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In this paper we consider the nonlinear Schr\"o\-din\-ger equation $i u_t+\Delta u +\kappa |u|^\alpha u=0$. We prove that if $\alpha <\frac {2} {N}$ and$\Im \kappa <0$, then every nontrivial $H^1$-solution blows up in finite orinfinite time. In the case $\alpha >\frac {2} {N}$ and $\kappa \in {\mathbbC}$, we improve the existing low energy scattering results in dimensions $N\ge7$. More precisely, we prove that if $ \frac {8} {N + \sqrt{ N^2 +16N }} <\alpha \le \frac {4} {N} $, then small data give rise to global, scatteringsolutions in $H^1$.

A Fujita-type blowup result and low energy scattering for a nonlinear Schr\"o\-din\-ger equation