Local well-posedness for Kawahara equation

Takamori Kato

Abstract

We consider the Cauchy problem for the Kawahara equation, which is a fifth-order KdV equation. This paper establishes the local well-posedness with initial data given in the Sobolev space $H^s(\mathbb{R})$. Previously, Chen, Li, Miao, and Wu (2009) proved the local well-posedness for $s>-7/4$, which has been improved to $s \geq -7/4$ by Chen and Guo. We improve this result to $s \geq -2$. The main idea is to modify the Bourgain space. Similar arguments are used by Bejenaru and Tao (2006). Moreover, we prove ill-posedness for $s<-2$ by using the argument by Bejenaru and Tao (2006).

Article information

Source
Adv. Differential Equations Volume 16, Number 3/4 (2011), 257-287.

Dates
First available in Project Euclid: 18 December 2012

Mathematical Reviews number (MathSciNet)
MR2767079

Zentralblatt MATH identifier
1298.35176

Citation

Kato, Takamori. Local well-posedness for Kawahara equation. Adv. Differential Equations 16 (2011), no. 3/4, 257--287. https://projecteuclid.org/euclid.ade/1355854309.