Global Well-Posedness for NLS with a Class of $H^s$-Supercritical Data

Abstract

We study the Cauchy problem for NLS with a class of $H^s$-super-critical data \begin{align} & {\rm i}u_t +Δu+ λ|u|^{2κ} u =0, \quad u(0)=u_0 \label{NLSabstract} \end{align} and show that \eqref{NLSabstract} is globally well-posed and scattering in $α$-modulation spaces $M^{s,α}_{2,1}$ ($α\in [0,1), \ s> dα/2-α/κ$, $κ\in \mathbb{N}$ and $κ\geq 2/d$) for the sufficiently small data. Moreover, NLS is ill-posed in $M^{s,α}_{2,1}$ if $s< dα/2-α/κ$. In particular, we obtain a class of initial data $u_0$ satisfying for any $M\gg 1$, \begin{align} \|u_0\|_2 \sim M^{1/κ-d/2 }, \ \ \|u_0\|_\infty \ =\infty , \ \ \|u_0\|_{M^{s,α}_{2,1}} \geq M^{(1-α)/κ}, \ \ \ \|u_0\|_{B^{s(κ)}_{2,\infty}} =\infty \nonumber \end{align} such that NLS is globally well-posed in $M^{s,α}_{2,1}$ if $κ>2/d, \ α\in [0,1)\ dα/2-α/κ<s < s(κ):= d/2-1/κ$. Such a kind of data are super-critical in $H^{s(κ)}$ and have infinite amplitude.