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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 timeblow-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 timeblow-up under radially symmetric assumption when $d=5,6$ and $\kappa \neq 1/2$and we show blow-up or grow-up when $d=4$.

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