A small initial data criterion of global existence for the damped nonlinear Schrödinger equation

This paper is concerned with the damped nonlinear Schrödinger equation. Through analyzing the characteristics of the equation and the effect of the damping on the global existence, we construct a variational problem. Then combining the variational problem, we establish a crucial invariant evolution flow to derive an explicit and computed criterion to answer: how small are the initial data such that the solutions of the system globally exist? Moreover, the small initial data criterion can be applied in the nonlinear Schrödinger equation with any positive damped parameter.