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

We consider the focusing mass-critical inhomogeneous biharmonic nonlinear Schrödinger equation $$i{\mathrm{\partial }}_{t}u={\mathrm{\Delta }}^{2}u-\mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\frac{8-2b}{N}}u,0<b<\min \{4,N\mathrm{/}2\}\mathrm{.}$$ 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.