On the Scattering in Gevrey Classes for the Subcriticai Hartree and Schrödinger Equations

We study the scattering problem and asymptotics for large time of solutions to the Cauchy problem for the subcriticai cubic nonlinear Schrödinger and Hartree type equations (Equation Presented) where the nonlinear interaction term in (Equation Presented). We suppose that the initial data MO are such that (Equation Presented) and the norm (Equation Presented) is sufficiently small. Then we prove the sharp decay estimate for the solution of the Cauchy problem (Equation Presented), for all t ≥ 1 and for every 2 < p < ∞. Furthermore we show that for 1/2 < δ < 1 there exists a unique final state u+ ∊ L² such that as t → ∞ (Equation Presented) and uniformly with respect to x (Equation Presented) where 0 denotes the Fourier transform of ϕ.