Proceedings of the Japan Academy, Series A, Mathematical Sciences

Analyticity and smoothing effect for the fifth order KdV type equation

Kyoko Tomoeda

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].

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 86, Number 7 (2010), 101-106.

Dates
First available in Project Euclid: 21 July 2010

https://projecteuclid.org/euclid.pja/1279719309

Digital Object Identifier
doi:10.3792/pjaa.86.101

Mathematical Reviews number (MathSciNet)
MR2663650

Zentralblatt MATH identifier
1205.35276

Subjects

Citation

Tomoeda, Kyoko. Analyticity and smoothing effect for the fifth order KdV type equation. Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 7, 101--106. doi:10.3792/pjaa.86.101. https://projecteuclid.org/euclid.pja/1279719309

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