On the Cauchy problem of the nonlinear Schrodinger equation without gauge invariance

This paper is dedicated to study the Cauchy problem of the nonlinear Schrodinger equation without gauge invariance where and . When , we first prove local well-posedness of the equation in . If in addition, , we prove the global well-posedness with small initial data in . Under a suitable condition on the initial data and , we prove that the -norm of the solution would blow up in finite time although the initial data are arbitrarily small. Meanwhile, we also give a large initial data blow-up result when in . Finally, we show the non-existence of local weak solution for some -data with when 1 + 4/n-2m.