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

Let $T_{ϵ}$ be the lifespan for the solution to the Schr"odinger equation on ${\mathbb{R}}^d$ with a power nonlinearity $\lambda \mathrm{\mid }u{\mathrm{\mid }}^{2\theta \mathrm{/}d}u$ ($\lambda \in \mathbb{C}$, $0<\theta <1$) and the initial data in the form $ϵ\phi (x)$. We provide a sharp lower bound estimate for $T_{ϵ}$ as $ϵ\to +0$ which can be written explicitly by $\lambda$, $d$, $\theta$, $\phi$ and $ϵ$. This is an improvement of the previous result by H.Sasaki [Adv. Diff. Eq. 14 (2009), 1021–1039].