The dynamics of the 3D radial NLS with the combined terms

TL;DR

This paper investigates the scattering and blow-up behavior of radial solutions to a 3D nonlinear Schrödinger equation with combined terms, establishing thresholds for energy below which solutions exhibit specific dynamics.

Contribution

It introduces a new radial profile decomposition accounting for scaling, addressing the lack of invariance, and shows the perturbation does not alter the scattering threshold.

Findings

Established scattering and blow-up results below energy threshold

Developed a new radial profile decomposition with scaling parameter

Perturbation does not affect the scattering threshold in energy space

Abstract

In this paper, we show the scattering and blow-up result of the radial solution with the energy below the threshold for the nonlinear Schr\"{o}dinger equation (NLS) with the combined terms iu_t + \Delta u = -|u|^4u + |u|^2u \tag{CNLS} in the energy space $H^1(\R^3)$. The threshold is given by the ground state $W$ for the energy-critical NLS: $iu_t + \Delta u = -|u|^4u$. This problem was proposed by Tao, Visan and Zhang in \cite{TaoVZ:NLS:combined}. The main difficulty is the lack of the scaling invariance. Illuminated by \cite{IbrMN:f:NLKG}, we need give the new radial profile decomposition with the scaling parameter, then apply it into the scattering theory. Our result shows that the defocusing, $\dot H^1$-subcritical perturbation $|u|^2u$ does not affect the determination of the threshold of the scattering solution of (CNLS) in the energy space.