Ngoc Uyen Cong Nguyen | Search arXiv

arXiv2026

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

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.

math.AP↗

arXiv2026

Finite-time blow-up for the four-dimensional mass-critical quadratic nonlinear Schrödinger system without mass resonance

We study the focusing quadratic nonlinear Schrödinger system \[ \begin{cases} i\partial_t u+Δu=-2v\overline{u}, \\ i\partial_t v+κΔv=-u^2, \end{cases} \qquad (t,x)\in I\times\mathbb R^4, \] where $κ>0$. In the non-mass-resonant case $κ\neq \frac12$, previous works of Inui--Kishimoto--Nishimura and Dinh--Forcella showed that radial solutions with negative energy must either blow up in finite time or exist globally while their $H^1$-norm grows without bound. In this paper, we prove that every radial $H^1\times H^1$ solution with negative energy blows up in finite time, both forward and backward in time. No finite-variance assumption is required. The main ingredient is a localized virial argument based on the bounded exponential weight \[ \nablaϕ_R(x)=2x e^{-|x|^2/R^2}. \] A radial weighted interpolation estimate allows us to control the nonlinear error terms by the corresponding weighted kinetic term, up to an $O(R^{-2})$ error depending only on the conserved mass. Moreover, the localized virial quantity itself can be bounded directly in terms of the same weighted kinetic defect. Combining these estimates yields a superlinear Riccati-type differential inequality, which cannot persist for all time and therefore forces finite-time blow-up.

math.AP↗