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arXiv · 2110.00194

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

Xuan Liu·Ting Zhang
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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.

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Xuan Liu, Ting Zhang. 2021-10-01. Global existence and asymptotics for the modified two-dimensional Schr\"{o}dinger equation in the critical regime. https://arxiv.org/abs/2110.00194

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