Blow-up construction and instability for mass-critical half-wave equation with slightly superthreshold mass

TL;DR

This paper constructs and analyzes finite-time blow-up solutions for the mass-critical half-wave equation with slightly superthreshold mass, extending known results from local NLS models to a nonlocal dispersive PDE.

Contribution

It introduces the first finite-time blow-up solutions with superthreshold mass for the half-wave equation and studies their asymptotic behavior and instability, overcoming nonlocal challenges.

Findings

Constructed finite-time blow-up solutions with superthreshold mass.

Described the asymptotic behavior and modulation dynamics near blow-up.

Demonstrated the instability of these solutions by nearby non-blow-up solutions.

Abstract

We study the blow-up dynamics for the $L^2$-critical focusing half-wave equation on the real line, a nonlocal dispersive PDE arising in various physical models. As in other mass-critical models, the ground state solution becomes a threshold between the global well-posedness and the existence of a blow-up. The first blow-up construction is due to Krieger, Lenzmann and Rapha\"el, in which they constructed the minimal mass blow-up solution at the threshold mass. In this paper, we construct finite-time blow-up solutions with mass slightly exceeding the threshold. This is inspired by similar results in the mass-critical NLS by Bourgain and Wang, and their instability by Merle, Rapha\"el and Szeftel. We exhibit a blow-up profile driven by the rescaled ground state, with a decoupled dispersive radiation component. We rigorously describe the asymptotic behavior of such solutions near the blow-up time, including sharp modulation dynamics. Furthermore, we demonstrate the instability of these solutions by constructing non-blow-up solutions that are arbitrarily close to the blow-up solutions. The main contribution of this work is to overcome the nonlocal setting of half-wave and to extend insights from the mass-critical NLS to a setting lacking pseudo-conformal symmetry.