The Cauchy problem for a fifth order KdV equation in weighted Sobolev spaces

Abstract

In this work we study the initial value problem (IVP) for the fifth order KdV equations, \begin{align*} \partial_{t}u+\partial_{x}^{5}u+u^k\partial_{x}u=0,\text{} & \quad x,t\in \mathbb R, \quad k=1,2, \end{align*} in weighted Sobolev spaces $H^s(\mathbb{R})\cap L^2(⟨x{⟩}^{2r}dx)$. We prove local and global results. In the case $k=2$ we point out the relation between decay and regularity of the solution of the IVP.

24 pages

References in corpus (3)

• The IVP for the dispersion generalized Benjamin-Ono equation in weighted Sobolev spaces
• On decay properties of solutions of the $k$-generalized KdV equations
• Well-posedness results for the dispersion generalized Benjamin-Ono equation via the contraction principle