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

Abstract

In this paper we consider the nonlinear Schrö\-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 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 $H^1$.

Cited by in corpus (4)

• Modified scattering for the critical nonlinear Schrödinger equation
• Local existence, global existence, and scattering for the nonlinear Schrödinger equation
• Asymptotic behavior for a dissipative nonlinear Schrödinger equation
• Blowup on an arbitrary compact set for a Schödinger equation with nonlinear source term