Uniqueness of inverse source problems for general evolution equations

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