Finite-time blow-up without finite variance or radiality

Determine whether a solution with initial data u_0\in H^1_a(\mathbb{R}^N) that is neither radial nor of finite variance, satisfies E_a(u_0)<m_a(c) and Q(u_0)<0, necessarily blows up in finite time.

Background

The paper proves finite-time blow-up below m_a(c) when the Pohozaev functional is negative and the initial datum has finite variance. It also discusses existing results for radial data and for the mass-critical equation without finite-variance assumptions.

The unresolved problem is the full nonradial, infinite-variance case in the intercritical regime. A solution would require localized virial or another mechanism capable of replacing the global variance argument.

References

The question is the following. Let u_0\inH1_a(RN) be neither radial nor of finite variance, with E_a(u_0)<m_a(c) and Q(u_0)<0. Does the solution blow up in finite time? Localized virial arguments go back to Ogawa and Tsutsumi . The decay of the weight |x|{-b} makes them available without symmetry assumptions, as in . For the classical equation a=b=0, this question is open in general.

— Normalized ground states and a mass-constrained scattering threshold for the inhomogeneous NLS with an inverse-square potential  (2610.02933 - Majdoub et al., 2 Oct 2026) in Section “Complements and open problems,” subsection “Open problems,” item (v)