Blow-up or Grow-up for the threshold solutions to the nonlinear Schr\"{o}dinger equation

TL;DR

This paper investigates the long-term behavior of solutions to a nonlinear Schrödinger equation, establishing conditions under which solutions blow up or grow up without requiring finite variance.

Contribution

It extends previous results by removing the finite-variance assumption, proving blow-up or grow-up for solutions with the same mass and energy as the ground state.

Findings

Proves blow-up or grow-up without finite variance assumption.

Extends previous blow-up results to broader class of solutions.

Provides a comprehensive understanding of solution dynamics in the specified regime.

Abstract

We consider the nonlinear Schr\"{o}dinger equation with $L^{2}$-supercritical and $H^{1}$-subcritical power type nonlinearity. Duyckaerts and Roudenko and Campos, Farah, and Roudenko studied the global dynamics of the solutions with same mass and energy as that of the ground state. In these papers, finite variance is assumed to show the finite time blow-up. In the present paper, we remove the finite-variance assumption and prove a blow-up or grow-up result.