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Local well-posedness and blow-up in the energy space for the 2D NLS with point interaction

Part of: Elliptic equations and systems Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  11 July 2025

Luigi Forcella Open the ORCID record for Luigi Forcella [Opens in a new window]  and
Vladimir Georgiev
Luigi Forcella*
Affiliation:
Dipartimento di Matematica, Università Degli Studi di Pisa, Largo Bruno Pontecorvo, 5, Pisa, Italy (luigi.forcella@unipi.it)
Vladimir Georgiev
Affiliation:
Dipartimento di Matematica, Università Degli Studi di Pisa, Largo Bruno Pontecorvo, 5, Pisa, Italy Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo, Japan Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. Georgi Bonchev Str., Block 8, Sofia, Bulgaria (vladimir.simeonov.gueorguiev@unipi.it)
*
*Corresponding author.
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Abstract

We consider the two-dimensional nonlinear Schrödinger equation with point interaction and we establish a local well-posedness theory, including blow-up alternative and continuous dependence on the initial data in the energy space. We provide proof by employing Kato’s method along with Hardy inequalities with logarithmic correction. Moreover, we establish finite time blow-up for solutions with positive energy and infinite variance.

Keywords

blow-upLWP theorynonlinear Schrödinger equationpoint interactions

MSC classification

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger)
Secondary: 35J10: Schrödinger operator 35A21: Propagation of singularities

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 20
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

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