Blow up of solutions to nonlinear Neumann boundary value problem for Schrödinger equations

Abstract

Our purpose is to consider the life span of solutions to the nonlinear Neumann boundary value problem for one dimensional nonlinear Schrödinger equation on a half-line \begin{equation*} \begin{cases} i\partial _{t}u+\tfrac{1}{2} \partial _{x}^{2}u=0, & t > 0, x\in \mathbb{R}_{+}, \\ u(0,x)=u_{0}(x), & x \in \mathbb{R}_{+}, \\ -\partial _{x}u( t,0) = g( t,0) , & t > 0, \end{cases} \end{equation*} where $g( t,0) =e^{-\frac{\pi }{4}i} | u ( t,0 ) | ^{q}.$ We prove that for any $q > 2$, and $r > 2( q-1) $, there exists an initial function $u_{0}\in \dot{H}_{x}^{- ( \frac{2}{r}-\frac{1}{2} ) }$ such that the maximal existence time is finite.