Blow-up of the radially symmetric solutions for the quadratic nonlinear Schrödinger system without mass-resonance

Abstract

We consider the quadratic nonlinear Schrödinger system \begin{align*} \begin{cases} i\partial_t u +Δu =v \overline{u},\\ i\partial_t v +κΔv =u^2, \end{cases} \text{ on } I \times \mathbb{R}^d, \end{align*} where $1\le d\le 6$ 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 $\kappa \ne 1/2$ and we show blow-up or grow-up when $d=4$.