Well-posedness and global solutions to the higher order Camassa-Holm equations with fractional inertia operator in Besov space

Abstract

In this paper, we study well-posedness and the global solutions to the higher-order Camassa-Holm equations with fractional inertia operator in Besov space. When a ∈ [ 1 2 , 1), we prove the existence of the solutions in space B s p,1 (R) with s ≥ 1 + 1 p and p < 1 a− 1 2 , the existence and uniqueness of the solutions in space B s p,1 (R) with s ≥ 1 + 2a − min{ 1 p , 1 p ′ }, and the local well-posedness in space B s p,1 (R) with s > 1 + 2a − min{ 1 p , 1 p ′ }. When a > 1, we obtain the existence of the solutions in space B s p,1 (R) with s ≥ a + max{ 1 p , 1 2 } and the local well-posedness in space B s p,1 (R) with s ≥ 1 + a + max{ 1 p , 1 2 }. Moreover, we obtain two results about the global solutions.