Lower bounds of the lifespan of small data solutions to the nonlinear Schrödinger equations

Abstract

Let $T_{\varepsilon}$ be the lifespan of solutions to the initial value problem for the one dimensional, derivative nonlinear Schrödinger equations with small initial data of size $O(\varepsilon)$. If the nonlinear term is cubic and gauge invariant, it is known that $\liminf_{\varepsilon \to +0} \varepsilon^{2} \log T_\varepsilon$ is positive. In this paper we obtain a sharp estimate of this lower limit, which is explicitly computed from the initial data and the nonlinear term.