Global existence and asymptotics for the modified two-dimensional Schr\"{o}dinger equation in the critical regime

TL;DR

This paper investigates the global existence, decay, and asymptotic behavior of solutions to a modified 2D Schrödinger equation with critical nonlinearity, using advanced analytical techniques.

Contribution

It establishes global solutions and detailed asymptotics for the critical 2D Schrödinger equation with complex nonlinearity, combining vector fields and semiclassical analysis.

Findings

Solutions exist globally for small initial data.

Derived pointwise decay estimates for solutions.

Provided large-time asymptotic formulas.

Abstract

We study the asymptotic behavior of the modified two-dimensional Schr\"{o}dinger equation $ (D_t -F(D))u=\lambda|u| u$ in the critical regime, where $\lambda \in \mathbb{C}$ with $\text{Im} \lambda \ge0$ and $F(\xi)$ is a second order constant coefficients elliptic symbol. For any smooth initial datum of size $\varepsilon\ll1$, 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.