A Fujita-type blowup result and low energy scattering for a nonlinear Schrödinger equation

Abstract

In this paper we consider the nonlinear Schrödinger equation $i u_t +\Delta u +\kappa |u|^\alpha u=0$. We prove that if $\alpha <\frac{2}{N}$ and $\mathfrak {I}\kappa <0$, then every nontrivial $H^1$-solution blows up in finite or infinite time. In the case $\alpha >\frac{2}{N}$ and $\kappa \in \mathbb {C}$, we improve the existing low energy scattering results in dimensions $N\ge 7$. 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, scattering solutions in $H^1$.