  We focus on open questions regarding the uniqueness of distributional
solutions of the fast diffusion equation (FDE) with a given source term. When
the source is sufficiently smooth, the uniqueness follows from standard
results. Assuming that the source term is a measure, the existence of different
classes of solutions is known, but in many cases, their uniqueness is an open
problem. In our work, we focus on the supercritical FDE and prove the
uniqueness of distributional solutions with a Dirac source term that moves
along a prescribed curve. Moreover, we extend a uniqueness results for the
subcritical FDE from standing to moving singularities.
