Long time dynamics for the focusing nonlinear Schr\"odinger equation with exponential nonlinearities

TL;DR

This paper investigates the long-term behavior of solutions to a focusing nonlinear Schrödinger equation with exponential nonlinearities in two dimensions, establishing thresholds for global existence, blow-up, and scattering.

Contribution

It derives sharp criteria for global existence and blow-up, and analyzes long-time dynamics, including scattering and vanishing solutions, for the exponential NLS.

Findings

Sharp thresholds for global existence and blow-up.

Existence of solutions that vanish locally at infinity.

Identification of scattering behavior for global solutions.

Abstract

In this paper, we study the focusing nonlinear Schr\"odinger equation with exponential nonlinearities \[ i \partial_t u + \Delta u = - \left(e^{4\pi |u|^2} - 1 - 4\pi \mu |u|^2 \right) u, \quad u(0) = u_0 \in H^1, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^2, \] where $\mu \in \{0, 1\}$. By using variational arguments, we first derive invariant sets where the global existence and finite time blow-up occur. In particular, we obtain sharp thresholds for global existence and finite time blow-up. In the case $\mu=1$, by adapting a recent argument of Arora-Dodson-Murphy \cite{ADM}, we study the long time dynamics of global solutions. It turns out that either there exist $t_n\rightarrow +\infty$ and $R_n \rightarrow \infty$ such that $u(t_n)$ vanishes inside $B(0,R_n)$ for all $n\geq 1$ or the solution scatters in $H^1$.