On a time-space fractional diffusion equation with a semilinear source of exponential type

In the current paper, we are concerned with the existence and uniqueness of mild solutions to a Cauchy problem involving a time-space fractional diffusion equation with an exponential semilinear source. By using the iteration method and some Lp - Lq-type estimates of fundamental solutions associated with the Mittag-Leffler function, we study the well-posedness of the problem in two differ-ent cases corresponding to two assumptions on the Cauchy data. On the one hand, when considering initial data in Lp(RN) boolean AND L infinity(RN), the problem possesses a local-in-time solution. On the other hand, we obtain a global existence result for a mild solution with small data in an Orlicz space.