Local existence and nonexistence for fractional semilinear parabolic systems with singular initial data

We study the fractional semilinear parabolic system$ \begin{cases} \partial^\alpha_t u+(-\Delta)^{\frac{\theta_1}{2}}u = |v|^{\frac{\theta_2}{N}}v & \text{in}\ {\mathbb{R}^N}\times(0,T),\\ \partial^\alpha_t v+(-\Delta)^{\frac{\theta_2}{2}}v = |u|^{\frac{\theta_1}{N}}u & \text{in}\ {\mathbb{R}^N}\times(0,T),\\ u(x,0) = u_0 (x), v(x,0) = v_0 (x) & \text{in}\ {\mathbb{R}^N}, \end{cases} $where $ 0<\alpha\le1 $, $ 0<\theta_1, \theta_2\le2 $, $ N\ge1 $, and $ T>0 $. When $ 0<\alpha<1 $, the fractional derivative $ \partial^{\alpha}_t $ is meant in a generalized Caputo sense. We mainly consider the case where $ u_0 $ and $ v_0 $ have a singularity. We obtain integrability conditions on $ (u_0,v_0) $ which explicitly determine local in time existence/nonexistence of a solution.