Scattering and blow-up dichotomy of the energy-critical nonlinear Schr\"odinger equation with the inverse-square potential

TL;DR

This paper studies the long-term behavior of radial solutions to the energy-critical nonlinear Schrödinger equation with a repulsive inverse-square potential, focusing on scattering and blow-up phenomena for specific initial data.

Contribution

It provides a detailed analysis of the scattering and blow-up dichotomy for solutions with initial energy matching the static solution's energy.

Findings

Characterizes conditions for scattering and blow-up.

Identifies the role of the inverse-square potential in solution dynamics.

Analyzes radial solutions with critical initial energy.

Abstract

In this paper, we consider the energy critical nonlinear Schr\"odinger equation with a repulsive inverse square potential. In particular, we deal with radial initial data, whose energy is equal to the energy of static solution to the corresponding nonlinear Schr\"odinger equation without a potential. We investigate time behavior of the radial solutions with such initial data.