Blowup for the defocusing septic complex-valued nonlinear wave equation in ℝ⁴⁺¹
Abstract
In this paper, we prove blowup for the defocusing septic complex-valued nonlinear wave equation in R 4 + 1 \mathbb {R}^{4+1} . This work builds on the earlier results of Shao, Wei, and Zhang [Forum Math. Pi 13 (2025); Self-similar imploding solutions of the relativistic Euler equations , arXiv:2403.11471, 2024], reducing the order of the nonlinearity from 29 29 to 7 7 in R 4 + 1 \mathbb {R}^{4+1} . As by Shao, Wei, and Zhang [Forum Math. Pi 13 (2025); Self-similar imploding solutions of the relativistic Euler equations , arXiv:2403.11471, 2024], the proof hinges on a connection between solutions to the nonlinear wave equation and the relativistic Euler equations via a front compression blowup mechanism. More specifically, the problem is reduced to constructing smooth, radially symmetric, self-similar imploding profiles for the relativistic Euler equations. As with implosion for the compressible Euler equations, the relativistic analogue admits a countable family of smooth imploding profiles. The result of Shao, Wei, and Zhang [ Self-similar imploding solutions of the relativistic Euler equations , arXiv:2403. 11471, 2024] represents the construction of the first profile in this family. In this paper, we construct a sequence of solutions corresponding to the higher-order profiles in the family. This allows us to saturate the inequalities necessary to show blowup for the defocusing complex-valued nonlinear wave equation with an integer order of nonlinearity and radial symmetry via this mechanism.
Related
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BibTeX
@article{buckmaster2026blowup,
title = {{Blowup for the defocusing septic complex-valued nonlinear wave equation in ℝ⁴⁺¹}},
author = {Tristan Buckmaster and Jiajie Chen},
journal = {Communications of the American Mathematical Society},
year = {2026},
month = jul,
doi = {10.1090/cams/71},
url = {https://doi.org/10.1090/cams/71},
note = {CC BY-NC-ND},
}
RIS
TY - JOUR
TI - Blowup for the defocusing septic complex-valued nonlinear wave equation in ℝ⁴⁺¹
AU - Tristan Buckmaster
AU - Jiajie Chen
PY - 2026
DA - 2026/07/14
JO - Communications of the American Mathematical Society
DO - 10.1090/cams/71
UR - https://doi.org/10.1090/cams/71
AB - In this paper, we prove blowup for the defocusing septic complex-valued nonlinear wave equation in R 4 + 1 \mathbb {R}^{4+1} . This work builds on the earlier results of Shao, Wei, and Zhang [Forum Math. Pi 13 (2025); Self-similar imploding solutions of the relativistic Euler equations , arXiv:2403.11471, 2024], reducing the order of the nonlinearity from 29 29 to 7 7 in R 4 + 1 \mathbb {R}^{4+1} . As by Shao, Wei, and Zhang [Forum Math. Pi 13 (2025); Self-similar imploding solutions of the relativistic Euler equations , arXiv:2403.11471, 2024], the proof hinges on a connection between solutions to the nonlinear wave equation and the relativistic Euler equations via a front compression blowup mechanism. More specifically, the problem is reduced to constructing smooth, radially symmetric, self-similar imploding profiles for the relativistic Euler equations. As with implosion for the compressible Euler equations, the relativistic analogue admits a countable family of smooth imploding profiles. The result of Shao, Wei, and Zhang [ Self-similar imploding solutions of the relativistic Euler equations , arXiv:2403. 11471, 2024] represents the construction of the first profile in this family. In this paper, we construct a sequence of solutions corresponding to the higher-order profiles in the family. This allows us to saturate the inequalities necessary to show blowup for the defocusing complex-valued nonlinear wave equation with an integer order of nonlinearity and radial symmetry via this mechanism.
SN - 2692-3688
ER -