Stable self similar blow up dynamics for slightly L^2 supercritical NLS equations

TL;DR

This paper constructs and analyzes stable self-similar blow-up solutions for slightly supercritical focusing nonlinear Schrödinger equations in low dimensions, providing insights into singularity formation.

Contribution

It demonstrates the existence and stability of self-similar blow-up solutions for slightly supercritical NLS equations in dimensions 1 to 5, advancing understanding of singularity dynamics.

Findings

Existence of stable self-similar blow-up solutions

Qualitative description of singularity formation

Stability in the energy space $H^1$

Abstract

We consider the focusing nonlinear Schr\"odinger equations $i\partial_t u+\Delta u +u|u|^{p-1}=0$ in dimension $1\leq N\leq 5$ and for slightly $L^2$ supercritical nonlinearities $p_c<p<(1+\e)p_c$ with $p_c=1+\frac{4}{N}$ and $0<\e\ll 1$. We prove the existence and stability in the energy space $H^1$ of a self similar finite time blow up dynamics and provide a qualitative description of the singularity formation near the blow up time