Uniqueness and non-uniqueness for an inverse source problem of a distributed order time-fractional diffusion equation

We investigate an inverse source problem for a distributed order time-fractional diffusion equation involving a fractional elliptic operator. The objective is to determine simultaneously the initial datum and a spatially dependent source term. We show that a single measurement at a fixed time is insufficient to guarantee uniqueness, even when observations are available over the entire spatial domain. In contrast, we establish a conditional uniqueness result showing that two measurements taken at distinct time instants uniquely determine both unknowns from a proper subdomain, under a non-resonance condition and a unique continuation assumption on the states at the observation times. The results highlight the essential role of temporal diversity in restoring identifiability under partial spatial observations.