PreprintCC BY 4.0arXiv
Effective threshold dynamics for the cubic nonlinear Klein-Gordon equation in three dimensions
Abstract
We consider the focussing cubic Klein-Gordon equation on ℝ3+1 in the radial setting. Foundational work by Nakanishi-Schlag gives a complete qualitative characterisation of the dynamics near the ground state Q, in particular revealing a threshold phenomenon, whereby a co-dimension one Lipschitz manifold separates finite time blow up from scattering solutions in forward time. Our aim in this paper is to derive an effective bound for the scattering norm |u|Lt3([0,∞);Lx6(ℝ3)), for scattering energy class solutions u with data sufficiently near Q and close to the threshold manifold. This requires the development of an induction on energy strategy in a setting where an alternative to the usual Morawetz approach appears necessary.
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BibTeX
@misc{gomez2026effective,
title = {{Effective threshold dynamics for the cubic nonlinear Klein-Gordon equation in three dimensions}},
author = {Jaime Gómez and Joachim Krieger},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41718},
eprint = {2610.41718},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41718},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Effective threshold dynamics for the cubic nonlinear Klein-Gordon equation in three dimensions
AU - Jaime Gómez
AU - Joachim Krieger
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41718
UR - https://arxiv.org/abs/2610.41718
AB - We consider the focussing cubic Klein-Gordon equation on ℝ^3+1 in the radial setting. Foundational work by Nakanishi-Schlag gives a complete qualitative characterisation of the dynamics near the ground state Q, in particular revealing a threshold phenomenon, whereby a co-dimension one Lipschitz manifold separates finite time blow up from scattering solutions in forward time. Our aim in this paper is to derive an effective bound for the scattering norm |u|_L_t^3([0,∞);L_x^6(ℝ^3)), for scattering energy class solutions u with data sufficiently near Q and close to the threshold manifold. This requires the development of an induction on energy strategy in a setting where an alternative to the usual Morawetz approach appears necessary.
ER -