On the life span of the Schr\"{o}dinger equation with sub-critical power nonlinearity

Abstract

We discuss the life span of the Cauchy problem for the one-dimensional Schr\"{o}dinger 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 $\varepsilon\varphi$. Here, $\varepsilon$ stands for the size of the data. It is not difficult to see that the life span $T(\varepsilon)$ is estimated by $C_0 \varepsilon^{-2(p-1)/(3-p)}$ from below, provided $\varepsilon$ is sufficiently small. In this paper, we consider a more precise estimate for $T(\varepsilon)$ and we prove that $\liminf_{\varepsilon\to 0}\varepsilon^{2(p-1)/(3-p)}T(\varepsilon)$ is larger than some positive constant expressed only by $p$, $\mathrm{Im}\lambda$ and $\varphi$.