Dynamics of Partial Differential Equations

Volume 17 (2020)

Number 4

A 2D Schrödinger equation with time-oscillating exponential nonlinearity

Pages: 307 – 327

DOI: https://dx.doi.org/10.4310/DPDE.2020.v17.n4.a1

Authors

A. Bensouilah (Department of Mathematics, New York University in Abu Dhabi, United Arab Emirates)

D. Draouil (Faculté des Sciences de Tunis, Département de Mathématiques, Laboratoire équations aux dérivées partielles (LR03ES04), Université de Tunis El Manar, Tunis, Tunisia)

M. 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

This paper deals with the 2-D Schrödinger equation with time-oscillating exponential nonlinearity $i \partial_t u + \Delta u = \theta (\omega t) \left (e^{{4 \pi \lvert u \rvert}^2} -1 \right)$, where $\theta$ is a periodic $C^1$-function. We prove that for a class of initial data $u_0 \in H^1 (\mathbb{R}^2)$, the solution $u_\omega$ converges, as $\lvert \omega \rvert$ tends to infinity to the solution $U$ of the limiting equation $i\partial_t U + \Delta U = I( \theta ) \left (e^{{4 \pi \lvert U \rvert}^2} -1 \right)$ with the same initial data, where $I(\theta)$ is the average of $\theta$.

Keywords

Schrödinger’s equation, time-oscillating, energy-critical regime, convergence, well-posedness, Moser–Trudinger inequalities

2010 Mathematics Subject Classification

Primary 35Q41. Secondary 35B20.

Full Text (PDF format)

Received 12 January 2020

Published 16 November 2020