Scattering For three waves Nonlinear Schr\"odinger System with mass-resonance in 5D

In this paper, we study the dynamics behavior of the NLS system with three waves interaction in the energy space $H^1(\mathbb{R}^5) \times H^1(\mathbb{R}^5)\times H^1(\mathbb{R}^5)$. Inspired by B. Dodson and J. Murphy in \cite{Dodson2018}, we establish an interaction Morawetz estimate for the NLS system, together with the criterion which proved by Tao-Dodson--Murphy we can get the scattering under the ground state in energy space with mass-resonance. Under the radial assumption, we can remove the mass-resonance condition.