begin{aligned} \Delta u(x) = -\sum_{i=1}^{n} \frac{\partial }{\partial x}_{i} \biggl( \Delta _{ij}(x) \frac{\partial }{\partial x_{j}}u(x) \biggr) + l(x)u(x,t),\quad x \in \overline{\Omega }. \end{aligned}$$

With assumption \(l(x)\in C(\overline{\Omega },[0,\infty))\), \(\Delta _{ij}\in C^{1}( \overline{\Omega })\), \(\Delta _{ij}= \Delta _{ji}\), \(1 \leq i, j \leq n \), and there exists a positive constant \(\widetilde{\Delta }>0\) for \(x\in \overline{\Omega }\), \(z = (z_{1}, z_{2}, \ldots , z_{n})\subset \mathbb{R}^{n}\), such that

$$ \widetilde{\Delta }\sum_{i=1}^{n}z_{i}^{2} \leq \sum_{1\leq i, j\leq n}z_{i} \Delta _{ij}(x)z_{j}, $$

see e.g. [16]. The constant \(\alpha \in (1,2)\) is the fractional order and \(\partial ^{\alpha }_{t}\) denotes the left-sided Caputo fractional derivative involving t, which is defined by

$$ \partial ^{\alpha }_{t} u(x,t):=\frac{1}{\Gamma (2-\alpha )} \int _{0}^{t}(t- \eta )^{1-\alpha } \frac{\partial {u}^{2}}{\partial \eta }(x,\eta ) \,d \eta , $$

where Γ is the gamma function.

If \(\alpha = 2 \), then \(\partial ^{2}_{t} \) is interpreted as a derivative of normal time. The second equation in the (1.1) called the fractional pseudo-parabolic equation has many practical applications, for example, the permeability of a homogeneous liquid through a cracked rock [17], one-way propagation of nonlinear dispersion long waves [1820], and populations of [21] populations. Let Ω be an open and bounded domain \(\mathbb{R}^{n} \) with the boundary Ω. The functions \(\mathcal{F}\), \(\mathcal{G}\), f, g satisfy some assumptions to be specified later.

In practice, many problems with space-time fraction equations are based on fractional parameters, that is, the order of fractions. However, these fractional parameters were not known during the modeling process. Therefore, the continuity of the solution on these parameters is very important for modeling purposes. To the best of our knowledge, there have been no results investigating continuity related to the fractional order of a system of pseudo-nonlinear parabolic equations.

This article is organized as follows. Part 2 provides some basic and preliminary definitions. In Part 3, we give the formula of a mild solution and some lemmas that may be related to the next section. In Part 4, we apply the results of Part 3 to establish the existence, uniqueness, and continuity of the solution to problem (1.1) in fractional order. Finally, we give an example to test the theory.

2 Preliminaries

2.1 Stability on the parameters of the Mittag-Leffler function

Consider the Mittag-Leffler function, which is defined by

$$\begin{aligned} E_{\alpha ,\beta }(\xi ) = \sum_{n=1}^{\infty } \frac{\xi ^{n}}{\Gamma (n\alpha +\beta )}\quad \text{for } \xi \in \mathbb{C}, \alpha > 0, \text{ and } \beta \in \mathbb{R}. \end{aligned}$$

We call to mind the following lemmas (see for example [3, 22, 23]), which will be useful for the main analysis of Sect. 3 and Sect. 4.

Lemma 2.1

If \(1< \alpha < 2\), then, for all \(\xi >0\), where M is a positive constant depending only on α,

$$\begin{aligned} \bigl\vert E_{\alpha ,1}(-\xi ) \bigr\vert \le \mathrm {M}, \qquad \bigl\vert E_{\alpha ,\alpha }(-\xi ) \bigr\vert \le \mathrm {M}. \end{aligned}$$

Now, we have the following lemmas.

Lemma 2.2

Let \(\lambda > 0\) and \(1<\alpha <2\). Then, for all \(\xi > 0\), the following identities hold:

$$\begin{aligned} &\partial _{\xi }E_{\alpha ,1}\bigl(-\lambda \xi ^{\alpha } \bigr) = -\lambda \xi ^{ \alpha -1}E_{\alpha ,\alpha }\bigl(-\lambda \xi ^{\alpha }\bigr), \\ &\partial _{\xi }\bigl(\xi ^{\alpha -1}E_{\alpha ,\alpha }\bigl(- \lambda \xi ^{\alpha }\bigr)\bigr)= -\xi ^{\alpha -2}E_{\alpha ,\alpha -1} \bigl(-\lambda \xi ^{\alpha }\bigr). \end{aligned}$$


Apply Lemma 2.2 in [24]. □

Lemma 2.3

(see [25])

Let \(1<\alpha <2\). If T is large enough, then

$$\begin{aligned} E_{\alpha ,1}\bigl(-\lambda _{j} T^{\alpha } \bigr) \neq 0 \end{aligned}$$

for all \(j\in \mathbb{N}\), then there exist two constants \(\mathrm {m}_{\alpha }\) and \(\mathrm {M}_{\alpha }\) such that

$$\begin{aligned} \frac{\mathrm {m}_{\alpha }}{1+ \lambda _{j}T^{\alpha }} \le \bigl\vert E_{ \alpha ,1}\bigl(- \lambda _{j}T^{\alpha }\bigr) \bigr\vert \le \frac{\mathrm {M}_{\alpha }}{1+\lambda _{j}T^{\alpha }}. \end{aligned}$$

From Lemma 2.3 in [26], we have the following lemmas.

Lemma 2.4

Let \(1<\alpha _{1} <\alpha _{2}< 2\) and \(\alpha \in (\alpha _{1},\alpha _{2})\). There exist two positive constants \(\mathrm {M}_{1}\), \(\mathrm {M}_{2}\), and \(\mathrm {M}_{3}\) which just rely upon \(\alpha _{1}\), \(\alpha _{2}\) such that, for any \(\xi \geq 0\), we get

$$\begin{aligned} &\frac{\mathrm {M}_{1}(\alpha _{1},\alpha _{2})}{1+\xi }\leq \bigl\vert E_{ \alpha ,1}(-\xi ) \bigr\vert \leq \frac{\mathrm {M}_{2}(\alpha _{1},\alpha _{2})}{1+\xi } \\ &\textit{if }\xi \textit{ is large enough and } \bigl\vert E_{\alpha ,\alpha }(-\xi ) \bigr\vert \leq \frac{\mathrm {M}_{3}(\alpha _{1},\alpha _{2})}{1+\xi }. \end{aligned}$$

Lemma 2.5

Let \(0 < \alpha _{1} < \alpha < \alpha ' < \alpha _{2}\) and \(0 <\xi \leq T\). For any \(\epsilon > 0\) independent of α, there always exists \(\mathrm {M}_{\epsilon }\) such that

$$\begin{aligned} \bigl\vert \xi ^{\alpha }-\xi ^{\alpha '} \bigr\vert \leq \max \bigl(T^{\alpha _{2}+2\epsilon },1\bigr) \mathrm {M}_{\epsilon }\bigl(\alpha '-\alpha \bigr)^{\epsilon }\xi ^{\alpha - \epsilon }. \end{aligned}$$


See Lemma 3.2 in [24]. □

Using Lemmas 3.3 and 3.4, Section 3 in [24], we have the following lemmas.

Lemma 2.6

Assume that \(1< \alpha _{1} < \alpha < {\alpha }'< \alpha _{2} <2\) and \(\epsilon >0\). Then there exists a positive constant \(\mathrm {A}(\alpha _{1}, \alpha _{2},\epsilon , \nu _{0},T)\)

$$\begin{aligned} & \bigl\vert E_{\alpha ,1}\bigl(-\lambda _{j}\xi ^{\alpha }\bigr)-E_{{\alpha }',1}\bigl(- \lambda _{j}\xi ^{{\alpha }'}\bigr) \bigr\vert \\ &\quad \leq \mathrm {A}(\alpha _{1},\alpha _{2},\epsilon , \nu _{0},T) \lambda _{j}^{\nu _{0}-1}\xi ^{-\alpha _{2}(1-\nu _{0})-\epsilon } \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'-\alpha \bigr) \bigr] \end{aligned}$$

for any \(0\leq \nu _{0}\leq 1\) and \(0 < \xi \leq T\).


See Lemma 3.3 for stability on parameters of the Mittag-Leffler function in [24]. □

Lemma 2.7

Assume that \(1< \alpha _{1} < \alpha < {\alpha }' < \alpha _{2} < 2\). For any \(0 \leq \nu _{0} \leq 1\) and \(\epsilon > 0\), there exists a positive constant \(\mathrm {B}(\alpha _{1},\alpha _{2}, \epsilon , \nu _{0}, T)\)

$$\begin{aligned} & \bigl\vert \xi ^{\alpha -1}E_{\alpha ,\alpha }\bigl(-\lambda _{j}t^{\alpha }\bigr)-\xi ^{{ \alpha }'-1}E_{{\alpha }',{\alpha }'}\bigl(- \lambda _{j}\xi ^{{\alpha }'}\bigr) \bigr\vert \\ &\quad \leq \mathrm {B}(\alpha _{1},\alpha _{2},\epsilon , \nu _{0},T) \lambda _{j}^{\nu _{0}-1}\xi ^{\alpha _{1}\nu _{0}-\epsilon -1} \bigl[\bigl({ \alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'-\alpha \bigr) \bigr]. \end{aligned}$$


Use Lemma 3.4 from Section 3 in [24]. □

2.2 Some Sobolev spaces

In this section, we present some appropriate Sobolev space. Let the operator Δ be considered on \(\mathbb{L}^{2}(\Omega )\) with domain \(\mathbb{H}^{2}(\Omega )\cap \mathbb{H}^{1}_{0}(\Omega )\). Then the spectrum of Δ is a non-diminishing arrangement of positive real numbers \(\{\lambda _{j}\}_{j\geq 1}\) which satisfy that \(\lim_{j\rightarrow \infty }\lambda _{j} = \infty \). Let us denote by \(\{\varphi _{j}\}_{j\geq 1}\) in \(\mathbb{H}^{2}(\Omega )\cap \mathbb{H}_{0}^{1}(\Omega )\) the set of orthonormal eigenfunctions of Δ, which means that \(\Delta \varphi _{j} = \lambda _{j}\varphi _{j}\). The sequence forms an orthonormal basis of \(\mathbb{L}^{2}(\Omega )\), see e.g. [27]. For all \(\gamma \geq 0\), the operator \(\Delta ^{\gamma }\) has the following representation:

$$\begin{aligned} \Delta ^{\gamma }v:=\sum_{ j=1}^{\infty } \langle v, \varphi _{j} \rangle \lambda _{j}^{\gamma } \varphi _{j},\quad v \in \mathbb{D}\bigl( \Delta ^{\gamma }\bigr) = \Biggl\{ v\in \mathbb{L}^{2}(\Omega ): \sum _{ j=1}^{\infty } \bigl\vert \langle v, \varphi _{j}\rangle \bigr\vert ^{2}\lambda _{j}^{2{ \gamma }}< \infty \Biggr\} . \end{aligned}$$

The domain \(\mathbb{D}(\Delta ^{\gamma })\) is Banach spaces equipped with the norm

$$\begin{aligned} \Vert v \Vert ^{2}_{\mathbb{D}(\Delta ^{\gamma })} : = \sum _{j=1}^{\infty } \lambda _{j}^{2{\gamma }} \bigl\vert \langle v,\varphi _{j}\rangle \bigr\vert ^{2}. \end{aligned}$$

If \(\gamma = 1\), we have \(\mathbb{D}(\Delta ^{1}) = \mathbb{H}^{2}(\Omega )\).

For a given number \(\gamma \geq 0\), the Hilbert space

$$\begin{aligned} \mathcal{H}^{\gamma }(\Omega ) = \Biggl\{ v\in \mathbb{L}^{2}( \Omega ): \sum_{ j=1}^{\infty } \bigl\vert \langle v, \varphi _{j}\rangle \bigr\vert ^{2} \lambda _{j}^{2{\gamma }}< \infty \Biggr\} \end{aligned}$$

is endowed with the norm as follows:

$$\begin{aligned} \Vert v \Vert ^{2}_{\mathcal{H}^{\gamma }} : = \sum _{j=1}^{\infty }\lambda _{j}^{2{ \gamma }} \bigl\vert \langle v,\varphi _{j}\rangle \bigr\vert ^{2}. \end{aligned}$$

If \(\gamma =0\), then \(\mathcal{H}^{0}(\Omega ) = \mathbb{L}^{2}(\Omega )\). We identified a norm for \(w(u,v)\in \mathscr{H}^{\gamma }(\Omega )=\mathcal{H}^{\gamma }(\Omega ) \times \mathcal{H}^{\gamma }(\Omega )\) as follows:

$$\begin{aligned} \Vert w \Vert _{\mathscr{H}^{\gamma }(\Omega )}=\sqrt{ \Vert u \Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )}+ \Vert v \Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )}}. \end{aligned}$$

Let us denote by \(\mathbb{C}((0,T]; \mathcal{H}^{\gamma }(\Omega ))\) a space of all continuous functions with the map \((0,T]\rightarrow \mathcal{H}^{\gamma }(\Omega )\). For a given number \(0<\beta <1\), we define by \(\mathbb{C}^{\beta }((0,T]; \mathcal{H}^{\gamma }(\Omega ))\) such that

$$\begin{aligned} \sup_{0 < t \le T} {t^{\beta }} { \bigl\Vert {f(t)} \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )}}< \infty ; \quad f\in \mathbb{C}(\bigl(0,T]; \mathcal{H}^{\gamma }(\Omega )\bigr), \end{aligned}$$

in which (see [26])

$$\begin{aligned} \Vert f \Vert _{\mathbb{C}^{\beta }((0,T]; \mathcal{H}^{\gamma }(\Omega ))}:= \sup_{0 < t \le T}t^{\beta } \Vert f \Vert _{\mathcal{H}^{\gamma }( \Omega )}. \end{aligned}$$

The product space \(\mathscr{C}^{\beta }(0,T,\mathcal{H}^{\gamma }(\Omega )) = \mathbb{C}^{ \beta }{((0,T],\mathcal{H}^{\gamma }(\Omega ))}\times \mathbb{C}^{\beta }{((0,T], \mathcal{H}^{\gamma }(\Omega ))}\) is also a Banach space endowed with the norm

$$\begin{aligned} \Vert w \Vert _{\mathscr{C}^{\beta }((0,T],\mathcal{H}^{\gamma }(\Omega ))} = \sqrt{ \Vert u \Vert ^{2}_{\mathbb{C}^{\beta }((0,T],\mathcal{H}^{\gamma }( \Omega ))} + \Vert v \Vert ^{2}_{\mathbb{C}^{\beta }((0,T],\mathcal{H}^{\gamma }( \Omega ))}} \end{aligned}$$

for \(w = (u,v)\in \mathscr{C}^{\beta }((0,T],\mathcal{H}^{\gamma }(\Omega ))\). For a given positive real number p, \(\mathbb{L}_{p}^{\infty }(0,T, \mathcal{H}^{\gamma }(\Omega ))\) is a Banach space with the norm

$$\begin{aligned} \Vert f \Vert _{\mathbb{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }(\Omega ))}: = ess \sup e^{-pt} \Vert f \Vert _{\mathcal{H}^{\gamma }(\Omega )}. \end{aligned}$$

The product space \(\mathscr{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }(\Omega )) = \mathbb{L}_{p}^{\infty }{(0,T,\mathcal{H}^{\gamma }(\Omega ))}\times \mathbb{L}_{p}^{\infty }{(0,T,\mathcal{H}^{\gamma }(\Omega ))}\) is a Banach space, and we also identified a norm for \(w = (u,v)\in \mathscr{L}_{p}^{\infty }(0,T,\mathbb{H}^{\gamma }( \Omega ))\) as follows:

$$\begin{aligned} \Vert w \Vert _{\mathscr{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }(\Omega ))} = \sqrt{ \Vert u \Vert ^{2}_{\mathbb{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }( \Omega ))} + \Vert v \Vert ^{2}_{\mathbb{L}_{p}^{\infty }(0,T,\mathcal{H}^{ \gamma }(\Omega ))}}. \end{aligned}$$

3 Relevant notations and a representation of solution

In perception of spectral decomposition

$$\begin{aligned} u(x,t) = \sum_{j=1}^{\infty } \bigl\langle u(\cdot ,t),\varphi _{j}( \cdot )\bigr\rangle \varphi _{j}(x) \quad \text{and}\quad v(x,t) = \sum_{j=1}^{\infty } \bigl\langle v(\cdot ,t),\varphi _{j}(\cdot )\bigr\rangle \varphi _{j}(x). \end{aligned}$$

We can transform the first two equations of (1.1) into

$$ \textstyle\begin{cases} \langle \partial ^{\alpha }_{t}(u(\cdot ,t),\varphi _{j} \rangle +a \langle \partial ^{\alpha }_{t}\Delta u(\cdot ,t), \varphi _{j} \rangle + \langle \Delta u(\cdot ,t),\varphi _{j} \rangle = \langle \mathcal{F}(u(\cdot ,t),v(\cdot ,t)), \varphi _{j} \rangle , \\ \langle \partial ^{\alpha }_{t}(v(\cdot ,t),\varphi _{j} \rangle +a \langle \partial ^{\alpha }_{t}\Delta v(\cdot ,t), \varphi _{j} \rangle + \langle \Delta v(\cdot ,t),\varphi _{j} \rangle = \langle \mathcal{G}(u(\cdot ,t),v(\cdot ,t)), \varphi _{j} \rangle . \end{cases} $$

Using the formula \(\Delta \varphi _{j}=\lambda _{j}\varphi _{j}\), we obtain

$$ \textstyle\begin{cases} (1+a\lambda _{j}) \langle \partial ^{\alpha }_{t}(u(\cdot ,t), \varphi _{j} \rangle + \lambda _{j} \langle u(\cdot ,t), \varphi _{j} \rangle = \langle \mathcal{F}(u(\cdot ,t),v( \cdot ,t)),\varphi _{j} \rangle , \\ (1+a\lambda _{j}) \langle \partial ^{\alpha }_{t}(v(\cdot ,t), \varphi _{j} \rangle + \lambda _{j} \langle v(\cdot ,t), \varphi _{j} \rangle = \langle \mathcal{G}(u(\cdot ,t),v( \cdot ,t)),\varphi _{j} \rangle . \end{cases} $$

The theory of fractional ordinary differential equations (see [3, 22, 23]) gives a unique function \(u_{j}\), \(v_{j}\) as follows:

$$ \textstyle\begin{cases} u_{j}(t) = E_{\alpha ,1} ( \frac{-\lambda _{j}t^{\alpha }}{1+ a \lambda _{j}} ) f_{j} \\ \hphantom{u_{j}(t) =}{}+ \frac{1}{1+a \lambda _{j}} \int _{0}^{t} [(t-s)^{ \alpha -1} E_{\alpha ,\alpha } ( \frac{-\lambda _{j}(t-s)^{\alpha }}{1+ a \lambda _{j}} ) \mathcal{F}_{j}(s) ] \,\mathrm{d}s, \\ v_{j}(t) = E_{\alpha ,1} ( \frac{-\lambda _{j}t^{\alpha }}{1+ a \lambda _{j}} ) g_{j} \\ \hphantom{v_{j}(t) =}{}+ \frac{1}{1+a \lambda _{j}} \int _{0}^{t} [(t-s)^{ \alpha -1} E_{\alpha ,\alpha } ( \frac{-\lambda _{j}(t-s)^{\alpha }}{1+ a \lambda _{j}} ) \mathcal{G}_{j}(s) ] \,\mathrm{d}s. \end{cases} $$

Here, we denote \(f_{j} := \langle f,\varphi _{j}\rangle \), \(g_{j}:= \langle g,\varphi _{j} \rangle \), \(\mathcal{F}_{j} := \langle \mathcal{F} (u(\cdot ,t),v(\cdot ,t)), \varphi _{j}\rangle \) and \(\mathcal{G}_{j}:= \langle \mathcal{G}(u(\cdot ,t), v(\cdot ,t)), \varphi _{j}\rangle \). Hence solution (1.1) can be described as by Fourier series (3.1) and then given by

$$ \textstyle\begin{cases} u(x,t) =\sum_{ j=1}^{\infty }E_{\alpha ,1} ( \frac{-\lambda _{j}t^{\alpha }}{1+ a \lambda _{j}} ) f_{j} \varphi _{j} \\ \hphantom{u(x,t) =}{} + \sum_{ j=1}^{\infty }\frac{1}{1+a \lambda _{j}} [\int _{0}^{t} (t-s)^{\alpha -1} E_{\alpha ,\alpha } ( \frac{-\lambda _{j}(t-s)^{\alpha }}{1+ a \lambda _{j}} ) \mathcal{F}_{j}(s)\,\mathrm{d}s ]\varphi _{j}, \\ v(x,t) =\sum_{ j=1}^{\infty }E_{\alpha ,1} ( \frac{-\lambda _{j}t^{\alpha }}{1+ a \lambda _{j}} ) g_{j}{ \varphi _{j}} \\ \hphantom{v(x,t) =}{}+ \sum_{ j=1}^{\infty }\frac{1}{1+a \lambda _{j}} [\int _{0}^{t} (t-s)^{\alpha -1} E_{\alpha ,\alpha } ( \frac{-\lambda _{j}(t-s)^{\alpha }}{1+ a \lambda _{j}} ) \mathcal{G}_{j}(s)\,\mathrm{d}s ] { \varphi _{j}}. \end{cases} $$

It is obvious to see that the mild solution (1.1) is given by

$$ \textstyle\begin{cases} {u}_{\alpha }(\cdot ,t) = \mathcal{P}_{\Delta ,\alpha }(t)f + \int _{0}^{t}\mathcal{Q}_{\Delta ,\alpha }(t-s)\mathcal{F}(u,v)( \cdot ,s)\,ds, \\ {v}_{\alpha }(\cdot ,t) = \mathcal{P}_{\Delta , \alpha }(t)g + \int _{0}^{t}\mathcal{Q}_{\Delta , \alpha }(t-s)\mathcal{G}(u,v)( \cdot ,s)\,ds, \end{cases} $$


$$\begin{aligned} &\mathcal{P}_{\Delta , \alpha }(t)h := \sum_{j=1}^{\infty } E_{ \alpha ,1} \biggl(\frac{-\lambda _{j}t^{\alpha }}{1+a\lambda _{j}} \biggr) \langle h,\varphi _{j}\rangle \varphi _{j}, \\ &\mathcal{Q}_{\Delta , \alpha }(t-s)h:= \sum_{j=1}^{\infty } \frac{1}{1+a \lambda _{j}}(t-s)^{\alpha - 1}E_{\alpha ,\alpha } \biggl( \frac{-\lambda _{j}(t-s)^{\alpha }}{1+a\lambda _{j}} \biggr)\langle h, \varphi _{j}\rangle \varphi _{j}. \end{aligned}$$

Therefore, with \(1< \alpha <\alpha '<2\), we also get

$$ \textstyle\begin{cases} {u}_{\alpha '}(\cdot ,t) = \mathcal{P}_{\Delta ,\alpha '}(t)f + \int _{0}^{t}\mathcal{Q}_{\Delta ,\alpha '}(t-s)\mathcal{F}(u,v)( \cdot ,s)\,ds, \\ {v}_{\alpha '}(\cdot ,t) = \mathcal{P}_{\Delta , \alpha '}(t)g + \int _{0}^{t}\mathcal{Q}_{\Delta , \alpha '}(t-s)\mathcal{G}(u,v)( \cdot ,s)\,ds. \end{cases} $$

Next, we give several lemmas related to Sect. 4 as follows.

Lemma 3.1

Let \(1<\alpha _{1}<\alpha <\alpha _{2}<2\), \(\gamma \geq 0\), and \(\mathrm{w}\in \mathcal{H}^{\gamma }(\Omega )\). The following inequalities hold:

$$\begin{aligned} & \bigl\Vert \mathcal{P}_{\Delta ,\alpha }(t)\mathrm{w} \bigr\Vert _{\mathcal{H}^{\gamma }( \Omega )}\leq \overline{\mathrm {M}}_{2}(\alpha _{1},\alpha _{2},a, \mu _{0})t^{-\alpha _{1} \mu _{0}} \Vert w \Vert _{\mathcal{H}^{\gamma }(\Omega )}, \end{aligned}$$
$$\begin{aligned} & \bigl\Vert \mathcal{Q}_{\alpha }(t-s)\mathrm{w} \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )} \leq \overline{\mathrm {M}}_{3}(\alpha _{1},\alpha _{2},a,\mu _{0}) (t-s)^{ \alpha _{1}-1-\alpha _{1}\mu _{0}} \Vert w \Vert _{\mathcal{H}^{\gamma }(\Omega )}, \end{aligned}$$

where \(\mu _{0}\) is a positive number satisfying \(0<\mu _{0}<1\).


Using Lemma 2.4, we get

$$\begin{aligned} \bigl\Vert \mathcal{P}_{\Delta ,\alpha }(t)\mathrm{w} \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }( \Omega )} &= \sum_{ j=1}^{\infty } \lambda _{j}^{2\gamma }E^{2}_{ \alpha ,1} \biggl( \frac{-\lambda _{j}t^{\alpha }}{1+a\lambda _{j}} \biggr) \bigl\vert \langle \mathrm{w}, \varphi _{j}\rangle \bigr\vert ^{2} \\ &\leq \sum_{j=1}^{\infty }\lambda _{j}^{2\gamma } \biggl( \frac{\mathrm {M}_{2}(\alpha _{1},\alpha _{2})}{1+\frac{\lambda _{j}t^{\alpha }}{1+a\lambda _{j}}} \biggr)^{2} \bigl\vert \langle \mathrm{w}, \varphi _{j}\rangle \bigr\vert ^{2} \\ &\leq \sum_{j=1}^{\infty }\lambda _{j}^{2\gamma } \frac{\mathrm {M}^{2}_{2}(\alpha _{1},\alpha _{2})}{ (1+\frac{\lambda _{j}t^{\alpha }}{1+a\lambda _{j}} )^{\mu _{0}}} \bigl\vert \langle \mathrm{w}, \varphi _{j}\rangle \bigr\vert ^{2} \\ &\leq \sum_{j=1}^{\infty }\lambda _{j}^{2\gamma }\mathrm {M}^{2}_{2}( \alpha _{1},\alpha _{2}){ \biggl( \frac{\lambda _{j}t^{\alpha }}{1+a\lambda _{j}} \biggr)^{-2\mu _{0}}} \bigl\vert \langle \mathrm{w}, \varphi _{j} \rangle \bigr\vert ^{2} \\ &\leq \mathrm {M}^{2}_{2}(\alpha _{1},\alpha _{2})t^{-2\alpha \mu _{0}}\bigl(a+ \lambda ^{-1}_{1} \bigr)^{2\mu _{0}}\sum_{ j=1}^{\infty }\lambda _{j}^{2 \gamma } \bigl\vert \langle \mathrm{w},\varphi _{j}\rangle \bigr\vert ^{2} \\ &\leq \mathrm {M}^{2}_{2}(\alpha _{1},\alpha _{2})t^{-2\alpha _{1} \mu _{0}}\bigl(\lambda ^{-1}_{1}+a \bigr)^{2\mu _{0}} \Vert \mathrm{w} \Vert ^{2}_{ \mathcal{H}^{\gamma }(\Omega )}. \end{aligned}$$

Therefore, with \(\overline{\mathrm {M}}_{2}(\alpha _{1},\alpha _{2},a,\mu _{0}) := \mathrm {M}_{2}(\alpha _{1},\alpha _{2})(\lambda ^{-1}_{1}+a)^{\mu _{0}}\), we have the following estimate:

$$\begin{aligned} \bigl\Vert \mathcal{P}_{\Delta ,\alpha }(t)\mathrm{w} \bigr\Vert _{\mathbb{H}^{\gamma }( \Omega )}&\leq \overline{\mathrm {M}}_{2}(\alpha _{1},\alpha _{2},a, \mu _{0})t^{-\alpha _{1} \mu _{0}} \Vert \mathrm{w} \Vert _{\mathcal{H}^{\gamma }( \Omega )}. \end{aligned}$$

Likewise, utilizing Lemma 2.4, we can get the following estimation:

$$\begin{aligned} & \bigl\Vert \mathcal{Q}_{\Delta ,\alpha }(t-s)\mathrm{w} \bigr\Vert ^{2}_{\mathcal{H}^{ \gamma }(\Omega )} \\ &\quad = \sum_{ j=1}^{\infty } \frac{\lambda _{j}^{2\gamma }}{(1+a\lambda _{j})^{2}}(t-s)^{2(\alpha -1)}E^{2}_{ \alpha ,\alpha } \biggl( \frac{-\lambda _{j}(t-s)^{\alpha }}{1+a\lambda _{j}} \biggr) \bigl\vert \langle \mathrm{w}, \varphi _{j}\rangle \bigr\vert ^{2} \\ &\quad \leq \sum_{j=1}^{\infty } \frac{\lambda _{j}^{2\gamma }}{(1+a\lambda _{j})^{2}}(t-s)^{2(\alpha -1)} \biggl( \frac{\mathrm {M}_{3}(\alpha _{1},\alpha _{2})}{1+\frac{\lambda _{j}(t-s)^{\alpha }}{1+a\lambda _{j}}} \biggr)^{2} \bigl\vert \langle \mathrm{w}, \varphi _{j} \rangle \bigr\vert ^{2} \\ &\quad \leq (t-s)^{2(\alpha -1-\alpha \mu _{0})}\mathrm {M}^{2}_{3}( \alpha _{1},\alpha _{2})\sum_{j=1}^{\infty } \lambda _{j}^{2 \gamma } \biggl({\frac{1}{\lambda _{j}}+a} \biggr)^{2(\mu _{0}-1)}\lambda _{j}^{-2} \bigl\vert \langle \mathrm{w}, \varphi _{j}\rangle \bigr\vert ^{2} \\ &\quad \leq (t-s)^{2(\alpha -1-\alpha \mu _{0})}\mathrm {M}^{2}_{3}( \alpha _{1},\alpha _{2}) \bigl({\lambda ^{-1}_{1}+a} \bigr)^{2(\mu _{0}-1)} \lambda _{1}^{-2}\sum _{j=1}^{\infty }\lambda _{j}^{2\gamma } \bigl\vert \langle \mathrm{w}, \varphi _{j}\rangle \bigr\vert ^{2} \\ &\quad \leq (t-s)^{2(\alpha _{1}-1-\alpha _{1} \mu _{0})}\mathrm {M}^{2}_{3}( \alpha _{1},\alpha _{2}) \bigl({\lambda ^{-1}_{1}+a} \bigr)^{2(\mu _{0}-1)} \lambda _{1}^{-2} \Vert \mathrm{w} \Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )}. \end{aligned}$$

Therefore, we deduce

$$\begin{aligned} \bigl\Vert \mathcal{Q}_{\alpha }(t-s)\mathrm{w} \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )}& \leq \overline{\mathrm {M}}_{3}(\alpha _{1},\alpha _{2},a,\mu _{0}) (t-s)^{ \alpha _{1}-1-\alpha _{1} \mu _{0}} \Vert \mathrm{w} \Vert _{\mathcal{H}^{\gamma }( \Omega )}, \end{aligned}$$

where \(\overline{\mathrm {M}}_{3}(\alpha _{1},\alpha _{2},a,\mu _{0}):= \mathrm {M}_{3}(\alpha _{1},\alpha _{2}) ({\lambda ^{-1}_{1}+a} )^{\mu _{0}-1}\lambda _{1}^{-1}\).

Thus, we complete the proof of Lemma 3.1. □

Lemma 3.2

Let \(1 <\alpha _{1}< \alpha <\alpha '<\alpha _{2} < 2\), \(\gamma \geq 0\) with \(0\leq \nu _{0}\leq 1\) and \(\mathrm{w} \in \mathcal{H}^{\gamma }(\Omega )\). The following inequalities hold:

$$\begin{aligned} & \bigl\Vert \bigl[\mathcal{P}_{\Delta ,\alpha '}(t)- \mathcal{P}_{\Delta , \alpha }(t) \bigr]\mathrm{w} \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )} \\ &\quad \leq \overline{\mathrm {A}}(\alpha _{1},\alpha _{2},\epsilon ,\nu _{0},a,T) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'- \alpha \bigr) \bigr]t^{- \alpha _{2}(1-\nu _{0})-\epsilon } \Vert \mathrm{w} \Vert _{\mathcal{H}^{\gamma }( \Omega )}. \end{aligned}$$

Besides, we have

$$\begin{aligned} & \bigl\Vert \bigl[\mathcal{Q}_{\Delta ,\alpha '}(t-s)- \mathcal{Q}_{ \Delta ,\omega }(t-s) \bigr]\mathrm{w} \bigr\Vert _{\mathcal{H}^{\gamma }( \Omega )} \\ &\quad \leq \overline{\mathrm {B}}(\alpha _{1},\alpha _{2},\epsilon ,\nu _{0},a,T) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'- \alpha \bigr) \bigr](t-s)^{ \alpha _{1}\nu _{0}-\epsilon -1} \Vert \mathrm{w} \Vert _{\mathcal{H}^{\gamma }( \Omega )}. \end{aligned}$$


We get

$$\begin{aligned} & \bigl\Vert \bigl[\mathcal{P}_{\Delta ,\alpha '}(t)- \mathcal{P}_{\Delta , \alpha }(t) \bigr]\mathrm{w} \bigr\Vert ^{2}_{\mathbb{H}^{\gamma }(\Omega )} \\ &\quad = \sum_{ j=1}^{\infty }\lambda _{j}^{2\gamma } \biggl[E_{ \alpha ',1} \biggl( \frac{-\lambda _{j}t^{\alpha '}}{1+a\lambda _{j}} \biggr)- E_{\alpha ,1} \biggl( \frac{-\lambda _{j}t^{\alpha }}{1+a\lambda _{j}} \biggr) \biggr]^{2} \bigl\vert \langle \mathrm{w},\varphi _{n} \rangle \bigr\vert ^{2}. \end{aligned}$$

By using Lemma 2.6, we obtain the following estimates:

$$\begin{aligned} & \bigl\Vert \bigl[\mathcal{P}_{\Delta ,\alpha '}(t)- \mathcal{P}_{\Delta , \alpha }(t) \bigr]\mathrm{w} \bigr\Vert ^{2}_{\mathcal{H}^{\alpha }(\Omega )} \\ &\quad \leq \sum_{j=1}^{\infty }\lambda _{j}^{2\gamma }\mathrm {A}^{2}( \alpha _{1},\alpha _{2},\epsilon ,\nu _{0},T) \bigl[ \bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'-\alpha \bigr) \bigr]^{2} \\ & \qquad {} \times \biggl(\frac{\lambda _{j}}{1+a\lambda _{j}} \biggr)^{2(\nu _{0}-1)}t^{-2 \alpha _{2}(1-\nu _{0})-2\epsilon } \bigl\vert \langle \mathrm{w}, \varphi _{j} \rangle \bigr\vert ^{2} \\ &\quad \leq \mathrm {A}^{2}(\alpha _{1},\alpha _{2},\epsilon ,\nu _{0},T) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'- \alpha \bigr) \bigr]^{2} \\ & \qquad {} \times t^{-2\alpha _{2}(1-\nu _{0})-2\epsilon }\bigl(\lambda _{1}^{-1}+a \bigr)^{2(1- \nu _{0})}\sum_{j=1}^{\infty }\lambda _{j}^{2\gamma } \bigl\vert \langle \mathrm{w}, \varphi _{j}\rangle \bigr\vert ^{2} \\ &\quad \leq \mathrm {A}^{2}(\alpha _{1},\alpha _{2},\epsilon ,\nu _{0},T) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'- \alpha \bigr) \bigr]^{2} \\ & \qquad {} \times t^{-2\alpha _{2}(1-\nu _{0})-2\epsilon }\bigl(\lambda _{1}^{-1}+a \bigr)^{2(1- \nu _{0})} \Vert \mathrm{w} \Vert _{\mathcal{H}^{\gamma }(\Omega )}^{2}. \end{aligned}$$

Therefore, we obtain

$$\begin{aligned} & \bigl\Vert \bigl[\mathcal{P}_{\Delta ,\alpha '}(t)-\mathcal{P}_{\Delta , \alpha }(t) \bigr]\mathrm{w} \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )} \\ &\quad \leq \overline{\mathrm {A}}(\alpha _{1},\alpha _{2},\epsilon ,\nu _{0},a,T) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'- \alpha \bigr) \bigr]t^{- \alpha _{2}(1-\nu _{0})-\epsilon } \Vert \mathrm{w} \Vert _{\mathcal{H}^{\gamma }( \Omega )}, \end{aligned}$$

where \(\overline{\mathrm {A}}(\alpha _{1},\alpha _{2},\epsilon ,\nu _{0},a,T):= \mathrm {A}(\alpha _{1},\alpha _{2},\epsilon ,\nu _{0},T)(\lambda _{1}^{-1}+a)^{1- \nu _{0}} \).

Similarly, by applying Lemma 2.7, we also get

$$\begin{aligned} & \bigl\Vert \bigl[\mathcal{Q}_{\Delta ,\alpha '}(t-s) -\mathcal{Q}_{\Delta , \alpha }(t-s) \bigr]\mathrm{w} \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \\ &\quad = \sum_{j=1}^{\infty } \frac{\lambda _{j}^{2\gamma }}{(1+a\lambda _{j})^{2}} \biggl[(t-s)^{{ \alpha }' - 1}E_{{\alpha }',{\alpha }'} \biggl( \frac{-\lambda _{j}(t-s)^{\alpha '}}{1+a\lambda _{j}} \biggr) \\ &\qquad {} - (t-s)^{\alpha - 1}E_{\alpha ,\alpha } \biggl( \frac{-\lambda _{j}(t-s)^{\alpha }}{1+a\lambda _{j}} \biggr) \biggr]^{2} \bigl\vert \langle \mathrm{w}, \varphi _{j}\rangle \bigr\vert ^{2} \\ &\quad \leq \sum_{j=1}^{\infty } \mathrm {B}^{2}(\alpha _{1}, \alpha _{2},\epsilon ,\nu _{0},T) \biggl( \frac{\lambda _{j}}{1+a\lambda _{j}} \biggr)^{2(\nu _{0}-1)}(t-s)^{2 \alpha _{1}\nu _{0}-2\epsilon -2} \\ &\qquad {} \times \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'-\alpha \bigr) \bigr]^{2} \frac{\lambda _{j}^{2\gamma }}{(1+a\lambda _{j})^{2}} \bigl\vert \langle \mathrm{w},\varphi _{j}\rangle \bigr\vert ^{2} \\ &\quad \leq \mathrm {B}^{2}(\alpha _{1},\alpha _{2},\epsilon ,\nu _{0},T) \bigl[\bigl({\alpha }'-\alpha _{2}\bigr)^{\epsilon }+\bigl({\alpha }'-\alpha \bigr) \bigr]^{2} \lambda _{1}^{-2}(t-s)^{2\alpha _{1}\nu _{0}-2\epsilon -2} \\ & \qquad {} \times \sum_{j=1}^{\infty } \biggl( \frac{1}{\lambda _{j}}+a \biggr)^{-2 \nu _{0}} \lambda _{j}^{2\gamma } \bigl\vert \langle \mathrm{w} ,\varphi _{j} \rangle \bigr\vert ^{2} \\ &\quad \leq \mathrm {B}^{2}(\alpha _{1},\alpha _{2},\epsilon ,\nu _{0},T) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'- \alpha \bigr) \bigr]^{2} \lambda _{1}^{-2}a^{-2\nu _{0}}(t-s)^{2\alpha _{1}\nu _{0}-2\epsilon -2} \Vert \mathrm{w} \Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )}. \end{aligned}$$

We denote \(\overline{\mathrm {B}}(\alpha _{1},\alpha _{2},\epsilon ,\nu _{0},a,T):= \mathrm {B}(\alpha _{1},\alpha _{2},\epsilon ,\nu _{0},T)\lambda _{1}^{-1}a^{- \nu _{0}} \). Therefore, we obtain

$$\begin{aligned} & \bigl\Vert \bigl[\mathcal{Q}_{\Delta ,\alpha '}(t-s)- \mathcal{Q}_{ \Delta ,\omega }(t-s) \bigr]\mathrm{w} \bigr\Vert _{\mathcal{H}^{\gamma }( \Omega )} \\ &\quad \leq \overline{\mathrm {B}}(\alpha _{1},\alpha _{2},\epsilon ,\nu _{0},a,T) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'- \alpha \bigr) \bigr](t-s)^{ \alpha _{1}\nu _{0}-\epsilon -1} \Vert \mathrm{w} \Vert _{\mathcal{H}^{\gamma }( \Omega )}. \end{aligned}$$

We get all estimates of Lemma 3.2. This completes the proof. □

4 Stability of the fractional order of problem (1.1)

In this section, we are interested in studying the existence of a mild solution and the continuous dependence of the solution of problem (1.1) with input (the fractional-order α, \(\alpha '\) and the initial condition f, g). We assume that \(\mathcal{F}\), \(\mathcal{G}\) satisfy the following assumptions:


$$\begin{aligned} &\bigl\Vert \mathcal{F}(u,v) (\cdot ,t) \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )}\leq \mathcal{C}_{1} \bigl(1+ \bigl\Vert u(\cdot ,t) \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert v(\cdot ,t) \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )} \bigr), \end{aligned}$$
$$\begin{aligned} &\bigl\Vert \mathcal{G}(u,v) (\cdot ,t) \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )}\leq \mathcal{C}_{2} \bigl(1+ \bigl\Vert u(\cdot ,t) \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert v(\cdot ,t) \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )} \bigr), \end{aligned}$$

where \((u,v)\in \mathscr{H}^{\gamma }(\Omega )=\mathcal{H}^{\gamma }(\Omega ) \times \mathcal{H}^{\gamma }(\Omega )\).


$$\begin{aligned} & \bigl\Vert \mathcal{F}(u_{1},v_{1}) (\cdot ,t) - \mathcal{F}(u_{2},v_{2}) ( \cdot ,t) \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )}\leq \mathcal{K}_{1} \bigl( \Vert u_{1}-u_{2} \Vert _{\mathcal{H}^{\gamma }(\Omega )}+ \Vert v_{1}-u_{2} \Vert _{ \mathcal{H}^{\gamma }(\Omega )} \bigr), \end{aligned}$$
$$\begin{aligned} & \bigl\Vert \mathcal{G}(u_{1},v_{1}) (\cdot ,t) - \mathcal{G}(u_{2},v_{2}) ( \cdot ,t) \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )}\leq \mathcal{K}_{2} \bigl( \Vert u_{1}-u_{2} \Vert _{\mathcal{H}^{\gamma }(\Omega )}+ \Vert v_{1}-u_{2} \Vert _{ \mathcal{H}^{\gamma }(\Omega )} \bigr), \end{aligned}$$

where \((u_{1},v_{1})\in \mathscr{H}^{\gamma }(\Omega )\), \((u_{2},v_{2})\in \mathscr{H}^{\gamma }(\Omega )\).

Definition 4.1

\(\mathrm{w}= (u(\cdot ,t),v(\cdot ,t) )\in \mathscr{L}_{p}^{ \infty }(0,T,\mathcal{H}^{\gamma }(\Omega )) = \mathbb{L}_{p}^{\infty }{(0,T, \mathcal{H}^{\gamma }(\Omega ))}\times \mathbb{L}_{p}^{\infty }(0,T, \mathcal{H}^{\gamma }(\Omega ))\) is called a mild solution of problem (1.1) if it satisfies system (3.6).

Theorem 4.1

Let \(w_{0}(f,g)\in \mathscr{H}^{\gamma }(\Omega )\). Assume that \(1<\alpha _{1}<\alpha <\alpha '<\alpha _{2}<2\) and \(0<\nu _{0}<1\). The nonlinear integral equation (1.1) has a unique solution \(\mathrm{w}(u,v)\in \mathscr{L}_{p}^{\infty }(0,T;\mathcal{H}^{\gamma }( \Omega ))\). Let \(\mathrm{w}_{\alpha }\in \mathscr{L}_{p}^{\infty }(0,T;\mathcal{H}^{ \gamma }(\Omega ))\) and \(\mathrm{w}_{\alpha '}\in \mathscr{L}_{p}^{\infty }(0,T;\mathcal{H}^{ \gamma }(\Omega ))\) be two solutions of (1.1) with fractional order α and \(\alpha '\), respectively. If there exist numbers \(\mu _{0}\), ϵ satisfying \(0 < \epsilon <\min ({ \frac{1}{2}-\alpha _{2}+\alpha _{2}\nu _{0}, \alpha _{1}\nu _{0}-\frac{1}{2}} )\) and \(0<\mu _{0}<1-\frac{1}{2\alpha _{1}}\), then

$$\begin{aligned} & \Vert \mathbf{w} \Vert _{\mathscr{C}^{\alpha _{1}\mu _{0}}((0,T],\mathcal{H}^{ \gamma }(\Omega ))} \\ &\quad \leq \sqrt{ \mathscr{M}_{1}\bigl(\alpha _{1},\alpha _{2},a,\mu _{0},T, \Vert w_{0} \Vert _{\mathscr{H}^{\gamma }(\Omega )}\bigr)}\exp { \biggl( \frac{\mathscr{M}_{2}(\alpha _{1},\alpha _{2},a,\mu _{0},T)T)}{2} \biggr)}, \end{aligned}$$


$$\begin{aligned} & \Vert \mathbf{w}_{\alpha '}- \mathbf{w}_{\alpha } \Vert _{\mathscr{C}^{\alpha _{2}(1- \nu _{0})+\epsilon }((0,T],\mathcal{H}^{\gamma }(\Omega ))} \\ &\quad \leq \sqrt{\mathscr{M}^{\mathcal{C}_{1},\mathcal{C}_{2}}_{\mu _{0}, \nu _{0}}\bigl( \alpha _{1},\alpha _{2},\epsilon , a,T, \Vert w_{0} \Vert _{ \mathscr{H}^{\gamma }(\Omega )}\bigr)} \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({ \alpha }'- \alpha \bigr) \bigr] \\ &\qquad {} \times \exp \biggl( \frac{\mathscr{M}^{\mathcal{K}_{1},\mathcal{K}_{2}}_{\mu _{0},\nu _{0}}(\alpha _{1},\alpha _{2},\epsilon ,a,T)T}{2} \biggr). \end{aligned}$$

Proof of Theorem 4.1

We divide the proof into three parts.

Part 1. The existence and uniqueness of the solution of the nonlinear fractional pseudo-parabolic equation systems (1.1). For \(\mathbf{w} \in \mathscr{L}_{p}^{\infty }(0,T;\mathcal{H}^{\gamma }( \Omega ))\), we consider the following function \(\mathscr{H}\mathbf{w} := (\mathscr{H}_{\Delta ,\alpha }{u}( \cdot ,t), \mathscr{H}_{\Delta ,\alpha }{v}(\cdot ,t) ) \nonumber \), where

$$\begin{aligned} \textstyle\begin{cases} \mathscr{H}_{\Delta ,\alpha }{u}(\cdot ,t)=\mathcal{P}_{\Delta , \alpha }(t)f + \int _{0}^{t}\mathcal{Q}_{\Delta ,\alpha }(t-s) \mathcal{F}(u,v)(\cdot ,s)\,\mathrm{ds}, \\ \mathscr{H}_{\Delta ,\alpha }{v}(\cdot ,t)=\mathcal{P}_{\Delta , \alpha }(t)g + \int _{0}^{t}\mathcal{Q}_{\Delta ,\alpha }(t-s) \mathcal{G}(u,v)(\cdot ,s)\,\mathrm{ds}. \end{cases}\displaystyle \end{aligned}$$

Let \(\mathbf{w}_{1} ( u_{1}(\cdot ,t),v_{1}(\cdot ,t) )\in \mathscr{L}_{p}^{\infty }(0,T;\mathcal{H}^{\gamma }(\Omega ))\), \(\mathbf{w}_{2} (u_{2}(\cdot ,t),v_{2}(\cdot ,t) )\in \mathscr{L}_{p}^{ \infty }(0,T;\mathcal{H}^{\gamma }(\Omega ))\), we have

$$\begin{aligned} & \mathscr{H}_{\Delta ,\alpha }{u}_{1}(\cdot ,t) - \mathscr{H}_{ \Delta ,\alpha }{u}_{2}( \cdot ,t) = \int _{0}^{t}\mathcal{Q}_{ \Delta ,\alpha }(t-s) \bigl[ \mathcal{F}(u_{1},v_{1}) (\cdot ,s) - \mathcal{F}(u_{2},v_{2}) (\cdot ,s) \bigr]\,\mathrm{ds}, \\ &\mathscr{H}_{\Delta ,\alpha }{v}_{1}(\cdot ,t) - \mathscr{H}_{\Delta , \alpha }v_{2}( \cdot ,t) = \int _{0}^{t}\mathcal{Q}_{\Delta , \alpha }(t-s) \bigl[ \mathcal{G}(u_{1},v_{1}) (\cdot ,s) - \mathcal{G}(u_{2},v_{2}) ( \cdot ,s) \bigr]\,\mathrm{ds}. \end{aligned}$$

With \(p>0\), we get the following estimate:

$$\begin{aligned} & \bigl\Vert e^{-pt} \bigl(\mathscr{H}_{\Delta ,\alpha }{{u}}_{1}( \cdot ,t)- \mathscr{H}_{\Delta ,\alpha }{{u}}_{2}(\cdot ,t) \bigr) \bigr\Vert _{ \mathcal{H}^{\gamma }(\Omega )} \\ &\quad = \biggl\Vert \int _{0}^{t}\mathcal{Q}_{\Delta ,\alpha }(t-s)e^{-pt} \bigl[\mathcal{F}(u_{1},v_{1}) (\cdot ,s)- \mathcal{F}(u_{2},v_{2}) ( \cdot ,s) \bigr]{ds} \biggr\Vert _{\mathcal{H}^{\gamma }(\Omega )}. \end{aligned}$$

Applying Lemma 3.1, we get the following estimate:

$$\begin{aligned} & \bigl\Vert e^{-pt} \bigl(\mathscr{H}_{\Delta ,\alpha }{{u}}_{1}( \cdot ,t)- \mathscr{H}_{\Delta , \alpha }{{u}}_{2}(\cdot ,t) \bigr) \bigr\Vert ^{2}_{ \mathcal{H}^{\gamma }(\Omega )} \\ &\quad \leq \biggl[ \int _{0}^{t}\overline{\mathrm {M}}_{3}( \alpha _{1}, \alpha _{2},a,\mu _{0}) (t-s)^{\alpha _{1}-1-\alpha _{1} \mu _{0}}e^{-p(t-s)} \\ & \qquad {}\times \bigl[e^{-ps} \bigl\Vert \mathcal{F}(u_{1},v_{1}) (\cdot ,s)- \mathcal{F}(u_{2},v_{2}) (\cdot ,s) \bigr\Vert _{\mathcal{H}^{\alpha }( \Omega )} \bigr]\,ds \biggr]^{2} \\ &\quad \leq \overline{\mathrm {M}}^{2}_{3}(\alpha _{1},\alpha _{2},a,\mu _{0}) \biggl[ \int _{0}^{t}(t-s)^{\alpha _{1}-1-\alpha _{1} \mu _{0}}e^{-p(t-s)} \\ & \qquad {}\times \bigl[e^{-ps} \bigl\Vert \mathcal{F}(u_{1},v_{1}) (\cdot ,s)- \mathcal{F}(u_{2},v_{2}) (\cdot ,s) \bigr\Vert _{\mathcal{H}^{\gamma }( \Omega )} \bigr]\,ds \biggr]^{2} . \end{aligned}$$

Using (4.3) and the inequality \((m+n)^{2}\leq 2(m^{2}+n^{2})\), we obtain

$$\begin{aligned} & \bigl\Vert e^{-pt} \bigl(\mathscr{H}_{\Delta ,\alpha }{{u}}_{1}( \cdot ,t)- \mathscr{H}_{\Delta ,\alpha }{{u}}_{2}(\cdot ,t) \bigr) \bigr\Vert ^{2}_{ \mathcal{H}^{\gamma }(\Omega )} \\ &\quad \leq \mathcal{K}^{2}_{1}\overline{ \mathrm {M}}^{2}_{3}(\alpha _{1}, \alpha _{2},a,\mu _{0}) \biggl[ \int _{0}^{t}e^{-ps} \bigl( \Vert {u}_{1}-{u}_{2} \Vert _{\mathcal{H}^{\gamma }(\Omega )}+ \Vert {v}_{1}-{v}_{2} \Vert _{ \mathcal{H}^{\gamma }(\Omega )} \bigr) \\ & \qquad {}\times (t-s)^{\alpha _{1}- 1-\alpha _{1}\mu _{0}}e^{-p(t-s)}\,ds \biggr]^{2}. \end{aligned}$$

Using the inequality \((m+n)^{2}\leq 2(m^{2}+n^{2})\), we deduce that

$$\begin{aligned} & \Vert \mathscr{H}_{\Delta , \alpha }{{u}}_{1}-\mathscr{H}_{\Delta , \alpha }{{u}}_{2} \Vert ^{2}_{\mathbb{L}_{p}^{\infty }(0,T,\mathcal{H}^{ \gamma }(\Omega ))} \\ &\quad \leq \mathcal{K}^{2}_{1}\overline{ \mathrm {M}}^{2}_{3}(\alpha _{1}, \alpha _{2},a,\mu _{0}) \bigl( \Vert {u}_{1}-{u}_{2} \Vert _{\mathbb{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }(\Omega ))}+ \Vert {v}_{1}-{v}_{2} \Vert _{\mathbb{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }(\Omega ))} \bigr)^{2} \\ &\qquad {}\times \biggl[ \int _{0}^{t}(t-s)^{\alpha _{1}- 1-\alpha _{1} \mu _{0}}e^{-p(t-s)} \,ds \biggr]^{2} \\ &\quad \leq 2\mathcal{K}^{2}_{1}\overline{ \mathrm {M}}^{2}_{3}(\alpha _{1}, \alpha _{2},a,\mu _{0}) \bigl( \Vert {u}_{1}-{u}_{2} \Vert ^{2}_{ \mathbb{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }(\Omega ))}+ \Vert {v}_{1}-{v}_{2} \Vert ^{2}_{\mathbb{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }(\Omega ))} \bigr) \\ &\qquad {}\times \biggl[ \int _{0}^{t}(t-s)^{\alpha _{1}- 1-\alpha _{1} \mu _{0}}e^{-p(t-s)} \,ds \biggr]^{2} \\ &\quad \leq 2\mathcal{K}^{2}_{1}\overline{ \mathrm {M}}^{2}_{3}(\alpha _{1}, \alpha _{2},a,\mu _{0}) \Vert \mathbf{w}_{1}- \mathbf{w}_{2} \Vert ^{2}_{ \mathscr{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }(\Omega ))} \\ &\qquad {}\times \biggl[ \int _{0}^{t}(t-s)^{\alpha _{1}- 1-\alpha _{1} \mu _{0}}e^{-p(t-s)} \,ds \biggr]^{2}. \end{aligned}$$

Applying Hölder’s inequality with assumption \(\mu _{0}<1-\frac{1}{2\alpha _{1}}\), we obtain

$$\begin{aligned} & \Vert \mathscr{H}_{\Delta , \alpha }{{u}}_{1}-\mathscr{H}_{\Delta , \alpha }{{u}}_{2} \Vert ^{2}_{\mathbb{L}_{p}^{\infty }(0,T,\mathcal{H}^{ \gamma }(\Omega ))} \\ &\quad \leq 2\mathcal{K}^{2}_{1}\overline{ \mathrm {M}}^{2}_{3}(\alpha _{1}, \alpha _{2},a,\mu _{0}) \Vert \mathbf{w}_{1}- \mathbf{w}_{2} \Vert ^{2}_{ \mathscr{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }(\Omega ))} \\ &\qquad {}\times \int _{0}^{t}(t-s)^{2\alpha _{1}-2-2\alpha _{1}\mu _{0}} \,\mathrm{d}s \int _{0}^{t} e^{-2p(t-s)}\,\mathrm{d}s \\ &\quad \leq 2\mathcal{K}^{2}_{1}\overline{ \mathrm {M}}^{2}_{3}(\alpha _{1}, \alpha _{2},a,\mu _{0}) \Vert \mathbf{w}_{1}- \mathbf{w}_{2} \Vert ^{2}_{ \mathscr{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }(\Omega ))} \frac{t^{2\alpha _{1}-1-2\alpha _{1}\mu _{0}}}{ 2\alpha _{1}-1-2\alpha _{1}\mu _{0}} \frac{1-{e}^{-2pt}}{2p}. \end{aligned}$$


$$\begin{aligned} & \Vert \mathscr{H}_{\Delta , \alpha }{{u}}_{1}- \mathscr{H}_{\Delta , \alpha }{{u}}_{2} \Vert ^{2}_{\mathbb{L}_{p}^{\infty }(0,T,\mathcal{H}^{ \gamma }(\Omega ))} \\ &\quad \leq \frac{2\mathcal{K}^{2}_{1}\overline{\mathrm {M}}^{2}_{3}(\alpha _{1},\alpha _{2},a,\mu _{0})}{2p(2\alpha _{1}-2\alpha _{1}\mu _{0}-1)}T^{2 \alpha _{1}-2\alpha _{1}\mu _{0}-1}\bigl(1-{e}^{-2pT}\bigr) \Vert \mathbf{w}_{1}- \mathbf{w}_{2} \Vert ^{2}_{\mathscr{L}_{p}^{\infty }(0,T,\mathcal{H}^{ \gamma }(\Omega ))}. \end{aligned}$$

We can obtain a similar estimate

$$\begin{aligned} & \Vert \mathscr{H}_{\Delta , \alpha }\mathrm{v}_{1}- \mathscr{H}_{ \Delta ,\alpha }\mathrm{v}_{2} \Vert ^{2}_{\mathbb{L}_{p}^{\infty }(0,T, \mathcal{H}^{\gamma }(\Omega ))} \\ &\quad \leq \frac{2\mathcal{K}^{2}_{2}\overline{\mathrm {M}}^{2}_{3}(\alpha _{1},\alpha _{2},a,\mu _{0})}{2p(2\alpha _{1}-2\alpha _{1}\mu _{0}-1)}{ T^{2\alpha _{1}-2\alpha _{1}\mu _{0}-1}}\bigl(1-{e}^{-2pT} \bigr) \Vert \mathbf{w}_{1}-\mathbf{w}_{2} \Vert ^{2}_{\mathscr{L}_{p}^{\infty }(0,T, \mathcal{H}^{\gamma }(\Omega ))}. \end{aligned}$$

From (4.10)–(4.11), we find that

$$\begin{aligned} & \Vert \mathscr{H}\mathbf{w}_{1}-\mathscr{H}\mathbf{w}_{2} \Vert _{\mathscr{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }(\Omega ))} \\ &\quad = \sqrt{ \Vert \mathscr{H} {{u}}_{1}-\mathscr{H} {{u}}_{2} \Vert ^{2}_{ \mathbb{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }(\Omega ))}+ \Vert \mathscr{H} {{v}}_{1}-\mathscr{H} {{v}}_{2} \Vert ^{2}_{\mathbb{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }(\Omega ))}} \\ &\quad \leq \sqrt{ \frac{2(\mathcal{K}^{2}_{1}+\mathcal{K}^{2}_{2})\overline{\mathrm {M}}^{2}_{3}(\alpha _{1},\alpha _{2},a,\mu _{0})}{2p(2\alpha _{1}-2\alpha _{1}\mu _{0}-1)}T^{2 \alpha _{1}-\alpha _{1}\mu _{0}-1} \bigl(1-{e}^{-2pT}\bigr)} \\ &\qquad {}\times \Vert \mathbf{w}_{1}- \mathbf{w}_{2} \Vert _{\mathscr{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }( \Omega ))}. \end{aligned}$$

If \(\mathbf{w}_{2}=0\), then for any \(\mathbf{w} \in \mathscr{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }( \Omega ))\)

$$\begin{aligned} & \Vert \mathscr{H}\mathbf{w} \Vert _{\mathscr{L}_{p}^{\infty }(0,T, \mathcal{H}^{\gamma }(\Omega ))} \\ &\quad \le \Vert \mathscr{H} \mathbf{w}- \mathscr{H}\mathbf{w}_{2} \Vert _{\mathscr{L}_{p}^{\infty }(0,T, \mathcal{H}^{\gamma }(\Omega ))} + \Vert \mathscr{H}\mathbf{w}_{2} \Vert _{\mathscr{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }(\Omega ))} \\ &\quad \le \sqrt{ \frac{2(\mathcal{K}^{2}_{1}+\mathcal{K}^{2}_{2})\overline{\mathrm {M}}^{2}_{3}(\alpha _{1},\alpha _{2},a,\mu _{0})}{2p(2\alpha _{1}-2\alpha _{1}\mu _{0}-1)}T^{2 \alpha _{1}-\alpha _{1}\mu _{0}-1}\bigl(1-{e}^{-2pT} \bigr)} \Vert \mathbf{w} \Vert _{\mathscr{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }(\Omega ))} \\ &\qquad {}+ \Vert \mathscr{H}\mathbf{w}_{2} \Vert _{\mathscr{L}_{p}^{\infty }(0,T, \mathcal{H}^{\gamma }(\Omega ))}. \end{aligned}$$

Note that \(\mathscr{H}\mathbf{w}_{2} := (\mathscr{H}_{\Delta ,\alpha }{u}_{2}( \cdot ,t), \mathscr{H}_{\Delta ,\alpha }{v}_{2}(\cdot ,t)), \nonumber \) where \({u}_{2}= {v}_{2}=0\) and

$$\begin{aligned} \textstyle\begin{cases} \mathscr{H}_{\Delta ,\alpha }{u}_{2}(\cdot ,t)=\mathcal{P}_{\Delta , \alpha }(t)f, \\ \mathscr{H}_{\Delta ,\alpha }{v}_{2}(\cdot ,t)=\mathcal{P}_{\Delta , \alpha }(t)g. \end{cases}\displaystyle \end{aligned}$$

By applying Lemma 3.1, we can deduce that \(\mathscr{H}\mathbf{w}_{2} \in \mathscr{L}_{p}^{\infty }(0,T, \mathcal{H}^{\gamma }(\Omega )) \). Therefore, we conclude that if any \(\mathbf{w} \in \mathscr{L}_{p}^{\infty }(0,T,\mathcal{H}^{\gamma }( \Omega ))\), then \(\mathscr{H}\mathbf{w}\) is bounded.

Part 2. From (3.6) and applying the inequality \((a+b)^{2} \leq 2(a^{2}+b^{2})\), we have the following estimate:

$$\begin{aligned} \bigl\Vert {u}(\cdot ,t) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} &\leq 2 \bigl\Vert \mathcal{P}_{\Delta ,\alpha }(t)f \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} + 2 \biggl\Vert \int _{0}^{t}\mathcal{Q}_{\Delta ,\alpha }(t-s) \mathcal{F}(u,v) (\cdot ,s)\,ds \biggr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )}. \end{aligned}$$

Using Lemma 3.1, we get

$$\begin{aligned} \bigl\Vert {u}(\cdot ,t) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )}&\leq 2 \overline{\mathrm {M}}_{2}(\alpha _{1},\alpha _{2},a,\mu _{0})t^{-2 \alpha _{1} \mu _{0}} \Vert f \Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \\ &\quad{}+2 \biggl( \int _{0}^{t}\overline{\mathrm {M}}_{3}( \alpha _{1}, \alpha _{2},a,\mu _{0}) (t-s)^{\alpha _{1}-1-\alpha _{1} \mu _{0}} \bigl\Vert \mathcal{F}\bigl((u,v) (\cdot ,s)\bigr) \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )} \,\mathrm{d}s \biggr)^{2}. \end{aligned}$$

Multiplying both sides by \(t^{2\alpha _{1}\mu _{0}}\) and using Hölder’s inequality with assumption (4.1), we can find that

$$\begin{aligned} & \bigl(t^{2\alpha _{1}\mu _{0}} \bigl\Vert {u}(\cdot ,t) \bigr\Vert _{\mathcal{H}^{\gamma }( \Omega )} \bigr)^{2} \\ &\quad \leq 2\overline{\mathrm {M}}_{2}( \alpha _{1}, \alpha _{2},a,\mu _{0}) \Vert f \Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )}+2 \overline{\mathrm {M}}_{3}( \alpha _{1},\alpha _{2},a,\mu _{0})t^{2 \alpha _{1}\mu _{0}} \\ &\qquad{}\times \biggl( \int _{0}^{t} \bigl(1+ \bigl\Vert {u}(\cdot ,s) \bigr\Vert _{ \mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert {v}(\cdot ,s) \bigr\Vert _{\mathcal{H}^{\gamma }( \Omega )} \bigr)s^{\alpha _{1}\mu _{0}}s^{-\alpha _{1}\mu _{0}}(t-s)^{ \alpha _{1}-1-\alpha _{1}\mu _{0}}\, \mathrm{ds} \biggr)^{2} \\ &\quad \leq 2\overline{\mathrm {M}}_{2}(\alpha _{1},\alpha _{2},a,\mu _{0}) \Vert f \Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )}+2 \overline{\mathrm {M}}_{3}( \alpha _{1},\alpha _{2},a,\mu _{0})t^{2\alpha _{1}\mu _{0}} \\ &\qquad{}\times \int _{0}^{t} \bigl(1+ \bigl\Vert {u}(\cdot ,s) \bigr\Vert _{\mathcal{H}^{ \gamma }(\Omega )}+ \bigl\Vert {v}(\cdot ,s) \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )} \bigr)^{2}s^{2\alpha _{1}\mu _{0}}\,\mathrm{ds} \int _{0}^{t}s^{-2 \alpha _{1}\mu _{0}}(t-s)^{2\alpha _{1}-2-2\alpha _{1}\mu _{0}} \, \mathrm{ds} . \end{aligned}$$

Using the beta function property \(\int _{0}^{t} s^{\theta _{1}-1}(t-s)^{\vartheta _{1}-1}\,ds=t^{ \theta _{1}+\vartheta _{1}-1}\mathbf{B}(\theta _{1},\vartheta _{1})\), \(\theta _{1} > 0\), \(\vartheta _{1} > 0\) with assumption \(0<\mu _{0}<1-\frac{1}{2\alpha _{1}}\), then \(2\alpha _{1}-1-2\alpha _{1}\mu _{0}>0\) and \(-2\alpha _{1}\mu _{0}+1>0\), we obtain

$$\begin{aligned} \bigl(t^{2\alpha _{1}\mu _{0}} \bigl\Vert {\mathbf{u}}(\cdot ,t) \bigr\Vert _{\mathcal{H}^{ \gamma }(\Omega )} \bigr)^{2} &\leq 2\overline{\mathrm {M}}_{2}( \alpha _{1}, \alpha _{2},a,\mu _{0}) \Vert f \Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \\ &\quad{}+2\overline{\mathrm {M}}_{3}(\alpha _{1},\alpha _{2},a,\mu _{0}t^{2 \alpha _{1}(1-\mu _{0})}{ \mathbf{B}(\theta _{1},\vartheta _{1})} \\ & \quad {} \times \int _{0}^{t} \bigl(1+ \bigl\Vert {u}(\cdot ,s) \bigr\Vert _{\mathcal{H}^{ \gamma }(\Omega )}+ \bigl\Vert {v}(\cdot ,s) \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )} \bigr)^{2}s^{2\alpha _{1}\mu _{0}}\,\mathrm{ds}, \end{aligned}$$

where \(\theta _{1}:=-2\alpha _{1}\mu _{0}+1\), \(\vartheta _{1}:= 2\alpha _{1}-1-2 \alpha _{1}\mu _{0}\). Using the inequality \((m+n+p)^{2}\leq 3(m^{2}+n^{2}+p^{2})\), we get

$$\begin{aligned} & \bigl(t^{2\alpha _{1}\mu _{0}} \bigl\Vert {u}(\cdot ,t) \bigr\Vert _{\mathbb{H}^{\eta }( \Omega )} \bigr)^{2} \\ &\quad \leq 2\overline{\mathrm {M}}_{2}( \alpha _{1}, \alpha _{2},a,\mu _{0}) \Vert f \Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \\ &\qquad{}+6\overline{\mathrm {M}}_{3}(\alpha _{1},\alpha _{2},a,\mu _{0})t^{2 \alpha _{1}(1-\mu _{0})}\mathbf{B}(\theta _{1},\vartheta _{1}) \\ & \qquad {}\times \int _{0}^{t} \bigl(1+ \bigl\Vert {u}(\cdot ,s) \bigr\Vert ^{2}_{\mathcal{H}^{ \gamma }(\Omega )}+ \bigl\Vert {v}(\cdot ,s) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \bigr)s^{2\alpha _{1}\mu _{0}}\,\mathrm{ds} \\ &\quad \leq 2\overline{\mathrm {M}}_{2}(\alpha _{1},\alpha _{2},a,\mu _{0}) \Vert f \Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )}+ \frac{6\overline{\mathrm {M}}_{3}(\alpha _{1},\alpha _{2},a,\mu _{0})t^{2\alpha _{1}\mu _{0}+1}\mathbf{B}(\theta _{1},\vartheta _{1})}{2\alpha _{1}\mu _{0}+1} \\ &\qquad{}+6\overline{\mathrm {M}}_{3}(\alpha _{1},\alpha _{2},a,\mu _{0})t^{2 \alpha _{1}(1-\mu _{0})}\mathbf{B}(\theta _{1},\vartheta _{1}) \\ & \qquad {}\times \int _{0}^{t} \bigl( \bigl\Vert {u}(\cdot ,s) \bigr\Vert ^{2}_{\mathbb{H}^{ \eta }(\Omega )}+ \bigl\Vert {v}(\cdot ,s) \bigr\Vert ^{2}_{\mathbb{H}^{\eta }(\Omega )} \bigr)s^{2\alpha _{1}\mu _{0}}\,\mathrm{ds}. \end{aligned}$$

Similarly, we can also obtain

$$\begin{aligned} & \bigl(t^{2\alpha _{1}\mu _{0}} \bigl\Vert {v}(\cdot ,t) \bigr\Vert _{\mathcal{H}^{\gamma }( \Omega )} \bigr)^{2} \\ &\quad \leq 2\overline{\mathrm {M}}_{2}(\alpha _{1},\alpha _{2},a,\mu _{0}) \Vert g \Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )}+ \frac{6\overline{\mathrm {M}}_{3}(\alpha _{1},\alpha _{2},a,\mu _{0})t^{2\alpha _{1}\mu _{0}+1}\mathbf{B}(\theta _{1},\vartheta _{1})}{2\alpha _{1}\mu _{0}+1} \\ &\qquad{}+6\overline{\mathrm {M}}_{3}(\alpha _{1},\alpha _{2},a,\mu _{0})t^{2 \alpha _{1}(1-\mu _{0})}\mathbf{B}(\theta _{1},\vartheta _{1}) \\ &\qquad {}\times\int _{0}^{t} \bigl( \bigl\Vert {u}(\cdot ,s) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }( \Omega )}+ \bigl\Vert {v}(\cdot ,s) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \bigr)s^{2 \alpha _{1}\mu _{0}}\,\mathrm{ds}. \end{aligned}$$

From (4.15) and (4.16), we arrive at

$$\begin{aligned} & \bigl( \bigl\Vert {u}(\cdot ,t) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert {v}( \cdot ,t) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \bigr)t^{2\alpha _{1} \mu _{0}} \\ &\quad \leq { \mathscr{M}_{1}\bigl(\alpha _{1}, \alpha _{2},a,\mu _{0},T, \Vert w_{0} \Vert _{\mathscr{H}^{\gamma }(\Omega )}\bigr) } \\ &\qquad{}+\mathscr{M}_{2}(\alpha _{1},\alpha _{2},a,\mu _{0},T) \int _{0}^{t} \bigl( \bigl\Vert {u}(\cdot ,s) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert {v}( \cdot ,s) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \bigr)s^{2\alpha _{1} \mu _{0}}\,ds, \end{aligned}$$


$$\begin{aligned} & \begin{aligned} \mathscr{M}_{1}\bigl(\alpha _{1},\alpha _{2},a, \mu _{0},T, \Vert w_{0} \Vert _{ \mathscr{H}^{\gamma }(\Omega )}\bigr)&:=2 \overline{\mathrm {M}}_{2}(\alpha _{1}, \alpha _{2},a,\mu _{0}) \Vert w_{0} \Vert ^{2}_{\mathscr{H}^{\gamma }(\Omega )} \\ & \quad {}+ \frac{12\overline{\mathrm {M}}_{3}(\alpha _{1},\alpha _{2},a,\mu _{0})T^{2\alpha _{1}\mu _{0}+1}\mathbf{B}(\theta _{1},\vartheta _{1})}{2\alpha _{1}\mu _{0}+1}, \end{aligned} \\ & \mathscr{M}_{2}(\alpha _{1},\alpha _{2},a,\mu _{0},T):=12 \overline{\mathrm {M}}_{3}(\alpha _{1},\alpha _{2},a,\mu _{0})T^{2 \alpha _{1}(1-\mu _{0})} \mathbf{B}(\theta _{1},\vartheta _{1}). \end{aligned}$$

Applying \(\|w_{0}\|^{2}_{\mathscr{H}^{\gamma }(\Omega )}= \|f\|^{2}_{ \mathcal{H}^{\gamma }(\Omega )}+\|g\|^{2}_{\mathcal{H}^{\gamma }( \Omega )}\) and Gronwall’s inequality, we get

$$\begin{aligned} & \Vert u \Vert ^{2}_{\mathbb{C}^{\alpha _{1}\mu _{0}}((0,T],\mathcal{H}^{ \gamma }(\Omega ))} + \Vert {v} \Vert ^{2}_{\mathbb{C}^{\alpha _{1}\mu _{0}}((0,T], \mathcal{H}^{\gamma }(\Omega ))} \\ &\quad \leq \mathscr{M}_{1} \bigl(\alpha _{1},\alpha _{2},a,\mu _{0},T, \Vert w_{0} \Vert _{\mathscr{H}^{\gamma }(\Omega )} \bigr)\exp { \bigl(\mathscr{M}_{2}( \alpha _{1},\alpha _{2},a,\mu _{0},T)T \bigr)}. \end{aligned}$$

Therefore, we get

$$\begin{aligned} & \Vert \mathbf{w} \Vert _{\mathscr{C}^{\alpha _{1}\mu _{0}}((0,T],\mathcal{H}^{ \gamma }(\Omega ))} \\ &\quad = \sqrt{ \Vert u \Vert ^{2}_{\mathbb{C}^{\alpha _{1}\mu _{0}}((0,T], \mathcal{H}^{\gamma }(\Omega ))} + \Vert {v} \Vert ^{2}_{\mathbb{C}^{\alpha _{1} \mu _{0}}((0,T],\mathcal{H}^{\gamma }(\Omega ))}} \\ &\quad \leq \sqrt{ \mathscr{M}_{1} \bigl(\alpha _{1},\alpha _{2},a,\mu _{0},T, \Vert w_{0} \Vert _{\mathscr{H}^{\gamma }(\Omega )} \bigr)}\exp { \biggl( \frac{\mathscr{M}_{2}(\alpha _{1},\alpha _{2},a,\mu _{0},T)T)}{2} \biggr)}. \end{aligned}$$

Part 3. From equation (3.6), we then obtain

$$ \textstyle\begin{cases} {u}_{\alpha }(\cdot ,t) = \mathcal{P}_{\Delta ,\alpha }(t)f + \int _{0}^{t}\mathcal{Q}_{\Delta ,\alpha }(t-s)\mathcal{F}(u_{ \alpha },v_{\alpha })(\cdot ,s)\,\mathrm{ds}, \\ {v}_{\alpha }(\cdot ,t) = \mathcal{P}_{\Delta ,\alpha }(t)g + \int _{0}^{t}\mathcal{Q}_{\Delta ,\alpha }(t-s)\mathcal{G}(u_{ \alpha },v_{\alpha })(\cdot ,s)\,\mathrm{ds}, \end{cases} $$


$$ \textstyle\begin{cases} {u}_{\alpha '}(\cdot ,t) = \mathcal{P}_{\Delta ,\alpha '}(t)f + \int _{0}^{t}\mathcal{Q}_{\Delta ,\alpha '}(t-s)\mathcal{F}(u_{ \alpha '},v_{\alpha '})(\cdot ,s)\,\mathrm{ds}, \\ {v}_{\alpha '}(\cdot ,t) = \mathcal{P}_{\Delta ,\alpha '}(t)g + \int _{0}^{t}\mathcal{Q}_{\Delta ,\alpha '}(t-s)\mathcal{G}(u_{ \alpha '},v_{\alpha '})(\cdot ,s)\,\mathrm{ds}. \end{cases} $$

Using (4.19) and (4.20), we get

$$\begin{aligned} &{u}_{\alpha '}(\cdot ,t) - {u}_{\alpha }(\cdot ,t) \\ &\quad = \bigl[\mathcal{P}_{\Delta ,\alpha '}(t) -\mathcal{P}_{\Delta , \alpha }(t) \bigr]f + \int _{0}^{t} \bigl[\mathcal{Q}_{\Delta , \alpha '}(t-s)- \mathcal{Q}_{\Delta ,\alpha }(t-s) \bigr]\mathcal{F}(u_{ \alpha '},v_{\alpha '}) (\cdot ,s)\,\mathrm{ds} \\ &\qquad {}+ \int _{0}^{t}\mathcal{Q}_{\Delta ,\alpha }(t-s) \bigl[ \mathcal{F}(u_{\alpha '},v_{\alpha '}) (\cdot ,s)-\mathcal{F}(u_{ \alpha },v_{\alpha }) (\cdot ,s) \bigr]\,\mathrm{ds}, \end{aligned}$$
$$\begin{aligned} &{v}_{\alpha '}(\cdot ,t) - {v}_{\alpha }(\cdot ,t) \\ &\quad = \bigl[\mathcal{P}_{\Delta ,\alpha '}(t) -\mathcal{P}_{\Delta , \alpha }(t) \bigr]g + \int _{0}^{t} \bigl[\mathcal{Q}_{\Delta , \alpha '}(t-s)- \mathcal{Q}_{\Delta ,\alpha }(t-s) \bigr]\mathcal{G}(u_{ \alpha '},v_{\alpha '}) (\cdot ,s)\,\mathrm{ds} \\ &\qquad {}+ \int _{0}^{t}\mathcal{Q}_{\Delta ,\alpha }(t-s) \bigl[ \mathcal{G}(u_{\alpha '},v_{\alpha '}) (\cdot ,s)-\mathcal{G}(u_{ \alpha },v_{\alpha }) (\cdot ,s) \bigr]\,\mathrm{ds}. \end{aligned}$$

Applying the inequality \((m+n+p)^{2}\le 3(m^{2}+n^{2}+p^{2})\), we get the following estimate:

$$\begin{aligned} & \bigl\Vert {u}_{\alpha '}(\cdot ,t) - {u}_{\alpha }(\cdot ,t) \bigr\Vert ^{2}_{ \mathcal{H}^{\gamma }(\Omega )} \\ &\quad \leq 3 \bigl\Vert \bigl[ \mathcal{P}_{\Delta , \alpha '}(t) -\mathcal{P}_{\Delta ,\alpha }(t) \bigr]f \bigr\Vert ^{2}_{ \mathcal{H}^{\gamma }(\Omega )} \\ &\qquad {} +3 \biggl\Vert \int _{0}^{t} \bigl[\mathcal{Q}_{\Delta ,\alpha '}(t-s)- \mathcal{Q}_{\Delta ,\alpha }(t-s) \bigr]\mathcal{F}(u_{\alpha '},v_{ \alpha '}) (\cdot ,s)\,\mathrm{ds} \biggr\Vert ^{2}_{\mathcal{H}^{\gamma }( \Omega )} \\ &\qquad {}+3 \biggl\Vert \int _{0}^{t}\mathcal{Q}_{\Delta ,\alpha }(t-s) \bigl[ \mathcal{F}(u_{\alpha '},v_{\alpha '}) (\cdot ,s)-\mathcal{F}(u_{ \alpha },v_{\alpha }) (\cdot ,s) \bigr]\,\mathrm{ds} \biggr\Vert ^{2}_{\mathcal{H}^{ \gamma }(\Omega )}. \end{aligned}$$

Using Lemmas 3.1 and 3.2, we obtain

$$\begin{aligned} & \bigl\Vert {u}_{\alpha '}(\cdot ,t) - {u}_{\alpha }(\cdot ,t) \bigr\Vert ^{2}_{ \mathcal{H}^{\gamma }(\Omega )} \\ &\quad \leq 3 \bigl(\overline{\mathrm {A}}(\alpha _{1},\alpha _{2},\epsilon , \nu _{0},a,T) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'- \alpha \bigr) \bigr]t^{-\alpha _{2}(1-\nu _{0})-\epsilon } \Vert f \Vert _{\mathcal{H}^{\gamma }( \Omega )} \bigr)^{2} \\ &\qquad{}+ 3\overline{\mathrm {B}}^{2}(\alpha _{1},\alpha _{2},\epsilon , \nu _{0},a,T) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'- \alpha \bigr) \bigr] \\ & \qquad {}\times \biggl( \int _{0}^{t}(t-s)^{\alpha _{1}\nu _{0}- \epsilon -1} \bigl\Vert \mathcal{F}(u_{\alpha '},v_{\alpha '}) (\cdot ,s) \bigr\Vert _{ \mathcal{H}^{\gamma }(\Omega )}\,\mathrm{ds} \biggr)^{2} \\ &\qquad{}+ 3\overline{\mathrm {M}}^{2}_{3}(\alpha _{1},\alpha _{2},a,\mu _{0}) \\ &\qquad{}\times \biggl( \int _{0}^{t}(t-s)^{\alpha _{1}-1-\alpha _{1} \mu _{0}} \bigl\Vert \mathcal{F}(u_{\alpha '},v_{\alpha '}) (\cdot ,s)- \mathcal{F}(u_{\alpha },v_{\alpha }) (\cdot ,s) \bigr\Vert _{\mathcal{H}^{\gamma }( \Omega )}\,\mathrm{ds} \biggr)^{2}. \end{aligned}$$

Using assumptions (4.1) and (4.3), we obtain

$$\begin{aligned} & \bigl\Vert {u}_{\alpha '}(\cdot ,t)- {u}_{\alpha }(\cdot ,t) \bigr\Vert ^{2}_{\mathbb{H}^{ \eta }(\Omega )} \\ &\quad \leq 3 \bigl(\overline{\mathrm {A}}(\alpha _{1},\alpha _{2},\epsilon , \nu _{0},a,T) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'- \alpha \bigr) \bigr]t^{-\alpha _{2}(1-\nu _{0})-\epsilon } \Vert f \Vert _{\mathcal{H}^{\gamma }( \Omega )} \bigr)^{2} \\ &\qquad{}+3\mathcal{C}^{2}_{1}\overline{ \mathrm {B}}^{2}(\alpha _{1},\alpha _{2}, \epsilon ,\nu _{0},a,T) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'- \alpha \bigr) \bigr]^{2}I_{1} \\ &\qquad{}+ 3\mathcal{K}^{2}_{1}\overline{ \mathrm {M}}^{2}_{3}(\alpha _{1}, \alpha _{2},a,\mu _{0})I_{2}, \end{aligned}$$


$$\begin{aligned}& I_{1}:= \biggl( \int _{0}^{t}(t-s)^{\alpha _{1}\nu _{0}- \epsilon -1} \bigl(1+ \bigl\Vert u_{\alpha '}(\cdot ,t) \bigr\Vert _{\mathcal{H}^{\gamma }( \Omega )}+ \bigl\Vert v_{\alpha '}(\cdot ,t) \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )} \bigr)\,\mathrm{ds} \biggr)^{2}, \end{aligned}$$
$$\begin{aligned}& I_{2}: = \biggl( \int _{0}^{t}(t-s)^{\alpha _{1}-1-\alpha _{1} \mu _{0}} \bigl( \Vert u_{\alpha '}-u_{\alpha } \Vert _{\mathcal{H}^{\gamma }( \Omega )}+ \Vert v_{\alpha '}-v_{\alpha } \Vert _{\mathcal{H}^{\gamma }(\Omega )} \bigr)\,\mathrm{ds} \biggr)^{2}. \end{aligned}$$

Multiplying both sides by \(t^{2\alpha _{2}(1-\nu _{0})+2\epsilon }\), we get

$$\begin{aligned} &t^{2\alpha _{2}(1-\nu _{0})+2\epsilon } \bigl\Vert {u}_{\alpha '}(\cdot ,t)- {u}_{ \alpha }(\cdot ,t) \bigr\Vert ^{2}_{\mathbb{H}^{\eta }(\Omega )} \\ &\quad \leq 3 \bigl(\overline{\mathrm {A}}(\alpha _{1},\alpha _{2},\epsilon , \nu _{0},a,T) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'- \alpha \bigr) \bigr] \Vert f \Vert _{\mathcal{H}^{\gamma }(\Omega )} \bigr)^{2} \\ &\qquad{}+3\mathcal{K}^{2}_{1}\overline{ \mathrm {M}}^{2}_{3}(\alpha _{1}, \alpha _{2},a,\mu _{0})t^{2\alpha _{2}(1-\mu _{0})+2\epsilon }I_{2} \\ &\qquad{}+3\mathcal{C}^{2}_{1}\overline{ \mathrm {B}}^{2}(\alpha _{1},\alpha _{2}, \epsilon ,\nu _{0},a,T) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'- \alpha \bigr) \bigr]^{2}t^{2\alpha _{2}(1-\nu _{0})+2\epsilon }I_{1}. \end{aligned}$$

Now we estimate \(I_{1}\), from (4.26) we get

$$\begin{aligned} I_{1} &= \biggl( \int _{0}^{t}(t-s)^{\alpha _{1}\nu _{0}- \epsilon -1}s^{-\alpha _{1}\mu _{0}} \\ &\quad {}\times s^{\alpha _{1}\mu _{0}} \bigl(1+ \bigl\Vert u_{\alpha '}(\cdot ,t) \bigr\Vert _{ \mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert v_{\alpha '}(\cdot ,t) \bigr\Vert _{ \mathcal{H}^{\gamma }(\Omega )} \bigr)\,\mathrm{ds} \biggr)^{2}. \end{aligned}$$

We assume that \(0<\epsilon <\alpha _{1}\nu _{0}-\frac{1}{2}\) and \(\mu _{0}<\frac{1}{2\alpha _{1}}\), then \(2\alpha _{1}\nu _{0}-2\epsilon -1>0\), \(1-2\alpha _{1}\mu _{0}>0\). Using Hölder’s inequality and the properties of beta function, we obtain that

$$\begin{aligned} I_{1}&\leq \int _{0}^{t}(t-s)^{2\alpha _{1}\nu _{0}-2 \epsilon -2}s^{-2\alpha _{1}\mu _{0}}\, \mathrm{ds} \\ &\quad {}\times \int _{0}^{t}s^{2\alpha _{1}\mu _{0}} \bigl(1+ \bigl\Vert u_{ \alpha '}(\cdot ,t) \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert v_{\alpha '}( \cdot ,t) \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )} \bigr)^{2}\, \mathrm{ds} \\ &\leq t^{2\alpha _{1}\nu _{0}-2\alpha _{1}\mu _{0}-2\epsilon } \mathbf{B}(\theta _{2},\vartheta _{2}) \\ &\quad {}\times \int _{0}^{t}s^{2\alpha _{1}\mu _{0}} \bigl(1+ \bigl\Vert u_{ \alpha '}(\cdot ,t) \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert v_{\alpha '}( \cdot ,t) \bigr\Vert _{\mathcal{H}^{\gamma }(\Omega )} \bigr)^{2}\, \mathrm{ds}, \end{aligned}$$

where \(\theta _{2}:=2\alpha _{1}\nu _{0}-2\epsilon -1\), \(\vartheta _{2}:=1-2 \alpha _{1}\mu _{0} \).

Applying the inequality \((m+n+p)^{2}\leq 3(m^{2}+n^{2}+p^{2})\), we can deduce

$$\begin{aligned} I_{1}&\leq 3t^{2\alpha _{1}\nu _{0}-2\alpha _{1}\mu _{0}-2\epsilon } \mathbf{B}(\theta _{2}, \vartheta _{2}) \int _{0}^{t}s^{2\alpha _{1} \mu _{0}} \bigl(1+ \Vert u_{\alpha '} \Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega ))} + \Vert { v_{\alpha '}} \Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega ))} \bigr) \,\mathrm{ds} \\ &\leq 3t^{2\alpha _{1}\nu _{0}-2\alpha _{1}\mu _{0}-2\epsilon } \mathbf{B}(\theta _{2},\vartheta _{2}) \\ &\quad{}\times \int _{0}^{t} \bigl(s^{2\alpha _{1}\mu _{0}}+ \Vert u_{ \alpha '} \Vert ^{2}_{\mathbb{C}^{\alpha _{1}\mu _{0}}((0,T],\mathcal{H}^{ \gamma }(\Omega ))} + \Vert { v_{\alpha '}} \Vert ^{2}_{\mathbb{C}^{\alpha _{1} \mu _{0}}((0,T],\mathcal{H}^{\gamma }(\Omega ))} \bigr)\,\mathrm{ds}. \end{aligned}$$

Therefore, from (4.18), we have the following estimate:

$$\begin{aligned} I_{1} &\leq \frac{3t^{2\alpha _{1}\nu _{0}+1-2\epsilon }\mathbf{B}(\theta _{2},\vartheta _{2})}{2\alpha _{1}\mu _{0}+1} \\ &\quad {}+3t^{2\alpha _{1}\nu _{0}-2\alpha _{1}\mu _{0}-2\epsilon +1} \mathbf{B}(\theta _{2},\vartheta _{2}) \Vert \mathbf{w}_{\alpha '} \Vert ^{2}_{ \mathscr{C}^{\alpha _{1}\mu _{0}}((0,T],\mathcal{H}^{\gamma }(\Omega ))} \\ &\leq \frac{3T^{2\alpha _{1}\nu _{0}+1-2\epsilon }\mathbf{B}(\theta _{2},\vartheta _{2})}{2\alpha _{1}\mu _{0}+1}+3T^{2 \alpha _{1}\nu _{0}-2\alpha _{1}\mu _{0}-2\epsilon +1}\mathbf{B}( \theta _{2}, \vartheta _{2}) \\ &\quad {} \times \mathscr{M}_{1}\bigl(\alpha _{1},\alpha _{2},a,\mu _{0},T, \Vert w_{0} \Vert _{ \mathscr{H}^{\gamma }(\Omega )}\bigr)\exp { \bigl(\mathscr{M}_{2}( \alpha _{1}, \alpha _{2},a,\mu _{0},T)T \bigr)}. \end{aligned}$$

To facilitate the calculation, we set

$$\begin{aligned} &\mathscr{M}\bigl(\alpha _{1},\alpha _{2},a,\mu _{0},\nu _{0},T, \Vert w_{0} \Vert _{ \mathscr{H}^{\gamma }(\Omega )}\bigr)\\ &\quad := \frac{3T^{2\alpha _{1}\nu _{0}+1-2\epsilon }\mathbf{B}(\theta _{2},\vartheta _{2})}{2\alpha _{1}\mu _{0}+1} \\ &\qquad{}+3T^{2\alpha _{1}\nu _{0}-2\alpha _{1}\mu _{0}-2\epsilon +1} \mathbf{B}(\theta _{2},\vartheta _{2}) \\ &\qquad {}\times \mathscr{M}_{1}\bigl(\alpha _{1},\alpha _{2},a,\mu _{0},T, \Vert w_{0} \Vert _{ \mathscr{H}^{\gamma }(\Omega )}\bigr)\exp { \bigl(\mathscr{M}_{2}( \alpha _{1}, \alpha _{2},a,\mu _{0},T)T \bigr)}. \end{aligned}$$


$$\begin{aligned} I_{1} \leq \mathscr{M}\bigl(\alpha _{1}, \alpha _{2},a,\mu _{0},\nu _{0},T, \Vert w_{0} \Vert _{\mathscr{H}^{\gamma }(\Omega )}\bigr). \end{aligned}$$

Next, estimate \(I_{2}\). From (4.27), applying Hölder’s inequality, we deduce

$$\begin{aligned} I_{2}&= \biggl( \int _{0}^{t}(t-s)^{\alpha _{1}-1-\alpha _{1} \mu _{0}}s^{-\alpha _{2}(1-\nu _{0})-\epsilon } \\ &\quad {} \times s^{\alpha _{2}(1-\nu _{0})+\epsilon } \bigl( \Vert u_{\alpha '}-u_{ \alpha } \Vert _{\mathcal{H}^{\gamma }(\Omega )}+ \Vert v_{\alpha '}-v_{\alpha } \Vert _{ \mathcal{H}^{\gamma }(\Omega )} \bigr)\,\mathrm{ds} \biggr)^{2} \\ &\leq \int _{0}^{t}(t-s)^{2\alpha _{1}-2-2\alpha _{1} \mu _{0}}s^{-2 \alpha _{2}(1-\nu _{0})-2\epsilon }\, \mathrm{ds} \\ &\quad{}\times \int _{0}^{t}s^{2\alpha _{2}(1-\nu _{0})+2\epsilon } \bigl( \bigl\Vert {u}_{\alpha '}(\cdot ,s)-{u}_{\alpha }(\cdot ,s) \bigr\Vert _{\mathcal{H}^{ \gamma }(\Omega )}+ \bigl\Vert {v}_{\alpha '}(\cdot ,s)-{v}_{\alpha }( \cdot ,s) \bigr\Vert _{ \mathcal{H}^{\gamma }(\Omega )} \bigr)^{2}\,\mathrm{ds} \\ &\leq \int _{0}^{t}(t-s)^{2\alpha _{1}-2-2\alpha _{1} \mu _{0}}s^{-2 \alpha _{2}(1-\nu _{0})-2\epsilon }\, \mathrm{ds} \\ &\quad{}\times 2 \int _{0}^{t}s^{2\alpha _{2}(1-\nu _{0})+2\epsilon } \bigl( \bigl\Vert {u}_{\alpha '}(\cdot ,s)-{u}_{\alpha }(\cdot ,s) \bigr\Vert ^{2}_{ \mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert {v}_{\alpha '}(\cdot ,s)-{v}_{\alpha }( \cdot ,s) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \bigr)\,\mathrm{ds} . \end{aligned}$$

Using the beta function property with assumption \(\mu _{0}<1-\frac{1}{2\alpha _{1}}\) and \(0 < \epsilon <\min ( \frac{1}{2}-\alpha _{2}+\alpha _{2}\nu _{0}, \alpha _{1}\nu _{0}-\frac{1}{2} )\), we obtain

$$\begin{aligned} I_{2}&\leq 2t^{2\alpha _{1}-2\alpha _{1}\mu _{0}-2\alpha _{2}+2 \alpha _{2}\nu _{0}-2\epsilon }\mathbf{B}(\theta _{3}, \vartheta _{3}) \\ &\quad{}\times \int _{0}^{t}s^{2\alpha _{2}(1-\nu _{0})+2\epsilon } \bigl( \bigl\Vert {u}_{\alpha '}(\cdot ,s)-{u}_{\alpha }(\cdot ,s) \bigr\Vert ^{2}_{ \mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert {v}_{\alpha '}(\cdot ,s)-{v}_{\alpha }( \cdot ,s) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \bigr)\,\mathrm{ds}, \end{aligned}$$


$$ \theta _{3}:=-2\alpha _{1}-1-2\alpha _{1}\mu _{0},\qquad \vartheta _{3}:=1-2 \alpha _{1}(1-\nu _{0})-2\epsilon . $$

And so, we get

$$\begin{aligned} I_{2}&\leq 2t^{2\alpha _{1}-2\alpha _{1}\mu _{0}-2\alpha _{2}+2 \alpha _{2}\nu _{0}-2\epsilon }\mathbf{B}(\theta _{3},\vartheta _{3}) \\ &\quad{}\times \int _{0}^{t}s^{2\alpha _{2}(1-\nu _{0})+2\epsilon } \bigl( \bigl\Vert {u}_{\alpha '}(\cdot ,s)-{u}_{\alpha }(\cdot ,s) \bigr\Vert ^{2}_{ \mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert {v}_{\alpha '}(\cdot ,s)-{v}_{\alpha }( \cdot ,s) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \bigr)\,\mathrm{ds}. \end{aligned}$$

From (4.28)–(4.32), we obtain

$$\begin{aligned} &t^{2\alpha _{2}(1-\nu _{0})+2\epsilon } \bigl\Vert {u}_{\alpha '}(\cdot ,t)- {u}_{ \alpha }(\cdot ,t) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \\ &\quad \leq 3 \bigl(\overline{\mathrm {A}}(\alpha _{1},\alpha _{2},\epsilon , \nu _{0},a,T) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'- \alpha \bigr) \bigr] \Vert f \Vert _{\mathcal{H}^{\gamma }(\Omega )} \bigr)^{2} \\ &\qquad{}+3\mathcal{C}^{2}_{1}\overline{ \mathrm {B}}^{2}(\alpha _{1},\alpha _{2}, \epsilon ,\nu _{0},a,T) \mathscr{{M}}(\alpha _{1},\alpha _{2},a,\mu _{0},T) \bigl[\bigl({\alpha }'- \alpha \bigr)^{\epsilon }+\bigl({\alpha }'-\alpha \bigr) \bigr]^{2}t^{2 \alpha _{2}(1-\nu _{0})+2\epsilon } \\ &\qquad{}+6\mathcal{K}^{2}_{1}\overline{ \mathrm {M}}^{2}_{3}(\alpha _{1}, \alpha _{2},a,\mu _{0})t^{2\alpha _{2}(1-\mu _{0})+2\epsilon }2t^{2 \alpha _{1}-2\alpha _{1}\mu _{0}-2\alpha _{2}+2\alpha _{2}\nu _{0}-2 \epsilon } \mathbf{B}(\theta _{3},\vartheta _{3}) \\ &\qquad{}\times \int _{0}^{t}s^{2\alpha _{2}(1-\nu _{0})+2\epsilon } \bigl( \bigl\Vert {u}_{\alpha '}(\cdot ,s)-{u}_{\alpha }(\cdot ,s) \bigr\Vert ^{2}_{ \mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert {v}_{\alpha '}(\cdot ,s)-{v}_{\alpha }( \cdot ,s) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \bigr)\,\mathrm{ds}. \end{aligned}$$

Hence, we get the following estimate:

$$\begin{aligned} &t^{2\alpha _{2}(1-\nu _{0})+2\epsilon } \bigl\Vert {u}_{\alpha '}(\cdot ,t)- {u}_{ \alpha }(\cdot ,t) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \\ &\quad \leq 3 \bigl(\overline{\mathrm {A}}(\alpha _{1},\alpha _{2},\epsilon , \nu _{0},a,T) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'- \alpha \bigr) \bigr] \Vert f \Vert _{\mathcal{H}^{\gamma }(\Omega )} \bigr)^{2} \\ &\qquad{}+3\mathcal{C}^{2}_{1}\overline{ \mathrm {B}}^{2}(\alpha _{1},\alpha _{2}, \epsilon ,\nu _{0},a,T) \mathscr{{M}}(\alpha _{1},\alpha _{2},a,\mu _{0},T) \bigl[\bigl({\alpha }'- \alpha \bigr)^{\epsilon }+\bigl({\alpha }'-\alpha \bigr) \bigr]^{2}T^{2 \alpha _{2}(1-\nu _{0})+2\epsilon } \\ &\qquad{}+6\mathcal{K}^{2}_{1}\overline{ \mathrm {M}}^{2}_{3}(\alpha _{1}, \alpha _{2},a,\mu _{0})T^{2\alpha _{1}(1-\mu _{0})+2\alpha _{2}(\nu _{0}- \mu _{0})}\mathbf{B}(\theta _{3},\vartheta _{3}) \\ &\qquad{}\times \int _{0}^{t}s^{2\alpha _{2}(1-\nu _{0})+2\epsilon } \bigl( \bigl\Vert {u}_{\alpha '}(\cdot ,s)-{u}_{\alpha }(\cdot ,s) \bigr\Vert ^{2}_{ \mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert {v}_{\alpha '}(\cdot ,s)-{v}_{\alpha }( \cdot ,s) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \bigr)\,\mathrm{ds}. \end{aligned}$$

In the same way as above, we obtain

$$\begin{aligned} &t^{2\alpha _{2}(1-\nu _{0})+2\epsilon } \bigl\Vert {v}_{\alpha '}(\cdot ,t)- {v}_{ \alpha }(\cdot ,t) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \\ &\quad \leq 3 \bigl(\overline{\mathrm {A}}(\alpha _{1},\alpha _{2},\epsilon , \nu _{0},a,T) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'- \alpha \bigr) \bigr] \Vert g \Vert _{\mathcal{H}^{\gamma }(\Omega )} \bigr)^{2} \\ &\qquad{}+3\mathcal{C}^{2}_{2}\overline{ \mathrm {B}}^{2}(\alpha _{1},\alpha _{2}, \epsilon ,\nu _{0},a,T) \mathscr{{M}}(\alpha _{1},\alpha _{2},a,\mu _{0},T) \bigl[\bigl({\alpha }'- \alpha \bigr)^{\epsilon }+\bigl({\alpha }'-\alpha \bigr) \bigr]^{2}T^{2 \alpha _{2}(1-\nu _{0})+2\epsilon } \\ &\qquad{}+6\mathcal{K}^{2}_{2}\overline{ \mathrm {M}}^{2}_{3}(\alpha _{1}, \alpha _{2},a,\mu _{0})T^{2\alpha _{1}(1-\mu _{0})+2\alpha _{2}(\nu _{0}- \mu _{0})}\mathbf{B}(\theta _{3},\vartheta _{3}) \\ &\qquad{}\times \int _{0}^{t}s^{2\alpha _{2}(1-\nu _{0})+2\epsilon } \bigl( \bigl\Vert {u}_{\alpha '}(\cdot ,s)-{u}_{\alpha }(\cdot ,s) \bigr\Vert ^{2}_{ \mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert {v}_{\alpha '}(\cdot ,s)-{v}_{\alpha }( \cdot ,s) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \bigr)\,\mathrm{ds}. \end{aligned}$$

For simplicity to some math formulas, one should put

$$\begin{aligned} &\mathscr{M}^{\mathcal{C}_{1},\mathcal{C}_{2}}_{\mu _{0},\nu _{0}}\bigl( \alpha _{1},\alpha _{2},\epsilon , a,T, \Vert w_{0} \Vert _{\mathscr{H}^{\gamma }( \Omega )} \bigr) \\ &\quad :=3\overline{\mathrm {A}}(\alpha _{1},\alpha _{2}, \epsilon ,\nu _{0},a,T) \Vert w_{0} \Vert ^{2}_{\mathscr{H}^{\gamma }(\Omega )} \\ &\qquad{}+ 3\bigl(\mathcal{C}^{2}_{1}+\mathcal{C}^{2}_{2} \bigr)\overline{\mathrm {B}}^{2}( \alpha _{1},\alpha _{2},\epsilon ,\nu _{0},a,T) \mathscr{{M}}(\alpha _{1}, \alpha _{2},a,\mu _{0},T)T^{2\alpha _{2}(1-\nu _{0})+2\epsilon } \end{aligned}$$


$$\begin{aligned} &\mathscr{M}^{\mathcal{K}_{1},\mathcal{K}_{2}}_{\mu _{0},\nu _{0}}( \alpha _{1},\alpha _{2},\epsilon , a,T) \\ &\quad :=3\bigl(\mathcal{K}^{2}_{1}+\mathcal{K}^{2}_{2} \bigr)\overline{\mathrm {M}}^{2}_{3}( \alpha _{1},\alpha _{2},a,\mu _{0})T^{2\alpha _{1}(1-\mu _{0})+2 \alpha _{2}(\nu _{0}-\mu _{0})} \mathbf{B}(\theta _{3},\vartheta _{3}). \end{aligned}$$

Combining estimates (4.33) and (4.35), we have

$$\begin{aligned} &t^{2\alpha _{2}(1-\nu _{0})+2\epsilon } \bigl( \bigl\Vert {u}_{\alpha '}(\cdot ,t)- {u}_{\alpha }(\cdot ,t) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert {v}_{ \alpha '}(\cdot ,t)- {v}_{\alpha }(\cdot ,t) \bigr\Vert ^{2}_{\mathcal{H}^{ \gamma }(\Omega )} \bigr) \\ &\quad \leq 3 \bigl(\overline{\mathrm {A}}(\alpha _{1},\alpha _{2},\epsilon , \nu _{0},a,T) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'- \alpha \bigr) \bigr] \bigr)^{2} \bigl( \Vert f \Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )}+ \Vert g \Vert ^{2}_{ \mathcal{H}^{\gamma }(\Omega )} \bigr) \\ &\qquad {}+ 3\bigl(\mathcal{C}^{2}_{1}+\mathcal{C}^{2}_{2} \bigr)\overline{\mathrm {B}}^{2}( \alpha _{1},\alpha _{2},\epsilon ,\nu _{0},a,T) \\ &\qquad \times \mathscr{{M}}(\alpha _{1},\alpha _{2},a,\mu _{0},T)T^{2 \alpha _{2}(1-\nu _{0})+2\epsilon } \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({ \alpha }'-\alpha \bigr) \bigr]^{2} \\ &\qquad {}+3\bigl(\mathcal{K}^{2}_{1}+\mathcal{K}^{2}_{2} \bigr)\overline{\mathrm {M}}^{2}_{3}( \alpha _{1},\alpha _{2},a,\mu _{0})T^{2\alpha _{1}(1-\mu _{0})+2 \alpha _{2}(\nu _{0}-\mu _{0})} \mathbf{B}(\theta _{3},\vartheta _{3}) \\ &\qquad {} \times \int _{0}^{t}s^{2\alpha _{2}(1-\nu _{0})+2\epsilon } \bigl( \bigl\Vert {u}_{\alpha '}(\cdot ,s)-{u}_{\alpha }(\cdot ,s) \bigr\Vert ^{2}_{ \mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert {v}_{\alpha '}(\cdot ,s)-{v}_{\alpha }( \cdot ,s) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \bigr)\,\mathrm{ds}. \end{aligned}$$

Hence, we get the following estimate:

$$\begin{aligned} &t^{2\alpha _{2}(1-\nu _{0})+2\epsilon } \bigl( \bigl\Vert {u}_{\alpha '}(\cdot ,t)- {u}_{\alpha }(\cdot ,t) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert {v}_{ \alpha '}(\cdot ,t)- {v}_{\alpha }(\cdot ,t) \bigr\Vert ^{2}_{\mathcal{H}^{ \gamma }(\Omega )} \bigr) \\ &\quad \leq \mathscr{M}^{\mathcal{C}_{1},\mathcal{C}_{2}}_{\mu _{0},\nu _{0}}\bigl( \alpha _{1},\alpha _{2},\epsilon , a,T, \Vert w_{0} \Vert _{\mathscr{H}^{\gamma }( \Omega )}\bigr) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'-\alpha \bigr) \bigr]^{2} \\ &\qquad{}+ \mathscr{M}^{\mathcal{K}_{1},\mathcal{K}_{2}}_{\mu _{0},\nu _{0}}( \alpha _{1}, \alpha _{2},\epsilon ,a,T) \\ &\qquad{}\times \int _{0}^{t}s^{2\alpha _{2}(1-\nu _{0})+2\epsilon } \bigl( \bigl\Vert {u}_{\alpha '}(\cdot ,s)-{u}_{\alpha }(\cdot ,s) \bigr\Vert ^{2}_{ \mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert {v}_{\alpha '}(\cdot ,s)-{v}_{\alpha }( \cdot ,s) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )} \bigr)\,\mathrm{ds}. \end{aligned}$$

Applying Gronwall’s inequality, we have the following estimate:

$$\begin{aligned} &t^{2\alpha _{2}(1-\nu _{0})+2\epsilon } \bigl( \bigl\Vert {u}_{\alpha '}(\cdot ,t)- {u}_{\alpha }(\cdot ,t) \bigr\Vert ^{2}_{\mathcal{H}^{\gamma }(\Omega )}+ \bigl\Vert {v}_{ \alpha '}(\cdot ,t)- {v}_{\alpha }(\cdot ,t) \bigr\Vert ^{2}_{\mathcal{H}^{ \gamma }(\Omega )} \bigr) \\ &\quad \leq \mathscr{M}^{\mathcal{C}_{1},\mathcal{C}_{2}}_{\mu _{0},\nu _{0}}\bigl( \alpha _{1},\alpha _{2},\epsilon , a,T, \Vert w_{0} \Vert _{\mathscr{H}^{\gamma }( \Omega )}\bigr) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'-\alpha \bigr) \bigr]^{2} \\ &\qquad {}\times \exp \bigl(\mathscr{M}^{\mathcal{K}_{1},\mathcal{K}_{2}}_{\mu _{0}, \nu _{0}}( \alpha _{1},\alpha _{2},\epsilon ,a,T)t \bigr). \end{aligned}$$

Therefore, we obtain that

$$\begin{aligned} & \Vert u_{\alpha '}- u_{\alpha } \Vert _{\mathbb{C}^{\alpha _{2}(1-\nu _{0})+ \epsilon }((0,T],\mathcal{H}^{\gamma }(\Omega ))}^{2}+ \Vert v_{\alpha '}- v_{ \alpha } \Vert _{\mathbb{C}^{\alpha _{2}(1-\nu _{0})+\epsilon }((0,T], \mathcal{H}^{\gamma }(\Omega ))}^{2} \\ &\quad \leq \mathscr{M}^{\mathcal{C}_{1},\mathcal{C}_{2}}_{\mu _{0},\nu _{0}}\bigl( \alpha _{1},\alpha _{2},\epsilon , a,T, \Vert w_{0} \Vert _{\mathscr{H}^{\gamma }( \Omega )}\bigr) \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({\alpha }'-\alpha \bigr) \bigr]^{2} \\ &\qquad {}\times \exp \bigl(\mathscr{M}^{\mathcal{K}_{1},\mathcal{K}_{2}}_{\mu _{0}, \nu _{0}}( \alpha _{1},\alpha _{2},\epsilon ,a,T)T \bigr). \end{aligned}$$

Finally, we obtain

$$\begin{aligned} & \Vert \mathbf{w}_{\alpha '}- \mathbf{w}_{\alpha } \Vert _{\mathscr{C}^{\alpha _{2}(1- \nu _{0})+\epsilon }((0,T],\mathcal{H}^{\gamma }(\Omega ))} \\ &\quad \leq \sqrt{\mathscr{M}^{\mathcal{C}_{1},\mathcal{C}_{2}}_{\mu _{0}, \nu _{0}}\bigl( \alpha _{1},\alpha _{2},\epsilon , a,T, \Vert w_{0} \Vert _{ \mathscr{H}^{\gamma }(\Omega )}\bigr)} \bigl[\bigl({\alpha }'-\alpha \bigr)^{\epsilon }+\bigl({ \alpha }'- \alpha \bigr) \bigr] \\ &\qquad {}\times \exp \biggl( \frac{\mathscr{M}^{\mathcal{K}_{1},\mathcal{K}_{2}}_{\mu _{0},\nu _{0}}(\alpha _{1},\alpha _{2},\epsilon ,a,T)T}{2} \biggr). \end{aligned}$$

This completes the proof. □

5 Numerical results

In this section, we show an example which shows the effectiveness of our method. Let the operator −Δ on the domain \(\Omega = (0,\pi )\) with the Dirichlet boundary condition and \(t \in [0,1]\), \(a=1\), we have the eigenvalues of −Δ given by \(\lambda _{j}=j^{2} \) (\(j\in \mathbb{Z}^{+}\)) and \(\varphi _{j}(x) = \sqrt{\frac{2}{\pi }}\sin (jx)\), respectively. We consider the problem to find \((u(x,t),v(x,t))\) as follows:

$$ \textstyle\begin{cases} \partial ^{\alpha }_{t} (u(x,t) + \partial ^{2}_{xx} u(x,t) ) + \partial ^{2}_{xx} u(x,t) = \mathcal{F}(u,v,x,t),& (x,t) \in (0,\pi ) \times (0,1), \\ \partial ^{\alpha }_{t} (v(x,t) + \partial ^{2}_{xx} v(x,t) ) + \partial ^{2}_{xx} v(x,t)= \mathcal{G}(u,v,x,t),& (x,t) \in (0,\pi ) \times (0,1), \end{cases} $$

where \((x,t) \in (0,\pi ) \times (0,1)\), the source functions are given by

$$ \textstyle\begin{cases} \mathcal{F}(u,v,x,t)= v^{2} - 4u - \frac{6 t^{2-\alpha } \sin (2x) }{ \Gamma (3-\alpha ) } + ( \frac{t^{6}}{2}-t^{3} + \frac{1}{2} ) (\cos (8x)-1 ), \\ \mathcal{G}(u,v,x,t) = u^{2} - 16v - \frac{90 t^{3-\alpha } \sin (4x) }{ \Gamma (4-\alpha ) } - ( \frac{t^{4}}{2}- t^{2} + \frac{1}{2} ) (1-\cos (4x) ), \end{cases} $$

and the Dirichlet boundary condition as follows:

$$ \textstyle\begin{cases} u(0,t)=u(\pi ,t)=0,& t \in (0,1), \\ v(0,t)=v(\pi ,t)=0,& t \in (0,1). \end{cases} $$

Assume that the values of u, v at the initial time \(t=0\) are given by

$$ \textstyle\begin{cases} u(x,0)= f(x) := -\sin (2x), \quad u_{t}(x,0)=0, & x \in (0,\pi ), \\ v(x,0)= g(x) := \sin (4x), \quad v_{t}(x,0)=0,& x \in (0,\pi ). \end{cases} $$

Then the exact solution of problem (5.1)–(5.4) is given by

$$\begin{aligned} \textstyle\begin{cases} u(x,t) = (t^{2}-1)\sin (2x), \\ v(x,t) = (1-t^{3}) \sin (3x). \end{cases}\displaystyle \end{aligned}$$

At the discretization level, a uniform grid of mesh-points \((x_{m}, t_{n})\) is used to discretize the space and time intervals

$$\begin{aligned} x_{m} = \frac{(m-1)\pi }{h_{x}},\quad m = \overline{1,h_{x}+1},\quad \text{and}\quad t_{n} = \frac{n-1}{h_{t}},\quad n = \overline{1,h_{t}+1}. \end{aligned}$$

Next, by using Simpson’s rule of numerical integration, we have the following approximate integration of \(z \in L^{2}(0,\pi )\):

$$\begin{aligned} \int _{0}^{\pi } z(s) \,\mathrm{d}s \approx \frac{\Delta s}{3} z(s_{1}) + \frac{2\Delta s}{3} \sum _{l=1}^{(N+1)/2-1} z(s_{2l}) + \frac{4\Delta s}{3} \sum_{l=1}^{(N+1)/2}z(s_{2l-1}) + \frac{\Delta s}{3} z(s_{N+1}). \end{aligned}$$

In code Matlab, we have the solution of problem (5.1) which can be written in a matrix form as follows:

$$ U = \begin{bmatrix} u(x_{1},t_{1}) &u(x_{2},t_{1})&u(x_{3},t_{1})&\cdots &u(x_{m+1},t_{1}) \\ u(x_{1},t_{2}) &u(x_{2},t_{2})&u(x_{3},t_{2})&\cdots &u(x_{m+1},t_{2}) \\ u(x_{1},t_{3}) &u(x_{2},t_{3})&u(x_{3},t_{3})&\cdots &u(x_{m+1},t_{3}) \\ \vdots & \vdots & \vdots & \vdots \\ u(x_{1},t_{n+1}) &u(x_{2},t_{n+1})&u(x_{3},t_{n+1})&\cdots &u(x_{m+1},t_{n+1}) \end{bmatrix}_{(n+1)\times (m+1)}, $$


$$ V= \begin{bmatrix} v(x_{1},t_{1}) &v(x_{2},t_{1})&v(x_{3},t_{1})&\cdots &v(x_{m+1},t_{1}) \\ v(x_{1},t_{2}) &v(x_{2},t_{2})&v(x_{3},t_{2})&\cdots &v(x_{m+1},t_{2}) \\ v(x_{1},t_{3}) &v(x_{2},t_{3})&v(x_{3},t_{3})&\cdots &v(x_{m+1},t_{3}) \\ \vdots & \vdots & \vdots & \vdots \\ v(x_{1},t_{n+1}) &v(x_{2},t_{n+1})&v(x_{3},t_{n+1})&\cdots &v(x_{m+1},t_{n+1}) \end{bmatrix}_{(n+1)\times (m+1)}, $$

where the numerical solutions are presented in the following (J is a truncation parameter of series):

$$\begin{aligned} &u(x_{m},t_{n}) \\ &\quad = {\frac{2}{\pi }} \sum _{j=1}^{ \mathbf{J}} E_{\alpha ,1} \biggl( \frac{-j^{2} t^{\alpha }_{n}}{1+j^{2}} \biggr) \int _{0}^{\pi } f(\zeta )\sin (j\zeta ) \,\mathrm{d} \zeta \sin (jx_{m}) \\ &\qquad {} + {\frac{2}{\pi }} \sum_{j=1}^{\mathbf{J}} \frac{1}{1+ j^{2}} \int _{0}^{t_{n}}(t_{n}-s)^{\alpha -1} E_{ \alpha ,\alpha } \biggl(\frac{-j^{2}(t_{n}-s)^{\alpha }}{1+j^{2}} \biggr) \int _{0}^{\pi } \mathcal{F}(\zeta )\sin (j \zeta ) \,\mathrm{d} \zeta \,\mathrm{d} s \sin (jx_{m}) \end{aligned}$$


$$\begin{aligned} &v(x_{m},t_{n}) \\ &\quad = {\frac{2}{\pi }} \sum _{j=1}^{ \mathbf{J}} E_{\alpha ,1} \biggl( \frac{-j^{2} t^{\alpha }_{n}}{1+j^{2}} \biggr) \int _{0}^{\pi } g(\zeta )\sin (j\zeta ) \,\mathrm{d} \zeta \sin (jx_{m}) \\ &\qquad {} + {\frac{2}{\pi }} \sum_{j=1}^{\mathbf{J}} \frac{1}{1+ j^{2}} \int _{0}^{t_{n}}(t_{n}-s)^{\alpha -1} E_{ \alpha ,\alpha } \biggl(\frac{-j^{2}(t_{n}-s)^{\alpha }}{1+j^{2}} \biggr) \int _{0}^{\pi } \mathcal{G}(\zeta )\sin (j \zeta ) \,\mathrm{d} \zeta \,\mathrm{d} s \sin (jx_{m}). \end{aligned}$$

By fixing t, we consider the following estimations with fractional derivative orders α and \(\alpha ^{*}\) to compare the regularity of the solution, which are given by

$$\begin{aligned} &\mathrm{ESTU}_{\alpha }^{\alpha ^{*}}(t) = \Biggl({ \sum _{m=1}^{{h_{x}}+1} \bigl\vert u_{\alpha }(x_{m},t) - u_{\alpha ^{*}}(x_{m},t) \bigr\vert ^{2}} /(h_{x}+1) \Biggr)^{1/2}, \end{aligned}$$
$$\begin{aligned} &\mathrm{ESTV}_{\alpha }^{\alpha ^{*}}(t) = \Biggl({ \sum _{m=1}^{{h_{x}}+1} \bigl\vert v_{\alpha }(x_{m},t) - v_{\alpha ^{*}}(x_{m},t) \bigr\vert ^{2}} /(h_{x}+1) \Biggr)^{1/2}, \end{aligned}$$

where \((u_{\alpha ^{*}}, v_{\alpha ^{*}})\) and \((u_{\alpha }, v_{\alpha })\) are the solutions in the case \(\alpha ^{*}\) and α, respectively.

Tables 13 present the error estimates between the solutions for α and \(\alpha ^{*}\). It clearly shows that the solution for \(\alpha ^{*}\) converges to the solution for α as the deflection of the fractional order tends to zero. For a more intuitive look, we can see the graphs of the solutions in Figs. 1, 2, and 3.

Figure 1
figure 1

The solutions at \(t=0.2\) for \(\alpha = 1.1\), \(\alpha ^{*} \in \{1.11, 1.12, 1.13, 1.14, 1.15 \}\)

Figure 2
figure 2

The solutions at \(t=0.4\) for \(\alpha = 1.5\), \(\alpha ^{*} \in \{1.51, 1.52, 1.53, 1.54, 1.55 \}\)

Figure 3
figure 3

The solutions at \(t=0.8\) for \(\alpha = 1.8\), \(\alpha ^{*} \in \{1.81, 1.82, 1.83, 1.84, 1.85 \}\)

Table 1 The error estimates at \(t=0.2\) for \(\alpha = 1.1\), \(\alpha ^{*} \in \{1.11, 1.12, 1.13, 1.14, 1.15 \}\)
Table 2 The error estimates at \(t=0.4\) for \(\alpha = 1.5\), \(\alpha ^{*} \in \{1.51, 1.52, 1.53, 1.54, 1.55 \}\)
Table 3 The error estimates at \(t=0.8\) for \(\alpha = 1.8\), \(\alpha ^{*} \in \{1.81, 1.82, 1.83, 1.84, 1.85 \}\)

6 Conclusion

In this paper, we considered the initial value problem for a system of nonlinear parabolic pseudo equations with Caputo fractional derivative. Here, we discuss the continuity which is related to a fractional order derivative. To illustrate the theoretical results, we gave an example in some cases. In the future work, we will expand the problem to the case of other derivative definitions such as Riemann–Liouville and conformable. Especially, the operators affecting the spatial variable of two equations are different, this is an open and difficult problem.