On the Cauchy problem for the nonlinear Schrödinger equations with time-dependent linear loss/gain

This paper is devoted to the Cauchy problem for the nonlinear Schrödinger equation with time-dependent loss/gain which reads iut+Δu+λ|u|αu+ia(t)u=0. This equation appears in the recent studies of Bose-Einstein condensates and optical systems. We obtain some global existence and blow-up results which depend on the size of the loss/gain coefficient. In particular, we prove the global existence for the energy critical nonlinearity. By scaling and compactness arguments, we also discuss asymptotic profiles and concentration properties of blow-up solutions.