Blow-up of the radially symmetric solutions for the quadratic nonlinear Schr\"{o}dinger system without mass-resonance

TL;DR

This paper investigates finite time blow-up phenomena for radially symmetric solutions of a quadratic nonlinear Schrödinger system in dimensions 4 to 6, extending known results beyond the mass-resonance case where blow-up was previously established.

Contribution

It proves finite time blow-up for radially symmetric solutions in dimensions 5 and 6 without the mass-resonance condition and explores blow-up or grow-up in dimension 4.

Findings

Finite time blow-up in dimensions 5 and 6 without mass-resonance.

Blow-up or grow-up behavior in dimension 4.

Extension of blow-up results beyond the mass-resonance condition.

Abstract

We consider the quadratic nonlinear Schr\"{o}dinger system \begin{align*} \begin{cases} i\partial_t u +\Delta u =v \overline{u},\\ i\partial_t v +\kappa \Delta v =u^2, \end{cases} \text{ on } I \times \mathbb{R}^d, \end{align*} where $1\leq d \leq 6$ and $\kappa>0$. In the lower dimensional case $d=1,2,3$, it is known that the $H^1$-solution is global in time. On the other hand, there are finite time blow-up solutions when $d=4,5,6$ and $\kappa=1/2$. The condition of $\kappa=1/2$ is called mass-resonance. In this paper, we prove finite time blow-up under radially symmetric assumption when $d=5,6$ and $\kappa \neq 1/2$ and we show blow-up or grow-up when $d=4$.