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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.

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