The Solution Properties for a Four-component Degenerated Camassa-Holm Equation

As a model for surface water waves in the shallow water regime, the Camassa-Holm equation attracts much attention and interest from scholars both at home and abroad. This paper deals with the Cauchy problem of a four-component Camassa-Holm equation. First, a blow-up scenario is established by making use of the Hölder's inequality, Young inequality and Gronwall's inequality. Then the persistence property of the strong solution within its lifespan is investigated. Especially, we prove that the solution decays exponentially with the same exponent as that of the initial data.