Lifespan of solutions for the nonlinear Schr\"odinger equation without gauge invariance

Abstract

We study the lifespan of solutions for the nonlinear Schr\"odinger equation id_{t}u+{\Delta}u={\lambda}|u|^{p}, (t,x)\in[0,T)\timesR^{n}, with the initial condition, where 1<p\leq 1+2/n and {\lambda}\in C. Our main aim in this paper is to prove an upper bound of the lifespan in the subcritical case 1<p<1+2/n.