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On the (generalized) Korteweg-de Vries equation. (English) Zbl 0795.35105
The well-posedness of the initial value problem for the Korteweg-de Vries equation $u\sb t+ u\sb{xxx}+ uu\sb x =0$ and its generalized form $u\sb t+ u\sb{xxx}+ a(u) u\sb x=0$ in the classical Sobolev spaces and the regularity of their solutions in $L\sb s\sp p$ spaces are studied. A global smoothing effect of the solutions of these equations is also proved. See also a paper by {\it T. Kato} [Studies in applied mathematics, Adv. Math., Suppl. Stud., Vol. 8, 93-128 (1983; Zbl 0508.00010)].

##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35B65 Smoothness and regularity of solutions of PDE
Full Text:
##### References:
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