A sharp lower bound for the lifespan of small solutions to the Schr\"odinger equation with a subcritical power nonlinearity

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].