Asymptotic behaviour of the solutions for a weakly damped anisotropic sixth-order Schrödinger type equation in $  \mathbb{R}^2 $

Brahim Alouini

doi:10.3934/dcdsb.2021032

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2022, Volume 27, Issue 1: 45-72. Doi: 10.3934/dcdsb.2021032

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Asymptotic behaviour of the solutions for a weakly damped anisotropic sixth-order Schrödinger type equation in $ \mathbb{R}^2 $

Brahim Alouini

University of Monastir, Faculty of Sciences, Research Laboratory: Analysis, Probability and Fractals, The Environment Avenue, 5019 Monastir, Tunisia

Received: June 2020

Revised: December 2020

Early access: February 2021

Published: January 2022

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Abstract

We study the long-time behaviour of the solutions to a nonlinear damped anisotropic sixth-order Schrödinger type equation in $ \mathbb{R}^2 $ that reads

$ u_t+i\Delta u-i \left(\partial_y^4 u-\partial_y^6 u\right)+ig(|u|^2)u+\gamma u = f\,,\;\;(t,(x,y))\in \mathbb{R}\times \mathbb{R}^2\,. $

We prove that this behaviour is described by the existence of regular global attractor in an anisotropic Sobolev space with finite fractal dimension.

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Mathematics Subject Classification: Primary: 35B40, 35Q55; Secondary: 76B03, 37L30.

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