Classification of Blow-ups and Monotonicity Formula for Half Laplacian Nonlinear Heat Equation

We consider the nonlinear half laplacian heat equation $$u_t+(-\Delta)^{\frac{1}{2}} u-|u|^{p-1}u=0,\quad \mathbb{R}^n\times (0, T).$$ We prove that all blows-up are type I, provided that $n \leq 4$ and $1<p<p_{*} (n)$ where $p_{*} (n)$ is an explicit exponent which is below $\frac{n+1}{n-1}$, the critical Sobolev exponent. Central to our proof is a Giga-Kohn type monotonicity formula for half laplacian and a Liouville type theorem for self-similar nonlinear heat equation. This is the first instance of a monotonicity formula at the level of the nonlocal equation, without invoking the extension to the half-space.