Classification of blow-ups and monotonicity formula for half-Laplacian nonlinear heat equation

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