Inverse problem of determining a time-dependent coefficient in the time-fractional subdiffusion equation

This paper explores the forward and inverse problems for a fractional subdiffusion equation characterized by time-dependent diffusion and reaction coefficients. Initially, the forward problem is examined, and its unique solvability is established. Subsequently, the inverse problem of identifying an unknown time-dependent reaction coefficient is addressed, with rigorous proofs of the existence and uniqueness of its solution. Both problems' existence and uniqueness are demonstrated using Banach's contraction mapping theorem. Notably, this work is the first to investigate direct and inverse problems for such equations with time-dependent coefficients.