Global regularity for defocusing energy-supercritical nonlinear wave equations

Establish global regularity and scattering for smooth solutions of defocusing energy-supercritical nonlinear wave equations beyond the conditional results based on bounded critical norms.

Background

The paper studies the defocusing nonlinear wave equation in the energy-supercritical regime, where the critical Sobolev regularity satisfies s_c>1. Although global smooth solutions are known in the energy-subcritical and energy-critical defocusing cases, the authors state that global regularity in the energy-supercritical case remains unresolved. Existing results provide conditional global existence and scattering under bounds on critical norms, but do not settle unconditional global regularity for arbitrary smooth data.

References

In the energy-supercritical case ($s_c>1$), global regularity of smooth solutions has been a well-known open problem.

— Finite-time blow-up for the real-valued defocusing energy-supercritical quintic NLW  (2610.06678 - Cadiot et al., 5 Oct 2026) in Section 1, subsection “The problem”