ON THE PROPAGATION OF REGULARITY OF SOLUTIONS TO THE NONLINEAR FIFTH ORDER EQUATION OF KDV TYPE

We investigate special regularity of solutions to the initial value problem associated to the nonlinear fifth order equation of KdV type. The main results show that for datum $u_{0} \in H^{s}(\mathbf{R}), F(u) \in C^{s+2}(\mathbf{R})$ with $s\geq5$, whose restriction belongs to $H^{l}((x_{0},\infty))$ and $H^{l+2}((x_{0},\infty))$ respectively, for some $l \in \mathbb{Z}^{+}$ and $x_{0}\in \mathbf{R}$, then the restriction of the corresponding solution $u(\cdot,t)$ belongs to $H^{l}((b,\infty))$ for any $b\in\mathbf{R}$ and any $t\in (0,T)$. Thus, this type of regularity travels with infinite speed to its left as time evolves. To a certain extent, our results complement the previous studies on the related aspects, and deepen the understanding of such properties for the dispersion equation.