Giga-Kohn-type results for the fully fractional heat equation

We consider the semilinear fully fractional heat equation $$(\partial_t-\Delta)^\sigma u = |u|^{p-1}u \quad \text{in } \mathbb{R}^n \times \mathbb{R}_{-}, \qquad 0 < \sigma < 1.$$ For $n\leq 2\sigma$ or $1<p\leq \frac{n+2\sigma}{n-2\sigma}$, we generalize the monotonicity formula and Liouville-type theorem when $\sigma=1$ proved by Giga and Kohn. In order to overcome the difficulty that this equation is nonlocal, we give a new interpretation of the classical Giga-Kohn's Pohozaev identity in terms of Hermite expansion. This insight is new and interesting even for $\sigma=1$. We further establish a space-time nonlocal monotonicity formula for the self-similar equation. As far as we are concerned, this is the first monotonicity formula for space-time nonlocal equations without using an extension by Stinga and Torrea.