On the supercritical Schr\"{o}dinger equation on the exterior of a ball

TL;DR

This paper proves global existence, bounded Sobolev norms, and uniqueness for the defocusing supercritical Schrödinger equation on the exterior of a ball, for small non-radial perturbations of large radial initial data.

Contribution

It establishes global well-posedness and norm control for supercritical nonlinear Schrödinger equations with non-radial initial data outside a ball.

Findings

Solutions exist globally for all times.

Sobolev norms remain bounded.

Unique solutions in the energy class.

Abstract

We consider the mixed problem on the exterior of the unit ball in $\mathbb{R}^{n}$, $n\ge2$, for a defocusing Schr\"{o}dinger equation with a power nonlinearity $|u|^{p-1}u$, with zero boundary data. Assuming that the initial data are non radial, sufficiently small perturbations of \emph{large} radial initial data, we prove that for all powers $p>n+6$ the solution exists for all times, its Sobolev norms do not inflate, and the solution is unique in the energy class.