Long time dynamics for the focusing nonlinear Schrödinger equation with exponential nonlinearities

In this paper, we study the focusing nonlinear Schrödinger equation with exponential nonlinearities $i \partial_t u + Δu = - \left(e^{4π|u|^2} - 1 - 4πμ|u|^2 \right) u, \quad u(0) = u_0 \in H^1, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^2,$ where $μ\in \{0, 1\}$. 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 $μ=1$, 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 $H^1$.

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