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 $u_t-\frac{i}{\rho }|{\partial }_x{|}^{\rho }u=0,(t,x)\in R\times R,$where $\rho \ge 2$. We obtain some estimates of solutions of linear problem and apply them to nonlinear problems with power nonlinearities of order $p\ge 3$. The nonexistence of wave operator and existence of the modified wave operator for the critical nonlinearity $i\lambda |u{|}^{2}u$ are studied.