Abstract
For s < 3/2, it is shown that the Cauchy problem for the Degasperis-Procesi equation (DP) is ill-posed in Sobolev spaces H s . If 1/2 β€ s < 3/2, then ill-posedness is due to norm inflation. This means that there exist DP solutions who are initially arbitrarily small and eventually arbitrarily large with respect to the H s norm, in an arbitrarily short time. Since DP solutions conserve a quantity equivalent to the L 2-norm, there is no norm inflation in H 0 for these solutions. In this case, ill-posedness is caused by failure of uniqueness. For all other s < 1/2, the situation is similar to H 0. Considering that DP is locally well-posed in H s for s > 3/2, this work establishes 3/2 as the critical index of well-posedness in Sobolev spaces.
2010 Mathematics Subject Classfication:
Acknowledgements
The authors would like to thank John Derwent for constructive suggestions.