Nonradial Finite-Time Blow-up for the Mass-Critical Inhomogeneous Biharmonic NLS

Abstract

We consider the focusing mass-critical inhomogeneous biharmonic nonlinear Schrödinger equation \[ i\partial_tu=Δ^2u-|x|^{-b}|u|^{\frac{8-2b}{N}}u, \quad 0<b<\min\{4,N/2\}. \] We prove that every $H^2({\mathbb{R}}^N)$ solution with negative energy blows up in finite time, both forward and backward. No radial symmetry or finite-variance assumption is required. In particular, this rules out the global grow-up scenario in the previously known negative-energy blow-up alternative and establishes finite-time blow-up in the genuinely nonradial setting. The proof relies on a new localized virial argument exploiting the spatial inhomogeneity to control the localization errors.