Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 01, pp. 1-15.
Title: Well-posedness and ill-posedness of the fifth-order
modified KdV equation
Author: Soonsik Kwon (Univ. of California, Los Angeles, CA, USA)
Abstract:
We consider the initial value problem of the fifth-order modified
KdV equation on the Sobolev spaces.
$$\displaylines{
\partial_t u - \partial_x^5u + c_1\partial_x^3(u^3)
+ c_2u\partial_x u\partial_x^2 u + c_3uu\partial_x^3 u =0\cr
u(x,0)= u_0(x)
}$$
where $u:\mathbb{R}\times\mathbb{R} \to \mathbb{R} $ and $c_j$'s
are real. We show the local well-posedness in $H^s(\mathbb{R})$
for $s\geq 3/4$ via the contraction principle on $X^{s,b}$
space. Also, we show that the solution map from data to the
solutions fails to be uniformly continuous below
$H^{3/4}(\mathbb{R})$. The counter example is obtained by
approximating the fifth order mKdV equation by the cubic NLS
equation.
Submitted November 7, 2007. Published January 02, 2008.
Math Subject Classifications: 35Q53.
Key Words: Local well-posedness; ill-posedness; mKdV hierarchy.