Scattering norm estimate near the threshold for the energy-subcritical NLS

We consider the focusing energy-subcritical Schr\"odinger equations. In earlier works by Holmer-Roudenko \cite{holmer}, Duyckaerts-Holmer-Roudenko \cite{duyckaerts2}, Akahori-Nawa \cite{akahori}, Fang-Xie-Cazenave \cite{fang}, Guevara \cite{guevara} and later by Dodson-Murphy \cite{dodson1,dodson2} and Arora-Dodson-Murphy \cite{arora}, they proved that scattering is the only dynamical behavior if the $H^1$ initial data satisfies $M(u_0)^{(1-s_c)/s_c}E(u_0)<M(Q)^{(1-s_c)/s_c}E(Q)$ and $\| u\|^{(1-s_c)/s_c}_{L^2}\| u\|_{\dot{H}^1}<\| Q\|^{(1-s_c)/s_c}_{L^2}\|Q\|_{\dot{H}^1}$, where $Q$ is the ground state. In this paper, we establish asymptotic estimates for the upper bound of the scattering norms as $M(u_0)^{(1-s_c)/s_c}E(u_0)$ approaches the threshold mass-energy threshold $M(Q)^{(1-s_c)/s_c}E(Q)$, which generalizes the work of Duyckaerts-Merle \cite{duyckaerts} on the energy-critical Schr\"odinger equation($s_c=1$).