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
Publié le : 2010-07-15
Classification:
Analytic,
smoothing effect,
fifth order KdV equation,
KdV hierarchy,
35Q53
@article{1279719309,
author = {Tomoeda, Kyoko},
title = {Analyticity and smoothing effect for the fifth order KdV type equation},
journal = {Proc. Japan Acad. Ser. A Math. Sci.},
volume = {86},
number = {1},
year = {2010},
pages = { 101-106},
language = {en},
url = {http://dml.mathdoc.fr/item/1279719309}
}
Tomoeda, Kyoko. Analyticity and smoothing effect for the fifth order KdV type equation. Proc. Japan Acad. Ser. A Math. Sci., Tome 86 (2010) no. 1, pp. 101-106. http://gdmltest.u-ga.fr/item/1279719309/