Analysis of a Time-Fractional Substantial Diffusion Equation of Variable Order

: A time-fractional substantial diffusion equation of variable order is investigated, in which the variable-order fractional substantial derivative accommodates the memory effects and the structure change of the surroundings of the physical processes with respect to time. The existence and uniqueness of the solutions to the proposed model are proved, based on which the weighted high-order regularity of the solutions, in which the weight function characterizes the singularity of the solutions, are analyzed.


Introduction
Fractional problems with substantial derivatives have attracted increasing amounts of attention due to their applications in, e.g., characterizing the functionals of anomalous diffusion and non-Brownian motions [1,2]. There are several investigations in mathematical analysis and numerical methods to such kinds of problems [3][4][5][6][7][8][9], while the corresponding studies for the variable-order fractional substantial derivative models, in which the variable fractional order accommodates, e.g., the structure change of the surroundings with respect to time, are rarely found in the literature.
In some recent works, the following Caputo variable-order time-fractional diffusion equations are investigated mathematically and numerically [10][11][12][13][14][15][16] Here, the Caputo variable-order fractional derivative operator is defined as [17][18][19] ∂ α(·,t) t The Caputo (variable-order) fractional derivative is widely used in various applications that exhibit the power law type memory effects due to the power function integral kernel. The Caputo variable-order fractional substantial derivative operator σ ∂ α(·,t) t , which is defined in the following for some σ ≥ 0 and 0 ≤ α(s, t) < 1 [1][2][3] to the complicated form of the variable-order fractional substantial derivative, existing analysis techniques for investigating the Caputo variable-order fractional derivative problems may not apply directly for problems involving (1), which motivates the current work.
In this paper, we analyze the following time-fractional diffusion equation [20][21][22][23][24][25][26][27][28] equipped with variable-order fractional substantial derivative where q ≥ 0, T > 0, Ω is a d-dimensional domain for 1 ≤ d ≤ 3 with a smooth boundary ∂Ω. In the following two sections, we rigorously prove the well-posedness and high-order regularity of the solutions to problem (2), respectively, which provide a theoretical basis for this model. In particular, a weight function is introduced in the estimate of high-order smoothing properties to characterize the initial singularity of the solutions.
In the rest of the paper, let H 1 (0, T) be the Hilbert space of functions with weak derivatives up to order 1 in L 2 (0, T), the space of square-integrable functions on [0, T]. The corresponding norms are defined by H 1 (0, T; D) for some Banach space D stands for the space of the functions in H 1 (0, T) with respect to the norm of D. Q denotes a generic constant that may assume different values at different cases.

Existence and Uniqueness
We prove the existence and uniqueness of the solutions to problem (2) in this section. Throughout the paper, we assume that there exists a constant 0 < α * < 1, such that Theorem 1. If f ∈ L 2 (0, T; L 2 (Ω)) and u 0 ∈Ȟ 1 (Ω), problem (2) has a unique solution u ∈ H 1 (0, T; L 2 (Ω)) ∩ L 2 (0, T;Ȟ 2 (Ω)) and where Q is independent from the data and the solutions and the norm · Ȟp (Ω) for p ≥ 0 is defined in terms of the eigenpairs satisfy the following ordinary differential equation where w, λ, h and w 0 refer to u i , λ i , ( f , φ i ) and (u 0 , φ i ), respectively, for i ≥ 1. We consider the fractional term as part of the right-hand side term in (3) to obtain An application of the variation-of-constants formula yields where * stands for the convolution on [0, t] as follows Then, we differentiate (4) with respect to t, multiply e −κt for some κ ≥ 0 on the resulting equation and reformulate e −κt · σ ∂ α(·,t) t as a sum of a linear operator S and a function H(t) to obtain an integral equation in terms ofw := e −κt ∂ t w as follows In order to prove the well-posedness of this integral equation in L 2 (0, T), it suffices to show that the linear operator −qS + λqe −(λ+κ)t * S is a contraction in L 2 (0, T). For v ∈ L 2 (0, T), we apply and Young's convolution inequality to bound Sv by The operator λe −(λ+κ)t * S is then accordingly bounded by We incorporate the above two equations to conclude that for κ large enough, −qS + λqe −(λ+κ)t * S is a contraction in L 2 (0, T) and, thus, the integral Equation (5) has a unique solutionw ∈ L 2 (0, T). We further apply this contractability on (5) to obtain a stability estimate Then, the w defined by satisfies the differential Equation (3). The unique H 1 solution to (3) follows from that of (5).

Weighted Regularity
We prove the weighted regularity of the solutions to problem (2) in the following theorem.
Proof. We differentiate (4) twice in time to obtain The ∂ t σ ∂ α(·,t) t w could be simplified as As we multiply t δ on both sides of (7) and apply we apply e −κt · L 2 (0,T) for some κ > 0 on (8) to find We, therefore, multiply t δ on both sides of (6), apply e −κt · L 2 (0,T) on the resulting equation, choosing κ large enough and employ (9) and (tλ) δ e −λt/2 ≤ Q, as well as similar (and simpler) proofs as above to obtain We employ this result to obtain the estimate of ∂ 2 t u as which completes the proof.

Conclusions
In this paper we rigorously prove the well-posedness and high-order regularity of the solutions to the time-fractional diffusion equation equipped with variable-order fractional substantial derivative. In particular, a weight function is introduced in the estimate of high-order smoothing properties to characterize the initial singularity of the solutions. Based on these theoretical estimates, we will carry out numerical analysis for the proposed model in the near future in order to provide supports for practical implementations.