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.
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