Large global solutions for energy-critical nonlinear Schrödinger equation

Abstract

In this work, we consider the 3D defocusing energy-critical nonlinear Schrödinger equation $i{\partial }_{t}u+\Delta u=|u{|}^{4}u,(t,x)\in \mathbb{R}\times {\mathbb{R}}^3$. Applying the outgoing and incoming decomposition presented in the recent work \cite{BECEANU-DENG-SOFFER-WU-2021}, we prove that any radial function $f$ with ${\chi }_{\le 1}f\in H^1$ and ${\chi }_{\ge 1}f\in H^{s_0}$ with $\frac{5}{6}<s_0<1$, there exists an outgoing component $f_{+}$ (or incoming component $f_{-}$) of $f$, such that when the initial data is $f_{+}$, then the corresponding solution is globally well-posed and scatters forward in time; when the initial data is $f_{-}$, then the corresponding solution is globally well-posed and scatters backward in time.

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