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Compact attractor for weakly damped driven Korteweg-de Vries equations on the real line. (English) Zbl 0928.35145

The long-time behaviour of solutions to the equation \[ u_t + u u_x + u_{xxx} + \gamma u = f \] for \(x \in\mathbb{R}\) is studied, where \(\gamma > 0\). If \(f \in H^2(\mathbb{R})\), the existence of a maximal compact attractor in \(H^2(\mathbb{R})\) is proved.
The main difficulty of the problem, namely, the lack of compactness due to the invariance of \(\mathbb{R}\) with respect to the noncompact group of translations, is overcome by means of the splitting method combined with the technique of weighted Sobolev spaces.
Reviewer: E.Feireisl (Praha)

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B41 Attractors
35B40 Asymptotic behavior of solutions to PDEs
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References:

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