Remark on the global existence and large time asymptotics of solutions for the quadratic NLS

We study the global in time existence of small solutions to the nonlinear Schrödinger equation with quadratic interactions equation(0.1){i∂tu+12Δu=|u|2,t>0,x∈R4,u(0,x)=u0(x),x∈R4. We prove that if the initial data u0u0 satisfy smallness conditions in the weighted Sobolev norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore, we prove the existence of the usual scattering states and find the large time asymptotics of the solutions.