Propagation of regularity and decay of solutions to the $k$-generalized Korteweg-de Vries equation

Abstract

We study special regularity and decay properties of solutions to the IVP associated to the $k$-generalized KdV equations. In particular, for datum $u_0\in H^{3/4^{+}}(\mathbb{R})$ whose restriction belongs to $H^l((b,\infty ))$ for some $l\in {\mathbb{Z}}^{+}$ and $b\in \mathbb{R}$ we prove that the restriction of the corresponding solution $u(\cdot ,t)$ belongs to $H^l((\beta ,\infty ))$ for any $\beta \in \mathbb{R}$ and any $t\in (0,T)$. Thus, this type of regularity propagates with infinite speed to its left as time evolves.

References in corpus (1)

• On persistence properties in fractional weighted spaces