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On the generalized Korteweg-de Vries type equations. (English) Zbl 0891.35135

The author investigates the Cauchy problem in \(\mathbb R^1\) for the generalized Korteweg-de Vries (KdV) equation with nonlinear term of the form \(F(u)\partial _x u\), and for an equation of mixed KdV and Schrödinger type. For the former equation the author proves the local well-posedeness assuming \(F(u)=u^2 g(u)\) with \(g(u)\) smooth, or global existence for \(F(u)\) smooth enough and small in \(H^s\). For the latter equation, local well-posedness is proved. In all cases mentioned here, the solution is shown to be (at least) continuous in time with values in an appropriate fractional Sobolev space \(H^s\). The local well-posedness means existence, uniqueness, and continuous dependence on the initial condition with the time of existence depending on the \(H^s\)-norm of \(u_0\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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