Blowup dynamics for inhomogeneous mass critical half-wave equation

TL;DR

This paper investigates the blowup behavior of solutions to the inhomogeneous mass critical half-wave equation in one dimension, establishing existence and rate of blowup for solutions with ground state mass.

Contribution

It demonstrates the existence of radial blowup solutions with ground state mass and determines their blowup rate in the inhomogeneous mass critical half-wave equation.

Findings

Existence of radial blowup solutions with ground state mass.

Blowup rate of solutions is proportional to 1/|t| as t approaches zero.

Characterization of blowup dynamics in the inhomogeneous setting.

Abstract

We consider the focusing inhomogeneous mass critical half-wave equation in one dimension. Under the mild conditions of the inhomogeneous factor, we show that the existence of the radial blowup solutions with ground state mass $\|u_0\|_2=\|Q\|_2$, where $Q$ is the unique positive ground state solution of equation $DQ+Q=Q^3$ and obtain the blowup rate $\|D^{\frac{1}{2}}u(t)\|_{L^2}\sim\frac{1}{|t|}$ as $t\nearrow 0^-$.