ABSTRACT. We study special regularity and decay properties of solutions to the IVP associated to the k-generalized KdV equations. In particular, for datum u 0 ∈ H 3/4 + (R) whose restriction belongs to H l ((b, ∞)) for some l ∈ Z + and b ∈ R we prove that the restriction of the corresponding solution u(·,t) belongs to H l ((β , ∞)) for any β ∈ R and any t ∈ (0, T ). Thus, this type of regularity propagates with infinite speed to its left as time evolves.

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We prove special decay properties of solutions to the initial value problem associated to the k-generalized Korteweg-de Vries equation. These are related with persistence properties of the solution flow in weighted Sobolev spaces and with sharp unique continuation properties of solutions to this equation. As application of our method we also obtain results concerning the decay behavior of perturbations of the traveling wave solutions as well as results for solutions corresponding to special data.1991 Mathematics Subject Classification. Primary: 35Q53. Secondary: 35B05.

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