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

Abstract

We study the asymptotic behaviour of the modified two-dimensional Schrödinger equation in the critical regime, where with and is a second order constant coefficients elliptic symbol. For any smooth initial datum of size , 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.