Blow-up solutions for non-scale-invariant nonlinear Schr\"odinger equation in one dimension

TL;DR

This paper extends the understanding of blow-up solutions in the one-dimensional mass-critical nonlinear Schrödinger equation, especially when a linear potential is present, by modifying existing proof techniques.

Contribution

It introduces a modified proof method to establish blow-up solutions for the nonlinear Schrödinger equation with a linear potential, beyond previous scaling-based approaches.

Findings

Proved blow-up solutions exist with linear potential

Extended blow-up results to non-scale-invariant cases

Modified proof technique applicable to linear potential scenarios

Abstract

In this paper, we consider the mass-critical nonlinear Schr\"odinger equation in one dimension. Ogawa--Tsutsumi [19] proved a blow-up result for negative energy solution by using a scaling argument for initial data. By the reason, the method cannot be used to an equation with a linear potential. So, we modify the proof and get that for the equation with the linear potential.