On the life span of the Schrödinger equation with sub-critical power nonlinearity

Abstract

We discuss the life span of the Cauchy problem for the one-dimensional Schrödinger equation with a single power nonlinearity $\lambda |u{|}^{p-1}u$ ($\lambda \in \mathbb{C}$, $2\le p<3$) prescribed an initial data of the form $\epsilon \phi$. Here, $\epsilon$ stands for the size of the data. It is not difficult to see that the life span $T(\epsilon )$ is estimated by $C_0{\epsilon }^{-2(p-1)/(3-p)}$ from below, provided $\epsilon$ is sufficiently small. In this paper, we consider a more precise estimate for $T(\epsilon )$ and we prove that ${\mathrm{liminf}}_{\epsilon \to 0}{\epsilon }^{2(p-1)/(3-p)}T(\epsilon )$ is larger than some positive constant expressed only by $p$, $\mathrm{Im}\lambda$ and $\phi$.