Asymptotic properties of solutions to dispersive equation of Schrödinger type

Abstract

We find the asymptotic behavior for large time of solutions to the dispersive equations of Schrödinger type
u_t-$¥frac{i}{¥rho}$ |∂_x|^ρu = 0, (t,x) ∈ R × R,
where ρ ≥ 2. We obtain some estimates of solutions of linear problem and apply them to nonlinear problems with power nonlinearities of order p ≥ 3. The nonexistence of wave operator and existence of the modified wave operator for the critical nonlinearity i λ |u|²u are studied.