We study the scattering problem and asymptotics for large time of solutions to the Cauchy problem for the nonlinear schrödinger and Hartree type equations wit h subcritical nonlinearities [Formula] where the nonlinear interaction term is F(|u|2) = V * |u|2, V(x) = λ|x|− δ, λ ∈ R, 0 < δ < 1 in the Hartree type case, or F(|u|2) = λ|t|1 −δ|u|2 in the case of the cubic nonlinear Schrödinger equation. We suppose that the initial data eβ|x|u0 ∈ L2, β > 0 with sufficiently small norm ε = ||eβ|x|u0||L2. Then we prove the sharp decay estimate ||u(t)||Lp ≤ C ϵt 1/p − 1/2, for all t ≥ 1 and for every 2 ≤ p ≤ ∞. Furthermore we show that for 1/2 < δ < 1 there exists a unique final state û + ∈ L2 such that for all t ≥ 1 [Formula] and uniformly with respect to x [formula] where ϕ ^ denotes the Fourier transform of ϕ. Our results show that the regularity condition on the initial data which was assumed in the previous paper  is not needed. Also a smoothing effect for the solutions in an analytic function space is discussed.
|Number of pages||17|
|Journal||Hokkaido Mathematical Journal|
|Publication status||Published - 1998|
- Nonlinear Schrödinger
- Subcritical case
ASJC Scopus subject areas