Calculus of Variations and Geometric Measure Theory

D. Finco - L. Tentarelli - A. Teta

Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity

created by tentarelli on 04 Oct 2022
modified on 30 Dec 2023


Published Paper

Inserted: 4 oct 2022
Last Updated: 30 dec 2023

Journal: Nonlinearity
Volume: 37
Number: 1
Pages: art.015009, 48p
Year: 2024
Doi: 10.1088/1361-6544/ad0aac

ArXiv: 2209.13504 PDF


We discuss strong local and global well-posedness for the three-dimensional NLS equation with nonlinearity concentrated on $\mathbb{S}^2$. Precisely, local well-posedness is proved for any $C^2$ power-nonlinearity, while global well-posedness is obtained either for small data or in the defocusing case under some growth assumptions. With respect to point-concentrated NLS models, widely studied in the literature, here the dimension of the support of the nonlinearity does not allow a direct extension of the known techniques and calls for new ideas.