1 Introduction

Time-fractional reaction–diffusion equation attracts widely attentions in applications like viscoelastic behaviors and anomalous diffusion with extensive investigations for both the direct and inverse problems [4,5,6, 9, 11,12,13,14,15, 18, 21, 23]. In particular, the time-fractional problem with Mittag-Leffler kernel has been applied in various fields such as the groundwater pollution problem [3]. Nevertheless, a constant fractional order could not accommodate the variability of the memory with respect to time in practical problems, which motivates the application of the variable fractional order [16, 20, 27]. For this reason, in this work we consider the following variable-order time-fractional diffusion equation with Mittag-Leffler kernel

$$\begin{aligned} {}_0\partial _t^{\alpha (t)} u(\varvec{x},t)-K\Delta u(\varvec{x},t)=f(\varvec{x},t),~~(\varvec{x},t)\in \Omega \times (0,T],\nonumber \\ u(\varvec{x},t)=0,~~(\varvec{x},t)\in \partial \Omega \times [0,T],~~u(\varvec{x},0)=u_0(\varvec{x}),~~\varvec{x}\in \Omega . \end{aligned}$$
(1)

Here \(\Omega \subset {\mathbb {R}}^d\) (\(d=1,2,3\)) is a bounded domain with a smooth boundary \(\partial \Omega \), \(\varvec{x}=(x_1,\ldots ,x_d)\in \Omega \),K represents the diffusivity coefficient, f represents the source term, and the fractional derivative \({}_0\partial _t^{\alpha } z\) with variable-order Mittag-Leffler kernel of order \(0<\alpha _*\le \alpha (t)\le \alpha ^*<1\) for some \(0<\alpha _*,\alpha ^*<1\) is defined by [2, 7, 25]

$$\begin{aligned} {}_0\partial _t^{\alpha (t)}z(t)=\int \limits _0^t\mathcal Q(\alpha (t),t-s) z'(s)\textrm{d}s \end{aligned}$$

where the function form of the variable-order Mittag-Leffler kernel is given as

$$\begin{aligned}\mathcal Q(z,t):=\frac{M(z)}{1-z}E_{z,1}\Big (-\frac{z}{1-z}t^{z}\Big ) \end{aligned}$$

Here \(M(\alpha (t))>0\) is a normalization function satisfying \(M(0)=M(1)=1\), and \(E_{\mu ,\nu }(t)\) (\(\mu > 0\) and \(\nu \in \mathbb R\)) represents the two-parametric Mittag-Leffler function given by [8]

$$\begin{aligned}E_{\mu ,\nu }(t)=\sum _{k=0}^{\infty }\frac{t^k}{\Gamma (\mu k+\nu )}.\end{aligned}$$

Despite growing applications of the time-fractional problems involving the variable-order Mittag-Leffler kernel, the analysis for the corresponding forward and backward problems is still not well developed. In [25] the well-posedness of model (1) was studied. In this work, we continue to prove the solution regularity of model (1) and the corresponding inverse problem of unique determination of the variable fractional order in model (1) based on the observations of the solutions over a small time-space interval, which provide theoretical supports for applications of the model. The main difficulty we overcome lies in the technical analysis of the variable-order Mittag-Leffler function and its derivatives via special functions. The derived methods could also be applied to analyze the analogous problems involving the variable-order Caputo–Fabrizio fractional operators via a simpler manner.

The rest of the paper is organized as follows. In Sect. 2, we prove the solution regularity of model (1), based on which we prove the unique determination of the variable fractional order in model (1) in Sect. 3. Throughout this paper, \(Q>0\) represents a generic constant that may assume different values in difference circumstances.

2 Preliminaries

Let \(L^2(\Omega )\) be the space of square integrable functions on \(\Omega \) and \(H^{m}(\Omega )\) (\(m \in \mathbb {N}\)) be the Hilbert spaces of functions with weak derivatives of order m in \(L^2(\Omega )\). For non-integer \(s\ge 0\), the fractional Sobolev spaces \(H^s(\Omega )\) are defined via interpolation. All the spaces are equipped with the standard norms [1].

Let \(\{\lambda _i \}_{i=1}^\infty \) be the eigenvalues of Strum-Liouville problem

$$\begin{aligned} -K\Delta \phi _i(\varvec{x}) = \lambda _i \phi _i(\varvec{x}), ~\varvec{x}\in \Omega ; \quad \phi _i(\varvec{x}) = 0, ~\varvec{x}\in \partial \Omega . \end{aligned}$$
(2)

By [17], the eigenfunctions \(\{\phi _i \}_{i=1}^\infty \) form an orthonormal basis in \(L^2(\Omega )\) and \(0 <\lambda _1\le \lambda _2\le \ldots \rightarrow \infty \). Based on (2), we define the Sobolev space for \(\gamma \ge 0\) by [10]

$$\begin{aligned}\begin{array}{c} \check{H}^{\gamma }(\Omega ):= \left\{ v \in L^2(\Omega ): | v |_{\check{H}^\gamma }^2: = \sum _{i=1}^{\infty } \lambda _i^{\gamma } (v,\phi _i)^2 < \infty \right\} . \end{array}\end{aligned}$$

Besides, for a Banach space \({\mathcal X}\), let \(C([0,T];\mathcal X)\) be the space of continuous functions on the interval [0, T] with respect to the norm \(\Vert \cdot \Vert _{\mathcal X}\) [1].

We then refer to the Gronwall inequality for future use [22].

Lemma 1

Let \(0 \le C_0(t) \in L_{loc}(a,b)\) and \(C_1\) be a non-negative constant. Suppose \(0 \le g(t) \in L_{loc}(a,b)\) satisfies

$$\begin{aligned} g(t)\le C_0(t)+C_1\,{}_aI_t^\gamma g(t),~~\forall t \in (a,b),~~0<\gamma <1, \end{aligned}$$

where the fractional integral operator \({}_aI_t^\gamma \) is defined as

$$\begin{aligned}{}_aI_t^\gamma g(t):=\frac{1}{\Gamma (\gamma )}\int \limits _a^t\frac{g(s)\textrm{d}s}{(t-s)^{1-\gamma }}.\end{aligned}$$

Then g can be bounded by

$$\begin{aligned} g(t)\le C_0(t)+ \sum _{n=1}^\infty (C_1\Gamma (\gamma ))^n{}_aI_t^{n\gamma }C_0(t),~~\forall t \in (a,b). \end{aligned}$$

In particular, if \(C_0(t)\) is non-decreasing, then

$$\begin{aligned} g(t)\le C_0(t)E_{\gamma ,1}\big (C_1\Gamma (\gamma )(t-a)^\gamma \big ),~~~\forall t ~\in (a,b). \end{aligned}$$

We finally refer the well-posedness result of model (1) from [25].

Theorem 1

Suppose \(\alpha \in C[0,T]\), \(u_0\in \check{H}^\gamma (\Omega )\), \(f\in H^\kappa (0,T;\check{H}^\gamma (\Omega ))\) for some \(\kappa >1/2\) and \(\gamma \ge 2\), and \(-K\Delta u_0(\varvec{x})=f(\varvec{x},0)\). Then problem (1) has a unique mild solution in \(C([0,T];\check{H}^\gamma (\Omega ))\) with the following stability estimate

$$\begin{aligned} \Vert u\Vert _{C([0,T];\check{H}^s(\Omega ))}\le Q \left( \Vert u_0\Vert _{\check{H}^{s}(\Omega )} + \Vert f\Vert _{\check{H}^\kappa (0,T;\check{H}^s(\Omega )) } \right) , \quad 0\le s \le \gamma . \end{aligned}$$

3 Solution regularity

We prove the regularity of the solution in the following theorem.

Theorem 2

Suppose M and \(\alpha \) are continuously differentiable functions and \(\partial _t f\in H^\kappa (0,T;\check{H}^{s-2}(\Omega ))\) for some \(s\ge 2\) and \(\kappa >1/2\). Then the following regularity estimate holds

$$\begin{aligned} \Vert \partial _t u\Vert _{C([0,T];\check{H}^s(\Omega ))}\le Q \Vert \partial _t f\Vert _{\check{H}^\kappa (0,T;\check{H}^{s-2}(\Omega )) }. \end{aligned}$$

Proof

Let \(\{u_i(t),f_i(t)\}_{i=1}^\infty \) be the Fourier coefficients of u and f, respectively, expanded under the eigenfunctions \(\{\phi _i\}_{i=1}^\infty \), that is,

$$\begin{aligned}u=\sum _{i=1}^\infty u_i(t)\phi _i(x),~~f=\sum _{i=1}^\infty f_i(t)\phi _i(x). \end{aligned}$$

Then problem (1) could be reduced to the following system of ordinary differential equations

$$\begin{aligned} \begin{array}{c} {}_0\partial _t^{\alpha (t)} u_i(t) +\lambda _i u_i(t)=f_{i}(t), ~~t \in (0,T], \quad u_i(0) = u_{0,i}:= (u_0,\phi _i). \end{array}\end{aligned}$$
(3)

Differentiating this equation with respect to t leads to a second-kind Volterra integral equation

$$\begin{aligned} \bigg (\frac{M(\alpha (t))}{1-\alpha (t)}+\lambda _i\bigg ) u_i'(t)+ \int \limits _0^t (\partial _t\mathcal Q(\alpha (t),t-s)) u_i'(s)\textrm{d}s=f'_i(t). \end{aligned}$$
(4)

The key component in \(\partial _t\mathcal Q(\alpha (t),t-s)\) is the following derivative of variable-order Mittag-Leffler function

$$\begin{aligned}{} & {} \partial _t E_{\alpha (t),1}\Big (-\frac{\alpha (t)}{1-\alpha (t)}(t-s)^{\alpha (t)}\Big ) \nonumber \\{} & {} = \partial _t\sum _{i=1}^\infty \Big (-\frac{\alpha (t)}{1-\alpha (t)}(t-s)^{\alpha (t)}\Big )^i\frac{1}{\Gamma (\alpha (t)i+1)} \nonumber \\{} & {} = \sum _{i=1}^\infty \bigg \{ i \Big (-\frac{\alpha (t)}{1-\alpha (t)}(t-s)^{\alpha (t)}\Big )^{i-1}\bigg [\frac{-\alpha '(t)(t-s)^{\alpha (t)}}{(1-\alpha (t))^2} \nonumber \\{} & {} \quad -\frac{\alpha (t)(t-s)^{\alpha (t)}}{1-\alpha (t)}\Big (\alpha '(t)\ln (t-s)+\frac{\alpha (t)}{t-s}\Big )\bigg ]\frac{1}{\Gamma (\alpha (t)i+1)} \nonumber \\{} & {} \quad -\Big (-\frac{\alpha (t)}{1-\alpha (t)}(t-s)^{\alpha (t)}\Big )^{i}\frac{\Gamma '(\alpha (t)i+1)}{\Gamma (\alpha (t)i+1)^2}i\alpha '(t)\bigg \}. \end{aligned}$$
(5)

Direct calculation shows that

$$\begin{aligned}{} & {} \sum _{i=1}^\infty i \Big (-\frac{\alpha (t)}{1-\alpha (t)}(t-s)^{\alpha (t)}\Big )^{i-1}\frac{1}{\Gamma (\alpha (t)i+1)}\nonumber \\{} & {} =\frac{1}{\alpha (t)}\sum _{i=1}^\infty \Big (-\frac{\alpha (t)}{1-\alpha (t)}(t-s)^{\alpha (t)}\Big )^{i-1}\frac{1}{\Gamma (\alpha (t)i)}\nonumber \\{} & {} =\frac{1}{\alpha (t)}\sum _{i=0}^\infty \Big (-\frac{\alpha (t)}{1-\alpha (t)}(t-s)^{\alpha (t)}\Big )^{i}\frac{1}{\Gamma (\alpha (t)i+\alpha (t))}\nonumber \\{} & {} =\frac{1}{\alpha (t)}E_{\alpha (t),\alpha (t)}\Big (-\frac{\alpha (t)}{1-\alpha (t)}(t-s)^{\alpha (t)}\Big ), \end{aligned}$$
(6)

the right-hand side of which is continuous with respect to s and t (cf. the proof of Lemma 2.2 in [25]). Thus all but the last right-hand side terms of (5) could be reformulated as

$$\begin{aligned}\begin{array}{l} \left[ \frac{-\alpha '(t)(t-s)^{\alpha (t)}}{\alpha (t)(1-\alpha (t))^2}-\frac{(t-s)^{\alpha (t)}}{1-\alpha (t)}\Big (\alpha '(t)\ln (t-s)+\frac{\alpha (t)}{t-s}\Big )\right] \\ \qquad \times E_{\alpha (t),\alpha (t)}\Big (-\frac{\alpha (t)}{1-\alpha (t)}(t-s)^{\alpha (t)}\Big ), \end{array}\end{aligned}$$

which could be bounded as \(Q(t-s)^{\alpha (t)-1}\) for some constant \(Q>0\). To estimate the last right-hand side term of (5), we apply the definition of the Polygamma function \(\psi (\cdot ):=\Gamma '(\cdot )/\Gamma (\cdot )\) and the asymptotic properties of \(\psi (\cdot )\) and \(\Gamma (\cdot )\) [19]

$$\begin{aligned}\psi (z)\sim \log z,~~\Gamma (z)\sim z^{z-1/2}e^{-z},~~z\rightarrow \infty \end{aligned}$$

to obtain for i large enough

$$\begin{aligned} \frac{\Gamma '(\alpha (t)i+1)}{\Gamma (\alpha (t)i+1)^2}i= & {} \frac{\psi (\alpha (t)i+1)}{\alpha (t)\Gamma (\alpha (t)i)}\sim \frac{\log (\alpha (t)i)}{(\alpha (t)i)^{\alpha (t)i-1/2}e^{-\alpha (t)i}}\\[0.1in]\le & {} \frac{Q}{(\alpha (t)i)^{\alpha (t)i-3/2}e^{-\alpha (t)i}}= \frac{Q}{ e^{(\alpha (t)i-3/2)\ln (\alpha (t)i)-\alpha (t)i}}\\\le & {} \frac{Q}{ e^{0.5\alpha (t)i\ln (\alpha (t)i)-\alpha (t)i}}=\frac{Q}{ e^{(0.5\alpha (t)\ln (\alpha (t)i)-\alpha (t))i}}. \end{aligned}$$

Consequently the ith summand (with i large enough) in the last right-hand side term of (5) could be bounded by

$$\begin{aligned}\bigg |\Big (-\frac{\alpha (t)}{1-\alpha (t)}(t-s)^{\alpha (t)}\Big )^{i}\frac{\Gamma '(\alpha (t)i+1)}{\Gamma (\alpha (t)i+1)^2}i\alpha '(t)\bigg |\le Q\Bigg (\frac{\frac{\alpha ^*}{1-\alpha ^*}\max \{1,T\}}{ e^{0.5\alpha _*\ln (\alpha _* i)-\alpha ^*}}\Bigg )^i. \end{aligned}$$

It is clear that for i large enough, the following estimate holds

$$\begin{aligned}\frac{\frac{\alpha ^*}{1-\alpha ^*}\max \{1,T\}}{ e^{0.5\alpha _*\ln (\alpha _* i)-\alpha ^*}}<1.\end{aligned}$$

Then we apply the Weierstrass M-test to conclude that the last right-hand side term of (5) is continuous and bounded. We incorporate the preceding estimates in (4) to obtain

$$\begin{aligned} \bigg (\frac{M(\alpha (t))}{1-\alpha (t)}+\lambda _i\bigg ) |u_i'(t)|\le Q\int \limits _0^t (t-s)^{\alpha _*-1}| u_i'(s)|\textrm{d}s+|f'_i(t)|, \end{aligned}$$
(7)

where we used the estimate

$$\begin{aligned}(t-s)^{\alpha (t)-1}=(t-s)^{\alpha _*-1}(t-s)^{\alpha (t)-\alpha _*}\le \max \{1,T\}(t-s)^{\alpha _*-1}.\end{aligned}$$

Then an application of the Gronwall inequality with singular kernel leads to

$$\begin{aligned}|u_i'(t)|\le Q\lambda _i^{-1}\Vert f'_i\Vert _{C[0,T]},\end{aligned}$$

which immediately yields

$$\begin{aligned} \Vert \partial _t u\Vert _{C([0,T];\check{H}^s(\Omega ))}^2\le & {} Q \sum _{i=1}^\infty \lambda _i^s \Vert u_i'\Vert _{C[0,T]}^2\\\le & {} Q \sum _{i=1}^\infty \lambda _i^{s-2} \Vert f_i' \Vert _{C[0,T]}^2 \le Q \Vert \partial _tf\Vert ^2_{H^\kappa (0,T;\check{H}^{s-2}(\Omega ))}. \end{aligned}$$

Thus the proof is completed. \(\square \)

4 Uniqueness of determining variable fractional order

We prove the uniqueness of determining the variable fractional order in model (1) based on the proved regularity results and the observation on the solution u over a small time-space interval. Here we require that the normalization function M(z) with \(\alpha _*\le z\le \alpha ^*\) satisfies

$$\begin{aligned} M'(z)(1-z)+M(z)\ge M_0>0 \end{aligned}$$
(8)

for some constant \(M_0>0\). In most literature \(M(z)\equiv 1\) while in, e.g. [7], a specific form \(M(z)=\frac{2}{2-z}\) is selected. It could be verified directly that for both cases the condition (8) is satisfied since for the later case

$$\begin{aligned} M'(z)(1-z)+M(z)= & {} \frac{2}{(2-z)^2}(1-z)+\frac{2}{2-z} \\\ge & {} \frac{2}{(2-\alpha _*)^2}(1-\alpha ^*)+\frac{2}{2-\alpha _*}>0. \end{aligned}$$

Theorem 3

Suppose M and \(\alpha \) are continuously differentiable functions with the condition (8), \(u_0\in \check{H}^{2+d/2+\varepsilon }(\Omega )\) and \(f\in H^\kappa (0,T;\check{H}^{2+d/2+\varepsilon }(\Omega ))\) for some \(0<\varepsilon \ll 1\) and \(\kappa >1/2\) with \(\partial _t f\in H^\kappa (0,T;L^2(\Omega ))\). Let u and \(\bar{u}\) be solutions to model (1) with respect to \(\alpha (t)\) and \(\bar{\alpha }(t)\), respectively, i.e. u and \(\bar{u}\) satisfy

$$\begin{aligned}{} & {} {}_0\partial _t^{\alpha (t)} u(\varvec{x},t)-K\Delta u(\varvec{x},t)=f(\varvec{x},t),~~(\varvec{x},t)\in \Omega \times (0,T],\nonumber \\{} & {} u(\varvec{x},t)=0,~~(\varvec{x},t)\in \partial \Omega \times [0,T],~~u(\varvec{x},0)=u_0(\varvec{x}),~~\varvec{x}\in \Omega . \end{aligned}$$
(9)

and

$$\begin{aligned}{} & {} {}_0\partial _t^{\bar{\alpha }(t)} \bar{u}(\varvec{x},t)-K\Delta \bar{u}(\varvec{x},t)=f(\varvec{x},t),~~(\varvec{x},t)\in \Omega \times (0,T],\nonumber \\{} & {} \bar{u}(\varvec{x},t)=0,~~(\varvec{x},t)\in \partial \Omega \times [0,T],~~\bar{u}(\varvec{x},0)=u_0(\varvec{x}),~~\varvec{x}\in \Omega . \end{aligned}$$
(10)

Suppose \(\alpha _*\le \alpha (t),\bar{\alpha }(t)\le \alpha ^*\) are analytic functions over [0, T] and \(u=\bar{u}\) over \(\Omega _0\times [0,\tau ]\) for some open set \(\Omega _0\subset \Omega \) and for some \(0<\tau \ll 1\) such that \(\partial _t u(\varvec{x}_0,0)\ne 0\) for some \(\varvec{x}_0\in \Omega _0\). Then \(\alpha (t)\equiv \bar{\alpha }(t)\) over [0, T].

Proof

By the smoothness assumptions on the data, we apply Theorems 1 and 2 and the Sobolev embedding \(H^{m+d/2+\varepsilon }(\Omega )\hookrightarrow C^{m}(\Omega )\) to conclude that \(u,\bar{u}\in C([0,T];C^2(\Omega ))\) with \(\partial _t u,\partial _t\bar{u}\in C([0,T];C(\Omega ))\). Since \(\partial _t u(\varvec{x}_0,0)(=\partial _t\bar{u}(\varvec{x}_0,0))\ne 0\) at \(\varvec{x}_0\in \Omega _0\), we assume \(\partial _t u(\varvec{x}_0,0)> 0\) on \(\Omega _0\) without loss of generality. By the smoothness of u and \(\bar{u}\), there exist \(0<\tau _1\le \tau \) and \(\Omega _1\subset \Omega _0\) such that \(\partial _t u=\partial _t \bar{u}\ge \beta >0\) over \(\Omega _1\times [0,\tau _1]\) with \(\Delta u=\Delta \bar{u}\) on \(\Omega _1\times [0,\tau _1]\). Thus we obtain by subtracting (9) from (10) that

$$\begin{aligned}{}_0\partial _t^{\alpha (t)} u(\varvec{x},t)-{}_0\partial _t^{\bar{\alpha }(t)} \bar{u}(\varvec{x},t)=0 \end{aligned}$$

over \(\Omega _1\times [0,\tau _1]\). As \(\partial _t u=\partial _t \bar{u}\ge \beta >0\) over \(\Omega _1\times [0,\tau _1]\), we get from this equation that

$$\begin{aligned}\int \limits _0^t \partial _s u(\varvec{x},s)\big (\mathcal Q(\alpha (t),t-s)-\mathcal Q(\bar{\alpha }(t),t-s)\big )\textrm{d}s=0, \end{aligned}$$

that is,

$$\begin{aligned} \int \limits _0^t \partial _s u(\varvec{x},s)\partial _z\mathcal Q(z,t-s)|_{z=\hat{\alpha }(s,t)}\textrm{d}s(\alpha (t)-\bar{\alpha }(t))=0, \end{aligned}$$
(11)

where \(\hat{\alpha }(s,t)\) lies in between \(\alpha (t)\) and \(\bar{\alpha }(t)\). Differentiating \(\mathcal Q(z,t-s)\) with respect to z and using a similar derivation as (6) yield

$$\begin{aligned}{} & {} \partial _z\mathcal Q(z,t-s)\nonumber \\{} & {} =\frac{M'(z)(1-z)+M(z)}{(1-z)^2}E_{z,1}\Big (-\frac{z}{1-z}(t-s)^{z}\Big )\nonumber \\{} & {} \quad +\frac{M(z)}{1-z}\bigg [\frac{1}{z}\bigg (\frac{-(t-s)^z}{(1-z)^2}+\frac{-z}{1-z}(t-s)^z\ln (t-s)\bigg ) E_{z,z}\bigg (\frac{-z}{1-z}(t-s)^z\bigg )\nonumber \\{} & {} \quad -\sum _{i=1}^\infty \bigg (\frac{-z}{1-z}(t-s)^z\bigg )^i \psi (zi+1) \frac{i}{\Gamma (zi+1)} \bigg ]. \end{aligned}$$
(12)

By the monotonicity of the Gamma function, we have

$$\begin{aligned}\frac{1}{\Gamma (zi+1)}\le \frac{Q}{\Gamma (\alpha _* i+1)}\end{aligned}$$

for some constant \(Q>0\), the following estimate holds for \(t\ll 1\)

$$\begin{aligned}{} & {} E_{z,1}\Big (-\frac{z}{1-z}(t-s)^{z}\Big )\in 1\pm \sum _{i=1}^\infty \Big (\frac{z}{1-z}(t-s)^{z}\Big )^i\frac{1}{\Gamma (zi+1)}\\{} & {} \subset 1\pm \sum _{i=1}^\infty \Big (\frac{\alpha ^*}{1-\alpha ^*}(t-s)^{\alpha _*}\Big )^i\frac{Q}{\Gamma (\alpha _* i+1)}. \end{aligned}$$

By the Weierstrass M-test, the right-hand side series of this equation is continuous with respect to \(t-s\), and consequently close to 0 for t small enough. We incorporate this with condition (8) to conclude that there exists a \(0<\tau _2\le \tau _1\) such that the first right-hand side term of (12) has a positive lower bound for \(t\le \tau _2\)

$$\begin{aligned}C_0:=\frac{1}{2}\frac{M_0}{(1-\alpha _*)^2}. \end{aligned}$$

For the other right-hand side terms of (12), similar analysis indicates that since they are exactly 0 for \(t-s=0\), there exists a \(0<\tau _3\le \tau _2\) such that these terms are bounded by \(C_0/2\) for \(t\le \tau _3\). We combine these estimates to conclude that

$$\begin{aligned}\partial _z\mathcal Q(z,t-s)\ge C_0/2,~~0\le t\le \tau _3.\end{aligned}$$

Since \(\partial _t u\ge \beta >0\) over \(\Omega _1\times [0,\tau _1]\), (11) implies that \(\alpha (t)=\bar{\alpha }(t)\) for \(t\in [0,\tau _3]\). Based on this result, we could follow the same proof around [26, Equation 13] to apply the analyticity assumption of \(\alpha (t)\) and \(\bar{\alpha }(t)\) in this theorem to reach \(\alpha (t)=\bar{\alpha }(t)\) for \(t\in [0,T]\).Thus the proof is completed. \(\square \)

5 Concluding remarks

In this work, we analyze the solution regularity of a variable-order time-fractional diffusion equation with Mittag-Leffler kernel, based on which we derive the uniqueness of determining the variable fractional order based on the observations of the solutions over a small time-space interval. These theoretical results provide concrete supports for the applications of the proposed model in both the forward and backward manner.

It is worth mentioning that by similar and simper calculations, one could also analyze the solution regularity and the same inverse problem for the variable-order time-fractional diffusion equation with Caputo–Fabrizio derivative, which replaces the definition of the kernel \(\mathcal Q\) as [24]

$$\begin{aligned}\mathcal Q(z,t):=\frac{M(z)}{1-z}\exp \Big (-\frac{z}{1-z}t\Big ). \end{aligned}$$

More generally, one could extend the developed ideas and methods to analyze more complicated nonlocal problems that will be investigated in the near future.