📝 SummarXiv

Abstract

We consider the focusing energy-subcritical Schr\"odinger equations. In earlier works by Holmer-Roudenko \cite{holmer}, Duyckaerts-Holmer-Roudenko \cite{duyckaerts2}, 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 $M(u_0)^{1-s_c}E(u_0)^{s_c}<M(Q)^{1-s_c}E(Q)^{s_c}$ and $\| u\|^{1-s_c}_{L^2}\| u\|^{s_c}_{\dot{H}^1}<\| Q\|^{1-s_c}_{L^2}\|Q\|^{s_c}_{\dot{H}^1}$, where $Q$ is the ground state. In this paper, we establish asymptotic estimates for the upper bound of the scattering norms as $M(u_0)^{1-s_c}E(u_0)^{s_c}$ approaches the threshold mass-energy threshold $M(Q)^{1-s_c}E(Q)^{s_c}$, which generalizes the work of Duyckaerts-Merle \cite{duyckaerts} on the energy-critical Schr\"odinger equation($s_c=1$).