Uniqueness in determining the temporal source factor and the fractional order with an unknown initial value

We consider inverse problems for a time-fractional diffusion equation with a second-order elliptic operator and the homogeneous Dirichlet boundary condition in a bounded domain of $\mathbb R^N$. The initial value is unknown and is not assumed to be zero. The main result concerns the simultaneous determination of the fractional order and the time-dependent source factor from one weighted spatial observation. For irrational orders in $(0,1)$ and bounded source factors satisfying an exponential integrability condition, we prove that a nonzero observation given for all positive times uniquely determines both unknowns. We also describe the information about the initial value determined by this observation. A counterexample shows that uniqueness of the source factor may fail at the classical order $α=1$, even when the observation is nonzero. In the second part, we establish two uniqueness results for the fractional order when the elliptic operators, initial values, and source terms in the compared equations may be different. The first result is proved by the Laplace transform, whereas the second uses short-time observations and the asymptotic behavior of the Mittag--Leffler functions.