The Cauchy problem for the generalized Degasperis-Procesi equation

In this paper, we investigate the Cauchy problem for the generalized Degasperis-Procesi equation in a Besov space. Firstly, we prove that the generalized Degasperis-Procesi equation is locally well posed in B-p,r(s) with s > 1+ 1/p (or s >= 1 + 1/p if r = 1 with p is an element of [1,+infinity)). Secondly, we prove that the generalized Degasperis-Procesi equation possesses the peaked solitary wave which is the weak solution to the generalized Degasperis-Procesi equation. Thirdly, we prove that the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in B-2,infinity(3/2).Fourthly, we prove that the data-to-solution map for the generalized Degasperis-Procesi equation is not uniformly continuous in H-s(R) with s < 3/ 2. Finally, we give a blow-up criterion.