This paper is devoted to studying a two-component peakon system with cubic nonlinearity, which extension of the Camassa–Holm equation. We first discuss local well-posedness for Cauchy problem system. Then, in light fine structure system, we present precise blow-up scenario strong solutions and derive new result respect initial data. Finally, are discussed as well.

Considered herein is the Cauchy problem of two-component Novikov system. In periodic case, we first constructed an approximate solution sequence that possesses nonuniform dependence property; then, by applying energy methods, managed to prove difference between and actual negligible, thus succeeding in proving result both supercritical Besov spaces Bp,rs(T)×Bp,rs(T) with s>max{32,1+1p},1≤p≤∞...