Small data blow-up of $L^2$-solution for the nonlinear Schrödinger equation without gauge invariance

We study the initial-value problem for the nonlinear Schrödinger equation $$ i\partial _{t}u+\Delta u=\lambda\vert u\vert ^{p}, \quad\left( t,x\right) \in \left[ 0,T\right) \times \mathbf{R}^{n}, $$ where $1 < p$ and $\lambda\in\mathbf{C}\setminus\{0\}$. The local well-posedness is well known in $L^2$ if $1 < p < 1+4/n$. In this paper, we study the global behavior of the solutions, and we will prove a small-data blow-up result of an $L^2$-solution when $1 < p\le 1+2/n$.