Blowup dynamics for Mass Critical Half-wave equation in 3D

TL;DR

This paper studies the blowup behavior of solutions to the mass-critical half-wave equation in three dimensions, constructing minimal mass blowup solutions with a specific blowup rate for radially symmetric initial data.

Contribution

It introduces a new class of minimal mass blowup solutions for the 3D mass-critical half-wave equation with explicit blowup rate characterization.

Findings

Constructed minimal mass blowup solutions with rate |t|^{-1/4}

Established blowup behavior for radially symmetric initial data

Contributed to understanding singularity formation in nonlocal dispersive equations

Abstract

We consider the half-wave equation $i u_t=Du-|u|^{\frac{2}{3}}u$ in three dimension and in the mass critical. For initial data $u(t_0,x)=u_0(x)\in H^{1/2}_{rad}(\mathbb{R}^3)$ with radial symmetry, we construct a new class of minimal mass blowup solutions with the blow up rate $\|D^{\frac{1}{2}}u(t)\|_2\sim\frac{C(u_0)}{|t|^{\frac{1}{4}}}$ as $t\rightarrow0^-$.