On 2D NLS on non-trapping exterior domains

TL;DR

This paper extends global existence and scattering results for the 2D nonlinear Schrödinger equation to more general non-trapping exterior domains and nonlinearities with powers greater than quartic, broadening previous known cases.

Contribution

It generalizes global existence to all non-trapping obstacles and proves scattering for almost star-shaped obstacles with nonlinearities above quartic power.

Findings

Global existence for all non-trapping obstacles.

Scattering established for almost star-shaped obstacles.

Applicable to nonlinearities with power greater than quartic.

Abstract

Global existence and scattering for the nonlinear defocusing Schr\"odinger equation in 2 dimensions are known for domains exterior to star-shaped obstacles and for nonlinearities that grow at least as the quintic power. In this paper, we extend the global existence result for all non-trapping obstacles and for nonlinearities with power strictly greater than quartic. For such nonlinearities, we also prove scattering for a class of so-called almost star-shaped obstacles.