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

Lei Zhang, Bin Liu

TL;DR

The paper addresses global solvability and finite-time blow-up for a class of weakly dissipative two-component Camassa-Holm-type equations in Besov spaces, extending results beyond Sobolev spaces and removing sign constraints on initial momentum. By reformulating the system through a damped, time-dependent coefficient framework with $\widetilde{m}=e^{\lambda t}m$ and $\widetilde{\sigma}=e^{\lambda t}\sigma$, and employing a nonstandard Friedrichs-type iterative scheme, the authors establish global well-posedness in Besov spaces under smallness conditions on the integrated time-dependent parameters. They also derive two blow-up criteria, connecting dissipation and coefficient dynamics to singularity formation, and provide a precise blow-up condition under sign assumptions on the coefficients, illustrating how damping governs wave-breaking behavior. Together, these results advance the understanding of dissipative CH-type systems by providing Besov-space well-posedness, explicit damping-dependent bounds, and sharp blow-up characterizations with implications for stability and long-time dynamics.

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 $λ>0$. 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 $λ$ influences the singularity formation of strong solutions.

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

TL;DR

The paper addresses global solvability and finite-time blow-up for a class of weakly dissipative two-component Camassa-Holm-type equations in Besov spaces, extending results beyond Sobolev spaces and removing sign constraints on initial momentum. By reformulating the system through a damped, time-dependent coefficient framework with and , and employing a nonstandard Friedrichs-type iterative scheme, the authors establish global well-posedness in Besov spaces under smallness conditions on the integrated time-dependent parameters. They also derive two blow-up criteria, connecting dissipation and coefficient dynamics to singularity formation, and provide a precise blow-up condition under sign assumptions on the coefficients, illustrating how damping governs wave-breaking behavior. Together, these results advance the understanding of dissipative CH-type systems by providing Besov-space well-posedness, explicit damping-dependent bounds, and sharp blow-up characterizations with implications for stability and long-time dynamics.

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 . 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 influences the singularity formation of strong solutions.

Paper Structure

This paper contains 4 sections, 11 theorems, 163 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Global well-posedness in Besov spaces
  4. Blow-up phenomena

Key Result

Theorem 1.1

Let $s>\max\{1+\frac{1}{p},\frac{3}{2}\}$ and $p,r\in [0,\infty]$. Assume that $(u_0,\rho_0)\in B_{p,r}^s(\mathbb{K})\times B_{p,r}^{s-1}(\mathbb{K})$, and the time-dependent parameters $\alpha,\beta,\gamma,\xi\in L^1([0,\infty);\mathbb{R})$ such that where Then for any $T> 0$, the system 1.6 has a unique global strong solution Moreover, the data-to-solution map $(u_0,\sigma_0)\mapsto (u,\sigma

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1: 64
  • Definition 2.2: 64
  • Lemma 2.3: 64
  • Lemma 2.4: 64
  • ...and 8 more