On Solutions to a Two‐Component Generalized Camassa‐Holm Equation

We consider a two‐component Camassa–Holm system which arises in shallow water theory. We analyze a wave breaking mechanism and the global existence of solutions. First, we discuss the local well posedness and a blow up mechanism, then establish some new blow up criteria for this system formulated either on the line or with space‐periodic initial conditions. Finally, the existence of global solutions is analyzed.