The Cauchy Problem for a Fifth-Order Dispersive Equation

Abstract

This paper is devoted to studying the Cauchy problem for a fifth-order equation. We prove that it is locally well-posed for the initial data in the Sobolev space $H^s(\mathbf{R})$ with $s\ge 1/4$. We also establish the ill-posedness for the initial data in $H^s(\mathbf{R})$ with . Thus, the regularity requirement for the fifth-order dispersive equations $s\ge 1/4$ is sharp.