Standing Ring Blow up Solutions to the N-Dimensional Quintic Nonlinear Schr�dinger Equation

Rapha�l, Pierre; Szeftel, J�r�mie
September 2009
Communications in Mathematical Physics;Sep2009, Vol. 290 Issue 3, p973
Academic Journal
We consider the quintic nonlinear Schr�dinger 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 in [29] and is the first result of description of a singularity formation in the energy supercritical class for (NLS) type problems. Our main tool is the proof of the propagation of regularity outside the blow up sphere in the presence a so-called log-log type singularity.


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