Global existence and asymptotics for the modified two-dimensional Schrödinger equation in the critical regime

Abstract

We study the asymptotic behavior of the modified two-dimensional Schrödinger equation $(D_t-F(D))u=\lambda |u|u$ in the critical regime, where $\lambda \in \mathbb{C}$ with $\text{Im}\lambda \ge 0$ and $F(\xi )$ is a second order constant coefficients elliptic symbol. For any smooth initial datum of size $\epsilon \ll 1$, we prove that the solution is global-in-time, combining the vector fields method and a semiclassical analysis method introduced by Delort. Moreover, we present the pointwise decay estimates and the large time asymptotic formulas of the solution.

References in corpus (1)

• Modified scattering for the critical nonlinear Schrödinger equation