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

TL;DR

This paper establishes blowup conditions for nonlinear Schrödinger equations with certain parameters and improves low energy scattering results for small data in higher dimensions.

Contribution

It proves a Fujita-type blowup result for specific nonlinear Schrödinger equations and enhances low energy scattering results for small data in dimensions N≥7.

Findings

Nontrivial solutions blow up when α < 2/N and Im κ < 0.

Improved low energy scattering results for α in (8/(N+√(N^2+16N)), 4/N] in dimensions N≥7.

Small data lead to global scattering solutions under specified conditions.

Abstract

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 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 $H^1$.