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

TL;DR

This paper refines the understanding of the lifespan of solutions to the one-dimensional Schrödinger equation with sub-critical power nonlinearity, providing a more precise lower bound involving initial data size and parameters.

Contribution

It offers a sharper estimate for the lifespan of solutions, explicitly relating it to the nonlinearity power, initial data, and the imaginary part of the coefficient.

Findings

Established a positive lower bound for the scaled lifespan as initial data size approaches zero.

Extended previous estimates by incorporating the imaginary part of the nonlinear coefficient.

Provided explicit dependence of lifespan on parameters $p$, $ ext{Im}\lambda$, and initial data.

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 $\varphi$.