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

Lei Zhang; Bin Liu

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arXiv · 2004.01843

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

Lei Zhang·Bin Liu

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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.

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Lei Zhang, Bin Liu. 2020-04-04. New results on the global solvability and blow-up for a class of weakly dissipative Camassa-Holm equations. https://arxiv.org/abs/2004.01843

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