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On the well-posedness and scattering for the transitional Benjamin-Ono equation. (English) Zbl 0854.35105

We consider the Cauchy problem for the transitional Benjamin-Ono equation \[ \partial_t u+ \sigma \partial^2_x u+ f(t) u\partial_x u= 0,\quad u(t_0, x)= \phi(x),\tag{1} \] where \(\sigma\) is the Hilbert transform, i.e., \((\sigma\varphi) (x)=(1/\pi)\) p.v. \(\int_{\mathbb{R}} {\varphi(y)\over y- x} dy\).
Our purpose is to investigate the local and the global well-posedness of the problem (1) in the Sobolev spaces \(H^s(R)\), when \(s\geq 3/2\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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