Standing Ring Blow up Solutions to the N-Dimensional Quintic Nonlinear Schrödinger Equation

@article{Raphal2009StandingRB,
  title={Standing Ring Blow up Solutions to the N-Dimensional Quintic Nonlinear Schr{\"o}dinger Equation},
  author={Pierre Rapha{\"e}l and J{\'e}r{\'e}mie Szeftel},
  journal={Communications in Mathematical Physics},
  year={2009},
  volume={290},
  pages={973-996}
}
We consider the quintic nonlinear Schrödinger equation $${i\partial_tu=-\Delta u-|u|^{4}u}$$ in dimension N ≥ 3. This problem is energy critical in dimension N = 3 and energy super critical for N ≥ 4. We prove the existence of a radially symmetric blow up mechanism with L2 concentration along the unit sphere of $${\mathbb{R}^{N}}$$. This singularity formation is moreover stable by smooth and radially symmetric perturbation of the initial data. This result extends the result obtained for N = 2… CONTINUE READING

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