Wave breaking and persistence property of solutions for a weakly dissipative 3-component Camassa–Holm system

This paper is concerned with the Cauchy problem for a weakly dissipative 3-component Camassa–Holm system. We first obtain the local well-posedness of solutions by using Kato’s theory. We then derive the wave breaking mechanism of solutions, and give two wave breaking criteria for solutions by the method of characteristic and convolution estimates. Finally, we prove that if the initial data with their derivatives of the system exponentially decay at infinity, then the corresponding solution also exponentially decays at infinity.