A sharp lower bound for the lifespan of small solutions to the Schrödinger equation with a subcritical power nonlinearity

Abstract

Let $T_{\varepsilon}$ be the lifespan for the solution to the Schrödinger 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 $\varepsilon \varphi(x)$. We provide a sharp lower bound estimate for $T_{\varepsilon}$ as $\varepsilon \to +0$ which can be written explicitly by $\lambda$, $d$, $\theta$, $\varphi$ and $\varepsilon$. This is an improvement of the previous result by H. Sasaki [Adv. Diff. Eq., 14 (2009), 1021-1039].