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Global existence for a coupled system of Schrödinger equations with power-type nonlinearities. (English) Zbl 1286.35230
The authors consider the Cauchy problem for a Schrödinger system with power-type nonlinearities. The 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, the global existence for small initial data is shown for the critical case \(p=1+2/N\). Finite-time blow-up, as well as the stability of solutions in the critical case, is discussed.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B44 Blow-up in context of PDEs
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