Wave breaking and global solutions of the weakly dissipative periodic Camassa-Holm type equation

In this paper, we mainly study several problems on the weakly dissipative periodic Camassa-Holm type equation with quadratic and cubic non-linearities. First, in the periodic setting, the local well-posedness of solutions in the Sobolev space is established via Kato's theory. Then, we establish a relation between the behavior of u(x) and wave breaking phenomena of solutions, and give a sufficient condition on the initial data to guarantee the occurrence of wave breaking. Finally, we obtain the global existence of solutions. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

In this paper, we focus on weakly dissipative periodic Camassa-Holm type equations with quadratic and cubic non-linearities. We establish local well-posedness of solutions in the Sobolev space, analyze the relation between the behavior of u(x) and wave breaking phenomena of solutions, and provide a sufficient condition on initial data for wave breaking to occur. Additionally, we prove the global existence of solutions.