Blow-up for a fully fractional heat equation

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