Applicability of the discretely self-similar blow-up mechanism across the supercritical range

Determine whether the exact discretely self-similar blow-up mechanism constructed for the defocusing quintic nonlinear wave equation in ten spatial dimensions can prove finite-time blow-up throughout the full energy-supercritical range s_c>1, or only for sufficiently supercritical equations.

Background

The paper constructs an exact real-valued discretely self-similar blow-up solution for the defocusing quintic wave equation in ten spatial dimensions, corresponding to s_c=9/2. The authors note that the analytic framework extends formally to other dimensions and odd powers, while the computer-assisted existence proof is carried out only for the specific pair (d,p)=(10,5). They explicitly leave unresolved whether the same mechanism can establish blow-up for every energy-supercritical equation or only for equations sufficiently far above the energy-critical threshold.

References

It is not clear to us if this same mechanism could be applied to prove blow-up in the full supercritical range $s_c>1$, or whether this is possible only for sufficiently supercritical problems.

— Finite-time blow-up for the real-valued defocusing energy-supercritical quintic NLW  (2610.06678 - Cadiot et al., 5 Oct 2026) in Remark 1.1(vi), “Exact discrete self-similarity”