Uniqueness for an inverse problem of determining order and temporal factor of the source for time-fractional evolution equations

Technical Summary: Uniqueness for an Inverse Problem of Determining Order and Temporal Factor of the Source for Time-Fractional Evolution Equations

Problem Formulation
This paper investigates an inverse problem associated with the Cauchy problem for a time-fractional evolution equation in a Hilbert space $X$. The forward problem is defined by the equation:
$$\partial_t^\alpha u(t) = -Au(t) + p(t)f$$
where $A$ is a positive definite self-adjoint operator with a compact inverse, $f \in X$ is a known source profile, and the unknowns are the fractional order $\alpha \in (0, 1) \cup (1, 2)$ and the time-dependent source factor $p(t) \in C[0, T]$. The initial condition is implicitly defined by the requirement that the solution $u$ belongs to the fractional Sobolev space $H^\alpha(0, T; X)$, which enforces $u(0)=0$ (and $u'(0)=0$ if $\alpha > 1$) in the sense of traces where applicable.

The overdetermination condition provided to solve the inverse problem is the scalar product of the solution with a fixed weight function $\psi \in D(A)$:
$$(u(t), \psi)_X = \Psi(t), \quad 0 < t < T.$$
The primary objective is to determine the uniqueness of simultaneously recovering both $\alpha$ and $p(t)$ from this data.

Methodology
The authors employ a spectral decomposition approach combined with asymptotic analysis near $t=0$. The methodology relies on the following key steps:

1. Eigenfunction Expansion: The solution $u(t)$ is expanded using the eigenfunctions of the operator $A$. This transforms the evolution equation into a series involving Mittag-Leffler functions $E_{\alpha, \beta}(z)$.
2. Asymptotic Decomposition: A central technique is the decomposition of the solution (or its scalar product with $\psi$) near $t=0$ into a "least smooth" leading component and a "smoother" remainder. Specifically, the authors utilize the property that the fractional integral $J^\alpha$ and the fractional derivative $\partial_t^\alpha$ allow for the separation of the dominant term, which behaves like $t^{\alpha+m}$ (where $m$ relates to the leading order of $p(t)$ at $t=0$), from higher-order terms.
3. Admissible Classes: The analysis restricts the unknown source factors $p(t)$ and $q(t)$ to specific admissible classes ($P$ and $\tilde{P}$). These classes require that the functions behave asymptotically as $p(t) \sim p_0 t^m$ near $t=0$ with specific smoothness conditions on the remainder.
4. Uniqueness via Asymptotics: By comparing the asymptotic expansions of the data $(u(t), \psi)_X$ for two potential solutions with parameters $(\alpha, p)$ and $(\beta, q)$, the authors derive contradictions unless the parameters are identical. This involves analyzing the leading powers of $t$ in the expansion.
5. Integral Equations and Stability: For the recovery of $p(t)$ when $\alpha$ is known, the problem is reduced to a Volterra integral equation. The authors utilize generalized Gronwall inequalities to establish existence, uniqueness, and Lipschitz stability estimates.

Key Contributions and Results

• Uniqueness of the Fractional Order ($\alpha$): The paper establishes that the fractional order $\alpha$ is uniquely determined by the data $(u(t), \psi)_X$ for $0 < t < T$, independent of the specific form of the operator $A$ and the source function $p(t)$, provided $p$ belongs to the class $P$ and $(f, \psi)_X \neq 0$. This holds even if the operator $A$ is unknown (i.e., comparing two systems with potentially different operators $\tilde{A}$).
• Uniqueness of the Source Factor ($p(t)$):
  • If $\alpha$ is known and $A$ is the same for both systems, $p(t)$ is uniquely determined.
  • If $\alpha$ is unknown, the paper proves that $p(t)$ is uniquely determined simultaneously with $\alpha$ under the condition that $p, q \in \tilde{P}$ (a narrower class requiring higher smoothness, $a \ge m+2$).
  • Theorem 1.1 and Theorem 1.2 assert that if the data matches for all $t \in (0, T)$, then $\alpha = \beta$ and $p = q$.
• Role of Initial Data: The paper highlights that the assumption $u(0)=0$ (implied by $u \in H^\alpha$) is critical. An example is provided showing that if the initial data is non-zero ($u(0) \neq 0$), uniqueness for $p(t)$ may fail.
• Refined Results for $\alpha \in (0, 1)$: For the case $\alpha \in (0, 1)$, the authors prove that if the data $(u(t), \psi)_X$ is sufficiently smooth (specifically, if $\partial_t^\alpha (u, \psi)_X \in H^\theta$ with $\theta > 1/2$), then the source factor $p(t)$ must be zero if the data is zero. This extends uniqueness results to cases where $p(t)$ does not necessarily vanish at $t=0$ but satisfies specific regularity conditions.
• Stability: Theorem 1.3 provides a two-sided Lipschitz stability estimate for the determination of $p(t)$ when $\alpha$ is known, relating the norm of $p$ to the norm of the fractional derivative of the observed data.

Significance and Claims
The paper claims to resolve the inverse problem of simultaneously identifying the fractional order and the time-dependent source factor without prior knowledge of the elliptic operator $A$. Unlike previous works that often rely on pointwise observations $u(x_0, t)$ or specific eigenfunctions, this work demonstrates that an arbitrary weight function $\psi \in D(A)$ suffices, provided the observation is continuous over the interval $(0, T)$.

The authors emphasize that their results rely on the decomposition of the solution near $t=0$ into a singular leading term and a smoother remainder. This technique allows them to extract the fractional order $\alpha$ directly from the asymptotic behavior of the data. The paper also notes that while the results are established for abstract Hilbert spaces, they are plausibly extendable to the Laplace operator in bounded domains with Dirichlet boundary conditions, though this remains a subject for future research. The work does not propose new experimental applications but rather provides a rigorous theoretical foundation for the uniqueness of these inverse problems in fractional calculus.