Global dynamics below a threshold for the nonlinear Schr\"odinger equations with the Kirchhoff boundary and the repulsive Dirac delta boundary on a star graph

TL;DR

This paper investigates the global behavior of solutions to nonlinear Schrödinger equations on star graphs with specific boundary conditions, establishing a scattering-blowup dichotomy below the ground state energy.

Contribution

It introduces a linear profile decomposition tailored for star graphs using symmetrical decomposition, advancing analysis techniques for such boundary conditions.

Findings

Established scattering-blowup dichotomy below ground state energy.

Developed a linear profile decomposition for star graphs.

Applied concentration compactness and rigidity methods.

Abstract

We consider the nonlinear Schr\"odinger equations on the star graph with the Kirchhoff boundary and the repulsive Dirac delta boundary at the origin. In the present paper, we show the scattering-blowup dichotomy result below the mass-energy of the ground state on the real line. The proof of the scattering part is based on a concentration compactness and rigidity argument. Our main contribution is to give a linear profile decomposition on the star graph by using a symmetrical decomposition.