Blow-up for a fully fractional heat equation

We study the existence and behaviour of blowing-up solutions to the fully fractional heat equation $$\mathcal{M} u=u^p,\qquad x\in\mathbb{R}^N,\;0<t<T$$ with $p>0$, where $\mathcal{M}$ is a nonlocal operator given by a space-time kernel $M(x,t)=c_{N,\sigma}t^{-\frac N2-1-\sigma}e^{-\frac{|x|^2}{4t}}{1}_{\{t>0\}}$, $0<\sigma<1$. This operator coincides with the fractional power of the heat operator, $\mathcal{M}=(\partial_t-\Delta)^{\sigma}$ defined through semigroup theory. We characterize the global existence exponent $p_0=1$ and the Fujita exponent $p_*=1+\frac{2\sigma}{N+2(1-\sigma)}$, and study the rate at which the blowing-up solutions below $p_*$ tend to infinity, $\|u(\cdot,t)\|_\infty\sim (T-t)^{-\frac\sigma{p-1}}$.