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

Abstract

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: $${\partial }_{t}m(t,x)={\partial }_x[m(t,x)(u(t,x)-{\partial }_{x}u(t,x))(u(-t,-x)+{\partial }_x(u(-t,-x)))],$$ where $m(t,x)=\left(1-{\partial }_x^2\right)u(t,x)$. 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.

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