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

Abstract

We consider the nonlinear half laplacian heat equation $$u_t+(-\Delta {)}^{\frac{1}{2}}u-|u{|}^{p-1}u=0,{\mathbb{R}}^n\times (0,T).$$ We prove that all blows-up are type I, provided that $n\le 4$ and $1<p<p_{\ast }(n)$ where $p_{\ast }(n)$ 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.

33 pages; comments welcome