The local well-posedness, blow-up phenomena and ill-posedness of a new fifth-order Camassa-Holm type equation

2025/10/15 by Y.-S. Lian, Zhaoyang Yin, Lian, Yiyao +1
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2510.13164

openalex publication_date 2025/10/15 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28

Abstract

In this paper, we study a new fifth-order Camassa-Holm type equation derived by Li \citeLi.Z. We firstly establish the local well-posedness in the sense of Hadamard for the Cauchy problem of the new fifth-order Camassa-Holm type equation in Besov spaces. Secondly, we obtain blow-up criteria. Building upon this, by utilizing the conservation laws and establishing local boundedness, we derive a blow-up result that precisely determines the blow-up time. Finally, the ill-posedness of the new fifth-order Camassa-Holm type equation in the critical Sobolev space H^(1)/(2) is established via a norm inflation argument.

Citations

• On the constants for some Sobolev imbeddings
• Multipeakons and the Classical Moment Problem
• Wave breaking for nonlinear nonlocal shallow water equations
• Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation
• Symplectic structures, their Bäcklund transformations and hereditary symmetries

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• Ill-posedness for the periodic Camassa--Holm type equations in the end-point critical Besov space B¹_∞,1