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

: 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 α ( 0 ) .


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 is represented as a modified form and has attracted extensive research [10,11].In Equation (1), Γ(•) refers to the Gamma function, and the probability density function (pdf) ρ(α) is nonnegative and satisfies 1 0 ρ(α)dα = 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 ρ changing over time.In other words, the pdf ρ depends on both α and t, which we denote by ρ(α, 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 distributedorder 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- Compared with Equation (1), the value of α varies on [0, α(s)](α(s) ≤ 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 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: Here ) is a simply connected bounded domain with smooth boundary ∂Ω, x = (x 1 , • • • , x d ), f (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, represents the subdiffusive transport of the absorbed particles, and k(t) represents the ratio of the particles of anomalous versus normal diffusion.
Let I ⊂ R be a bounded interval and m ∈ N, with C m (I) as the space of continuous functions with continuous derivatives up to an order m, which is equipped with We make the following assumptions on the variable order and the probability density function ρ(α, t): (i) There exists a constant 0 < α * < 1 which satisfies that 0 ≤ α(t) ≤ α * for t ∈ [0, T] and and ρ(α, t) ∈ C([0, T]; C[0, α * ]).
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.

Analysis of Hidden-Memory Variably Distributed-Order ODE
We study the following hidden-memory variably distributed-order ordinary diffusion equation (ODE): We move the fractional derivative term to the right-hand side to rewrite Equation (4) as Then, we integrate Equation ( 5) multiplied by e λt from 0 to t to obtain By differentiating the above formula and setting v(t) = ξ (t), we obtain We obtain ξ in terms of v as follows: Theorem 1. Suppose that the assumption (i) holds and k, g ∈ C[0, T].Then, Equation ( 4) has a unique solution ξ ∈ C 1 [0, T] with the following stability estimate: Proof.We define a functional sequence {v n } ∞ n=0 on [0, T] by Equation ( 6) as For n ≥ 1, we subtract v n from v n+1 to find By setting v −1 (t) ≡ 0, we have We assume that The, we substitute Equation (10) into Equation ( 8) to obtain Here, B(•, •) is the Beta function.Thus, the assumption in Equation ( 10) holds for n ≥ 1 by mathematical induction.The series with function terms defined by the right-hand side of Equation ( 10) can be bounded by where E 1−α * ,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 ∈ [0, T] as satisfies Equations ( 6) and (7).
Theorem 2. Suppose that assumptions (i) and (ii) hold and k, g ∈ C 1 [0, T].Then, we have ξ ∈ C 2 (0, T] with the following estimate: Proof.We differentiate Equation ( 6) on both sides to find The first four terms on the right-hand side of Equation ( 13) can be bounded by We bound 0 Thus, the fifth and sixth terms on the right-hand side of Equation ( 13) can be bounded by To estimate the last term, we denote If α(s) ≤ α(t), then we have ρ(α, s) ≡ 0 for α(s) < α ≤ α(t), and thus η(s) can be expressed in terms of Here, we note that the second term is equal to 0. Otherwise, if α(s) > α(t), then we decompose the support as [0, α(s)] = [0, α(t)] (α(t), α(s)], and Equation ( 16) holds too.Thus, the hidden-memory variably distributed-order fractional integral 0 I ρ(•,t) t v can be decomposed as follows: Next, we analyze I 1 and I 2 .As the upper bound of α in I 1 depends on t, we can exchange the integral order to rewrite I 1 as We differentiate Equation ( 17) on both sides to obtain By applying α (t) ∈ C[0, T] in assumption (ii), we have In addition, we can bound I 1 with To estimate I 2 , we rewrite I 2 into the following form: ) By differentiating on J 1 and J 2 , we obtain Therefore, J 1 and J 2 can be bounded by Similarly, we estimate J 3 and J 4 according to Theorem 1: We differentiate on J 5 to obtain We bound the first two terms on the right-hand side of Equation (22) with By assumptions (i) and (ii) and the mean value theorem, we have Therefore, the third term on the right-hand side of Equation ( 22) can be bounded by Thus, J 5 can be estimated by We submit the estimates of Equations ( 20)- (23) into Equation (19) to obtain the estimate of d dt I 2 , and hence ( 0 I ρ(•,t) t v) can be bounded by When combined with Equations ( 14)-( 15), we found that v (t) in Equation ( 13) satisfies By applying the generalized Gronwall inequality [29,30], we obtain This finishes the proof.

Discussion
In this paper, we discuss the well-posedness and regularities of a hidden-memory variably distributed-order tFDE in which the pdf ρ(α, 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 secondorder derivative with respect to the time of the order O(t −α(0) ).Analysis under weaker assumptions such as the pdf ρ as a Dirac delta function and numerical methods will be considered in our future work.