L2-decay estimate for the dissipative nonlinear Schrodinger equation in the Gevrey class

We consider the Cauchy problem for the dissipative nonlinear Schrodinger equation with a cubic nonlinear term lambda vertical bar u vertical bar(2)u where lambda is an element of C with Im lambda< 0. We prove the global existence of a unique solution and obtain the uniform estimate in the Gevrey class. Using the uniform regularity estimate, we show the L-2-decay rate for the solution which has the Gevrey regularity.