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

Abstract

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 $\kappa >0$. In the non-mass-resonant case $\kappa \ne \frac{1}{2}$, 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.

12 pages