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 $κ&gt;0$. 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 $H<sup>1$-norm grows without bound. In this paper, we prove that every radial $H^1\times H<sup>1$ 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 $O(R<sup>{-2})$ 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.

• The paper proves that every radial, negative-energy solution in the four-dimensional mass-critical quadratic NLS system blows up in finite time both forward and backward, closing the previously unresolved global grow-up branch.
• The proof combines an exponential localized virial multiplier with a radial weighted interpolation inequality to control nonlinear errors through a shared weighted kinetic defect.
• A Riccati argument converts the coercive virial estimate into quantitative blow-up-time bounds, while the result remains open for nonradial data because spatial translation complicates the defect mechanism.

Setting and prior results

The paper analyzes the focusing quadratic nonlinear Schrödinger system

$$i\partial_t u+\Delta u=-2v\overline{u},\qquad i\partial_t v+\kappa\Delta v=-u^2,$$

on $\mathbb{R}^4$ with $\kappa>0$, which is invariant under the mass-critical scaling $u_\lambda=\lambda^2 u(\lambda^2 t,\lambda x)$. The conserved mass is $M(u,v)=\|u\|_2^2+2\|v\|_2^2$ and the energy $$E=\frac12(\|\nabla u\|_2^2+\kappa\|\nabla v\|_2^2)-\mathrm{Re}\int v\overline{u}^2$$; local well-posedness in $H^1\times H^1$ follows from Hayashi–Ozawa–Tanaka. The case $\kappa=1/2$ is the mass-resonant regime, where stronger virial cancellations were already known to yield finite-time blow-up for negative-energy data.

In the non-resonant regime, Inui–Kishimoto–Nishimura proved that radial negative-energy solutions either blow up in finite time or exist globally while growing up along a sequence of times. Dinh–Forcella strengthened the grow-up branch: any hypothetical global solution satisfies a quadratic kinetic growth estimate, roughly $T(u(t),v(t))\gtrsim t^2$. Crucially, this leaves open an entire global branch with unbounded kinetic energy but finite $H^1$ norm at each finite time — precisely the branch the present paper eliminates.

Main theorem

Theorem (finite-time blow-up). For $\mathbb{R}^4$0, $\mathbb{R}^4$1, every radial $\mathbb{R}^4$2 with $\mathbb{R}^4$3 has both maximal lifespan endpoints finite, i.e. $\mathbb{R}^4$4 and $\mathbb{R}^4$5, with $\mathbb{R}^4$6 and analogously backward in time. Notably, no finite-variance condition ($\mathbb{R}^4$7) is required, since the multiplier gradient $\mathbb{R}^4$8 is bounded. The proof also does not use $\mathbb{R}^4$9, so the result extends verbatim to the resonant case.

The virial-defect mechanism

The argument is a localized virial method in the spirit of Ogawa–Tsutsumi, but with two structural innovations that together close the gap left by Dinh–Forcella's compactly truncated weight. First, the exponential weight $\kappa>0$0 enjoys a defect geometry linking three quantities through the same function $\kappa>0$1:

• Coercivity of the Hessian: as quadratic forms, $\kappa>0$2, so the principal kinetic term contributes $\kappa>0$3, where $\kappa>0$4.
• Nonlinear control: $\kappa>0$5, so the cubic error term $\kappa>0$6 is controlled by $\kappa>0$7-weighted quantities.
• Virial bound: $\kappa>0$8, giving $\kappa>0$9.

The paper emphasizes why the exponential profile is decisive: a flattened multiplier equaling $u_\lambda=\lambda^2 u(\lambda^2 t,\lambda x)$0 on a ball has zero Hessian defect there while $u_\lambda=\lambda^2 u(\lambda^2 t,\lambda x)$1, so the localized virial cannot be controlled by the same defect that appears favorably in its derivative. The exponential choice makes both quantities vanish quadratically at the origin at compatible rates and links them globally.

Second, a radial weighted interpolation inequality controls the cubic nonlinearity by the weighted kinetic defect up to a small error depending only on the conserved mass:

$u_\lambda=\lambda^2 u(\lambda^2 t,\lambda x)$2

The proof rescales on dyadic annuli via the exactly critical map $u_\lambda=\lambda^2 u(\lambda^2 t,\lambda x)$3, applies a one-dimensional Gagliardo–Nirenberg inequality locally, absorbs the gradient part by Young's inequality, and uses the uniform bound $u_\lambda=\lambda^2 u(\lambda^2 t,\lambda x)$4 plus bounded overlap of the enlarged annuli. A pointwise Young inequality $u_\lambda=\lambda^2 u(\lambda^2 t,\lambda x)$5 reduces the mixed term to this estimate.

Combining these ingredients yields the coercive localized virial inequality

$u_\lambda=\lambda^2 u(\lambda^2 t,\lambda x)$6

Fixing $u_\lambda=\lambda^2 u(\lambda^2 t,\lambda x)$7 large enough that the $u_\lambda=\lambda^2 u(\lambda^2 t,\lambda x)$8 term is absorbed by the negative energy converts this into $u_\lambda=\lambda^2 u(\lambda^2 t,\lambda x)$9.

Riccati closure

Once $M(u,v)=\|u\|_2^2+2\|v\|_2^2$0 becomes negative, setting $M(u,v)=\|u\|_2^2+2\|v\|_2^2$1, the chain

$M(u,v)=\|u\|_2^2+2\|v\|_2^2$2

yields $M(u,v)=\|u\|_2^2+2\|v\|_2^2$3 with $M(u,v)=\|u\|_2^2+2\|v\|_2^2$4. Integration gives $M(u,v)=\|u\|_2^2+2\|v\|_2^2$5, so $M(u,v)=\|u\|_2^2+2\|v\|_2^2$6 diverges no later than the explicit time $M(u,v)=\|u\|_2^2+2\|v\|_2^2$7. Since a global $M(u,v)=\|u\|_2^2+2\|v\|_2^2$8 solution would keep $M(u,v)=\|u\|_2^2+2\|v\|_2^2$9 bounded on $$E=\frac12(\|\nabla u\|_2^2+\kappa\|\nabla v\|_2^2)-\mathrm{Re}\int v\overline{u}^2$$0 ($$E=\frac12(\|\nabla u\|_2^2+\kappa\|\nabla v\|_2^2)-\mathrm{Re}\int v\overline{u}^2$$1), this contradicts global existence. The backward statement follows from the time-reversal symmetry $$E=\frac12(\|\nabla u\|_2^2+\kappa\|\nabla v\|_2^2)-\mathrm{Re}\int v\overline{u}^2$$2, which preserves mass, energy, and radiality.

The quantitative content is worth noting: the blow-up time is explicitly bounded in terms of the initial data via $$E=\frac12(\|\nabla u\|_2^2+\kappa\|\nabla v\|_2^2)-\mathrm{Re}\int v\overline{u}^2$$3, $$E=\frac12(\|\nabla u\|_2^2+\kappa\|\nabla v\|_2^2)-\mathrm{Re}\int v\overline{u}^2$$4, and the chosen radius $$E=\frac12(\|\nabla u\|_2^2+\kappa\|\nabla v\|_2^2)-\mathrm{Re}\int v\overline{u}^2$$5 satisfying $$E=\frac12(\|\nabla u\|_2^2+\kappa\|\nabla v\|_2^2)-\mathrm{Re}\int v\overline{u}^2$$6.

Scope and limitations

Two remarks delimit the reach of the argument. First, the extension beyond the specific system is asserted but not carried out in detail: the authors state that the same mechanism applies to general multi-component systems with homogeneous cubic interactions under the structural assumptions of Dinh–Forcella and Noguera–Pastor, using componentwise weighted defects, and they sketch the corresponding estimates rather than proving them. Second, the theorem is confined to radial data. The paper draws an analogy with the Holmer–Roudenko program for the three-dimensional focusing cubic NLS, where nonradial solutions without finite variance exhibit only grow-up along a sequence; the authors pose as an open question whether a translation-adapted version of the virial-defect mechanism could address the Holmer–Roudenko weak conjecture (continuous divergence of $$E=\frac12(\|\nabla u\|_2^2+\kappa\|\nabla v\|_2^2)-\mathrm{Re}\int v\overline{u}^2$$7) in the nonradial setting. Handling spatial translation is identified as the principal obstruction.

Conclusion

The paper closes the grow-up alternative for radial negative-energy solutions of the four-dimensional mass-critical quadratic NLS system in both resonant and non-resonant cases: every such solution blows up in finite time, forward and backward, without finite-variance assumptions. The proof rests on the compatibility of the exponential multiplier's Hessian defect, Laplacian defect, and gradient size through a single weight $$E=\frac12(\|\nabla u\|_2^2+\kappa\|\nabla v\|_2^2)-\mathrm{Re}\int v\overline{u}^2$$8, combined with a radial weighted interpolation inequality whose error depends only on conserved mass, producing a superlinear Riccati differential inequality incompatible with global existence. The remaining natural problem — removing radiality, where translation must be incorporated into the virial-defect coupling — is left open.