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.
Dinh Nguyen Duy Hai (Wed,) studied this question.