Critical nonlinear Schroedinger equations in higher space dimensions

摘要

We study the critical nonlinear Schroedinger equations i∂_tu+1/2∆u =λ|u|~(2)u, (t,x) ∈R~+ ×R~n, in space dimensions n ≥ 4, where λ∈ R. We prove the global in time existence of solutions to the Cauchy problem under the assumption that the absolute value of Fourier transform of the initial data is bounded below by a positive constant. Also we prove the two side sharp time decay estimates of solutions in the uniform norm.