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

We consider the semilinear fully fractional heat equation $$({\mathrm{\partial }}_t-\mathrm{\Delta }{)}^{\sigma }u=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u\text{in }{\mathbb{R}}^n\times {\mathbb{R}}_{-},0<\sigma <1.$$ For $n\le 2\sigma$ or $1<p\le \frac{n+2\sigma }{n-2\sigma }$, 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.