On the Cauchy problem for the Degasperis-Procesi equation

Abstract

Abstract. We establish here the local well-posedness for the Degasperis-Procesi equation in the Besov spaces. We also determine some blow-up criteria of the strong solutions and investigate the nonexistence of smooth solitary-wave solutions.

Remarks on the well-posedness of Camassa–Holm type equations in Besov spaces

“…According to [32] and Corollary 3.4, if r < ∞, the solution is continuous with respect to initial data u 0 ∈ B s p,r . Else if r = ∞, we have the solution is weak continuous with respect to initial data u 0 ∈ B s p,r .…”

“…Its traveling wave solutions was investigated in [38,47]. The local well-posedness of the Cauchy problem of the DP equation in Sobolev spaces and Besov spaces was studied in [32,35,54]. Similar to the CH equation, the DP equation has also global strong solutions [40,55,57] and finite time blow-up solutions [26,27,40,41,[54][55][56][57].…”

“…In most of papers the authors just proved the local existence and uniqueness of solutions of these equations when the initial data belongs to Besov spaces B s p,r (see [34,32,35,50]). For the continuity of the solution map of these equations with respect to the initial data, it was proved in the spaces C([0, T ]; B s p,r ), s < s.…”

Global weak solutions for a generalized Camassa–Holm equation

“…For the DP equation, the local well-posedness of the Cauchy problem in Sobolev spaces and Besov spaces was established in [24,25,27,44]. The DP equation also has global strong solutions [29,45,47] and finite time blow-up solutions [20,21,29,30,[44][45][46][47].…”

Non-uniform dependence for a generalized Degasperis–Procesi equation

“…Its traveling wave solutions were investigated in [31,41]. The local wellposedness of the Cauchy problem of the DP equation in Sobolev spaces and Besov spaces was established in [28,29,48]. Similar to the CH equation, the DP equation has also global strong solutions [34,49,51] and finite time blow-up solutions [25,26,34,35,48,49,50,51].…”