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

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  • 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.

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


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