Global existence for a coupled system of Schrödinger equations with power-type nonlinearities

Nghiem V. Nguyen, Rushun Tian, Bernard Deconinck, Natalie Sheils

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33 Scopus citations

Abstract

In this manuscript, we consider the Cauchy problem for a Schrödinger system with power-type nonlinearities where for uj uj: RN × R → C, Ψj0: RN → C for j = 1, 2, ... m and ajk = akjare 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 languageEnglish (US)
Article number011503
JournalJournal of Mathematical Physics
Volume54
Issue number1
DOIs
StatePublished - Jan 22 2013

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