In this short paper, we study time-fractional diffusion equations with time-dependent coefficients. The derivative operator that appears in the main equation is the Riemann–Liouville derivative. The main purpose of the paper is to prove the existence of a global solution. Due to the nonlocality of the derivative operator, we cannot represent the solution directly when the coefficient depends on time. Using some new transformations and techniques, we investigate the global solution. This paper can be considered as one of the first results on problems with time-dependent coefficients. Our main tool is the application of Fourier analysis methods combined with estimates involving Mittag–Leffler functions and certain Sobolev embeddings.