Finite-Time Blow-Up of Radial Negative-Energy Solutions for the Mass-Critical Biharmonic Nonlinear Schrödinger Equation

Abstract

We resolve an open question of Boulenger–Lenzmann for the focusing mass- critical biharmonic nonlinear Schrödinger equation i∂_tu = ∆^2 u − |u|^{8/N} u, (t, x) ∈ I × R^N , N ≥ 2. We prove that every radial solution with negative energy and initial data in H_2(R^N ) blows up in finite time, both forward and backward. This removes the previously unresolved possibility of blow-up occurring only at infinite time. The proof uses a new exponentially localized virial argument, together with a radial interpolation estimate adapted to the fourth-order dispersion, and leads to a quartic Riccati inequality forcing finite-time blow-up. We remark on the use of AI in this research, see Statement of AI Use.