### Abstract

We consider the blow-up of the solution in H^{1} for the following nonlinear Schrödinger equation: i ∂ ∂tu + Δu = -|u|^{p - 1}u, x ε{lunate} R^{n}, t ≥ 0, (*) u(0, x) = u_{0}(X), x ε{lunate} R^{n}, t = 0, where n ≥2 and 1 + 4/n ≤ p < min { (n + 2) (n - 2), 5}. We prove that if the initial data u_{0} in H^{1} are radially symmetric and have negative energy, then the solution of (*) in H^{1} blows up in finite time. We do not assume that xu_{0} ε{lunate} L^{2}, and therefore our result is the generalization of the results of Glassey [4] and M. Tsutsumi [18] for the radially symmetric case.

Original language | English |
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Pages (from-to) | 317-330 |

Number of pages | 14 |

Journal | Journal of Differential Equations |

Volume | 92 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1991 Aug |

Externally published | Yes |

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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## Cite this

Ogawa, T., & Tsutsumi, Y. (1991). Blow-up of H

^{1}solution for the nonlinear Schrödinger equation.*Journal of Differential Equations*,*92*(2), 317-330. https://doi.org/10.1016/0022-0396(91)90052-B