Existence of solutions to a nonlinear fractional diffusion equation with exponential growth

Abstract

In this paper, we study a Cauchy problem for a space–time fractional diffusion equation with exponential nonlinearity. Based on the standard L_p-L_q estimates of strongly continuous semigroup generated by fractional Laplace operator, we investigate the existence of global solutions for initial data with small norm in Orlicz space exp L^p(R_d) and a time weighted L^r(R_d) space. In the framework of the Hölder interpolation inequality, we also discuss the existence of local solutions without the Orlicz space.