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

Frank Merle; Pierre Raphael; Jeremie Szeftel

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

Analysis of PDEs 2009-07-24 v1

Authors: Frank Merle , Pierre Raphael , Jeremie Szeftel

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

Keywords

nonlinear Schrödinger equation self-similar blow up L^2 supercritical stability singularity formation

Cite

@article{arxiv.0907.4098,
  title  = {Stable self similar blow up dynamics for slightly L^2 supercritical NLS equations},
  author = {Frank Merle and Pierre Raphael and Jeremie Szeftel},
  journal= {arXiv preprint arXiv:0907.4098},
  year   = {2009}
}

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