In this paper we consider the nonlinear Schrödinger system (NLS) with quadratic interaction in five dimensions. We determine the global behavior of the solutions to the system with data below the ground state. Our proof of the scattering result is based on an argument by . In particular, the new part of this paper is to deal with asymmetric interaction. A blowing-up or growing-up result is proved by combining the argument by Du-Wu-Zhang in [6] and a variational characterization of minimizers. Moreover, we show a blowing-up result if the data has finite variance or is radial.
In this paper, we consider a nonlinear Schrödinger equation with a repulsive inversepower potential. It is known that the corresponding stationary problem has a "radial" ground state. Here, the "radial" ground state is a least energy solution among radial solutions to the stationary problem. We prove that if radial initial data below the "radial" ground state has positive virial functional, then the corresponding solution to the nonlinear Schrödinger equation scatters. In particular, we can treat not only short range potentials but also long range potentials. Contents 1. Introduction 1 2. Preliminaries 6 2.1. Notation and definition 6 2.2. Some tools 6 3. Parameter independence of the splitting below the "radial" ground state 8 4. Well-posedness 9 4.1. Local well-posedness 9 4.2. Small data theory 11 4.3. Stability 13 4.4. Final state problem 16 5. Linear profile decomposition 17 6. Scattering 21 Acknowledgements 28 References 29
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