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

Abstract

We consider the focusing nonlinear Schrödinger equations $i{\partial }_{t}u+\Delta u+u|u{|}^{p-1}=0$ in dimension $1\le N\le 5$ 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