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### Open Access Journals

DCDS

It is shown in both the periodic and the non-periodic cases that
the data-to-solution map for the Degasperis-Procesi (DP) equation
is not a uniformly continuous map on
bounded subsets of Sobolev spaces with exponent greater than 3/2.
This shows that continuous dependence on initial data of solutions to
the DP equation is sharp.
The proof is based on well-posedness results and approximate solutions.
It also exploits the fact that DP solutions conserve
a quantity which is equivalent to the $L^2$ norm.
Finally, it provides an outline of the local well-posedness proof
including the key estimates for the size of the solution and for the solution's lifespan that are
needed in the proof of the main result.

DCDS

It is shown that if a classical solution $(u, n)$ of
the modified Euler-Poisson equation (mEP) in one space dimension
is such that $u$, $u_x$ and $n$ are initially decaying exponentially
and for some later
time the first component $u$ is also decaying exponentially, then
$n$ must be identically equal to zero and
$u$ must be a solution to the Burgers equation. In particular, if $n$
and $u$ are initially compactly supported then $n$ can not be compactly
supported at any later time, unless $n$ is identically equal to zero and
$u$ is a solution to the Burgers equation.
It is also shown that the mEP equations are locally well-posed
in $H^s \times H^{s-1}$ for $s>5/2$.

PROC

It is shown that the solution to the Cauchy problem for the modified Korteweg-de Vries equation with initial data in an analytic Gevrey space $G^\sigma$, $\sigma \>= 1$, as a function of the spacial variable belongs to the same Gevrey space. However, considered as function of time the solution does not belong to
$G^\sigma$. In fact, it belong to $G^(3\sigma)$ and not to any Gevrey space $G^r$, 1 $\<= r$ < 3$\sigma$.

keywords:
Modied KdV equation
,
Sobolev spaces
,
periodic
,
Gevrey
regularity
,
Cauchy problem
,
mKdV

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