Well-posedness and persistence properties for two-component higher order Camassa–Holm systems with fractional inertia operator

In this paper, we study the Cauchy problem for a two-component higher order Camassa–Holm systems with fractional inertia operator A=(1−∂x2)r,r≥1, which was proposed by Escher and Lyons (2015). By the transport equation theory and Littlewood–Paley decomposition, we confirm the local well-posedness of solutions for the system in nonhomogeneous Besov spaces Bp,qs×Bp,qs−2r+1 with 1≤p,q≤+∞1≤p,q≤+∞ and the Besov index s>max{2r+1p,2r+1−1p}. Moreover, we demonstrate the local well-posedness in the critical Besov space B2,12r+12×B2,132. On the other hand, the propagation behavior of compactly supported solutions is examined, namely whether solutions which are initially compactly supported will retain this property throughout their time of evolution. Finally, we also establish the persistence properties of the solutions to the two-component Camassa–Holm equation with r=1r=1 in weighted Lϕp:=Lp(R,ϕp(x)dx) spaces for a large class of moderate weights.