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

Abstract

<p style='text-indent:20px;'>We study the long-time behaviour of the solutions to a nonlinear damped anisotropic sixth-order Schrödinger type equation in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> that reads</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ 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\,. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>We prove that this behaviour is described by the existence of regular global attractor in an anisotropic Sobolev space with finite fractal dimension.</p>