The focusing cubic NLS on exterior domains in three dimensions

TL;DR

This paper proves that the threshold for global existence and scattering of the focusing cubic nonlinear Schrödinger equation in three-dimensional exterior domains matches that of the Euclidean space, extending known results to exterior domains.

Contribution

It establishes the same sharp threshold for global existence and scattering in exterior domains as in Euclidean space for the focusing cubic NLS.

Findings

Global existence and scattering hold under the same threshold as in Euclidean space.

Solutions with energy and mass below the ground state threshold exist globally and scatter.

The results extend the understanding of NLS behavior to exterior convex domains.

Abstract

We consider the focusing cubic NLS in the exterior $\Omega$ of a smooth, compact, strictly convex obstacle in three dimensions. We prove that the threshold for global existence and scattering is the same as for the problem posed on Euclidean space. Specifically, we prove that if $E(u_0)M(u_0)<E(Q)M(Q)$ and $\|\nabla u_0\|_2\|u_0\|_2<\|\nabla Q\|_2\|Q\|_2$, the corresponding solution to the initial-value problem with Dirichlet boundary conditions exists globally and scatters to linear evolutions asymptotically in the future and in the past. Here, $Q(x)$ denotes the ground state for the focusing cubic NLS in $\mathbb{R}^3$.