Global attractor for the weakly damped driven Schrödinger equation in
$ H^2 (\mathbb{R}) $ |
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Authors: | Nikos I Karachalios Nikos M Stavrakakis |
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Institution: | Department of Applied Mathematics, National Technical University, Zografou Campus 157, 80 Athens, Greece, GR
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Abstract: | We discuss the asymptotic behaviour of the Schrödinger equation ¶¶$ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, x \in \mathbb{R}, t \geq 0,\alpha,k>0 $ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, x \in \mathbb{R}, t \geq 0,\alpha,k>0 ¶¶ with the initial condition u(x,0) = u0 (x) u(x,0) = u_0 (x) . We prove existence of a global attractor in\ H2 (\mathbbR) H^2 (\mathbb{R}) , by using a decomposition of the semigroup in weighted Sobolev spaces to overcome the noncompactness of the classical Sobolev embeddings. |
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