Duke Mathematical Journal

Existence and stability of a solution blowing up on a sphere for an L2-supercritical nonlinear Schrödinger equation

Pierre Raphaël

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We consider the quintic two-dimensional focusing nonlinear Schrödinger equation iut=Δu|u|4u which is L2-supercritical. Even though the existence of finite-time blow-up solutions in the energy space H1 is known, very little is understood concerning the singularity formation. Numerics suggest the existence of a stable blow-up dynamic corresponding to a self-similar blowup at one point in space. We prove the existence of a different type of dynamic and exhibit an open set among the H1-radial distributions of initial data for which the corresponding solution blows up in finite time on a sphere. This is the first result of an explicit description of a blow-up dynamic in the L2-supercritical setting

Article information

Duke Math. J., Volume 134, Number 2 (2006), 199-258.

First available in Project Euclid: 8 August 2006

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Zentralblatt MATH identifier

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 35Q51: Soliton-like equations [See also 37K40] 35B05: Oscillation, zeros of solutions, mean value theorems, etc.


Raphaël, Pierre. Existence and stability of a solution blowing up on a sphere for an $L^2$ -supercritical nonlinear Schrödinger equation. Duke Math. J. 134 (2006), no. 2, 199--258. doi:10.1215/S0012-7094-06-13421-X. https://projecteuclid.org/euclid.dmj/1155045502

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