PreprintCC BY 4.0arXiv
Blow-up for energy supercritical defocusing nonlinear Schrödinger equations in dimensions three, four and five
Abstract
We prove finite time blow-up from smooth, radial, compactly supported initial data for the energy supercritical defocusing nonlinear Schrödinger equation i∂tu+Δu−|u|p−1u=0 in ℝd, for (d,p)∈{(3,27),(4,9),(5,7)}. The solutions are asymptotically self-similar, their L∞ norm grows at the scaling rate (T−t)−1/(p−1), and their critical Sobolev norm Ḣsc diverges. This closes the conjecture of Bourgain \cite{Bourgain2000} on global well-posedness and scattering for the defocusing energy supercritical equation in dimensions three and four. The proof relies on a shooting argument and is computer-assisted.
Related
- Orbital stability of standing waves for defocusing fractional nonlinear Schrödinger equations with a delta potential
- Singular evolution of Gaussian measures under energy-critical and supercritical nonlinear Schrödinger equations
- On blow-up for the supercritical defocusing nonlinear scalar wave equation
- Global well-posedness and scattering for the defocusing nonlinear Schrödinger equation below energy space
- Global well-posedness and scattering of energy-critical defocusing nonlinear Schrödinger equation on high-dimensional hyperbolic spaces
BibTeX
@misc{buck2026blowup,
title = {{Blow-up for energy supercritical defocusing nonlinear Schrödinger equations in dimensions three, four and five}},
author = {Kevin Buck and Alvaro Carballeira and Javier Gómez-Serrano and Jia Shi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23685},
eprint = {2609.23685},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23685},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Blow-up for energy supercritical defocusing nonlinear Schrödinger equations in dimensions three, four and five
AU - Kevin Buck
AU - Alvaro Carballeira
AU - Javier Gómez-Serrano
AU - Jia Shi
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23685
UR - https://arxiv.org/abs/2609.23685
AB - We prove finite time blow-up from smooth, radial, compactly supported initial data for the energy supercritical defocusing nonlinear Schrödinger equation i∂_tu+Δu−|u|^p−1u=0 in ℝ^d, for (d,p)∈{(3,27),(4,9),(5,7)}. The solutions are asymptotically self-similar, their L^∞ norm grows at the scaling rate (T−t)^−1/(p−1), and their critical Sobolev norm Ḣ^s_c diverges. This closes the conjecture of Bourgain \cite{Bourgain2000} on global well-posedness and scattering for the defocusing energy supercritical equation in dimensions three and four. The proof relies on a shooting argument and is computer-assisted.
ER -