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

Abstract

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

15 pages

Cited by in corpus (1)

• On the scattering problem for the nonlinear Schrödinger equation with a potential in 2D