## Abstract

In this manuscript, we consider the Cauchy problem for a Schrödinger system with power-type nonlinearities where for u_{j} u_{j}: R^{N} × R → C, Ψj0: R^{N} → C for j = 1, 2, ... m and a_{jk} = a_{kj}are positive real numbers. Global existence for the Cauchy problem is established for a certain range of p. A sharp form of a vector-valued Gagliardo-Nirenberg inequality is deduced, which yields the minimal embedding constant for the inequality. Using this minimal embedding constant, global existence for small initial data is shown for the critical case p = 1 + 2/N. Finite-time blow-up, as well as stability of solutions in the critical case, is discussed.

Original language | English (US) |
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Article number | 011503 |

Journal | Journal of Mathematical Physics |

Volume | 54 |

Issue number | 1 |

DOIs | |

State | Published - Jan 22 2013 |

### Bibliographical note

Funding Information:Bernard Deconinck and Natalie Sheils acknowledge support from the National Science Foundation (Grant No. NSF-DMS-1008001). Natalie Sheils also acknowledges this material is based upon work supported by the National Science Foundation (Grant No. NSF-DGE-0718124). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding sources.