Global smoothness for the real scalar defocusing septic wave equation
Consider the defocusing septic wave equation in three space dimensions, $$\partial_t^2 u - \Delta u + u^7 = 0, \qquad u(0, x) = u_0(x), \quad \partial_t u(0, x) = u_1(x),$$ for real-valued $u$ and real-valued data $(u_0, u_1) \in C_c^\infty(\mathbb{R}^3) \times C_c^\infty(\mathbb{R}^3)$. Smooth solutions conserve the energy $$E(u) = \int_{\mathbb{R}^3} \left( \tfrac12 |\partial_t u|^2 + \tfrac12 |\nabla u|^2 + \tfrac18 u^8 \right) dx,$$ whose scaling is supercritical for this equation. For the energy-critical quintic equation $\partial_t^2 u - \Delta u + u^5 = 0$, Grillakis proved that all smooth data give global smooth solutions, and Tao proved the same for spherically symmetric data and the logarithmically supercritical nonlinearity $u^5 \log(2 + u^2)$. Tao constructed, in three space dimensions and for every $p > 5$, defocusing systems with vector-valued unknowns, together with smooth data whose solution blows up in finite time. Shao, Wei and Zhang constructed smooth complex-valued solutions of $\partial_t^2 u - \Delta u + |u|^{p-1} u = 0$ that blow up in finite time, for $d = 4$ and odd $p \ge 29$ and for $d \ge 5$ and odd $p \ge 17$. Cadiot, Ionescu and Palasek gave a computer-assisted construction of smooth, compactly supported, radial real data for the defocusing quintic equation in ten space dimensions whose solution blows up in finite time (2026 preprint).
Problem. Does every such pair $(u_0, u_1)$ give a smooth solution on every finite time interval? No symmetry or smallness is assumed; real data whose solution breaks down in finite time also answer the question.
Source: A long-standing question on energy-supercritical defocusing wave equations; see Tao (2007) and Shao, Wei and Zhang (2025). Stated for the real septic equation in three dimensions by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, energy-supercritical equations, finite-time blow-up, nonlinear wave equations
References
@article{tao2007,
author = {Tao, Terence},
title = {Global regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric data},
journal = {J. Hyperbolic Differ. Equ.},
volume = {4},
number = {2},
year = {2007},
pages = {259--265},
doi = {10.1142/S0219891607001124},
eprint = {math/0606145},
archivePrefix = {arXiv},
}
@article{shao2025,
author = {Shao, Feng and Wei, Dongyi and Zhang, Zhifei},
title = {On blow-up for the supercritical defocusing nonlinear wave equation},
journal = {Forum Math. Pi},
volume = {13},
year = {2025},
pages = {e15},
doi = {10.1017/fmp.2025.7},
eprint = {2405.19674},
archivePrefix = {arXiv},
}
@article{grillakis1990,
author = {Grillakis, Manoussos G.},
title = {Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity},
journal = {Ann. of Math. (2)},
volume = {132},
number = {3},
year = {1990},
pages = {485--509},
doi = {10.2307/1971427},
}
@article{tao2016,
author = {Tao, Terence},
title = {Finite-time blowup for a supercritical defocusing nonlinear wave system},
journal = {Anal. PDE},
volume = {9},
number = {8},
year = {2016},
pages = {1999--2030},
doi = {10.2140/apde.2016.9.1999},
eprint = {1602.08059},
archivePrefix = {arXiv},
}
@misc{cadiot2026,
author = {Cadiot, Matthieu and Ionescu, Alexandru D. and Palasek, Stan},
title = {Finite-time blow-up for the real-valued defocusing energy-supercritical quintic {NLW}},
year = {2026},
eprint = {2610.06678},
archivePrefix = {arXiv},
}
@misc{aim-scalar-supercritical-wave,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global smoothness for the real scalar defocusing septic wave equation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/077-scalar-supercritical-wave.md},
note = {Entry 077 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00640, "Global smoothness for the real scalar defocusing septic wave equation", clopen.solutions.