Finite-time blow-up for the real-valued defocusing energy-supercritical NLW
Abstract
We construct real-valued solutions of the defocusing energy-supercritical nonlinear wave equation ∂t2u−Δu+up=0 that blow up in finite time from smooth, compactly supported initial data. We consider three cases: for p=3 we construct blow-up solutions in all dimensions d≥ 9, for p=5 we treat all dimensions d≥ 7, while for p=7 we treat all dimensions d≥ 6. The solutions we construct here appear to be the first blow-up solutions of defocusing energy-supercritical nonlinear wave equations with smooth real-valued data. The main step is the construction of nonzero \emph{discretely self-similar profiles} that are periodic in self-similar time and real-analytic across the light cone. We find approximate profiles by numerical continuation from rotating self-similar solutions of the complex-valued equation and upgrade them to exact solutions by a computer-assisted Newton--Kantorovich argument in a Banach algebra of Fourier--Chebyshev coefficients. The main analytic ingredients are an exact description of the linear mode operators and uniform bounds for their inverses. The codes for the computer-assisted parts of the proofs, together with the approximate solutions, are available on GitHub \cite{code}.
Related
- Blow-up for energy supercritical defocusing nonlinear Schrödinger equations in dimensions three, four and five
- 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
- Long time stability of low--frequency packets of modes at finite total energy
BibTeX
@misc{cadiot2026finitetime,
title = {{Finite-time blow-up for the real-valued defocusing energy-supercritical NLW}},
author = {Matthieu Cadiot and Alexandru D. Ionescu and Stan Palasek},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06678},
eprint = {2610.06678},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06678},
}
RIS
TY - UNPB
TI - Finite-time blow-up for the real-valued defocusing energy-supercritical NLW
AU - Matthieu Cadiot
AU - Alexandru D. Ionescu
AU - Stan Palasek
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.06678
UR - https://arxiv.org/abs/2610.06678
AB - We construct real-valued solutions of the defocusing energy-supercritical nonlinear wave equation ∂_t^2u−Δu+u^p=0 that blow up in finite time from smooth, compactly supported initial data. We consider three cases: for p=3 we construct blow-up solutions in all dimensions d≥ 9, for p=5 we treat all dimensions d≥ 7, while for p=7 we treat all dimensions d≥ 6. The solutions we construct here appear to be the first blow-up solutions of defocusing energy-supercritical nonlinear wave equations with smooth real-valued data. The main step is the construction of nonzero \emph{discretely self-similar profiles} that are periodic in self-similar time and real-analytic across the light cone. We find approximate profiles by numerical continuation from rotating self-similar solutions of the complex-valued equation and upgrade them to exact solutions by a computer-assisted Newton--Kantorovich argument in a Banach algebra of Fourier--Chebyshev coefficients. The main analytic ingredients are an exact description of the linear mode operators and uniform bounds for their inverses. The codes for the computer-assisted parts of the proofs, together with the approximate solutions, are available on GitHub \cite{code}.
ER -