On the well-posedness of the Cauchy problem for the two-component peakon system in $C^k\cap W^{k,1}$

This study focuses on the Cauchy problem associated with the two-component peakon system featuring a cubic nonlinearity, constrained to the class $(m,n)\in C^k(\mathbb{R})\cap W^{k,1}(\mathbb{R})$ with $k\in \mathbb{N}\cup \{0\}$.This system extends the celebrated Fokas-Olver-Rosenau-Qiao equation, and the following nonlocal (two-place) counterpart proposed by Lou and Qiao: $${\mathrm{\partial }}_{t}m(t,x)={\mathrm{\partial }}_x[m(t,x)(u(t,x)-{\mathrm{\partial }}_{x}u(t,x))(u(-t,-x)+{\mathrm{\partial }}_x(u(-t,-x)))],$$ where $m(t,x)=(1-{\mathrm{\partial }}_x^2)u(t,x)$. Employing an approach based on Lagrangian coordinates, we establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map in the class $C^k\cap W^{k,1}$. Moreover, we derive criteria for blow-up of the local solution in this class.