On the well-posedness of the Cauchy problem for the two-component peakon system in Ck ∧ Wk,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 ) is an element of C-k boolean AND W-k,W-1 . This system extends the celebrated Fokas-Olver-Rosenau-Qiao equation and the following nonlocal (two-place) counterpart proposed by Lou and Qiao: partial derivative t m ( t , x ) = partial derivative x [ m ( t , x ) ( u ( t , x ) - partial derivative x u ( t , x ) ) ( u ( - t , - x ) + partial derivative x ( u ( - t , - x ) ) ) ] , where m ( t , x ) = 1 - partial derivative (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 boolean AND W-k,W-1 , Moreover, we derive criteria for blow-up of the local solution in this class.