Analyticity and smoothing effect for the fifth order KdV type equation

Abstract

We consider the initial value problem for the reduced fifth order KdV type equation: $\partial_{t}u-\partial_{x}^{5}u-10\partial_{x}(u^{3})+5\partial_{x}(\partial_{x}u)^{2}=0$ which is obtained by removing the nonlinear term $10\partial_{x}(u\partial_{x}^{2} u)$ from the fifth order KdV equation. We show the existence of the local solution which is real analytic in both time and space variables, if the initial data $\phi\in H^{s}(\mathbf{R})$ $(s>1/8)$ satisfies the condition \begin{equation*} ∑_{k=0}^{∞}\frac{A_{0}^{k}}{k!}{\|}(x\partial_{x})^{k}φ{\|}_{H^{s}}<{∞}, \end{equation*} for some constant $A_{0}(0<A_{0}<1)$. Moreover, the smoothing effect for this equation is obtained. The proof of our main result is based on the argument used in [5].