On the Propagation of Regularity and Decay of Solutions to thek-Generalized Korteweg-de Vries Equation

Abstract

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.

On a class of solutions to the generalized KdV type equation

“…By using an argument similar to that introduced by Cazenave and Naumkin [2] we establish the local well-posedness for a class of data in an appropriate weighted Sobolev space. Also, we show that the solutions obtained satisfy the propagation of regularity principle proven in [3] in solutions of the k-generalized KdV equation.…”

“…Note that the above formal computation is justified by arguing as in [3,Section 3]. The idea is to use the formula (3.30) and induction argument in l ∈ Z + to establish (1.14) and (1.15).…”

On the propagation of regularity and decay of solutions to the Benjamin equation

“…where The persistence of decay and regularity effects established in [12] can also be extended to the IVP (1.1). In fact, we have Theorem 1.2.…”

“…To justify the previous formal computations we refer the reader to [12] and we omit the details here.…”

“…The difference between [12] and [13] lies in the regularity of the initial data. For the k-generalized KdV equations, the initial value u 0 belongs to H 3/4 + (R), while u 0 lies in H 3/2 (R) for the Benjamin-Ono equation.…”