On the critical exponent for the Schrödinger equation with a nonlinear boundary condition

Abstract

We study the Schrödinger equation: $iu_t+u_{xx}=0,$ $ x\in {\bf R}_+,$ $ t>0$ with a nonlinear boundary condition $-u_x(0,t)=\vert u(0,t)\vert ^{p-1} u(0,t),$ $ t>0$. We show that if $1 <p <3,$ every solution is global in $H^1({\bf R}_+)$, while if $p\ge 3$, then nonglobal solutions exist.