In this work, we are interested in a fractional diffusion equation with a conformable derivative that contains the time dependent coefficients which occurs in many application models. By using some given assumptions, we consider the global solution to the problem. Moreover, the convergence of the mild solution when fractional order tends to 1 − is presented. This research can be considered as one of the first results on the topic related to conformable problem with time-dependent coefficients. In the simple case of coefficient, we show the global regularity for the mild solution in L p space. The main techniques of this work are to use Banach fixed point theorem, L p − L q heat semigroup and some complex evaluations and techniques.