Threshold dynamics for the 4$d$ mass-energy double critical NLS

TL;DR

This paper extends the understanding of the 4$d$ mass-energy double critical nonlinear Schrödinger equation by proving that the scattering/blowup dichotomy remains valid at the energy threshold, building on prior energy constraint results.

Contribution

It demonstrates that the scattering/blowup dichotomy persists at the energy threshold, advancing the analysis of critical NLS solutions.

Findings

The dichotomy holds at the energy threshold $E(u_0)=E^c(W)$.

Previous results were limited to $E(u_0)< E^c(W)$, now extended to equality.

The result confirms the robustness of the scattering/blowup behavior at critical energy levels.

Abstract

We consider the 4$d$ mass-energy double critical NLS \[ (i\partial_t+\Delta)u = -|u|^2 u + |u| u. \] In Luo (2024) and Cheng--Miao--Zhao (2016), the authors established a scattering/blowup dichotomy for solutions satisfying the energy constraint $E(u_0)< E^c(W)$, where $W$ is the energy-critical NLS ground state and $E^c$ is the energy for the underlying cubic NLS. We prove that the scattering/blowup dichotomy persists even at the energy threshold $E(u_0)=E^c(W)$.