## Duke Mathematical Journal

### Existence and stability of a solution blowing up on a sphere for an $L^2$-supercritical nonlinear Schrödinger equation

Pierre Raphaël

#### Abstract

We consider the quintic two-dimensional focusing nonlinear Schrödinger equation $iu_t=-\Delta u-|u|^{4}u$ which is $L^2$-supercritical. Even though the existence of finite-time blow-up solutions in the energy space $H^1$ 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 $H^1$-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 $L^2$-supercritical setting

#### Article information

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

Dates
First available in Project Euclid: 8 August 2006

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1155045502

Digital Object Identifier
doi:10.1215/S0012-7094-06-13421-X

Mathematical Reviews number (MathSciNet)
MR2248831

Zentralblatt MATH identifier
1117.35077

#### Citation

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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