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

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

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Acknowledgements

The authors are grateful to their friend and colleague Claudio Cacciapuoti for useful discussions.

Funding

D. N. acknowledges the support of EC grant IPaDEGAN (MSCA-RISE-778010), and D.F and D.N acknowledge the support of the Gruppo Nazionale di Fisica Matematica (GNFM-INdAM).

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Authors and Affiliations

  1. Facoltà di Ingegneria, Università Telematica Internazionale Uninettuno, Corso Vittorio Emanuele II 39, 00186, Roma, Italy

    Domenico Finco

  2. Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via Roberto Cozzi 55, 20126, Milano, Italy

    Diego Noja

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  1. Domenico Finco
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  2. Diego Noja
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Both DF and DN wrote and reviewed the manuscript.

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Correspondence to Domenico Finco.

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Finco, D., Noja, D. Blow-up and instability of standing waves for the NLS with a point interaction in dimension two. Z. Angew. Math. Phys. 74, 162 (2023). https://doi.org/10.1007/s00033-023-02056-z

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  • DOI: https://doi.org/10.1007/s00033-023-02056-z

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