Advances in Differential Equations

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

Hironobu Sasaki

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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 $\varphi$.

Article information

Adv. Differential Equations, Volume 14, Number 11/12 (2009), 1021-1039.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]


Sasaki, Hironobu. On the life span of the Schrödinger equation with sub-critical power nonlinearity. Adv. Differential Equations 14 (2009), no. 11/12, 1021--1039.

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