Inverse problem for a multi-term time-fractional diffusion equation with the Caputo derivatives

1. Problem Statement

The paper addresses an inverse source problem for a multi-term time-fractional diffusion equation defined on a bounded domain $\Omega \subset \mathbb{R}^N$ (or the torus $T^N$). The governing equation is:

$$\sum_{j=1}^{M} q_j \partial_t^{\rho_j} u(x, t) + A(x, D)u(x, t) = f(x)g(t), \quad x \in \Omega, \, 0 < t \leq T$$

Key Components:

• Operators: $\partial_t^{\rho_j}$ denotes the Caputo fractional derivative of order $0 < \rho_M < \dots < \rho_1 < 1$. $A(x, D)$ is a self-adjoint, positive, formally elliptic differential operator of even order $m \geq 2$.
• Source Term: The source is separable, $f(x)g(t)$, where the time-dependent part $g(t)$ is known, and the spatial component $f(x)$ is the unknown to be reconstructed.
• Conditions:
  • Initial condition: $u(x, 0) = \phi(x)$.
  • Boundary conditions: Homogeneous conditions $B_j u(x, t) = 0$ on $\partial \Omega$.
  • Overdetermination: An additional condition is given at a fixed interior time $t_0 \in (0, T]$: $u(x, t_0) = \Psi(x)$.

The goal is to determine the pair $(u(x, t), f(x))$ satisfying these conditions. The problem is inherently ill-posed due to the potential vanishing of the denominator in the reconstruction formula.

2. Methodology

The authors employ the Fourier method (spectral decomposition) combined with advanced asymptotic analysis of special functions.

1. Spectral Decomposition:

   • The problem is reduced to a sequence of ordinary differential equations by expanding the solution $u(x, t)$ and the source $f(x)$ in terms of the eigenfunctions $\{v_k(x)\}$ and eigenvalues $\{\lambda_k\}$ of the operator $A(x, D)$.
   • This transforms the PDE into a system of multi-term fractional ODEs for the Fourier coefficients $T_k(t)$ and $f_k$.

2. Solution of Fractional ODEs:

   • The solution to the time-fractional ODEs is expressed using the multinomial Mittag-Leffler function, denoted as $E_{\rho', \beta}(z_1, \dots, z_M)$.
   • The reconstruction formula for the unknown coefficient $f_k$ involves a denominator term $b_{k, \rho_1}(t_0)$, which is an integral involving $g(t)$ and the Mittag-Leffler function.

3. Asymptotic Analysis (Core Contribution):

   • The authors derive novel, precise asymptotic expansions for the multinomial Mittag-Leffler function for large arguments.
   • These expansions are used to establish uniform lower bounds for the denominator term $b_{k, \rho_1}(t_0)$. This step is critical for proving the convergence of the series and the existence of the solution.

4. Convergence Proofs:

   • Using the derived lower bounds and properties of the Mittag-Leffler function, the authors prove the absolute and uniform convergence of the Fourier series for both the solution $u(x, t)$ and the source $f(x)$ in appropriate Sobolev and continuous function spaces.

3. Key Contributions

• Novel Asymptotic Estimates: The paper provides the first precise asymptotic expansion for the specific form of the multinomial Mittag-Leffler function required for multi-term fractional diffusion equations. This allows for rigorous control over the spectral denominators.
• General Operator Framework: Unlike previous works restricted to the Laplacian or fractional Laplacian, this study handles a general self-adjoint elliptic operator $A(x, D)$ of order $m \geq 2$ with general boundary conditions.
• Handling Ill-Posedness: The authors rigorously address the ill-posed nature of the inverse problem. They identify conditions under which the problem is well-posed (unique solution) and characterize the non-uniqueness case (when the denominator vanishes).
• Existence and Uniqueness Theorems:
  • Existence: Proven under smoothness assumptions on initial data ($\phi, \Psi$) and the time function $g(t)$.
  • Uniqueness: Proven if the set of indices where the denominator vanishes is empty. If the set is non-empty, necessary and sufficient compatibility conditions are derived, and the non-uniqueness structure is fully described (infinite solutions exist).

4. Main Results

A. Asymptotic Behavior (Theorem 3.1)

For large $|z_1|$, the multinomial Mittag-Leffler function behaves as:
$$E_{\rho', \beta}(z_1, \dots, z_M) = -\frac{1}{z_1 \Gamma(\beta - \rho_1)} - \frac{1}{z_1^2} \left( \frac{1}{\Gamma(\beta - 2\rho_1)} - \sum \dots \right) + O(|z_1|^{-p-1})$$
This expansion is the foundation for the lower bound estimates.

B. Lower Bounds for the Denominator (Section 4)

The term $b_{k, \rho_1}(t_0)$ determines solvability.

• Case 1 (Sign-preserving $g(t)$): If $g(t)$ does not change sign, $|b_{k, \rho_1}(t_0)| \geq \frac{C}{\lambda_k}$. The denominator never vanishes, ensuring a unique solution.
• Case 2 (Sign-changing $g(t)$): If $g(0) \neq 0$ and $g(t)$ changes sign, the denominator may vanish for a finite number of indices $k$.
  • If the set of vanishing indices $K_{0, \rho_1}$ is empty, the solution is unique.
  • If $K_{0, \rho_1}$ is not empty, a solution exists if and only if the data satisfies orthogonality conditions: $\Psi_k = \phi_k [1 - \lambda_k t_0^{\rho_1} E_{\dots}(-\lambda_k t_0^{\rho_1})]$ for all $k \in K_{0, \rho_1}$. In this case, the solution is not unique (arbitrary coefficients can be added for $k \in K_{0, \rho_1}$).

C. Existence and Uniqueness (Theorems 5.1, 5.2, 6.1)

• Existence: A classical solution $(u, f)$ exists if the data $\phi, \Psi$ belong to the domain $D(\hat{A}^{\tau+1})$ with $\tau > N/2m$, and $g(t)$ satisfies the sign conditions.
• Uniqueness: The solution is unique if $b_{k, \rho_1}(t_0) \neq 0$ for all $k$. If zeros exist, uniqueness fails unless compatibility conditions are met, and even then, the solution is not unique.

D. Special Case: The Torus (Section 7)

The results are applied to the $N$-dimensional torus with constant coefficient operators. The domain of the fractional power of the operator is explicitly identified with Sobolev spaces $L^{\tau m}_2(T^N)$, providing a concrete functional analytic setting.

5. Significance

• Theoretical Advancement: The paper extends the theory of inverse problems from single-term to multi-term fractional equations, which are more realistic for modeling complex media with multiple relaxation mechanisms.
• Analytical Rigor: By deriving new asymptotics for the Mittag-Leffler function, the authors provide the necessary tools to handle the spectral denominators that typically cause divergence in such inverse problems.
• Practical Applicability: The results offer clear criteria for when an inverse source problem is solvable and unique. This is crucial for numerical implementations and experimental design, as it warns practitioners when data might lead to non-unique or non-existent solutions (degenerate cases).
• Generality: The framework applies to a wide class of elliptic operators and boundary conditions, making it a robust reference for future research in fractional calculus and inverse problems.

In summary, this paper provides a comprehensive analytical framework for solving inverse source problems in multi-term time-fractional diffusion, bridging the gap between abstract spectral theory and concrete solvability conditions through novel asymptotic analysis.