New results on the global solvability and blow-up for a class of weakly dissipative Camassa-Holm equations

TL;DR

This paper establishes global existence and blow-up criteria for a class of weakly dissipative Camassa-Holm equations in Besov spaces, expanding understanding of their solutions' behavior with respect to dissipation parameters.

Contribution

It introduces a novel approach transforming the equations into damped forms and extends results to Besov spaces without initial data sign restrictions.

Findings

Global strong solutions exist under certain dissipation conditions.

Two blow-up criteria are derived based on dissipation influence.

Results generalize previous Sobolev space findings to Besov spaces.

Abstract

In this paper, we consider the Cauchy problem for a class of weakly dissipative Camassa-Holm equations in nonhomogeneous Besov spaces. First, we prove that the Cauchy problem admits a unique global strong solution in Besov spaces with proper condition on the dissipation parameter $\lambda>0$. The novel ingredients in the proof lies in transforming the equations into a class of damped Camassa-Holm equations, and performing a non-standard iterative method. It is shown that our result holds for the damped equations with more general time-dependent parameters, which improves the existed results from Sobolev spaces to Besov spaces without assuming any sign condition on the initial data. Second, we derive two kinds of blow-up criteria in suitable Sobolev spaces, which in some sense inform us how the dissipation parameter $\lambda$ influences the singularity formation of strong solutions.