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On time-fractional diffusion equations with space-dependent variable order

We investigate diffusion equations with time-fractional derivatives of space-dependent variable order. We examine the well-posedness issue and prove that the space-dependent variable order coefficient is uniquely determined among other coefficients of these equations, by the knowledge of a suitable time-sequence of partial Dirichlet-to-Neumann maps.

In this paper, we pursue two goals. The first one is to establish the well-posedness of the initial-boundary value problem (1.4) for a suitable source term f and initial value u 0 . The second one is to analyse the uniqueness in an inverse problem of determining simultaneously the fractional order α and two coefficients ρ and q of the diffusion equation in (1.4) by partial Cauchy data.
1.2. Physical motivations. Anomalous diffusion in complex media is a rapidly growing field of academic research with multiple engineering applications in geophysics, environmental science and biology. The diffusion properties of homogeneous porous media are currently modeled, see e.g., [1,5], by constant order (CO) time-fractional diffusion processes where in (1.4) the mapping x → α(x) is maintained constant over Ω.
However, in complex media, the presence of heterogeneous regions causes variations of the permeability in different spatial positions, and in this case, the VO time-fractional model is more relevant for describing the diffusion process, and see e.g., [33]. This is a background for VO time-fractional diffusion equations.

1.3.
A short review of the mathematical literature of time-fractional diffusion equations. Ordinary and partial differential equations with fractional derivatives have been intensively studied over the two last decades. We refer to [21,27,28,32] for a general introduction to fractional calculus, and for example to [2,12,26] for more specific foci on partial differential equations with time fractional derivatives. The well-posedness 2 problem for CO time-fractional diffusion equations was addressed in [4,11,31,35] and see also [8], where the local existence, the uniqueness and the continuity on initial data of the solution to partial integrodifferential equations of parabolic type are discussed. The time decay of their solutions was studied in [17]. Recently, a new definition of the weak solution to these equations was introduced in [20], which allows for defining solutions to semi-linear fractional wave equations. Moreover, initial-boundary value problems for multi-terms time-fractional diffusion equations were studied by [4,24]. Notice that a De Giorgi-Nash Hölder regularity theorem was derived in [3] (see also [36]) for solutions to CO time-fractional equations with fractional diffusion in space. As for distributed order (DO) time-fractional diffusion equations, we refer to [23] for the analysis of the well-posedness problem, and to [23,25] for the study of the asymptotic behavior of the solution. However, in contrast with CO or DO time-fractional equations, for VO timefractional diffusion equations. to our best knowledge, there are no results available in the mathematical literature.
Quite similarly, there is only a small number of mathematical papers dealing with inverse problems associated with time-fractional diffusion processes, which are listed below. In the one-dimensional case, [6] proved simultaneous determination of the constant fractional differential order and the time-independent diffusion coefficient by Dirichlet boundary measurements for the solution. In dimensions 2 or larger, [13] determined a constant fractional order from measurements at one point of the solution over the entire time span. In [9,31], the time-varying factor in the source term or in the zeroth order coefficient of time-fractional equations was stably determined by pointwise observation of the solution. For half-order fractional diffusion equations, [7,34] proved stability in determining a zeroth order coefficient by means of a Carleman estimate. An inverse boundary value problem for diffusion equations with multiple fractional time derivatives is examined in [22] and the authors prove the uniqueness in determining the number of fractional time-derivative terms, the orders of the derivatives, and spatially varying coefficients. Finally, in [19], the zeroth and first order space-dependent coefficients defined on a Riemanian manifold, along with the Riemanian metric, are simultaneously determined by a partial Dirichlet-to-Neumann map taken at one arbitrarily fixed time.
Furthermore by A q we denote the self-adjoint realization in L 2 (Ω) of the operator A q with the homogeneous Dirichlet boundary condition and for p ∈ C \ R − , by (A q + ρ(x)p α(x) ) −1 the resolvent operator of A q + ρ(x)p α(x) .
Henceforth B(X, Y ) denotes the Banach space of all the bounded linear operators from a Banach space X to another Banach space Y , and we set B(X) = B(X, X).
Then the existence and uniqueness result of a weak solution to (1.4) is as follows.
We discuss the uniqueness in the inverse problem of determining the coefficients (α, ρ, q) from the knowledge of the boundary operators {N α,ρ,q (t n ); n ∈ N} associated with a time sequence t n , n ∈ N fulfilling the set {t n ; n ∈ N} has an accumulation point in (0, +∞). (1.12) Moreover we assume that Ω, S in and S out satisfy the following conditions.
The uniqueness result for our inverse coefficients problem is as follows.
Since there exists a constantc 0 > 0 such that by (1.1) and the Poincaré inequality, the operator A 0 is self-adjoint in L 2 (Ω) and acts Put r := 2κ/(κ − 2) and notice from the Hölder inequality that and so it follows from (2.1) that Thus, taking ε > 0 so small that εc 2 q 2 L κ (Ω) < 2c 2 0 , we see that the multiplier by q in In this article, we suppose (1.2) in such a way that A q c 0 in the operator sense, wherec 0 is the constant appearing in (2.1). This hypothesis is quite convenient for proving Proposition 2.1 below stated, which is essential for the proof of Theorem 1.1 and Proposition 3.1, but it could be removed at the price of greater unessential technical difficulties. Nevertheless, for simplicity, we shall not go further into this direction.
Proof. We shall prove (2.4) only for r ∈ [1, +∞), because the corresponding estimate for r ∈ (0, 1) can be derived in the same way.
We define a multiplication operator Then iU 2 β is the skew-adjoint part of the operator Hence the self-adjoint operator U β is bounded and boundedly invertible in L 2 (Ω) and Moreover, for each p = re iβ , it holds true that where B q,p := A q + ρ(x)r α(x) cos(βα(x)) denotes the self-adjoint part of the operator Indeed, the multiplication operator by ρ(x)r α(x) cos(βα(x)) is bounded by ρ M r α M in (Ω) and satisfies the estimate 1.
It follows from this and (2. . As a consequence, we infer from (2.7) that in the operator sense. Therefore B q,p is boundedly invertible in L 2 (Ω) and .
This and (2.9) entail that ( To this purpose, we introduce the closed sequilinear form 10 which is associated with the operator A q + ρ(x)p α(x) in L 2 (Ω). In light of (2.1) and (1.2), we have for all u ∈ H 1 0 (Ω), entailing that a q,p is sectorial for every p ∈ C \ R − . Here and henceforth R and I mean the real part and the imaginary part of a complex number under consideration, respectively.

Remark 3.
For all h ∈ C 1 (R + ) such that 1) Let us start with S 0 (t). To this purpose, we set
As a consequence, the mapping p → ∂ t g q (t, p) ∈ L 1 (γ(ε, θ); L 2 (Ω)) for any fixed t ∈ R + in virtue of (2.38). Now, applying the Fubini theorem to the right-hand side of the above identity, we obtain (2.37). This establishes that the restriction to Q of the function expressed by the right-hand side of (2.37), is a weak solution to (1.4) associated with u 0 = 0. Evidently such a function lies in C([0, T ]; L 2 (Ω)). Moreover it is unique from Definition 2.2.
Finally, by superposition, the desired result follows readily from 1) and 2).

Analysis of the inverse problem
In this section, we suppose that ∂Ω is C 1,1 and (1.9) holds, that is, A 0 = −∆ and We recall for further use that the norm in H 2 (Ω) is equivalent to the norm in D(A 0 ) or in D(A q ).
In terms of Lemma 3.2, we can complete the proof of Theorem 1.2.
Therefore, by (1.12), the set of the zeros of the analytic function h has accumulation point in (0, +∞), so that identically vanishes, and (3.9) follows.