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On the well-posedness for the generalized Ostrovsky, Stepanyants and Tsimring equation. (English) Zbl 1079.35088

Summary: We prove that the initial value problem associated with equation \[ u_t +u_{xxx}-\eta({\mathcal H}u_x+{\mathcal H}u_{xxx})+u^ku_x =0,\quad x\in\mathbb{R},\;t\geq 0, \] where \(\eta>0\) and \({\mathcal H}\) denotes the usual Hilbert transform, is locally well-posed in Sobolev spaces \(H^s(\mathbb{R})\) for \(s\geq 0\) and \(k=1,2,3\) and globally well posed in \(L^2(\mathbb{R})\).

MSC:

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