Sharp scattering for focusing intercritical NLS on high-dimensional waveguide manifolds

TL;DR

This paper extends the understanding of the focusing intercritical nonlinear Schrödinger equation on high-dimensional waveguide manifolds, establishing sharp scattering thresholds using adapted interaction Morawetz estimates.

Contribution

It introduces a novel adaptation of interaction Morawetz estimates to high-dimensional waveguides, enabling the proof of large data scattering for dimensions d ≥ 5.

Findings

Established sharp scattering thresholds characterized by ground states.

Proved large data scattering for all dimensions d ≥ 5.

Unified the scattering theory for focusing and defocusing cases.

Abstract

We study the focusing intercritical NLS \begin{align}\label{abstract_nls} i\pt_t u+\Delta_{x,y}u=-|u|^\alpha u\tag{NLS} \end{align} on the semiperiodic waveguide manifold $\R^d_x\times \T_y$ with $d\geq 5$ and $\alpha\in(\frac{4}{d},\frac{4}{d-1})$. In the case $d\leq 4$, with the aid of the semivirial vanishing theory \cite{Luo_inter}, the author was able to construct a sharp threshold, which being uniquely characterized by the ground state solutions, that sharply determines the bifurcation of global scattering and finite time blow-up solutions in dependence of the sign of the semivirial functional. As the derivative of the nonlinear potential is no longer Lipschitz in $d\geq 5$ and the underlying domain possesses an anisotropic nature, the proof in \cite{Luo_inter}, which makes use of the concentration compactness principle, can not be extended to higher dimensional models. In this paper, we exploit a well-tailored adaptation of the interaction Morawetz-Dodson-Murphy (IMDM) estimates, which were only known to be applicable on Euclidean spaces, into the waveguide setting, in order to prove that the large data scattering result formulated in \cite{Luo_inter} continues to hold for all $d\geq 5$. Together with Tzvetkov-Visciglia \cite{TzvetkovVisciglia2016} and the author \cite{Luo_inter}, we thus give a complete characterization of the large data scattering for \eqref{abstract_nls} in both defocusing and focusing case and in arbitrary dimension.