Analysis of a Hidden-Memory Variably Distributed-Order Time-Fractional Diffusion Equation

Abstract

We analyze the well-posedness and regularity of a variably distributed-order time-fractional diffusion equation (tFDE) with a hidden-memory fractional derivative, which provide a competitive means to describe the anomalously diffusive transport of particles in heterogeneous media. We prove that the solution of a variably distributed-order tFDE has weak singularity at the initial time $t=0$ which depends on the upper bound of a distributed order $\overline{\alpha }\left(0\right)$.

1. Introduction

Field tests showed that the diffusive transport of solutes in heterogeneous porous media usually exhibits highly skewed power law decays and cannot be accurately modeled by the integer-order diffusion equation, as its basic solution is exponentially decaying. The time-fractional diffusion equation (tFDE) was derived by using a continuous time random walk under the assumption that the mean waiting time of solute transport is power law decaying. Therefore, the tFDE provides a competitive means to model the anomalous diffusive transport in heterogeneous porous media [1,2,3,4,5,6,7]. The order of tFDE is related to the fractional dimension of porous materials via the Hurst index [8,9], and a determined scaler Hurst index is not sufficient to quantify the fractional dimension of a highly heterogeneous porous medium. The distributed-order tFDE with its distributed-order time fractional derivative defined by

$$\begin{array}{c}{\displaystyle {}_0^C{D}_t^{\alpha }u:={\int }_0^1\rho \left(\alpha \right){\partial }_t^{\alpha }ud\alpha ,{\partial }_t^{\alpha }u:=\frac{1}{\mathrm{\Gamma }(1-\alpha )}{\int }_0^t\frac{{\partial }_{s}u}{{(t-s)}^{\alpha }}ds}\hfill \end{array}$$

(1)

is represented as a modified form and has attracted extensive research [10,11]. In Equation (1), $\mathrm{\Gamma }(·)$ refers to the Gamma function, and the probability density function (pdf) $\rho \left(\alpha \right)$ is nonnegative and satisfies ${\int }_0^1\rho \left(\alpha \right)d\alpha =1$.

In many applications such as gas or oil recovery, the hydraulic fracturing technique is often used to increase the permeability of the porous medium [12,13,14,15], and so the structure of the porous medium changes, which leads to the pdf $\rho$ changing over time. In other words, the pdf $\rho$ depends on both $\alpha$ and t, which we denote by $\rho (\alpha ,t)$. Recently, analysis and numerical methods for tFDEs involving the above variably distributed-order derivative were investigated in [16,17], but analysis of the tFDE with a hidden-memory distributed-order fractional derivative, which can describe the fractional order state’s history memory itself, are rarely found in the literature. The Caputo hidden-memory variably distributed-order time fractional derivative ${}_0^C{\mathbb{D}}_t^{\rho (·,t)}u$ is given by

$$\begin{array}{cc}{\displaystyle {}_0^C{\mathbb{D}}_t^{\rho (·,t)}u}\hfill & {\displaystyle :={\int }_0^t{\partial }_{s}u\left(s\right){\int }_0^{\overline{\alpha }\left(s\right)}\frac{\rho (\alpha ,s)d\alpha }{\mathrm{\Gamma }(1-\alpha ){(t-s)}^{\alpha }}ds:={}_0{\mathbb{I}}_t^{\rho (·,t)}{\partial }_{t}u.}\hfill \end{array}$$

(2)

Compared with Equation (1), the value of $\alpha$ varies on $[0,\overline{\alpha }\left(s\right)](\overline{\alpha }\left(s\right)\le 1)$ with $0<s<t$ in Equation (2), which indicates the order itself can memorize the history [18,19,20,21,22].

However, it was proven that the first-order derivative of the solution of the tFDE has weak singularity and fails to capture the Fickian diffusion behavior at the initial time $t=0$ [23,24,25,26]. The reason for this is that the tFDE was derived as the diffusion limit of a continuous time random walk which holds for a large time $t\gg 1$, rather than all the way up to time $t=0$, as is often assumed in the literature. The mobile-immobile tFDE was developed in [27,28] to describe the subdiffusion in heterogeneous porous media, in which a large number of particles may be absorbed into the media and then get released later. Motivated by the above discussions, we consider the following initial boundary value tFDE:

$$\begin{array}{c}{\displaystyle u_t+k{\left(t\right)}_0^C{\mathbb{D}}_t^{\rho (·,t)}u-\Delta u=f(\mathit{x},t),(\mathit{x},t)\in \mathrm{\Omega }\times (0,T];}\hfill \\ {\displaystyle u(\mathit{x},0)=u_0\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };u(\mathit{x},t)=0,(\mathit{x},t)\in \partial \mathrm{\Omega }\times [0,T].}\hfill \end{array}$$

(3)

Here $\mathrm{\Omega }\subset R^d(d=1,2,3)$ is a simply connected bounded domain with smooth boundary $\partial \mathrm{\Omega }$, $\mathit{x}=(x_1,\cdots ,x_d)$, $f(\mathit{x},t)$ is the source or sink term, and $u_0$ is the initial data. In Equation (3), the term $u_t$ represents the normal diffusive transport of particles, ${}_0^C{\mathbb{D}}_t^{\rho (·,t)}$ represents the subdiffusive transport of the absorbed particles, and $k\left(t\right)$ represents the ratio of the particles of anomalous versus normal diffusion.

Let $\mathcal{I}\subset \mathbb{R}$ be a bounded interval and $m\in \mathbb{N}$, with $C^m\left(\mathcal{I}\right)$ as the space of continuous functions with continuous derivatives up to an order m, which is equipped with

$$\begin{array}{c}{\displaystyle {\parallel g\left(t\right)\parallel }_{C\left(\mathcal{I}\right)}:=\underset{t\in \mathcal{I}}{sup}{\left|g\left(t\right)\right|,\parallel g\parallel }_{C^m\left(\mathcal{I}\right)}:=\underset{0\le n\le m}{max}{\parallel g^{\left(n\right)}\left(t\right)\parallel }_{C\left(\mathcal{I}\right)}.}\hfill \end{array}$$

We make the following assumptions on the variable order and the probability density function $\rho (\alpha ,t)$:

(i)

There exists a constant $0<{\alpha }^{*}<1$ which satisfies that $0\le \overline{\alpha }\left(t\right)\le {\alpha }^{*}$ for $t\in [0,T]$ and ${\overline{\alpha }}^{\prime }\left(t\right)\in C[0,T]$.

(ii)

For $t\in [0,T]$, ${\partial }_s\rho (\alpha ,s)\in C[0,\overline{\alpha }\left(t\right)]$, and $\rho (\alpha ,t)\in C([0,T];C[0,{\alpha }^{*}])$.

In the following sections, we use Q to denote a generic positive constant which is independent of $u_0$ and f, and $M_0,Q_0$ and $Q_1$ denote some fixed positive constants.

2. Analysis of Hidden-Memory Variably Distributed-Order ODE

We study the following hidden-memory variably distributed-order ordinary diffusion equation (ODE):

$$\begin{array}{c}{\displaystyle {\xi }^{\prime }\left(t\right)+k{\left(t\right)}_0^C{\mathbb{D}}_t^{\rho (·,t)}\xi \left(t\right)+\lambda \xi \left(t\right)=g\left(t\right),t\in (0,T];\xi \left(0\right)={\xi }_0.}\hfill \end{array}$$

(4)

We move the fractional derivative term to the right-hand side to rewrite Equation (4) as

$$\begin{array}{c}{\displaystyle {\xi }^{\prime }\left(t\right)+\lambda \xi \left(t\right)=g\left(t\right)-k{\left(t\right)}_0^C{\mathbb{D}}_t^{\rho (·,t)}\xi \left(t\right),}\hfill \end{array}$$

(5)

Then, we integrate Equation (5) multiplied by $e^{\lambda t}$ from 0 to t to obtain

$$\begin{array}{cc}{\displaystyle \xi \left(t\right)}\hfill & {\displaystyle =-{\int }_0^{t}k\left(s\right)e^{-\lambda (t-s)}{}_0^C{\mathbb{D}}_s^{\rho (·,s)}\xi \left(s\right)ds+{\int }_0^t{e}^{-\lambda (t-s)}g\left(s\right)ds+e^{-\lambda t}{\xi }_0.}\hfill \end{array}$$

By differentiating the above formula and setting $v\left(t\right)={\xi }^{\prime }\left(t\right)$, we obtain

$$\begin{array}{cc}{\displaystyle v\left(t\right)}\hfill & {\displaystyle =-k{\left(t\right)}_0{\mathbb{I}}_t^{\rho (·,t)}v+\lambda {\int }_0^{t}k\left(s\right)e^{-\lambda (t-s)}{}_0{\mathbb{I}}_s^{\rho (·,s)}vds+g\left(t\right)}\hfill \\ \hfill & {\displaystyle -\lambda {\int }_0^t{e}^{-\lambda (t-s)}g\left(s\right)ds-\lambda e^{-\lambda t}{\xi }_0.}\hfill \end{array}$$

(6)

We obtain $\xi$ in terms of v as follows:

$$\begin{array}{c}{\displaystyle \xi \left(t\right)={\xi }_0+{\int }_0^{t}v\left(s\right)ds.}\hfill \end{array}$$

Theorem 1.

Suppose that the assumption (i) holds and $k,g\in C[0,T]$. Then, Equation (4) has a unique solution $\xi \in C^1[0,T]$ with the following stability estimate:

$$\begin{array}{c}{\displaystyle {\parallel \xi \parallel }_{C^1[0,T]}\le {Q(\parallel g\parallel }_{C[0,T]}+\lambda |{\xi }_0\left|\right).}\hfill \end{array}$$

(7)

Proof. 

We define a functional sequence ${\left\{v_n\right\}}_{n=0}^{\infty }$ on $[0,T]$ by Equation (6) as

$$\begin{array}{cc}{\displaystyle v_0\left(t\right):}\hfill & {\displaystyle =g\left(t\right)-\lambda {\int }_0^t{e}^{-\lambda (t-s)}g\left(s\right)ds-\lambda e^{-\lambda t}{\xi }_0,}\hfill \\ {\displaystyle v_n\left(t\right):}\hfill & {\displaystyle =v_0\left(t\right)-k{\left(t\right)}_0{\mathbb{I}}_t^{\rho (·,t)}{v}_{n-1}+\lambda {\int }_0^t{e}^{-\lambda (t-s)}k{\left(s\right)}_0{\mathbb{I}}_t^{\rho (·,s)}{v}_{n-1}ds.}\hfill \end{array}$$

We bound $v_0\left(t\right)$ with

$$\begin{array}{cc}{\displaystyle |v_0\left(t\right)|}\hfill & {\displaystyle =|g\left(t\right)-\lambda {\int }_0^{t}g\left(s\right)e^{-\lambda (t-s)}ds-\lambda e^{-\lambda t}{\xi }_0|}\hfill \\ & {\displaystyle \le {2\parallel g\parallel }_{C[0,T]}+\lambda \left|{\xi }_0\right|:=M_0.}\hfill \end{array}$$

For $n\ge 1$, we subtract $v_n$ from $v_{n+1}$ to find

$$\begin{array}{cc}{\displaystyle v_{n+1}-v_n}\hfill & {\displaystyle =-k{\left(t\right)}_0{\mathbb{I}}_t^{\rho (·,t)}(v_n-v_{n-1})}\hfill \\ \hfill & {\displaystyle +\lambda {\int }_0^{t}k\left(s\right)e^{-\lambda (t-s)}{}_0{\mathbb{I}}_t^{\rho (·,s)}\left(v_n\left(s\right)-v_{n-1}\left(s\right)\right)ds.}\hfill \end{array}$$

(8)

By setting $v_{-1}\left(t\right)\equiv 0$, we have

$$\begin{array}{cc}{\displaystyle |v_1-v_0|}\hfill & {\displaystyle \le {\parallel k\parallel }_{C[0,T]}\left(\left|{}_0{\mathbb{I}}_t^{\rho (·,t)}{v}_0\right|+\lambda \left|{\int }_0^t{e}^{-\lambda (t-s)}{}_0{\mathbb{I}}_t^{\rho (·,s)}{v}_{0}ds\right|\right)}\hfill \\ \hfill & {\displaystyle \le {\parallel k\parallel }_{C[0,T]}\left({\int }_0^t\left|v_0\left(s\right)\right|{\int }_0^{\overline{\alpha }\left(s\right)}\frac{\rho (\alpha ,s)d\alpha }{\mathrm{\Gamma }(1-\alpha ){(t-s)}^{\alpha }}ds\right|}\hfill \\ \hfill & {\displaystyle +\lambda {\int }_0^t{e}^{-\lambda (t-s)}{\int }_0^s\left|v_0\left(\theta \right)\right|{\int }_0^{\overline{\alpha }\left(\theta \right)}\frac{\rho (\alpha ,\theta )d\alpha }{\mathrm{\Gamma }(1-\alpha ){(s-\theta )}^{\alpha }}d\theta ds)}\hfill \\ \hfill & {\displaystyle \le \frac{Q_1{\parallel k\parallel }_{C[0,T]}{M}_0}{\mathrm{\Gamma }(1-{\alpha }^{*})}({\int }_0^t{(t-s)}^{-{\alpha }^{*}}{\int }_0^{\overline{\alpha }\left(s\right)}\rho (\alpha ,s)d\alpha ds}\hfill \\ \hfill & {\displaystyle +\lambda {\int }_0^t{e}^{-\lambda (t-s)}{\int }_0^s{(s-\theta )}^{-{\alpha }^{*}}{\int }_0^{\overline{\alpha }\left(\theta \right)}\rho (\alpha ,\theta )d\alpha d\theta ds)}\hfill \\ \hfill & {\displaystyle \le \frac{Q_0{\parallel k\parallel }_{C[0,T]}{M}_0{t}^{1-{\alpha }^{*}}}{\mathrm{\Gamma }(2-{\alpha }^{*})}.}\hfill \end{array}$$

(9)

We assume that

$$\begin{array}{c}{\displaystyle |v_n\left(t\right)-v_{n-1}\left(t\right)|\le \frac{Q_0^n{M}_0{\parallel k\parallel }_{C[0,T]}^n{t}^{n(1-{\alpha }^{*})}}{\mathrm{\Gamma }(1+n(1-{\alpha }^{*}))},n\ge 2,}\hfill \end{array}$$

(10)

The, we substitute Equation (10) into Equation (8) to obtain

$$\begin{array}{cc}{\displaystyle |v_{n+1}-v_n|}\hfill & {\displaystyle \le {\parallel k\parallel }_{C[0,T]}({\int }_0^t|v_n-v_{n-1}|{\int }_0^{\overline{\alpha }\left(s\right)}\frac{\rho (\alpha ,s)ds}{\mathrm{\Gamma }(1-\alpha ){(t-s)}^{\alpha }}ds}\hfill \\ \hfill & {\displaystyle +\lambda {\int }_0^t{e}^{-\lambda (t-s)}{\int }_0^s|v_n-v_{n-1}|{\int }_0^{\overline{\alpha }\left(\theta \right)}\frac{\rho (\alpha ,\theta )d\alpha }{\mathrm{\Gamma }(1-\alpha ){(s-\theta )}^{\alpha }}d\theta ds)}\hfill \\ \hfill & {\displaystyle \le \frac{Q_0^n{\parallel k\parallel }_{C[0,T]}^{n+1}{M}_0{Q}_1}{\mathrm{\Gamma }(1+n(1-{\alpha }^{*}))\mathrm{\Gamma }(1-{\alpha }^{*})}({\int }_0^t{s}^{n(1-{\alpha }^{*})}{(t-s)}^{-{\alpha }^{*}}ds}\hfill \\ \hfill & {\displaystyle +\lambda {\int }_0^t{e}^{-\lambda (t-s)}{\int }_0^s{\theta }^{n(1-{\alpha }^{*})}{(s-\theta )}^{-{\alpha }^{*}}d\theta ds)}\hfill \\ \hfill & {\displaystyle =\frac{Q_0^n{M}_0{\parallel k\parallel }_{C[0,T]}^{n+1}{Q}_1}{\mathrm{\Gamma }(1+n(1-{\alpha }^{*}))\mathrm{\Gamma }(1-{\alpha }^{*})}B(1+n(1-{\alpha }^{*}),1-{\alpha }^{*})}\hfill \\ \hfill & {\displaystyle \times \left(t^{(n+1)(1-{\alpha }^{*})}+\lambda {\int }_0^t{e}^{-\lambda (t-s)}{s}^{(n+1)(1-{\alpha }^{*})}ds\right)}\hfill \\ \hfill & {\displaystyle \le \frac{Q_0^{n+1}{\parallel k\parallel }_{C[0,T]}^{n+1}{M}_0}{\mathrm{\Gamma }(1+(n+1)(1-{\alpha }^{*}))}{t}^{(n+1)(1-{\alpha }^{*})},}\hfill \end{array}$$

(11)

Here, $B(·,·)$ is the Beta function. Thus, the assumption in Equation (10) holds for $n\ge 1$ by mathematical induction. The series with function terms defined by the right-hand side of Equation (10) can be bounded by

$$\begin{array}{c}{\displaystyle \sum _{n=1}^{\infty }\frac{Q_0^n{\parallel k\parallel }_{C[0,T]}^n{M}_0{T}^{n(1-{\alpha }^{*})}}{\mathrm{\Gamma }(1+n(1-{\alpha }^{*}))}=M_0{E}_{1-{\alpha }^{*},1}(Q_0{\parallel k\parallel }_{C[0,T]}{T}^{1-{\alpha }^{*}})<\infty ,}\hfill \end{array}$$

where $E_{1-{\alpha }^{*},1}(·)$ is the Mittag-Leffler function. We conclude that the left-hand side series of Equation (10) converges uniformly to its limiting function v on $t\in [0,T]$ as

$$\begin{array}{c}{\displaystyle v\left(t\right):=\underset{n\to \infty }{lim}{v}_n\left(t\right)=\sum _{n=1}^{\infty }(v_n\left(t\right)-v_{n-1}\left(t\right))+v_0\left(t\right)}\hfill \end{array}$$

satisfies Equations (6) and (7). □

Theorem 2.

Suppose that assumptions (i) and (ii) hold and $k,g\in C^1[0,T]$. Then, we have $\xi \in C^2(0,T]$ with the following estimate:

$$\begin{array}{c}{\displaystyle |{\xi }^{″}\left(t\right)|\le Q\left({\lambda }^2|{\xi }_0{|+\lambda \parallel g\parallel }_{C[0,T]}+{\parallel g^{\prime }\parallel }_{C[0,T]}\right)t^{-\overline{\alpha }\left(0\right)}.}\hfill \end{array}$$

(12)

Proof. 

We differentiate Equation (6) on both sides to find

$$\begin{array}{cc}{\displaystyle v^{\prime }\left(t\right)}\hfill & {\displaystyle ={\lambda }^2{e}^{-\lambda t}{\xi }_0+g^{\prime }\left(t\right)-\lambda g\left(t\right)+{\lambda }^2{\int }_0^t{e}^{-\lambda (t-s)}g\left(s\right)ds}\hfill \\ \hfill & {\displaystyle +\left(\lambda k\left(t\right)-k^{\prime }\left(t\right)\right){}_0{\mathbb{I}}_t^{\rho (·,t)}v\left(t\right)-{\lambda }^2{\int }_0^{t}k\left(s\right)e^{-\lambda (t-s)}{}_0{\mathbb{I}}_s^{\rho (·,s)}v\left(s\right)ds}\hfill \\ \hfill & {\displaystyle -k\left(t\right){\left({}_0{\mathbb{I}}_t^{\rho (·,t)}v\right)}^{\prime }.}\hfill \end{array}$$

(13)

The first four terms on the right-hand side of Equation (13) can be bounded by

$$\begin{array}{c}{\displaystyle |{\lambda }^2{e}^{-\lambda t}{\xi }_0+g^{\prime }\left(t\right)-\lambda g\left(t\right)+{\lambda }^2{\int }_0^t{e}^{-\lambda (t-s)}g\left(s\right)ds|}\hfill \\ {\displaystyle \le {\lambda }^2|{\xi }_0{|+\lambda \parallel g\parallel }_{C[0,T]}+{\parallel g^{\prime }\parallel }_{C[0,T]}.}\hfill \end{array}$$

(14)

We bound ${}_0{\mathbb{I}}_t^{\rho (·,t)}v$ with

$$\begin{array}{cc}{\displaystyle \left|{}_0{\mathbb{I}}_t^{\rho (·,t)}v\right|}\hfill & {\displaystyle =\left|{\int }_0^{t}v\left(s\right){\int }_0^{\overline{\alpha }\left(s\right)}\frac{\rho (\alpha ,s)d\alpha }{\mathrm{\Gamma }(1-\alpha ){(t-s)}^{\alpha }}ds\right|}\hfill \\ & {\displaystyle \le {Q\parallel v\parallel }_{C[0,T]}{\int }_0^t{(t-s)}^{-{\alpha }^{*}}ds\le Q{\parallel v\parallel }_{C[0,T]},}\hfill \end{array}$$

Thus, the fifth and sixth terms on the right-hand side of Equation (13) can be bounded by

$$\begin{array}{c}{\displaystyle |{(\lambda k-k^{\prime }\left(t\right))}_0{\mathbb{I}}_t^{\rho (·,t)}v-{\lambda }^2{\int }_0^{t}k\left(s\right)e^{-\lambda (t-s)}{}_0{\mathbb{I}}_s^{\rho (·,s)}v\left(s\right)ds|\le Q\lambda {\parallel v\parallel }_{C[0,T]}.}\hfill \end{array}$$

(15)

To estimate the last term, we denote

$$\begin{array}{c}{\displaystyle \eta \left(s\right):=v\left(s\right){\int }_0^{\overline{\alpha }\left(s\right)}\frac{\rho (\alpha ,s)d\alpha }{\mathrm{\Gamma }(1-\alpha ){(t-s)}^{\alpha }}.}\hfill \end{array}$$

If $\overline{\alpha }\left(s\right)\le \overline{\alpha }\left(t\right)$, then we have $\rho (\alpha ,s)\equiv 0$ for $\overline{\alpha }\left(s\right)<\alpha \le \overline{\alpha }\left(t\right)$, and thus $\eta \left(s\right)$ can be expressed in terms of

$$\begin{array}{c}{\displaystyle \eta \left(s\right)=v\left(s\right)\left({\int }_0^{\overline{\alpha }\left(t\right)}\frac{\rho (\alpha ,s)d\alpha }{\mathrm{\Gamma }(1-\alpha ){(t-s)}^{\alpha }}+{\int }_{\overline{\alpha }\left(t\right)}^{\overline{\alpha }\left(s\right)}\frac{\rho (\alpha ,s)d\alpha }{\mathrm{\Gamma }(1-\alpha ){(t-s)}^{\alpha }}\right),}\hfill \end{array}$$

(16)

Here, we note that the second term is equal to 0. Otherwise, if $\overline{\alpha }\left(s\right)>\overline{\alpha }\left(t\right)$, then we decompose the support as $[0,\overline{\alpha }\left(s\right)]=[0,\overline{\alpha }\left(t\right)]\bigcup (\overline{\alpha }\left(t\right),\overline{\alpha }\left(s\right)]$, and Equation (16) holds too. Thus, the hidden-memory variably distributed-order fractional integral ${}_0{\mathbb{I}}_t^{\rho (·,t)}v$ can be decomposed as follows:

$$\begin{array}{cc}{\displaystyle {}_0{\mathbb{I}}_t^{\rho (·,t)}v}\hfill & {\displaystyle ={\int }_0^{t}v\left(s\right){\int }_0^{\overline{\alpha }\left(t\right)}\frac{\rho (\alpha ,s)d\alpha }{\mathrm{\Gamma }(1-\alpha ){(t-s)}^{\alpha }}ds+{\int }_0^{t}v\left(s\right){\int }_{\overline{\alpha }\left(t\right)}^{\overline{\alpha }\left(s\right)}\frac{\rho (\alpha ,s)d\alpha }{\mathrm{\Gamma }(1-\alpha ){(t-s)}^{\alpha }}ds}\hfill \\ & {\displaystyle :=I_1+I_2.}\hfill \end{array}$$

Next, we analyze $I_1$ and $I_2$.

As the upper bound of $\alpha$ in $I_1$ depends on t, we can exchange the integral order to rewrite $I_1$ as

$$\begin{array}{cc}{\displaystyle I_1}\hfill & {\displaystyle ={\int }_0^{\overline{\alpha }\left(t\right)}\frac{1}{\mathrm{\Gamma }(1-\alpha )}{\int }_0^t\frac{v\left(s\right)\rho (\alpha ,s)ds}{{(t-s)}^{\alpha }}d\alpha }\hfill \\ \hfill & {\displaystyle =-{\int }_0^{\overline{\alpha }\left(t\right)}\frac{1}{\mathrm{\Gamma }(2-\alpha )}{\int }_0^{t}v\left(s\right)\rho (\alpha ,s)d{(t-s)}^{1-\alpha }d\alpha }\hfill \\ \hfill & {\displaystyle ={\int }_0^{\overline{\alpha }\left(t\right)}\frac{1}{\mathrm{\Gamma }(2-\alpha )}{\int }_0^t{(t-s)}^{1-\alpha }{\partial }_s\left(v\left(s\right)\rho (\alpha ,s)\right)dsd\alpha }\hfill \\ \hfill & {\displaystyle +v\left(0\right){\int }_0^{\overline{\alpha }\left(t\right)}\frac{\rho (\alpha ,0)t^{1-\alpha }}{\mathrm{\Gamma }(2-\alpha )}d\alpha .}\hfill \end{array}$$

(17)

We differentiate Equation (17) on both sides to obtain

$$\begin{array}{cc}{\displaystyle \frac{d}{dt}{I}_1}\hfill & {\displaystyle :=\frac{{\overline{\alpha }}^{\prime }\left(t\right)}{\mathrm{\Gamma }(2-\overline{\alpha }\left(t\right))}{\int }_0^t{(t-s)}^{1-\overline{\alpha }\left(t\right)}{\partial }_s\left(v\left(s\right)\rho (\overline{\alpha }\left(t\right),s)\right)ds}\hfill \\ & {\displaystyle +{\int }_0^{\overline{\alpha }\left(t\right)}\frac{1}{\mathrm{\Gamma }(1-\alpha )}{\int }_0^t{(t-s)}^{-\alpha }{\partial }_s\left(v\left(s\right)\rho (\alpha ,s)\right)dsd\alpha }\hfill \\ & {\displaystyle +v\left(0\right)\frac{{\overline{\alpha }}^{\prime }\left(t\right)\rho (\overline{\alpha }\left(t\right),0)t^{1-\overline{\alpha }\left(t\right)}}{\mathrm{\Gamma }(2-\overline{\alpha }\left(t\right))}+v\left(0\right){\int }_0^{\overline{\alpha }\left(t\right)}\frac{\rho (\alpha ,0)}{\mathrm{\Gamma }(1-\alpha )t^{\alpha }}d\alpha .}\hfill \end{array}$$

By applying ${\overline{\alpha }}^{\prime }\left(t\right)\in C[0,T]$ in assumption (ii), we have

$${\displaystyle t^{-\overline{\alpha }\left(t\right)}=t^{-\overline{\alpha }\left(0\right)}{t}^{\overline{\alpha }\left(0\right)-\overline{\alpha }\left(t\right)}\le t^{-\overline{\alpha }\left(0\right)}{e}^{-\parallel \overline{\alpha }{\parallel }_{C^1[0,T]}tlnt}\le Q{t}^{-\overline{\alpha }\left(0\right)},}$$

In addition, we can bound $I_1$ with

$$\begin{array}{cc}{\displaystyle \left|\frac{d}{dt}{I}_1\right|}\hfill & {\displaystyle \le |-\frac{{\overline{\alpha }}^{\prime }\left(t\right)}{\mathrm{\Gamma }(1-\overline{\alpha }\left(t\right))}{\int }_0^t\frac{v\left(s\right)\rho (\overline{\alpha }\left(t\right),s)ds}{{(t-s)}^{\overline{\alpha }\left(t\right)}}}\hfill \\ \hfill & {\displaystyle +{\int }_0^{\overline{\alpha }\left(t\right)}\frac{1}{\mathrm{\Gamma }(1-\alpha )}{\int }_0^t\frac{v^{\prime }\left(s\right)\rho (\alpha ,s)+v\left(s\right){\partial }_s\rho (\alpha ,s)}{{(t-s)}^{\alpha }}dsd\alpha }\hfill \\ \hfill & {\displaystyle +v\left(0\right){\int }_0^{\overline{\alpha }\left(t\right)}\frac{\rho (\alpha ,0)d\alpha }{\mathrm{\Gamma }(1-\alpha )t^{\alpha }}|}\hfill \\ \hfill & {\displaystyle \le Q\left|v\left(0\right)\right|t^{-\overline{\alpha }\left(0\right)}+Q{\int }_0^t\frac{|v^{\prime }\left(s\right)|ds}{{(t-s)}^{{\alpha }^{*}}}.}\hfill \end{array}$$

(18)

To estimate $I_2$, we rewrite $I_2$ into the following form:

$$\begin{array}{cc}{\displaystyle I_2}\hfill & {\displaystyle =-\frac{1}{1-\overline{\alpha }\left(t\right)}{\int }_0^{t}v\left(s\right){\int }_{\overline{\alpha }\left(t\right)}^{\overline{\alpha }\left(s\right)}\frac{\rho (\alpha ,s){(t-s)}^{\overline{\alpha }\left(t\right)-\alpha }d\alpha }{\mathrm{\Gamma }(1-\alpha )}d{(t-s)}^{1-\overline{\alpha }\left(t\right)}}\hfill \\ \hfill & {\displaystyle =\frac{1}{1-\overline{\alpha }\left(t\right)}(v\left(0\right){\int }_{\overline{\alpha }\left(t\right)}^{\overline{\alpha }\left(0\right)}\frac{\rho (\alpha ,0)t^{1-\alpha }}{\mathrm{\Gamma }(1-\alpha )}d\alpha +{\int }_0^t{v}^{\prime }\left(s\right){\int }_{\overline{\alpha }\left(t\right)}^{\overline{\alpha }\left(s\right)}\frac{\rho (\alpha ,s){(t-s)}^{1-\alpha }d\alpha }{\mathrm{\Gamma }(1-\alpha )}ds}\hfill \\ \hfill & {\displaystyle +{\int }_0^t\frac{v\left(s\right){\overline{\alpha }}^{\prime }\left(s\right)\rho (\overline{\alpha }\left(s\right),s){(t-s)}^{1-\overline{\alpha }\left(s\right)}}{\mathrm{\Gamma }(1-\overline{\alpha }\left(s\right))}ds+{\int }_0^{t}v\left(s\right){\int }_{\overline{\alpha }\left(t\right)}^{\overline{\alpha }\left(s\right)}\frac{{(t-s)}^{1-\alpha }{\partial }_s\rho (\alpha ,s)d\alpha }{\mathrm{\Gamma }(1-\alpha )}ds}\hfill \\ \hfill & {\displaystyle -{\int }_0^{t}v\left(s\right){\int }_{\overline{\alpha }\left(t\right)}^{\overline{\alpha }\left(s\right)}\frac{\rho (\alpha ,s)(\overline{\alpha }\left(t\right)-\alpha )}{\mathrm{\Gamma }(1-\alpha ){(t-s)}^{\alpha }}d\alpha ds)}\hfill \\ \hfill & {\displaystyle =J_1+J_2+J_3+J_4+J_5.}\hfill \end{array}$$

(19)

By differentiating on $J_1$ and $J_2$, we obtain

$$\begin{array}{cc}{\displaystyle \frac{d}{dt}{J}_1}\hfill & {\displaystyle =\frac{{\overline{\alpha }}^{\prime }\left(t\right)v\left(0\right)}{{(1-\overline{\alpha }\left(t\right))}^2}{\int }_{\overline{\alpha }\left(t\right)}^{\overline{\alpha }\left(0\right)}\frac{\rho (\alpha ,0)t^{1-\alpha }d\alpha }{\mathrm{\Gamma }(1-\alpha )}}\hfill \\ & {\displaystyle +\frac{v\left(0\right)}{1-\overline{\alpha }\left(t\right)}\left(-\frac{{\overline{\alpha }}^{\prime }\left(t\right)\rho (\overline{\alpha }\left(t\right),0)t^{1-\overline{\alpha }\left(t\right)}}{\mathrm{\Gamma }(1-\overline{\alpha }\left(t\right))}+{\int }_{\overline{\alpha }\left(t\right)}^{\overline{\alpha }\left(0\right)}\frac{\rho (\alpha ,0)(1-\alpha )d\alpha }{\mathrm{\Gamma }(1-\alpha )t^{\alpha }}\right),}\hfill \end{array}$$

and

$$\begin{array}{cc}{\displaystyle \frac{d}{dt}{J}_2}\hfill & {\displaystyle =\frac{{\overline{\alpha }}^{\prime }\left(t\right)}{{(1-\overline{\alpha }\left(t\right))}^2}{\int }_0^t{v}^{\prime }\left(s\right){\int }_{\overline{\alpha }\left(t\right)}^{\overline{\alpha }\left(s\right)}\frac{\rho (\alpha ,s){(t-s)}^{1-\alpha }d\alpha }{\mathrm{\Gamma }(1-\alpha )}ds}\hfill \\ & {\displaystyle -\frac{{\overline{\alpha }}^{\prime }\left(t\right)}{1-\overline{\alpha }\left(t\right)}{\int }_0^t\frac{v^{\prime }\left(s\right)\rho (\overline{\alpha }\left(t\right),s){(t-s)}^{1-\overline{\alpha }\left(t\right)}}{\mathrm{\Gamma }(1-\overline{\alpha }\left(t\right))}ds}\hfill \\ & {\displaystyle +\frac{1}{1-\overline{\alpha }\left(t\right)}{\int }_0^t{v}^{\prime }\left(s\right){\int }_{\overline{\alpha }\left(t\right)}^{\overline{\alpha }\left(s\right)}\frac{\rho (\alpha ,s)(1-\alpha )d\alpha }{\mathrm{\Gamma }(1-\alpha ){(t-s)}^{\alpha }}ds.}\hfill \end{array}$$

Therefore, $J_1$ and $J_2$ can be bounded by

$$\begin{array}{cc}{\displaystyle \left|\frac{d}{dt}{J}_1\right|}\hfill & {\displaystyle \le Q\left|v\left(0\right)\right|+Q\left|v\left(0\right)\right|t^{-max\{\overline{\alpha }\left(0\right),\overline{\alpha }\left(t\right)\}}\le Q\left|v\left(0\right)\right|t^{-\overline{\alpha }\left(0\right)},}\hfill \\ {\displaystyle \left|\frac{d}{dt}{J}_2\right|}\hfill & {\displaystyle \le Q{\int }_0^t\frac{|v^{\prime }\left(s\right)|ds}{{(t-s)}^{{\alpha }^{*}}}.}\hfill \end{array}$$

(20)

Similarly, we estimate $J_3$ and $J_4$ according to Theorem 1:

$$\begin{array}{c}{\displaystyle \left|\frac{d}{dt}{J}_3\right|+\left|\frac{d}{dt}{J}_4\right|\le Q{\parallel v\parallel }_{C[0,T]}.}\hfill \end{array}$$

(21)

We differentiate on $J_5$ to obtain

$$\begin{array}{cc}{\displaystyle \frac{d}{dt}{J}_5}\hfill & {\displaystyle =\frac{{\overline{\alpha }}^{\prime }\left(t\right)}{1-\overline{\alpha }\left(t\right)}{J}_5-\frac{1}{1-\overline{\alpha }\left(t\right)}({\int }_0^{t}v\left(s\right){\int }_{\overline{\alpha }\left(t\right)}^{\overline{\alpha }\left(s\right)}\frac{{\overline{\alpha }}^{\prime }\left(t\right)\rho (\alpha ,s)d\alpha }{\mathrm{\Gamma }(1-\alpha ){(t-s)}^{\alpha }}ds}\hfill \\ \hfill & {\displaystyle -{\int }_0^{t}v\left(s\right){\int }_{\overline{\alpha }\left(t\right)}^{\overline{\alpha }\left(s\right)}\frac{\alpha \rho (\alpha ,s)(\overline{\alpha }\left(t\right)-\alpha )d\alpha }{\mathrm{\Gamma }(1-\alpha ){(t-s)}^{1+\alpha }}ds).}\hfill \end{array}$$

(22)

We bound the first two terms on the right-hand side of Equation (22) with

$$\begin{array}{c}{\displaystyle \left|\frac{{\overline{\alpha }}^{\prime }\left(t\right)}{1-\overline{\alpha }\left(t\right)}{J}_5-\frac{1}{1-\overline{\alpha }\left(t\right)}{\int }_0^{t}v\left(s\right){\int }_{\overline{\alpha }\left(t\right)}^{\overline{\alpha }\left(s\right)}\frac{{\overline{\alpha }}^{\prime }\left(t\right)\rho (\alpha ,s)d\alpha }{x}\mathrm{\Gamma }(1-\alpha ){(t-s)}^{\alpha }ds\right|}\hfill \\ {\displaystyle \le {Q\parallel v\parallel }_{C[0,T]}{\int }_0^t{(t-s)}^{-{\alpha }^{*}}ds\le Q{\parallel v\parallel }_{C[0,T]}.}\hfill \end{array}$$

By assumptions (i) and (ii) and the mean value theorem, we have

$${\displaystyle |\overline{\alpha }\left(s\right)-\overline{\alpha }\left(t\right)|\le \parallel \overline{\alpha }{\parallel }_{C^1[0,T]}(t-s),}$$

Therefore, the third term on the right-hand side of Equation (22) can be bounded by

$$\begin{array}{c}{\displaystyle \left|\frac{1}{1-\overline{\alpha }\left(t\right)}{\int }_0^{t}v\left(s\right){\int }_{\overline{\alpha }\left(t\right)}^{\overline{\alpha }\left(s\right)}\frac{\alpha \rho (\alpha ,s)(\overline{\alpha }\left(t\right)-\alpha )d\alpha }{\mathrm{\Gamma }(1-\alpha ){(t-s)}^{1+\alpha }}ds\right|}\hfill \\ {\displaystyle \le {Q\parallel v\parallel }_{C[0,T]}{\int }_0^t\frac{|\overline{\alpha }\left(s\right)-\overline{\alpha }\left(t\right)|ds}{{(t-s)}^{1+{\alpha }^{*}}}}\hfill \\ {\displaystyle \le {Q\parallel v\parallel }_{C[0,T]}{\int }_0^t\frac{ds}{{(t-s)}^{{\alpha }^{*}}}\le Q{\parallel v\parallel }_{C[0,T]}.}\hfill \end{array}$$

Thus, $J_5$ can be estimated by

$$\begin{array}{c}{\displaystyle \left|\frac{d}{dt}{J}_5\right|\le Q{\parallel v\parallel }_{C[0,T]}.}\hfill \end{array}$$

(23)

We submit the estimates of Equations (20)–(23) into Equation (19) to obtain the estimate of ${\displaystyle \frac{d}{dt}{I}_2}$, and hence ${\left({}_0{\mathbb{I}}_t^{\rho (·,t)}v\right)}^{\prime }$ can be bounded by

$$\begin{array}{cc}{\displaystyle \left|{\left({}_0{\mathbb{I}}_t^{\rho (·,t)}v\right)}^{\prime }\right|}\hfill & {\displaystyle \le {Q\parallel v\parallel }_{C[0,T]}{t}^{-\overline{\alpha }\left(0\right)}+Q{\int }_0^t\frac{|v^{\prime }\left(s\right)|ds}{{(t-s)}^{{\alpha }^{*}}}}\hfill \\ & {\displaystyle \le Q\left(\lambda \right|{\xi }_0{|+\parallel g\parallel }_{C[0,T]})t^{-\overline{\alpha }\left(0\right)}+Q{\int }_0^t\frac{|v^{\prime }\left(s\right)|ds}{{(t-s)}^{{\alpha }^{*}}}.}\hfill \end{array}$$

When combined with Equations (14)–(15), we found that $v^{\prime }\left(t\right)$ in Equation (13) satisfies

$$\begin{array}{c}{\displaystyle |v^{\prime }\left(t\right)|\le Q\left({\lambda }^2|{\xi }_0{|+\lambda \parallel g\parallel }_{C[0,T]}+{\parallel g^{\prime }\parallel }_{C[0,T]}\right)t^{-\overline{\alpha }\left(0\right)}+Q{\int }_0^t\frac{|v^{\prime }\left(s\right)|ds}{{(t-s)}^{{\alpha }^{*}}}.}\hfill \end{array}$$

By applying the generalized Gronwall inequality [29,30], we obtain

$$\begin{array}{cc}{\displaystyle |v^{\prime }\left(t\right)|}\hfill & {\displaystyle \le \left({\lambda }^2|{\xi }_0{|+\lambda \parallel g\parallel }_{C[0,T]}+{\parallel g^{\prime }\parallel }_{C[0,T]}\right)(Q{t}^{-\overline{\alpha }\left(0\right)}+}\hfill \\ & {\displaystyle \sum _{n=1}^{\infty }\frac{{\left(Q\mathrm{\Gamma }(1-{\alpha }^{*})\right)}^n}{\mathrm{\Gamma }\left(n(1-{\alpha }^{*})\right)}{\int }_0^t{(t-s)}^{n(1-{\alpha }^{*})-1}{s}^{-\overline{\alpha }\left(0\right)}ds)}\hfill \\ & {\displaystyle \le Q\left({\lambda }^2|{\xi }_0{|+\lambda \parallel g\parallel }_{C[0,T]}+{\parallel g^{\prime }\parallel }_{C[0,T]}\right)t^{-\overline{\alpha }\left(0\right)},}\hfill \end{array}$$

This finishes the proof. □

3. Analysis of the Hidden-Memory Variably Distributed-Order PDE

Consider the eigenfunctions ${\left\{{\varphi }_i\right\}}_{i=1}^{\infty }$ of the Sturm–Liouville problem

$$\begin{array}{c}{\displaystyle -\Delta u={\lambda }_i{\varphi }_i\left(\mathit{x}\right),\mathit{x}\in \mathrm{\Omega };{\varphi }_i\left(\mathit{x}\right)=0,\mathit{x}\in \partial \mathrm{\Omega }}\hfill \end{array}$$

(24)

for an orthogonal basis in $L^2\left(\mathrm{\Omega }\right)$, where ${\left\{{\lambda }_i\right\}}_{i=1}^{\infty }$ are the corresponding positive eigenvalues which form a non-decreasing sequence that tends toward ∞ with $i\to \infty$ [31]. For any $\gamma \ge 0$, we define a Sobolev space as

$$\begin{array}{c}{\displaystyle {\check{H}}^{\gamma }\left(\mathrm{\Omega }\right):=\left\{v\in L^2\left(\mathrm{\Omega }\right):{\left|v\right|}_{{\check{H}}^{\gamma }\left(\mathrm{\Omega }\right)}^2:=({(-\Delta )}^{\gamma }v,v)=\sum _{i=1}^{\infty }{\lambda }_i^{\gamma }{(v,{\varphi }_i)}^2<\infty )\right\}}\hfill \end{array}$$

with the norm given by ${\parallel v\parallel }_{{\check{H}}^{\gamma }}:={(\parallel v\parallel }^2+{\left|v\right|}_{{\check{H}}^{\gamma }}^2{)}^{1/2}.$ Obviously, this satisfies ${\check{H}}^0\left(\mathrm{\Omega }\right)=L^2\left(\mathrm{\Omega }\right)$ and ${\check{H}}^2\left(\mathrm{\Omega }\right)=H^2\left(\mathrm{\Omega }\right)\bigcap H_0^1\left(\mathrm{\Omega }\right)$ [31].

Theorem 3.

Suppose that assumption (i) holds, $u_0\in {\check{H}}^{2+\gamma }\left(\mathrm{\Omega }\right)$ and $f\in H^{\nu }({\check{H}}^{\gamma })$ with $\gamma >d/2$ and $\nu >1/2$. Then, Equation (3) has a unique solution $u\in C^1([0,T];{\check{H}}^{\gamma }\left(\mathrm{\Omega }\right))$ with the estimate

$$\begin{array}{c}{\displaystyle {\parallel u\parallel }_{C^1([0,T];{\check{H}}^{\gamma }\left(\mathrm{\Omega }\right))}\le Q(\parallel u_0{\parallel }_{{\check{H}}^{2+\gamma }\left(\mathrm{\Omega }\right)}+{\parallel f\parallel }_{H^{\nu }\left({\check{H}}^{\gamma }\left(\mathrm{\Omega }\right)\right)}).}\hfill \end{array}$$

(25)

Proof. 

We express the solution $u(\mathit{x},t)$ and the source term $f(\mathit{x},t)$ in Equation (3) in terms of the orthogonal basis ${\left\{{\varphi }_i\right\}}_{i=1}^{\infty }$:

$$\begin{array}{c}{\displaystyle u(\mathit{x},t)=\sum _{i=1}^{\infty }{u}_i\left(t\right){\varphi }_i\left(\mathit{x}\right),u_i\left(t\right):=(u(·,t),{\varphi }_i),t\in [0,T],}\hfill \\ {\displaystyle f(\mathit{x},t)=\sum _{i=1}^{\infty }{f}_i\left(t\right){\varphi }_i\left(\mathit{x}\right),f_i\left(t\right):=(f(·,t),{\varphi }_i),t\in [0,T].}\hfill \end{array}$$

(26)

We plug the expansions in Equation (26) into Equation (3) to obtain

$$\begin{array}{c}{\displaystyle \sum _{i=1}^{\infty }{u}_i^{\prime }\left(t\right){\varphi }_i\left(\mathit{x}\right)+k\left(t\right)\sum _{i=1}^{\infty }{}_0^C{\mathbb{D}}_t^{\rho (·,t)}{u}_i\left(t\right){\varphi }_i\left(\mathit{x}\right)}\hfill \\ {\displaystyle -\sum _{i=1}^{\infty }{u}_i\left(t\right)\Delta {\varphi }_i\left(\mathit{x}\right)=\sum _{i=1}^{\infty }{f}_i\left(t\right){\varphi }_i\left(\mathit{x}\right),t\in (0,T].}\hfill \end{array}$$

(27)

Thus, $u(\mathit{x},t)$ is a solution to Equation (3) if and only if ${\left\{u_i\left(t\right)\right\}}_{i=1}^{\infty }$ are solutions to the following ODEs:

$$\begin{array}{c}{\displaystyle u_i^{\prime }\left(t\right)+k{\left(t\right)}_0^C{\mathbb{D}}_t^{\rho (·,t)}{u}_i\left(t\right)+{\lambda }_i{u}_i\left(t\right)=f_i\left(t\right),t\in (0,T],}\hfill \\ {\displaystyle u_i\left(0\right)=u_{0,i}:=(u_0,{\varphi }_i),i=1,2,\cdots .}\hfill \end{array}$$

(28)

We apply Theorem 1 by replacing $\xi ,\lambda ,g$ with $u_i,{\lambda }_i,f_i$ to conclude that Equation (28) has a unique solution $u_i\in C^1[0,T]$ with an estimate

$$\begin{array}{c}{\displaystyle \parallel u_i^{\prime }{\parallel }_{C[0,T]}\le Q\left({\lambda }_i|u_{i,0}|+\parallel f_i{\parallel }_{C[0,T]}\right).}\hfill \end{array}$$

By letting ${\displaystyle S_n=\sum _{i=1}^n{u}_i\left(t\right){\varphi }_i\left(\mathit{x}\right)}$, we use Sobolev embedding to conclude that for $n\to \infty$, the following is true:

$$\begin{array}{c}{\displaystyle \parallel S_{n+k}^{\prime }-S_n^{\prime }{\parallel }_{C([0,T];C\left(\overline{\mathrm{\Omega }}\right))}^2\le Q{\parallel \sum _{i=n+1}^{n+k}{u}_i^{\prime }\left(t\right){\varphi }_i\left(\mathit{x}\right)\parallel }_{C([0,T];{\check{H}}^{\gamma }\left(\mathrm{\Omega }\right))}^2}\hfill \\ {\displaystyle \le Q\sum _{i=n+1}^{n+k}{\lambda }_i^{\gamma }{\parallel u_i\parallel }_{C^1[0,T]}^2\le Q\sum _{i=n+1}^{n+k}{\lambda }_i^{\gamma }\left({\lambda }_i^2|u_{0,i}{|}^2+{\parallel f_i\parallel }_{C[0,T]}^2\right)\to 0.}\hfill \end{array}$$

Thus, the interchange of the differentiation and summation is justified, from which we conclude that $u\in C^1([0,T];{\check{H}}^{\gamma }\left(\mathrm{\Omega }\right))$ with the estimate

$$\begin{array}{c}{\displaystyle {\parallel u\parallel }_{C^1([0,T];{\check{H}}^{\gamma }\left(\mathrm{\Omega }\right))}^2\le Q\sum _{i=1}^{\infty }{\lambda }_i^{\gamma }{\parallel u_i\parallel }_{C^1[0,T]}^2}\hfill \\ {\displaystyle \le Q\sum _{i=1}^{\infty }{\lambda }_i^{\gamma }({\lambda }_i^2|u_{0,i}{|}^2+{\parallel f_i\parallel }_{C[0,T]}^2)\le Q(\parallel u_0{\parallel }_{{\check{H}}^{2+\gamma }}^2+{\parallel f\parallel }_{H^{\nu }\left({\check{H}}^{\gamma }\right)}^2).}\hfill \end{array}$$

□

Combining with Theorem 2, we obtain the high-order regularity of u in the following theorem:

Theorem 4.

Suppose that assumptions (i) and (ii) hold, $f\in H^{\nu }({\check{H}}^{2+\gamma })\bigcap H^{1+\nu }({\check{H}}^{\gamma })$ and $u_0\in {\check{H}}^{4+\gamma }$ for $\gamma >d/2$ and $\nu >1/2$. Then, the solution to Equation (3) belongs to $C^2((0,T];{\check{H}}^{\gamma })$ with the estimate

$$\begin{array}{c}{\displaystyle {\parallel u\parallel }_{C^2([t,T];{\check{H}}^{\gamma })}\le Q{t}^{-\overline{\alpha }\left(0\right)}(\parallel u_0{\parallel }_{{\check{H}}^{4+\gamma }}+{\parallel f\parallel }_{H^{\nu }\left({\check{H}}^{2+\gamma }\right)}+{\parallel f\parallel }_{H^{1+\nu }\left({\check{H}}^{\gamma }\right)}）.}\hfill \end{array}$$

Proof. 

We prove the estimate with the following:

$$\begin{array}{cc}{\displaystyle {\parallel u\parallel }_{C^2([t;T];{\check{H}}^{\gamma })}^2}\hfill & {\displaystyle \le Q\sum _{i=1}^{\infty }{\lambda }_i^{\gamma }{\parallel u_i\parallel }_{C^2[t,T]}^2}\hfill \\ & {\displaystyle \le Q\sum _{i=1}^{\infty }{\lambda }_i^{\gamma }\left({\lambda }^4\parallel u_{0,i}{\parallel }^2+{\lambda }^2\parallel f_i{\parallel }_{C[0,T]}^2+{\parallel f_i^{\prime }\parallel }_{C[0,T]}^2\right)t^{-2\overline{\alpha }\left(0\right)}}\hfill \\ & {\displaystyle \le Q\left(\parallel u_0{\parallel }_{H^{4+\gamma }}^2++{\parallel f\parallel }_{H^{\nu }(0,T;{\check{H}}^{2+\gamma })}^2+{\parallel f\parallel }_{H^{1+\nu }(0,T;{\check{H}}^{\gamma })}^2\right)t^{-2\overline{\alpha }\left(0\right)}.}\hfill \end{array}$$

□

4. Discussion

In this paper, we discuss the well-posedness and regularities of a hidden-memory variably distributed-order tFDE in which the pdf $\rho (\alpha ,t)$ has a history memory property. Under assumptions (i) and (ii), we proved that the mobile-immobile hidden memory distributed-order tFDE in Equation (3) has a unique weak singular solution with its second-order derivative with respect to the time of the order $O\left(t^{-\overline{\alpha }\left(0\right)}\right)$. Analysis under weaker assumptions such as the pdf $\rho$ as a Dirac delta function and numerical methods will be considered in our future work.