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

We study inverse problems for a time-fractional evolution equation with a positive self-adjoint operator in a Hilbert space. The unknowns are the fractional order \(α∈(0,1)∪(1,2)\) and the time-dependent factor \(p(t)\) in the source term, while the additional information is given by a single weighted observation \((u(t),ψ)_X\), \(0<t<T\). We first prove uniqueness of the fractional order when the two compared orders belong to the same interval, either \((0,1)\) or \((1,2)\), under mild assumptions on the behavior of the source factors near \(t=0\). Under stronger assumptions, the a priori restriction on the intervals containing the orders can be removed. When the operator and the spatial part of the source are fixed, we also obtain uniqueness of the temporal factor. For a prescribed order, we prove unique recovery of \(p(t)\) together with a two-sided Lipschitz stability estimate. The proofs are based on precise short-time decompositions of the observation into its leading fractional term and a smoother remainder. Finally, we construct an explicit counterexample showing that, when nonzero initial data are allowed, a vanishing weighted observation does not in general imply a vanishing source factor, even though \((f,ψ)_X≠0\). This demonstrates the essential role of the zero-initial-data assumption in the source identification results.