Critical nonlinear Schrödinger equations with data in homogeneous weighted [formula omitted] spaces

Abstract

We consider the initial value problem for nonlinear Schrödinger equations with the critical nonlinearities λ1|u|2nu, where Imλ1≤0, when the space dimension n=1,2. We prove the global existence of small solutions in homogeneous weighted L2(Rn) spaces. It is shown that the small solutions decay uniformly like t−n2 for t>1 if Imλ1=0. The higher uniform time decay rates t−n2(logt)−n2 for t>1 are obtained if Imλ1<0.