Global dynamics below the ground state for the quadratic Sch\"odinger system in 5d

In this paper we consider the nonlinear Schr"odinger 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 Kenig Merle [16]. 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.