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

Abstract

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 $ L^p-L^q $-type estimates of fundamental solutions associated with the Mittag-Leffler function, we study the well-posedness of the problem in two different cases corresponding to two assumptions on the Cauchy data. On the one hand, when considering initial data in $ L^p({\mathbb{R}}^N)\cap L^\infty({\mathbb{R}}^N) $, 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.

Keywords

• exponential nonlinearity
• time-space fractional diffusion
• caputo derivative
• global well-posedness