The inhomogeneous nonlinear Schrödinger equation with harmonic potential is discussed.By establishing the corresponding energy function and the variational structure and through applying the potential well argument,the concave function and the improved Gagliardo-Nirenberg interpolation inequation,the optimum conditions for the global solution and explosive solution for the Cauchy problem are obtained.In addition，the initial value for the existence of global solution is determined.

%U https://zkxb.jsu.edu.cn/EN/10.3969/j.cnki.jdxb.2016.03.002