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November/December 2009 On the life span of the Schrödinger equation with sub-critical power nonlinearity
Hironobu Sasaki
Adv. Differential Equations 14(11/12): 1021-1039 (November/December 2009).

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

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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.

Information

Published: November/December 2009
First available in Project Euclid: 18 December 2012

zbMATH: 1185.35266
MathSciNet: MR2560867

Subjects:
Primary: 35Q55

Rights: Copyright © 2009 Khayyam Publishing, Inc.

JOURNAL ARTICLE
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Vol.14 • No. 11/12 • November/December 2009
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