On the regularity of solutions to the $k$-generalized Korteweg-de Vries equation

Abstract

This work is concerned with special regularity properties of solutions to the $k$-generalized Korteweg-de Vries equation. In \cite{IsazaLinaresPonce} it was established that if the initial datun $u_0\in H^l((b,\infty ))$ for some $l\in {\mathbb{Z}}^{+}$ and $b\in \mathbb{R}$, then the corresponding solution $u(\cdot ,t)$ belongs to $H^l(({\hat{I}}^2,\infty ))$ for any ${\hat{I}}^2\in \mathbb{R}$ and any $t\in (0,T)$. Our goal here is to extend this result to the case where $l\in {\mathbb{R}}^{+}$.