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arXiv · 1810.09086

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

Luccas Campos·Mykael Cardoso
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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.

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Luccas Campos, Mykael Cardoso. 2018-10-22. On the critical norm concentration for the inhomogeneous nonlinear Schr\"odinger equation. https://arxiv.org/abs/1810.09086

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