  In this article, we investigate inverse source problems for a wide range of
PDEs of parabolic and hyperbolic types as well as time-fractional evolution
equations by partial interior observation. Restricting the source terms to the
form of separated variables, we establish uniqueness results for simultaneously
determining both temporal and spatial components without non-vanishing
assumptions at $t=0$, which seems novel to the best of our knowledge.
Remarkably, mostly we allow a rather flexible choice of the observation time
not necessarily starting from $t=0$, which fits into various situations in
practice. Our main approach is based on the combination of the Titchmarsh
convolution theorem with unique continuation properties and time-analyticity of
the PDEs under consideration.
