Blow-up and instability of standing waves for the NLS with a point interaction in dimension two

TL;DR

This paper investigates the blow-up behavior and instability of standing waves in a two-dimensional nonlinear Schrödinger equation with a point interaction, especially in the supercritical regime, revealing conditions for blow-up and strong instability.

Contribution

It introduces a nonstandard virial formula and demonstrates blow-up and strong instability of ground state standing waves in the supercritical regime.

Findings

Existence of initial data leading to blow-up.

Strong instability of ground state standing waves for high frequencies.

Development of a nonstandard virial formula for analysis.

Abstract

In the present note we study the NLS equation in dimension two with a point interaction and in the supercritical regime, showing two results. After obtaining the (nonstandard) virial formula, we exhibit a set of initial data that blow-up. Moreover we show the standing waves $e^{i\omega t} \varphi_\omega$ corresponding to ground states $\varphi_\omega$ of the action are strongly unstable, at least for sufficiently high $\omega$.