# Dynamics of Partial Differential Equations

## Volume 17 (2020)

### 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

Van Duong Dinh (Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, Villeneuve d’Ascq, France; and Department of Mathematics, HCMC University of Pedagogy, Ho Chi Minh, Vietnam)

Sahbi Keraani (Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, Villeneuve d’Ascq, France)

Mohamed Majdoub (Department of Mathematics, College of Science, Imam Abdulrahman Bin Faisal University, Dammam, Saudi Arabia; and Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University, Dammam, Saudi Arabia)

#### 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).