Additional Measurements for Semilinear Fractional Inverse Sources

Definitions and mathematical statement

The supplied problem asks which additional measurement suffices to recover an unknown source in a semilinear fractional partial differential equation. It does not specify the equation or its fractional operator, so a concrete mathematical recovery problem cannot yet be stated.

A conditional formulation of recovery as uniqueness is as follows. Once the forward equation, its domain, initial and boundary conditions, solution spaces, and admissible source class have been specified, let $\mathcal{A}$ denote the set of admissible solution--source pairs $(u,f)$ satisfying that model. Let $B(u)$ denote the observations already available, and let $M(u)$ denote an additional observation of the solution. Seek an admissible measurement operator $M$ such that \[ \forall (u_1,f_1),(u_2,f_2)\in\mathcal{A},\qquad \left(B(u_1)=B(u_2)\ \text{and}\ M(u_1)=M(u_2)\right) \Longrightarrow f_1=f_2. \] Here equality of sources must be interpreted in the source space eventually specified. This implication expresses unique identification among admissible sources; it does not assert existence, stability, or a reconstruction method.

The notation above is a formulation template, not an equation supplied by the source. The record specifies neither $\mathcal{A}$, $B$, nor the permitted class of additional measurements $M$. It also does not state whether the source depends on space, time, or both, whether it has a prescribed factorization, or whether the intended measurement is pointwise, boundary, terminal-time, or integral data. Consequently, no particular sufficient measurement or fully quantified model-specific claim can be extracted without further source information.

Short English statement

Determine an additional observation that uniquely identifies the unknown source in a specified semilinear fractional PDE. The supplied record omits the PDE, admissible sources, initial and boundary conditions, and existing observations, so the displayed uniqueness condition is only a conditional formulation requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004570$, source code catalogue entry, title: Additional Measurements for Semilinear Fractional Inverse Sources. Project registry references: data/unsolvedmath/problems.json and catalog/problems.json. No individual proposer is identified; the proposed year is recorded as 2020.
• Supplied report attribution: Computational Inverse Problems for Partial Differential Equations (2021), Oberwolfach Reports, pp. 1903--1954, DOI 10.4171/owr/2020/39. Also listed as "OWR 2020/39 report": https://doi.org/10.4171/owr/2020/39. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "Fractional inverse-source workshop paper": https://oa.tib.eu/renate/server/api/core/bitstreams/c39dce19-622a-43d3-ac6a-fede42f2e1ae/content.
• Supplied literature source, labeled "Hasanov and Slodicka, inverse source from point data": https://doi.org/10.1016/j.aml.2012.08.013.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that selected semilinear fractional inverse-source models admit uniqueness and stability mechanisms using additional measurements or integral operators, while no general choice of a single appropriate measurement covers the stated problem. This is supplied dated curation, not independently verified. The record itself requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its verification note reports manual semantic, typographic, and LaTeX review and describes the clean statement as self-contained. These are registry assertions, not independent verification of a precise mathematical formulation or its correctness.