Uniquely identifying the variable order of time-fractional partial differential equations on general multi-dimensional domains

ABSTRACT

We proved the unique determination of the variable order in a two-scale mobile–immobile variable-order time-fractional partial differential equation with a variable diffusivity tensor imposed on a general multi-dimensional domain, with the observations of the unknown solutions on any arbitrarily small spatial domain over a sufficiently small time interval. The proved theorem provides a guidance where the measurements should be performed and ensures that with these observations the uniqueness of the identification is theoretically guaranteed.

Keywords:

• Determination of variable order
• variable-order time-fractional diffusion equation
• inverse problem
• uniqueness
• variable diffusivity

2010 Mathematics Subject Classification:

• 35R30

1. Modelling issues

We consider the initial boundary value problem of the two-scale mobile–immobile variable-order time-fractional partial differential equation with variable diffusivity [1–4] (1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) Here $\mathrm{\Omega }\subset {\mathbb{R}}^d$ (d = 1, 2, 3) is a simply connected bounded domain with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$ and convex corners, $\mathit{x}:=(x_1,\dots ,x_d)$, $\mathrm{\nabla }:=(\mathrm{\partial }/\mathrm{\partial }{x}_1,\dots ,\mathrm{\partial }/\mathrm{\partial }{x}_d{)}^T$ and $\mathit{K}\left(\mathit{x}\right):=\left(k_{ij}\right(\mathit{x}){)}_{i,j=1}^d$ is the diffusion tensor. ${}_0{I}_t^{\alpha \left(t\right)}$ and ${\mathrm{\partial }}_t^{\alpha \left(t\right)}$ denote the variable-order fractional integral operator and the variable-order Caputo fractional differential operator, respectively [5–7] (2) $\begin{array}{rl}{}_0{I}_t^{\alpha \left(t\right)}g\left(t\right)& :=\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t\frac{g\left(s\right)}{(t-s{)}^{1-\alpha \left(t\right)}}\mathrm{d}s,\\ {\mathrm{\partial }}_t^{\alpha \left(t\right)}g\left(t\right)& :={}_0{I}_t^{1-\alpha \left(t\right)}{g}^{\prime }\left(t\right)=\frac{1}{\mathrm{\Gamma }(1-\alpha (t\left)\right)}{\int }_0^t\frac{g^{\prime }\left(s\right)}{(t-s{)}^{\alpha \left(t\right)}}\mathrm{d}s.\end{array}$(2) Equation (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) is a variable-order generalization [2] of the two-scale mobile–immobile time-fractional partial differential equation, which was developed in [1,3] to improve the modelling of the subdiffusive transport of solutes in heterogeneous porous media by the conventional time-fractional partial differential equations [8–11]. In this scenario, a large amount of solute particles may get absorbed to the aquifer and then get released at later time, leading to subdiffusive transport of solutes. This process is modelled by the $k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u$ term in (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ), in which $k\left(t\right)$ is the dynamic partition coefficient quantifying that $k/(1+k)$ portion of the total solute mass is adsorbed to the porous media and undergoes subdiffusive transport. The ${\mathrm{\partial }}_{t}u$ terms in (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) models the Fickian diffusive transport of the remaining $1/(1+k)$ portion of the total solute mass in the bulk fluid phase in the aquifer. In traditional integer-order partial differential equation models, the adsorption of the solutes is modelled via an algebraic constitutive relation that relates the amount of the absorpted solutes to the current amount of solutes in the bulk fluid phase. This yields an adsorption term in the integer-order partial differential equation model that has an extra (possibly nonlinear) retardation coefficient in front of the ${\mathrm{\partial }}_{t}u$ term in the governing integer-order partial differential equation. However, such a model does not accurately describe the power-law decaying memory effect of the adsorbed solutes to the porous media.

The two-scale mobile–immobile variable-order time-fractional partial differential equation (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) provides an accurate modelling of the diffusive transport of solutes in heterogeneous porous media over the entire time interval $[0,\mathrm{\infty })$, in contrast to the conventional time-fractional partial differential equation [9] that provides an accurate description of the long-term subdiffusive transport behaviour of solutes in heterogeneous porous media but fails to accurately describe the short-term Fickian diffusive transport behaviour of the solute transport, and so introduces nonphysical weak initial singularity of the solutions [8,10,12]. Moreover, Equation (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) incorporates the impact of the porous medium structure change in such applications as manufacturing of biomaterials in orthopedic implants [13], biological clogging of saturated soils and aquifer materials [14], modelling and design of shape memory polymer [15], nonconventional hydrocarbon or shale gas recovery [16], and modelling and design of viscoelastic materials [17]. The order of the fractional partial differential equation is determined by the fractal dimension of the porous materials via the Hurst index [18,19]. In this way, the dynamic variation of the order $\alpha \left(t\right)$ in the two-scale mobile–immobile time-fractional partial differential equation (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) incorporates the structure change of porous materials [5]. The numerical approximations and fast computations of variable-order fractional partial differential equations were studied and analysed in the literature [6,20–31].

However, the parameters in governing fractional partial differential equations, in particular, the fractional orders are usually unknown and often need to be determined based on the observed data, which are formulated as inverse problems attracting extensive investigations [32–53]. The theoretical analysis on the unique identification of the parameters is the key to guarantee the reliability of the experimentally inferred parameters in these fractional partial differential equations. In particular, the theoretical analysis on the uniqueness of the determination of the fractional order(s) in single-term and multi-term time-fractional diffusion equations was carried out in [54]. A Lipschitz stability analysis on the determination of the fractional order in time-fractional diffusion equations with respect to the measured data at an interior point was established in [55]. A uniqueness result on the determination of the fractional orders in space–time fractional diffusion equations was proved in [56]. A uniqueness result on the simultaneous determination of the fractional order and the diffusivity coefficients in time-fractional diffusion equations with variable-diffusivity based on boundary observations was studied in [57].

The aforementioned uniqueness results were for constant-order time-fractional diffusion equations. But the corresponding results on the unique determination of the variable fractional order of variable-order time-fractional partial differential equations are meager in the literature. One of the main obstacles lies in that a closed-form solution representation, which is crucial in the analysis of constant-order fractional partial differential equations, is not available in general in the context of variable-order fractional partial differential equations. In [58], the authors proved the wellposedness of variable-order time-fractional diffusion equations with a space-dependent variable fractional order by taking the full advantage of the fact that the analytic tools, such as the Laplace transform, can still be applied to find the solution representation of the problem. They are based on these results to prove that the space-dependent variable order can be uniquely determined by the knowledge of a suitable time sequence of partial Dirichlet-to-Neumann maps. In [59], the uniqueness of the determination of a time-dependent variable order of the one-dimensional simplification of problem (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) with constant diffusivity was proved via a spectral decomposition. However, the proof in [59] relies heavily on the fact that the eigenvalues of the Sturm–Liouville problem of the diffusion operator form a strictly increasing sequence, which is not true even in the context of a Laplacian operator on a unit square domain and not mentioning for problem (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) on a general (not necessarily tensor product) multi-dimensional domain with a variable diffusivity tensor.

In this paper, we prove the unique determination of the variable order α for problem (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ), with the observations of the unknown solutions on a small spatial domain (that is determined from the initial data $u_0\left(\mathit{x}\right)$, refer to Theorem 3.2) over a small time interval. The rest of the paper is organized as follows: in Section 2 we address the wellposedness and smoothing properties of problem (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ). In Section 3, we prove the uniqueness of the inverse problem of determining the variable order in problem (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) from some available observed values of the unknown solutions. In Section 4, we draw concluding remarks.

2. Wellposedness and smoothing properties of the variable-order two-scale mobile–immobile time-fractional partial differential equation (1)

In this section, we go over the wellposedness and smoothing properties of the variable-order time-fractional partial differential equation (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) [2], which will be used subsequently. Let $C[a,b]$ be the Banach space of continuous functions on $[a,b]$. Let $\mathbb{N}$ be the set of non-negative integers and $m\in \mathbb{N}$, let $C^m[a,b]$ be the Banach space of continuous functions with continuous derivatives up to order m on $[a,b]$. Let $L^2\left(\mathrm{\Omega }\right)$ be the Hilbert space of Lebesgue square integrable functions on Ω and $H^m\left(\mathrm{\Omega }\right)$ be the Sobolev space of Lebesgue square integrable functions with their weak derivatives of order m being in $L^2\left(\mathrm{\Omega }\right)$. For non-negative real number $s\ge 0$, the fractional Sobolev space $H^s\left(\mathrm{\Omega }\right)$ is defined by the interpolation. All the spaces are equipped with the standard norms [60].

For a Banach space $\mathcal{X}$ equipped with the norm $\parallel \cdot {\parallel }_{\mathcal{X}}$, let $C^m\left(\right[a,b];\mathcal{X})$ be the space-time Banach space equipped with the norm [60,61] $\begin{array}{rl}{C}^m\left(\right[a,b];\mathcal{X})& :=\{w(\mathit{x},t):\parallel {\mathrm{\partial }}_t^{l}w(\cdot ,t){\parallel }_{\mathcal{X}}\in C^l\left(\right[a,b\left]\right),l=0,1,\dots ,m\},\\ \parallel w{\parallel }_{C^m(\mathcal{I};\mathcal{X})}& :=\underset{0\le l\le m}{max}\underset{t\in [a,b]}{max}\parallel {\mathrm{\partial }}_t^{l}w(\cdot ,t){\parallel }_{\mathcal{X}}.\end{array}$Let $\{{\lambda }_i,{\varphi }_i{\}}_{i=1}^{\mathrm{\infty }}$ be the eigenvalues and eigenfunctions of the Sturm–Liouville problem $\begin{array}{rl}-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }{\varphi }_i\left(\mathit{x}\right))& ={\lambda }_i{\varphi }_i\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ {\varphi }_i\left(\mathit{x}\right)& =0,\mathit{x}\in \mathrm{\partial }\mathrm{\Omega },\end{array}$where $\mathit{K}\left(\mathit{x}\right)=\left(k_{ij}\right(\mathit{x}){)}_{i,j=1}^d$ satisfies $\mathit{K}={\mathit{K}}^{\mathrm{\top }}$, $k_{ij}\in C^1\left(\overline{\mathrm{\Omega }}\right)$ for $1\le i,j\le d$ and there exist positive constants $0<K_m\le K_M<\mathrm{\infty }$ such that for any $\mathbf{0}\ne \mathit{\xi }\in {\mathbb{R}}^d$, the following inequalities hold: $0<K_m\le \frac{{\mathit{\xi }}^{\mathrm{\top }}\mathit{K}\left(\mathit{x}\right)\mathit{\xi }}{{\mathit{\xi }}^{\mathrm{\top }}\mathit{\xi }}\le K_M<\mathrm{\infty },\mathrm{\forall }\mathit{x}\in \overline{\mathrm{\Omega }}.$It is known that $\{{\varphi }_i{\}}_{i=1}^{\mathrm{\infty }}$ form an orthonormal basis in $L^2\left(\mathrm{\Omega }\right)$ and the eigenvalues $\{{\lambda }_i{\}}_{i=1}^{\mathrm{\infty }}$ are positive and form a nondecreasing sequence that tends to ∞ with i [61]. Furthermore, for any $\gamma \ge 0$ the Sobolev space defined by $\begin{array}{rl}{\check{H}}^{\gamma }\left(\mathrm{\Omega }\right)& :=\{v\in L^2\left(\mathrm{\Omega }\right):\parallel v{\parallel }_{{\check{H}}^{\gamma }\left(\mathrm{\Omega }\right)}^2:=((-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }){)}^{\gamma }v\left(\mathit{x}\right),v)\\ & =\sum _{i=1}^{\mathrm{\infty }}{\lambda }_i^{\gamma }(v,{\varphi }_i{)}^2<\mathrm{\infty }\}\end{array}$is a subspace of the fractional Sobolev space $H^{\gamma }\left(\mathrm{\Omega }\right)$ [12,60,62].

The following result holds for the initial boundary value problem of variable-order two-scale mobile–immobile time-fractional partial differential equation (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) [30].

Theorem 2.1

Suppose the variable order $\alpha \left(t\right)$ satisfies (3) $\alpha \in C[0,T],0\le \alpha \left(t\right)\le {\alpha }_{\ast }<1,t\in [0,T]$(3) and the dynamic partition coefficient k belongs to $C[0,T]$. Further, $u_0\in {\check{H}}^{\gamma +2}\left(\mathrm{\Omega }\right)$ for some $d/2<\gamma \in {\mathbb{R}}^{+}$. Then problem (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) has a unique solution $u\in C^1\left(\right[0,T];{\check{H}}^{\gamma }(\mathrm{\Omega }\left)\right)$ and the following stability estimate holds: $\parallel u{\parallel }_{C^1\left(\right[0,T];{\check{H}}^s(\mathrm{\Omega }\left)\right)}\le Q\parallel u_0{\parallel }_{{\check{H}}^{2+s}\left(\mathrm{\Omega }\right)},0\le s\le \gamma .$

3. Uniquely determining the variable order of time-fractional partial differential equation (1) on general multi-dimensional domains

In this section, we prove the main result of this paper, the uniqueness of the inverse problem of determining the variable order $\alpha \left(t\right)$ in the variable-order time-fractional partial differential equation (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) with a variable diffusivity tensor, imposed on a general multi-dimensional domain based on the observations of the solution $u(\mathit{x},t)$ at a sufficiently small spatial domain, which is determined from the initial data $u_0\left(\mathit{x}\right)$, over a sufficiently small time interval.

Without loss of generality, we assume that the initial data $u_0\left(\mathit{x}\right)$ in problem (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) is not identically zero on the entire domain Ω. Otherwise, problem (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) has only the trivial solution $u(\mathit{x},t)\equiv 0$ by Theorem 2.1, for any given variable order $\alpha \left(t\right)$ that satisfies the assumption of the theorem, so there is no way to uniquely identify the variable order $\alpha \left(t\right)$ in the case $u_0\left(\mathit{x}\right)$ is identically zero in Ω. We begin with the following lemma.

Lemma 3.1

Suppose that the initial data $u_0\left(\mathit{x}\right)$ in problem (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) is not identically zero in the domain Ω and satisfies $u_0\in {\check{H}}^s\left(\mathrm{\Omega }\right)$ for $s>2+d/2$. Then there exists an open spatial domain $\mathrm{\Lambda }\subset \mathrm{\Omega }$ and a positive constant $\sigma >0$ such that either $-\mathrm{\nabla }\cdot \left(\mathit{K}\right(\mathit{x}\left)\mathrm{\nabla }{u}_0\right(\mathit{x}\left)\right)\ge \sigma$ or $-\mathrm{\nabla }\cdot \left(\mathit{K}\right(\mathit{x}\left)\mathrm{\nabla }{u}_0\right(\mathit{x}\left)\right)\le -\sigma$ for $\mathit{x}\in \mathrm{\Lambda }$.

Proof.

By the Sobolev embedding $H^{2+d/2+ϵ}\left(\mathrm{\Omega }\right)\hookrightarrow C^2\left(\mathrm{\Omega }\right)$ for any $ϵ>0$ and the fact that ${\check{H}}^s\left(\mathrm{\Omega }\right)$ is a subspace of $H^s\left(\mathrm{\Omega }\right)$, $u_0\in {\check{H}}^s\left(\mathrm{\Omega }\right)$ implies $u_0\in C^2\left(\mathrm{\Omega }\right)$. As $u_0\left(\mathit{x}\right)$ is not identically zero on Ω and $u_0\left(\mathit{x}\right)=0$ on $\mathit{x}\in \mathrm{\partial }\mathrm{\Omega }$, we claim that $-\mathrm{\nabla }\cdot \left(\mathit{K}\right(\mathit{x}\left)\mathrm{\nabla }{u}_0\right(\mathit{x}\left)\right)$ is not identically zero on Ω. Otherwise, the theory of second-order elliptic partial differential equation [61] concludes that the boundary-value problem $\begin{array}{rl}-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }{u}_0\left(\mathit{x}\right))& =0,\mathit{x}\in \mathrm{\Omega };\\ u_0\left(\mathit{x}\right)& =0,\mathit{x}\in \mathrm{\partial }\mathrm{\Omega }\end{array}$has only the trivial solution $u_0\left(\mathit{x}\right)\equiv 0$, which contradicts to the assumption that $u_0\left(\mathit{x}\right)$ is not identically zero on Ω.

Thus there exists a point ${\mathit{x}}_0\in \mathrm{\Omega }$ such that $-\mathrm{\nabla }\cdot \left(\mathit{K}\right({\mathit{x}}_0\left)\mathrm{\nabla }{u}_0\right({\mathit{x}}_0\left)\right)\ne 0$. Without loss of generality, we assume that $-\mathrm{\nabla }\cdot \left(\mathit{K}\right({\mathit{x}}_0\left)\mathrm{\nabla }{u}_0\right({\mathit{x}}_0\left)\right)>0$. Since $-\mathrm{\nabla }\cdot \left(\mathit{K}\right(\mathit{x}\left)\mathrm{\nabla }{u}_0\right(\mathit{x}\left)\right)\in C\left(\mathrm{\Omega }\right)$, there exists an open ball $B_{\delta }\left({\mathit{x}}_0\right)\subset \mathrm{\Omega }$, which is centred at ${\mathit{x}}_0$ with radius $\delta >0$, and a positive constant $\sigma >0$ such that $-\mathrm{\nabla }\cdot \left(\mathit{K}\right(\mathit{x}\left)\mathrm{\nabla }{u}_0\right(\mathit{x}\left)\right)\ge \sigma >0$ for $\mathit{x}\in B_{\delta }\left({\mathit{x}}_0\right)$. The case of $-\mathrm{\nabla }\cdot \left(\mathit{K}\right({\mathit{x}}_0\left)\mathrm{\nabla }{u}_0\right({\mathit{x}}_0\left)\right)<0$ can be proved by symmetry.

Remark 3.1

Lemma 3.1 ensures the existence of an open set $\mathrm{\Lambda }\subset \mathrm{\Omega }$ on which $-\mathrm{\nabla }\cdot \left(\mathit{K}\right(\mathit{x}\left)\mathrm{\nabla }{u}_0\right(\mathit{x}\left)\right)$ is strictly bounded away from 0. In practice, as the $u_0\left(\mathit{x}\right)$ is the given data, such an open set Λ can be identified by directly calculating $-\mathrm{\nabla }\cdot \left(\mathit{K}\right(\mathit{x}\left)\mathrm{\nabla }{u}_0\right(\mathit{x}\left)\right)$.

We prove the main result of this paper in the following theorem.

Theorem 3.2

Suppose that the assumptions of Theorem 2.1 hold. Assume that the dynamic partition coefficient $k\left(t\right)$ and the initial data $u_0\left(\mathit{x}\right)$ in problem (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) satisfy $k\left(0\right)\ne 0$ and $u_0\left(\mathit{x}\right)$ is not identically zero on the domain Ω with $u_0\in {\check{H}}^s\left(\mathrm{\Omega }\right)$ for some $s>4+d/2$. Let $\mathrm{\Lambda }\subset \mathrm{\Omega }$ be an open subset on which either $-\mathrm{\nabla }\cdot \left(\mathit{K}\right(\mathit{x}\left)\mathrm{\nabla }{u}_0\right(\mathit{x}\left)\right)\ge \sigma$ or $-\mathrm{\nabla }\cdot \left(\mathit{K}\right(\mathit{x}\left)\mathrm{\nabla }{u}_0\right(\mathit{x}\left)\right)\le -\sigma$ holds for some positive constant $\sigma >0$. Then the variable order $\alpha \left(t\right)$ in the initial boundary value problem of the two-scale mobile–immobile variable-order time-fractional partial differential equation (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) can be determined uniquely in the following admissible set: $\mathcal{A}:=\{\alpha \left(t\right):\alpha \left(t\right)\text{is analytic on}[0,T]\text{and satisfies}\left(3\right)\},$given the observations of the solution $u(\mathit{x},t)$ to problem (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) in Λ over a small time interval.

More precisely, let $\hat{u}(\mathit{x},t)$ be the solution to the initial boundary value problem of the two-scale mobile–immobile variable-order time-fractional partial differential equation (4) $\begin{array}{rl}& {\mathrm{\partial }}_t\hat{u}+k\left(t\right){\mathrm{\partial }}_t^{\hat{\alpha }\left(t\right)}\hat{u}-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }\hat{u}(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & \hat{u}(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega },\\ & \hat{u}(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T]\end{array}$(4) with the variable order $\hat{\alpha }\left(t\right)\in \mathcal{A}$ satisfies condition (3(3) $\alpha \in C[0,T],0\le \alpha \left(t\right)\le {\alpha }_{\ast }<1,t\in [0,T]$(3) ). If (5) $u(\mathit{x},t)=\hat{u}(\mathit{x},t),(\mathit{x},t)\in \mathrm{\Lambda }\times [0,\tau ]$(5) for some sufficiently small $\tau >0$, then we have (6) $\alpha \left(t\right)=\hat{\alpha }\left(t\right),t\in [0,T].$(6)

Proof.

By Theorem 2.1, $u_0\in {\check{H}}^s\left(\mathrm{\Omega }\right)$ implies $u\in C^1\left(\right[0,T];{\check{H}}^{s-2}(\mathrm{\Omega }\left)\right)$. As $s-2>2+d/2$, we apply the Sobolev embedding theorem [60] to conclude that $u\in C^1\left(\right[0,T];C^2(\mathrm{\Omega }\left)\right)$. By Lemma 3.1, there exists an open subset $\mathrm{\Lambda }\subset \mathrm{\Omega }$ such that either $-\mathrm{\nabla }\cdot \left(\mathit{K}\right(\mathit{x}\left)\mathrm{\nabla }{u}_0\right(\mathit{x}\left)\right)\ge \sigma$ or $-\mathrm{\nabla }\cdot \left(\mathit{K}\right(\mathit{x}\left)\mathrm{\nabla }{u}_0\right(\mathit{x}\left)\right)\le -\sigma$ for $\mathit{x}\in \mathrm{\Lambda }$.

Without loss of generality, we assume that the former holds. By the continuity of $-\mathrm{\nabla }\cdot \left(\mathit{K}\right(\mathit{x}\left)\mathrm{\nabla }u\right(\mathit{x},t\left)\right)$ in both space and time, there exists a time interval $[0,t_0]\subset [0,T]$ such that $-\mathrm{\nabla }\cdot \left(\mathit{K}\right(\mathit{x}\left)\mathrm{\nabla }u\right(\mathit{x},t\left)\right)\ge 3\sigma /4>0$ for $(\mathit{x},t)\in \mathrm{\Lambda }\times [0,t_0]$. We combine the facts that $k\in C[0,T]$, ${\mathrm{\partial }}_{t}u\in C\left(\right[0,T];C(\mathrm{\Omega }\left)\right)$ and the expression of the variable-order fractional differential operator (2(2) $\begin{array}{rl}{}_0{I}_t^{\alpha \left(t\right)}g\left(t\right)& :=\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t\frac{g\left(s\right)}{(t-s{)}^{1-\alpha \left(t\right)}}\mathrm{d}s,\\ {\mathrm{\partial }}_t^{\alpha \left(t\right)}g\left(t\right)& :={}_0{I}_t^{1-\alpha \left(t\right)}{g}^{\prime }\left(t\right)=\frac{1}{\mathrm{\Gamma }(1-\alpha (t\left)\right)}{\int }_0^t\frac{g^{\prime }\left(s\right)}{(t-s{)}^{\alpha \left(t\right)}}\mathrm{d}s.\end{array}$(2) ) with a weakly integrable kernel to conclude that $\underset{t\to 0^{+}}{lim}k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)=0$. Then there exists a positive time instant τ with $0<\tau \le t_0$ such that $|k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)|\le \sigma /4$ for $(\mathit{x},t)\in \mathrm{\Lambda }\times [0,\tau ]$. We incorporate the preceding estimates into the variable-order time-fractional partial differential equation (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) to conclude that (7) ${\mathrm{\partial }}_{t}u(\mathit{x},t)\le -\sigma /2<0,\mathrm{\forall }(\mathit{x},t)\in \mathrm{\Lambda }\times [0,\tau ].$(7) Since $u(\mathit{x},t)=\hat{u}(\mathit{x},t)$ on $(\mathit{x},t)\in \mathrm{\Lambda }\times [0,\tau ]$ and $u,\hat{u}\in C^1\left(\right[0,T];C^2(\mathrm{\Omega }\left)\right)$, we have $-\mathrm{\nabla }\cdot \left(\mathit{K}\right(\mathit{x}\left)\mathrm{\nabla }u\right(\mathit{x},t\left)\right)=-\mathrm{\nabla }\cdot \left(\mathit{K}\right(\mathit{x}\left)\mathrm{\nabla }\hat{u}\right(\mathit{x},t\left)\right)$ for all $(\mathit{x},t)\in \mathrm{\Lambda }\times [0,\tau ]$. We subtract Equation (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) from Equation (4(4) $\begin{array}{rl}& {\mathrm{\partial }}_t\hat{u}+k\left(t\right){\mathrm{\partial }}_t^{\hat{\alpha }\left(t\right)}\hat{u}-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }\hat{u}(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & \hat{u}(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega },\\ & \hat{u}(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T]\end{array}$(4) ) for $(\mathit{x},t)\in \mathrm{\Lambda }\times [0,\tau ]$ to get $k\left(t\right)({\mathrm{\partial }}_t^{\alpha \left(t\right)}-{\mathrm{\partial }}_t^{\hat{\alpha }\left(t\right)})u(\mathit{x},t)=0,\mathrm{\forall }(\mathit{x},t)\in \mathrm{\Lambda }\times [0,\tau ].$By the assumption that $k\left(0\right)\ne 0$ and $k\in C[0,T]$, there exists an $0<{ϵ}_1\le \tau$ such that (8) $({\mathrm{\partial }}_t^{\alpha \left(t\right)}-{\mathrm{\partial }}_t^{\hat{\alpha }\left(t\right)})u(\mathit{x},t)=0,\mathrm{\forall }(\mathit{x},t)\in \mathrm{\Lambda }\times (0,{ϵ}_1].$(8) For any fixed $t\in (0,{ϵ}_1]$, 0<s<t and $0\le \text{β}<1$, let $G\left(\text{β}\right):=\frac{(t-s{)}^{-\text{β}}}{\mathrm{\Gamma }(1-\text{β})}.$Then, for any 0<s<t and $t\in (0,{ϵ}_1]$ we have (9) $\begin{array}{rl}{G}^{\prime }\left(\text{β}\right)& =\frac{{\mathrm{\Gamma }}^{\prime }(1-\text{β})}{\mathrm{\Gamma }(1-\text{β}{)}^2}(t-s{)}^{-\text{β}}-\frac{(t-s{)}^{-\text{β}}\ln (t-s)}{\mathrm{\Gamma }(1-\text{β})}\\ & =\frac{(t-s{)}^{-\text{β}}}{\mathrm{\Gamma }(1-\text{β})}(\frac{{\mathrm{\Gamma }}^{\prime }(1-\text{β})}{\mathrm{\Gamma }(1-\text{β})}-\ln (t-s)).\end{array}$(9) We insert the expression (9(9) $\begin{array}{rl}{G}^{\prime }\left(\text{β}\right)& =\frac{{\mathrm{\Gamma }}^{\prime }(1-\text{β})}{\mathrm{\Gamma }(1-\text{β}{)}^2}(t-s{)}^{-\text{β}}-\frac{(t-s{)}^{-\text{β}}\ln (t-s)}{\mathrm{\Gamma }(1-\text{β})}\\ & =\frac{(t-s{)}^{-\text{β}}}{\mathrm{\Gamma }(1-\text{β})}(\frac{{\mathrm{\Gamma }}^{\prime }(1-\text{β})}{\mathrm{\Gamma }(1-\text{β})}-\ln (t-s)).\end{array}$(9) ) into Equation (8(8) $({\mathrm{\partial }}_t^{\alpha \left(t\right)}-{\mathrm{\partial }}_t^{\hat{\alpha }\left(t\right)})u(\mathit{x},t)=0,\mathrm{\forall }(\mathit{x},t)\in \mathrm{\Lambda }\times (0,{ϵ}_1].$(8) ) and use (2(2) $\begin{array}{rl}{}_0{I}_t^{\alpha \left(t\right)}g\left(t\right)& :=\frac{1}{\mathrm{\Gamma }\left(\alpha \right(t\left)\right)}{\int }_0^t\frac{g\left(s\right)}{(t-s{)}^{1-\alpha \left(t\right)}}\mathrm{d}s,\\ {\mathrm{\partial }}_t^{\alpha \left(t\right)}g\left(t\right)& :={}_0{I}_t^{1-\alpha \left(t\right)}{g}^{\prime }\left(t\right)=\frac{1}{\mathrm{\Gamma }(1-\alpha (t\left)\right)}{\int }_0^t\frac{g^{\prime }\left(s\right)}{(t-s{)}^{\alpha \left(t\right)}}\mathrm{d}s.\end{array}$(2) ) to rewrite Equation (8(8) $({\mathrm{\partial }}_t^{\alpha \left(t\right)}-{\mathrm{\partial }}_t^{\hat{\alpha }\left(t\right)})u(\mathit{x},t)=0,\mathrm{\forall }(\mathit{x},t)\in \mathrm{\Lambda }\times (0,{ϵ}_1].$(8) ) as follows: (10) $\begin{array}{rl}& ({\mathrm{\partial }}_t^{\alpha \left(t\right)}-{\mathrm{\partial }}_t^{\hat{\alpha }\left(t\right)})u(\mathit{x},t)\\ & ={\int }_0^t\left[\frac{1}{\mathrm{\Gamma }(1-\alpha (t\left)\right)(t-s{)}^{\alpha \left(t\right)}}-\frac{1}{\mathrm{\Gamma }(1-\hat{\alpha }(t\left)\right)(t-s{)}^{\hat{\alpha }\left(t\right)}}\right]{\mathrm{\partial }}_{t}u(\mathit{x},s)\mathrm{d}s\\ & ={\int }_0^t\frac{(t-s{)}^{-\overline{\alpha }(s,t)}}{\mathrm{\Gamma }(1-\overline{\alpha }(s,t\left)\right)}(\frac{{\mathrm{\Gamma }}^{\prime }(1-\overline{\alpha }(s,t\left)\right)}{\mathrm{\Gamma }(1-\overline{\alpha }(s,t\left)\right)}-\ln (t-s)){\mathrm{\partial }}_{s}u(\mathit{x},s)\mathrm{d}s\left(\alpha \right(t)-\hat{\alpha }(t\left)\right)\\ & =0,(\mathit{x},t)\in \mathrm{\Lambda }\times (0,{ϵ}_1],\end{array}$(10) where $\overline{\alpha }(s,t)$ lies between $\alpha \left(t\right)$ and $\hat{\alpha }\left(t\right)$ for any $0<t\le {ϵ}_1$ and 0<s<t.

By assumptions on $\alpha \left(t\right)$ (and $\hat{\alpha }\left(t\right)$), $\overline{\alpha }(s,t)$ lies between $\alpha \left(t\right)$ and $\hat{\alpha }\left(t\right)$ that are bounded between 0 and ${\alpha }_{\ast }<1$. This leads to the following estimate: (11) $|\frac{{\mathrm{\Gamma }}^{\prime }(1-\overline{\alpha }(s,t\left)\right)}{\mathrm{\Gamma }(1-\overline{\alpha }(s,t\left)\right)}|\le Q_0,t\in (0,{ϵ}_1],0<s<t.$(11) For the constant $Q_0$, there exists a positive ${ϵ}_2\le {ϵ}_1$ such that $\ln t<-2Q_0$ for $t\in (0,{ϵ}_2]$ such that (12) $\frac{{\mathrm{\Gamma }}^{\prime }(1-\overline{\alpha }(s,t\left)\right)}{\mathrm{\Gamma }(1-\overline{\alpha }(s,t\left)\right)}-\ln (t-s)>Q_0>0,0<s<t,t\in (0,{ϵ}_2].$(12) We incorporate the estimates (7(7) ${\mathrm{\partial }}_{t}u(\mathit{x},t)\le -\sigma /2<0,\mathrm{\forall }(\mathit{x},t)\in \mathrm{\Lambda }\times [0,\tau ].$(7) ) and (11(11) $|\frac{{\mathrm{\Gamma }}^{\prime }(1-\overline{\alpha }(s,t\left)\right)}{\mathrm{\Gamma }(1-\overline{\alpha }(s,t\left)\right)}|\le Q_0,t\in (0,{ϵ}_1],0<s<t.$(11) )–(12(12) $\frac{{\mathrm{\Gamma }}^{\prime }(1-\overline{\alpha }(s,t\left)\right)}{\mathrm{\Gamma }(1-\overline{\alpha }(s,t\left)\right)}-\ln (t-s)>Q_0>0,0<s<t,t\in (0,{ϵ}_2].$(12) ) to conclude that $\begin{array}{rl}& \frac{(t-s{)}^{-\overline{\alpha }(s,t)}}{\mathrm{\Gamma }(1-\overline{\alpha }(s,t\left)\right)}(\frac{{\mathrm{\Gamma }}^{\prime }(1-\overline{\alpha }(s,t\left)\right)}{\mathrm{\Gamma }(1-\overline{\alpha }(s,t\left)\right)}-\ln (t-s)){\mathrm{\partial }}_{s}u(\mathit{x},s)\\ & \le -\frac{\sigma Q_0(t-s{)}^{-\overline{\alpha }(s,t)}}{2\mathrm{\Gamma }(1-\overline{\alpha }(s,t\left)\right)},0<s<t,(\mathit{x},t)\in \mathrm{\Lambda }\times (0,{ϵ}_2].\end{array}$Consequently, we apply this and (10(10) $\begin{array}{rl}& ({\mathrm{\partial }}_t^{\alpha \left(t\right)}-{\mathrm{\partial }}_t^{\hat{\alpha }\left(t\right)})u(\mathit{x},t)\\ & ={\int }_0^t\left[\frac{1}{\mathrm{\Gamma }(1-\alpha (t\left)\right)(t-s{)}^{\alpha \left(t\right)}}-\frac{1}{\mathrm{\Gamma }(1-\hat{\alpha }(t\left)\right)(t-s{)}^{\hat{\alpha }\left(t\right)}}\right]{\mathrm{\partial }}_{t}u(\mathit{x},s)\mathrm{d}s\\ & ={\int }_0^t\frac{(t-s{)}^{-\overline{\alpha }(s,t)}}{\mathrm{\Gamma }(1-\overline{\alpha }(s,t\left)\right)}(\frac{{\mathrm{\Gamma }}^{\prime }(1-\overline{\alpha }(s,t\left)\right)}{\mathrm{\Gamma }(1-\overline{\alpha }(s,t\left)\right)}-\ln (t-s)){\mathrm{\partial }}_{s}u(\mathit{x},s)\mathrm{d}s\left(\alpha \right(t)-\hat{\alpha }(t\left)\right)\\ & =0,(\mathit{x},t)\in \mathrm{\Lambda }\times (0,{ϵ}_1],\end{array}$(10) ) to immediately obtain $\alpha \left(t\right)=\hat{\alpha }\left(t\right)$ on $t\in (0,{ϵ}_2]$.

We now utilize the assumption that α belongs to the admissible set $\mathcal{A}$ to prove that (6(6) $\alpha \left(t\right)=\hat{\alpha }\left(t\right),t\in [0,T].$(6) ) holds. In fact, let (13) $t_{\ast }:=max\{t:\alpha \left(s\right)=\hat{\alpha }\left(s\right),\mathrm{\forall }s\in [0,t]\}.$(13) We prove $t_{\ast }=T$. Suppose not, that is $t_{\ast }<T$. By assumption, $\alpha ,\hat{\alpha }\in \mathcal{A}$ are analytical on $[0,T]$ and $\alpha \left(t\right)\equiv \hat{\alpha }\left(t\right)$ on $[0,t_{\ast }]$. Hence, ${\alpha }^{\left(m\right)}\left(t_{\ast }\right)={\hat{\alpha }}^{\left(m\right)}\left(t_{\ast }\right)$ for $m\in \mathbb{N}$. Since α and $\hat{\alpha }$ are analytic at $t_{\ast }$, there is a $0<{ϵ}_3\ll 1$ such that $\alpha \left(t\right)\equiv \hat{\alpha }\left(t\right)$ for all $t\in [0,t_{\ast }+{ϵ}_3]$. This contradicts to the definition of $t_{\ast }$ in (13(13) $t_{\ast }:=max\{t:\alpha \left(s\right)=\hat{\alpha }\left(s\right),\mathrm{\forall }s\in [0,t]\}.$(13) ). Thus we conclude that $\alpha \left(t\right)=\hat{\alpha }\left(t\right)$ for all $t\in [0,T]$.

4. Concluding remarks

The two-scale mobile–immobile variable-order time-fractional partial differential equation (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) provides an accurate modelling of the diffusive transport of solutes in heterogeneous porous media over the entire time interval $[0,\mathrm{\infty })$, in contrast to the conventional time-fractional partial differential equation that provides an accurate description of the long-term subdiffusive transport behaviour of solutes in heterogeneous porous media but fails to accurately describe the short-term Fickian diffusive transport behaviour of the solute transport. Moreover, Equation (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) properly addresses the impact of the porous medium structure change in many applications.

In practice, the parameters in governing fractional partial differential equations are often unknown and need to be determined based on the observed data. Most of the results on the inverse problems of fractional partial differential equations were for constant-order fractional partial differential equations, and the corresponding results on the unique determination of the variable fractional order of variable-order time-fractional partial differential equations are meager in the literature (see, e.g.[58,59]).

In this paper, we proved the unique determination of the variable-order in the two-scale mobile–immobile variable-order time-fractional partial differential equation (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) with a variable diffusivity tensor imposed on a general multi-dimensional domain, with the observations of the unknown solutions on any arbitrarily small spatial domain Λ, which is identified by either $-\mathrm{\nabla }\cdot \left(\mathit{K}\right(\mathit{x}\left)\mathrm{\nabla }{u}_0\right(\mathit{x}\left)\right)\ge \sigma$ or $-\mathrm{\nabla }\cdot \left(\mathit{K}\right(\mathit{x}\left)\mathrm{\nabla }{u}_0\right(\mathit{x}\left)\right)\le -\sigma$ for $\mathit{x}\in \mathrm{\Lambda }$ for some $\sigma >0$, over a sufficiently small time interval. In other words, the theorem provides a theoretical guidance where the measurements should be performed. This work generalized the uniqueness result in [59] for the one-dimensional analogue of model (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) with constant diffusivity coefficient. Since the analysis in [59] relies heavily on the fact that the eigenvalues of the Sturm–Liouville problem of the diffusion operator form a strictly increasing sequence, which is not true even in the context of a Laplacian operator on a unit square domain, the proposed techniques do not apply for the multi-dimensional model (1(1) $\begin{array}{rl}& {\mathrm{\partial }}_{t}u(\mathit{x},t)+k\left(t\right){\mathrm{\partial }}_t^{\alpha \left(t\right)}u(\mathit{x},t)-\mathrm{\nabla }\cdot (\mathit{K}\left(\mathit{x}\right)\mathrm{\nabla }u(\mathit{x},t))=0,(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];\\ & u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };\\ & u(\mathit{x},t)=0,(\mathit{x},t)\in \mathrm{\partial }\mathrm{\Omega }\times [0,T].\end{array}$(1) ) with a variable diffusivity tensor. Therefore, a PDE-based approach is developed in the current work in contrast with the variable separation method used in [59], which represents a salient difference between these two works.

Acknowledgments

The authors would like to express their most sincere thanks to the referees for their very helpful comments and suggestions, which greatly improved the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially funded by the ARO MURI Grant W911NF-15-1-0562, by the National Science Foundation under Grants DMS-1620194 and DMS-2012291, and by a SPARC Graduate Research Grant from the Office of the Vice President for Research at the University of South Carolina.