Blow-up solutions of fractional diffusion equations with an exponential nonlinearity

The goal of this work is to investigate finite-time blow-up solutions to a class of time-space fractional diffusion equations with nonlinear exponential source terms. In contrast to the small critical data case, which leads to global solutions, we prove in this study that if the initial Schwartz data is large enough, our solutions will blow up in a finite time. The main idea of the analysis is based on the Fourier analytic approach and embeddings between Triebel-Lizorkin spaces and Besov spaces.