In the present note, we study the focusing NLS equation in dimension two with a point interaction in the supercritical regime, showing two results. After obtaining the (nonstandard) virial formula, we exhibit a set of initial data that shows blow-up. Moreover, we show that the standing waves eiωtφω corresponding to ground states φω of the action functional are strongly unstable, at least for sufficiently high ω .
Finco, D., Noja, D. (2023). Blow-up and instability of standing waves for the NLS with a point interaction in dimension two. ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK, 74(4), 1-17 [10.1007/s00033-023-02056-z].
Blow-up and instability of standing waves for the NLS with a point interaction in dimension two
Noja D.
2023
Abstract
In the present note, we study the focusing NLS equation in dimension two with a point interaction in the supercritical regime, showing two results. After obtaining the (nonstandard) virial formula, we exhibit a set of initial data that shows blow-up. Moreover, we show that the standing waves eiωtφω corresponding to ground states φω of the action functional are strongly unstable, at least for sufficiently high ω .
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