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- Title
Standing Ring Blow up Solutions to the N-Dimensional Quintic Nonlinear Schrödinger Equation.
- Authors
Raphaël, Pierre; Szeftel, Jérémie
- Abstract
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 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.
- Subjects
QUINTIC equations; SCHRODINGER equation; PERTURBATION theory; DATA; PARTIAL differential equations
- Publication
Communications in Mathematical Physics, 2009, Vol 290, Issue 3, p973
- ISSN
0010-3616
- Publication type
Article
- DOI
10.1007/s00220-009-0796-2