Inverse Problems with Unknown Boundary Conditions and Final Overdetermination for Time Fractional Diffusion-Wave Equations in Cylindrical Domains

: Inverse problems to reconstruct a solution of a time fractional diffusion-wave equation in a cylindrical domain are studied. The equation is complemented by initial and ﬁnal conditions and partly given boundary conditions. Two cases are considered: (1) full boundary data on a lateral hypersurface of the cylinder are given, but the boundary data on bases of the cylinder are speciﬁed in a neighborhood of a ﬁnal time; (2) boundary data on the whole boundary of the cylinder are speciﬁed in a neighborhood of the ﬁnal time, but the cylinder is either a cube or a circular cylinder. Uniqueness of solutions of the inverse problems is proved. Uniqueness for similar problems in an interval and a disk is established,

Often, parameters of equations are a priori unknown. A common way to determine them is to solve inverse problems that use measurements of states of the processes. Depending on the practical situation, the state can be measured at interior or boundary points of a medium over the time [2,[9][10][11][12] or at fixed time moments over the space, e.g., at a final time [13][14][15][16].
Inverse problems to reconstruct space-dependent parameters of fractional diffusion equations by means of final measurements have been studied in several papers. Many methods that are applicable to such problems in the usual integer case, such as the Fredholm alternative [15], extremum principles [17], Fourier method [2,16,[18][19][20], and Carleman estimates [21], can be successfully adjusted to the fractional case.
Recently, the author of the present paper proved that a time-dependent factor f (t) of a source term in the fractional diffusion-wave equation is uniquely recovered by the final data [22]. A character of this problem in the fractional case essentially differs from that one in the integer case. The key feature is a behavior at +∞ of a kernel contained in formula of a Fourier coefficient of a state function. In the case α = 1, this kernel is exponentially decaying, but, in a fractional case α ∈ (0, 1) ∪ (1, 2), it admits a power-type asymptotic expansion. Assuming that f (t) is a priori known in a left neighborhood (T − δ, T) of the final moment T, the aforementioned power-type expansion enables reduction of the inverse problem to a moment problem and, in turn, establishment of the uniqueness of f (t) for t ∈ (0, T − δ).
In this paper, we continue the study of inverse problems with time-dependent unknowns and over-specified final data. We consider two problems for the time fractional diffusion-wave equation in a cylindrical domain. In both problems, the initial and final conditions are given. In the first problem, the boundary conditions are prescribed for t ∈ (0, T) on the lateral surface and for t ∈ (T − δ, T) on both bases. In the second problem, the boundary conditions are prescribed for t ∈ (T − δ, T) on the whole boundary, and the domain is either a cube or a circular cylinder.
From the physical viewpoint, such problems may arise when the regime of energy exchange between the domain and a surrounding medium is partly or fully unknown before the time value T − δ.
We will prove that solutions of the mentioned problems are unique. As corollaries, we establish uniqueness for similar inverse problems on an interval and a disk, too. At the end of the paper, we briefly discuss some other questions: ill-posedness, open problems, etc.
Problems with final overdetermination and unknown boundary conditions are completely new and have not yet been studied neither in the fractional nor in the classical integer case.

Formulation of Problems
Let ω ⊂ R d , d ≥ 1, be an open bounded domain with the boundary ∂ω and Ω = ω × (0, 1). We use the following notation for points of Ω and their coordinates: (x, y) ∈ Ω, x = (x 1 , . . . , x d ) ∈ ω, y ∈ (0, 1), and the following abbreviated notation for partial derivatives: In case m = 1, we omit the superscript m. Moreover, we denote the Laplace operators in Ω and ω as follows: Let α ∈ (0, 1) ∪ (1, 2). Let · stand for the floor function and q ∈ R. We consider the following time fractional diffusion-wave equation in the domain Ω (cf. References [1,22]): Here, the symbol D β t stands for the Riemann-Liouville fractional derivative of the order β with respect to t that is defined as follows: We mention that the term D α− α t ∂ α t u − ∂ α t u| t=0 is the Caputo fractional derivative of the order α of u. Namely, in case u is α + 1 times differentiable with respect to t, this can be rewritten in the following form: where the right-hand side contains the Caputo derivative in the usual form. Let T > 0. We formulate the following (direct) problem for the function u: Here, F, ϕ j , j ∈ {0; α }, h, g 0 , g 1 are given functions, and the boundary operators B, B 0 , B 1 are defined as follows: where I is the unity operator, ν = ν(x) is the outer normal vector of ∂ω at x ∈ ∂ω, Now, let us proceed to inverse problems to reconstruct the function u in case of partly given boundary data. Suppose that either the pair g 0 , g 1 or the triplet g 0 , g 1 , h of boundary functions is given only in the time subinterval t ∈ (T − δ, T), where δ is some number between 0 and T. In order to recover u, we assume the following final condition: where ψ is a given function.
The corresponding inverse problems are as follows.
A method that we use in the analysis of IP2 requires additional restrictions on the domain Ω. We will will study this problem in special cases when Ω is either a cube or a circular cylinder.
We will also extend the uniqueness results to inverse problems on an interval and a disk (Corollaries 1 and 2). Formulations of the latter problems are given right before the mentioned corollaries.

Eigenvalues and Eigenfunctions
Let (µ k , v k )| k∈K be the set of solutions of the eigenvalue problem where v k , k ∈ K, form a complete orthogonal system in L 2 (ω). We assume that the domain ω is sufficiently regular to guarantee v k ∈ H 2 (ω), k ∈ K. For instance, this is the case if ∂ω ∈ C 2 (Reference [23], Thm. 3.10). Another example of such an ω is the hyperrectangle ω = ∏ d j=1 (0, a j ), a j > 0. Although K is countable, and it is possible to identify it with N, in some cases below, it is more convenient to use other forms of K.
The system v k,l , k ∈ K, l ∈ N, is complete in L 2 (Ω).

Abstract Functional Spaces
Let X be a complex Banach space and G ⊆ R. As usual, L p (G; X), p ∈ [1, ∞] denotes the abstract Lebesgue space, i.e., The space C(G; X) consists of functions that are continuous on G with values in X and C n (G; X) = { f : G → X : f (i) ∈ C(G; X), i = 0, . . . , n}, n ∈ N.
In addition, we introduce the spaces and F denotes the Fourier transform with the argument ξ. Moreover, we define the following space: In case X is the set of complex numbers, we omit it in the notations of H s p and 0 H s p . Next, we formulate a lemma that describes the domain of the operator D β t in L p . Lemma 1. Let X be a complex Hilbert space, β ∈ (0, 1) and p ∈ (1, ∞). The operator D β t is a bijection from 0 H β p ((0, T); X) to L p ((0, T); X) and its inverse is t β−1 Γ(β) * . (The function t β−1 Γ(β) * f is called the Riemann-Liouville fractional integral of the order β of a function f ).

Formula for Fourier Coefficients of Solution of Direct Problem
where e α is the α-exponential function that is defined via the two-parametric Mittag-Leffler function E α,α as follows: e α (λ; t) = t α−1 E α,α (λt α ).
Formula (11) is deduced, e.g., in Example 42.2 of Reference [25]. It can also be derived by means of some relations available for Mittag-Leffler functions. Let us show this. Let us denote the right-hand side of (11) by RHS. Bringing the operator d dt from the term D [26], p. 61) with γ = α − α and integrating by We will deal with solutions of the direct problem that belong to the following space: Then, the fractional derivative and Laplacian of u in the Equation (1) are Lebesgue integrable functions, and u has regular traces u| t=0 , Bu| x∈∂ω , B 0 u| y=0 , B 1 u| y=1 . Moreover, due to well-known trace theorems, u ∈ U α implies that the data of the direct problem necessarily satisfy the conditions h ∈ L p ((0, T); L 2 (∂ω × (0, 1))), g 0 , g 1 ∈ L p ((0, T); L 2 (ω)).
In the next proposition, we deduce a formula for Fourier coefficients of the solution of (1)-(4) in the case the equation and initial conditions are homogeneous.
. Let a function u belong to the space U α and solve (1)-(4) with F = 0 and ϕ j = 0, j ∈ {0; α }. Then, ·, · X denotes the inner product in a Hilbert space X, dS is the element of ∂ω, Proof. Due to the assumption u ∈ U α and (14), we have h l ∈ L p ((0, T); L 2 (∂ω)), g 0;k , g 1;k ∈ L p (0, T). Therefore, the right-hand side of the Formula (15) is well-defined. Moreover, the functions u k,l belong to C α [0, T] and satisfy u

Remark 1.
In subsequent proofs of uniqueness of solutions of IP1 and IP2, we need the formula for Fourier coefficients of the solution of (1)-(4) only in the case However, a corresponding formula in case of nonvanishing F and ϕ j can be derived in a similar manner, with some additions.

Two Basic Lemmas
The analysis of IP1 and IP2 essentially uses two lemmas that we state below.
This lemma, in a bit more general case, was proved in paper [22], pp. 1688-1690. A sketch of the proof is as follows. There exist n 0 , where a and b are some numbers, and g is expressed in terms of g by an explicit formula. This implies b a P(s) g(s)ds = 0 for any polynomial P and, due to the Weierstrass theorem, b a Z(s) g(s)ds = 0 for any Z ∈ C[a, b]. This yields g = 0 and, in turn, g = 0.

Uniqueness for IP1
We start this section with a technical lemma. Lemma 5. The following relations are valid: where the numbers p l are defined in (9) and (10). In the special case B 0 = ∂ y − γ 0 I, B 1 = ∂ y + γ 1 I, γ 0 = γ 1 = 0, it holds that p 1 = 0, and we equate the undetermined quotient Now, we formulate and prove the uniqueness statement for IP1.

Uniqueness for IP2
In this section, we consider IP2 in two cases: Ω is a d + 1-dimensional cube, and Ω is a circular cylinder. For the sake of simplicity, we will be limited to problems with Dirichlet boundary conditions.

Circular Cylinder
In this subsection, we will study IP2 when Ω is the circular cylinder Ω = ω × (0, 1), ω = {x ∈ R 2 : x 2 1 + x 2 2 < 1}. So, let ω = {x ∈ R 2 : x 2 1 + x 2 2 < 1} and B = I. We define K = N 2 × {1; 2} and express the solution of the eigenvalue problem (6) as follows: where J k 1 is the Bessel function of the first kind of the order k 1 , and z k 1 ,k 2 is the k 2 -s positive root of J k 1 .

Lemma 6.
The relations are valid for any fixed k 1 ∈ N.

Conclusions and Additional Remarks
We have proven the uniqueness for inverse problems to reconstruct solutions of the fractional diffusion-wave equation in a cylinder ω × (0, 1) in two cases: (1) the initial and final data and full boundary data on lateral hypersurface of the cylinder are given, but the boundary data on the bases of the cylinder are specified in a partial time interval (T − δ, T) (IP1); (2) the initial and final data are given and Dirichlet boundary data on the whole boundary of the cylinder are specified in (T − δ, T) (IP2). In IP2, we were limited to a cube and a circular cylinder.
We hope that the uniqueness results regarding IP2 can be extended to other cylinders of simple geometry, e.g., cylinders-based annuli, balls, and also to Neumann and Robin boundary conditions. However, the extension to the case of an arbitrary base ω seems difficult. The reason is that the method applied in Section 5 requires a decomposition of eigenfunctions v k into two factors, one of which forms a complete system in L 2 (∂ω), while the other enables the use of Lemmas 3 and 4. For instance, in the case of the circular cylinder (Section 5.2), the first factor is either cos(k 1 θ) or sin(k 1 θ), and the second one is J k 1 (z k 1 k 2 r); in case of the cube (Section 5.1), these factors are d ∏ j=1 j =s sin(k j πx j ) and sin(k s πx j ), respectively, where s is an index of a pair of parallel faces of the cube. The aforementioned decomposition is not possible in the case of a general ω. IP1 and IP2 are linear, and the uniqueness is valid for solutions that belong to the space U α defined in (12). The corresponding regularity conditions for the initial data, the source term and the boundary data are given in (13) and (14).
The proofs of uniqueness for IP1 and IP2 work only in case the fractional order α belongs to the set (0, 1) ∪ (1, 2). In the limiting cases α = 1 and α = 2, the Mittag-Leffler function E α,α loses the power-type asymptotics, and the basic Lemma 3 fails.
On the other hand, in the literature, one can find several treatments of inverse problems with unknown boundary conditions for the usual diffusion equation that use additional data that are different from the final data. A well-known problem is the Cauchy problem, where the over-specified data are given on a portion of the boundary and nothing is given on the remaining part of the boundary. The uniqueness can be proved by means of Carleman estimates [13].
An open question is the uniqueness for IP1 and IP2 in the case δ = 0. Our theory does not work in such a case. The basic Lemma 3 fails because the integrals at the right-hand side of (21) are singular.