Abstract: We consider the blow-up solutions to the following coupled nonlinear Schrödinger equations \begin{equation*} \left\{ \begin{aligned} &{\rm i}u_{t}+\Delta u+(|u|^{2p}+\beta|u|^{p-1}|v|^{p+1})u=0,\\ &{\rm i}v_{t}+\Delta v+(|v|^{2p}+\beta|v|^{p-1}|u|^{p+1})v=0,\\ &u(0,x)=u_{0}(x),\ \ \ \ v(0,x)=v_{0}(x),\ \ x\in \mathbb{R}^{N},\ t\geq0. \end{aligned} \right. \end{equation*} On the basis of the conservation of mass and energy, we establish two sufficient conditions to obtain the existence of a blow-up for radially symmetric solutions. These results improve the blow-up result of Li and Wu [10] by dropping the hypothesis of finite variance ($(|x|u_{0},|x|v_{0})\in L^{2}(\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$).

Key words: Schrö, dinger equations, radial symmetry, blow-up, virial identity