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# Dynamics of Partial Differential Equations

## Volume 17 (2020)

### Number 4

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

Pages: 329 – 360

DOI: https://dx.doi.org/10.4310/DPDE.2020.v17.n4.a2

#### Authors

#### Abstract

In this paper, we study the focusing nonlinear Schrödinger equation with exponential nonlinearities\[\left \{\begin{align}i \partial_t u + \Delta u & = - ( e^{{4 \pi \lvert u \rvert}^2} - 1 - {4 \pi \mu \lvert u \rvert}^2 ) u, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^2 , \\u(0) & = u_0 \in H^1 ,\end{align}\right .\]where $\mu \in \lbrace 0, 1 \rbrace$. By using variational arguments, we 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 $\mu = 1$, we show the asymptotic behavior or energy scattering of global solutions by using a recent argument of Arora–Dodson–Murphy [3].

#### Keywords

nonlinear Schrödinger equation, exponential nonlinearity, ground state, scattering, blow-up

#### 2010 Mathematics Subject Classification

Primary 35Q55. Secondary 35P25.

This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01). S. K. benefited from the support of the project ODA (ANR-18-CE40-0020-02).

Received 13 February 2020

Published 16 November 2020