A sharp lower bound for the lifespan of small solutions to the Schrödinger equation with a subcritical power nonlinearity
A sharp lower bound for the lifespan of small solutions to the Schrödinger equation with a subcritical power nonlinearity
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DOI：
10.57262/die/1528855435
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发表时间：
2017-03
影响因子：
1.4
通讯作者：
Yuji Sagawa;Hideaki Sunagawa;Shunsuke Yasuda
中科院分区：
文献类型：
--
作者：
Yuji Sagawa;Hideaki Sunagawa;Shunsuke Yasuda
设$T_{\displaystyle $\lambda}$为$\mathbb{R}^d$上具有幂非线性项$\lambda的Schr\“odinger方程解的寿命|u| ^{2\theta/d}u$（$\lambda \in \mathbb{C}$，$0<\theta<1$）和形式为$\lambda\varphi（x）$的初始数据。我们提供了$T_{\displaystyle T_{\mathrm}$的一个精确的下界估计为$\lambda$，$d$，$\theta$，$\varphi$和$\arphi $。这是H.Sasaki [Adv. Diff.当量14（2009），1021- 1039]。
Let $T_{\epsilon}$ be the lifespan for the solution to the Schr\"odinger equation on $\mathbb{R}^d$ with a power nonlinearity $\lambda |u|^{2\theta/d}u$ ($\lambda \in \mathbb{C}$, $0<\theta<1$) and the initial data in the form $\epsilon \varphi(x)$. We provide a sharp lower bound estimate for $T_{\epsilon}$ as $\epsilon \to +0$ which can be written explicitly by $\lambda$, $d$, $\theta$, $\varphi$ and $\epsilon$. This is an improvement of the previous result by H.Sasaki [Adv. Diff. Eq. 14 (2009), 1021--1039].