On the critical norm concentration for the inhomogeneous nonlinear Schr\"odinger equation

TL;DR

This paper investigates the concentration phenomena of solutions to the inhomogeneous nonlinear Schrödinger equation at blow-up, extending previous classifications and analyzing critical norm concentration in various regimes.

Contribution

It provides new results on norm concentration at blow-up for INLS and offers an alternative classification of minimal mass blow-up solutions.

Findings

Proves $L^2$-norm concentration for finite time blow-up in the $L^2$-critical case.

Classifies minimal mass blow-up solutions using an alternative approach.

Extends norm concentration results to $L^p$-critical regimes.

Abstract

We consider the inhomogeneous nonlinear Schr\"odiger equation (INLS) in $\mathbb{R}^N$ $$i \partial u_t + \Delta u + |x|^{-b} |u|^{2\sigma}u = 0,$$ and show the $L^2$-norm concentration for the finite time blow-up solutions in the $L^2$-critical case, $\sigma=\frac{2-b}{N}$. Moreover, we provide an alternative for the classification of minimal mass blow-up solutions first proved by Genoud and Combet [4]. For the case $\frac{2-b}{N} < \sigma < \frac{2-b}{N-2}$, we show results regarding the $L^p$-critical norm concentration, generalizing the argument of Holmer and Roudenko [16] to the INLS setting.