Abstract
We discuss the life span of the Cauchy problem for the one-dimensional Schrödinger equation with a single power nonlinearity $\lambda |u|^{p-1}u$ ($\lambda\in\mathbb{C}$, $2\le p < 3$) and initial data of the form $\varepsilon\varphi$ prescribed. 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 $\varphi$.
Citation
Hironobu Sasaki. "On the life span of the Schrödinger equation with sub-critical power nonlinearity." Adv. Differential Equations 14 (11/12) 1021 - 1039, November/December 2009. https://doi.org/10.57262/ade/1355854783
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