Geodesic
Well-posedness and ill-posedness of the fifth-order modified KdV equation
Electronic Journal of Differential Equations, Tome 2008 (2008).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Classification : 35Q53
Keywords: local well-posedness, ill-posedness, mkdv hierarchy 
@article{EJDE_2008__2008__a13,
     author = {Kwon, Soonsik},
     title = {Well-posedness and ill-posedness of the fifth-order modified {KdV} equation},
     journal = {Electronic Journal of Differential Equations},
     publisher = {mathdoc},
     volume = {2008},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2008__2008__a13/}
}
TY  - JOUR
AU  - Kwon, Soonsik
TI  - Well-posedness and ill-posedness of the fifth-order modified KdV equation
JO  - Electronic Journal of Differential Equations
PY  - 2008
VL  - 2008
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/EJDE_2008__2008__a13/
LA  - en
ID  - EJDE_2008__2008__a13
ER  - 
%0 Journal Article
%A Kwon, Soonsik
%T Well-posedness and ill-posedness of the fifth-order modified KdV equation
%J Electronic Journal of Differential Equations
%D 2008
%V 2008
%I mathdoc
%U http://geodesic.mathdoc.fr/item/EJDE_2008__2008__a13/
%G en
%F EJDE_2008__2008__a13
Kwon, Soonsik. Well-posedness and ill-posedness of the fifth-order modified KdV equation. Electronic Journal of Differential Equations, Tome 2008 (2008). http://geodesic.mathdoc.fr/item/EJDE_2008__2008__a13/