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| Main Authors: | , |
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| Format: | Preprint |
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2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2004.01843 |
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| _version_ | 1866908321038991360 |
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| author | Zhang, Lei Liu, Bin |
| author_facet | Zhang, Lei Liu, Bin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_01843 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | New results on the global solvability and blow-up for a class of weakly dissipative Camassa-Holm equations Zhang, Lei Liu, Bin Analysis of PDEs 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. |
| title | New results on the global solvability and blow-up for a class of weakly dissipative Camassa-Holm equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2004.01843 |