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 $\alpha$-modulation spaces $M_{2,1}^{s,\alpha }$ ($\alpha \in [0,1),s>d\alpha /2-\alpha /\kappa$, $\kappa \in \mathbb{N}$ and $\kappa \ge 2/d$) for the sufficiently small data. Moreover, NLS is ill-posed in $M_{2,1}^{s,\alpha }$ if $s<d\alpha /2-\alpha /\kappa$. 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 $M_{2,1}^{s,\alpha }$ if $\kappa >2/d,\alpha \in [0,1)d\alpha /2-\alpha /\kappa <s<s(\kappa ):=d/2-1/\kappa$. Such a kind of data are super-critical in $H^{s(\kappa )}$ and have infinite amplitude.

34 Pages

References in corpus (3)

• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
• Well-posedness and scattering for the KP-II equation in a critical space
• Large global solutions for energy supercritical nonlinear wave equations on \R^{3+1}