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, \;00 $, 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}} \mathbb{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)} $. We also 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}} $.