Advances in Mathematical Physics
Volume 2011 (2011), Article ID 238138, 39 pages
http://dx.doi.org/10.1155/2011/238138
Research Article

Local Analyticity in the Time and Space Variables and the Smoothing Effect for the Fifth-Order KdV-Type Equation

Kyoko Tomoeda

Graduate School of Humanities and Sciences, Nara Women's University, Nara 630-8506, Japan

Received 6 October 2010; Revised 23 December 2010; Accepted 26 January 2011

Academic Editor: M. Lakshmanan

Copyright © 2011 Kyoko Tomoeda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the initial value problem for the reduced fifth-order KdV-type equation: 𝜕 𝑡 𝑢 − 𝜕 5 𝑥 𝑢 − 1 0 𝜕 𝑥 ( 𝑢 3 ) + 1 0 𝜕 𝑥 ( 𝜕 𝑥 𝑢 ) 2 = 0 , 𝑡 , 𝑥 ∈ ℝ , 𝑢 ( 0 , 𝑥 ) = 𝜙 ( 𝑥 ) , 𝑥 ∈ ℝ . This equation is obtained by removing the nonlinear term 1 0 𝑢 𝜕 3 𝑥 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 𝜙 ∈ 𝐻 𝑠 ( ℝ ) ( 𝑠 > 1 / 8 ) satisfies the condition ∑ ∞ 𝑘 = 0 ( 𝐴 𝑘 0 / 𝑘 ! ) ‖ ( 𝑥 𝜕 𝑥 ) 𝑘 𝜙 ‖ 𝐻 𝑠 < ∞ , 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 contraction principle and the bootstrap argument used in the third-order KdV equation (K. Kato and Ogawa 2000). The key of the proof is to obtain the estimate of 𝜕 𝑥 ( 𝜕 𝑥 𝑢 ) 2 on the Bourgain space, which is accomplished by improving Kenig et al.'s method used in (Kenig et al. 1996).