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

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 $e^{i\omega t}{\phi }_{\omega }$ corresponding to ground states ${\phi }_{\omega }$ of the action are strongly unstable, at least for sufficiently high $\omega$.

12 pages, minor modifications

References in corpus (1)

• Ground states for the planar NLSE with a point defect as minimizers of the constrained energy

Cited by in corpus (7)

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• Two dimensional NLS ground states with attractive Coulomb potential and point interaction
• Local well-posedness and blow-up in the energy space for the 2D NLS with point interaction
• Normalized NLS ground states on a double plane hybrid