Blow-up phenomena for the generalized FORQ/MCH equation

The peakons and mulit-peakons for a generalized cubic–quintic Camassa–Holm type equation have been obtained by Weng et al. (Monatsh Math, 2022. https://doi.org/10.1007/s00605-022-01699-w). In this paper, by constructing certain Lyapunov functionals, we prove that the peakons were orbitally stable in the energy space. Furthermore, using energy argument and combining the method of the orbital stability of peakons with monotonicity of the local energy norm, we also prove that the sum of N sufficiently decoupled peakons is orbitally stable in the energy space.

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“…Moreover, the local well-posedness and the blow-up of Eq. (1.6) were considered [28,29]. The orbital stability of peakons and N-peakons of Eq.…”

Section: Introductionmentioning

confidence: 99%

Orbital Stability of Peakons and Multi-peakons for a Generalized Cubic–Quintic Camassa–Holm Type Equation

Deng

Chen

2022

J Nonlinear Math Phys

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“…The local well-posedness of the equation (1) in Besov spaces B s p,r with s > max{2 + 1 p , 5 2 } was studied in [13], where two blow-up criterions were also provided. The strong solutions to the gmCH equation (1) will blow up in finite time for some initial data; see [14]. Hence, it is important to find some proper spaces and study global weak solutions to the gmCH equation (1), which is also one of our purposes in this paper.…”

mentioning

confidence: 99%

“…Hence, (16) holds. For some initial data m 0 ∈ H s (R) with s > 1 2 , the classical solutions will blow up in finite time; see [14,Theorem 3.3]. Hence, the natural questions are in which space we can extend and how to extend the solutions globally after T max .…”

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confidence: 99%

Global weak solutions to the generalized mCH equation via characteristics

Zeng¹,

Gao²,

Xue³

2022

DCDS-B

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<p style='text-indent:20px;'>In this paper, we study the generalized modified Camassa-Holm (gmCH) equation via characteristics. We first change the gmCH equation for unknowns <inline-formula><tex-math id="M1">\begin{document}$ (u,m) $\end{document}</tex-math></inline-formula> into its Lagrangian dynamics for characteristics <inline-formula><tex-math id="M2">\begin{document}$ X(\xi,t) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ \xi\in\mathbb{R} $\end{document}</tex-math></inline-formula> is the Lagrangian label. When <inline-formula><tex-math id="M4">\begin{document}$ X_\xi(\xi,t)>0 $\end{document}</tex-math></inline-formula>, we use the solutions to the Lagrangian dynamics to recover the classical solutions with <inline-formula><tex-math id="M5">\begin{document}$ m(\cdot,t)\in C_0^k(\mathbb{R}) $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M6">\begin{document}$ k\in\mathbb{N},\; \; k\geq1 $\end{document}</tex-math></inline-formula>) to the gmCH equation. The classical solutions <inline-formula><tex-math id="M7">\begin{document}$ (u,m) $\end{document}</tex-math></inline-formula> to the gmCH equation will blow up if <inline-formula><tex-math id="M8">\begin{document}$ \inf_{\xi\in\mathbb{R}}X_\xi(\cdot,T_{\max}) = 0 $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M9">\begin{document}$ T_{\max}>0 $\end{document}</tex-math></inline-formula>. After the blow-up time <inline-formula><tex-math id="M10">\begin{document}$ T_{\max} $\end{document}</tex-math></inline-formula>, we use a double mollification method to mollify the Lagrangian dynamics and construct global weak solutions (with <inline-formula><tex-math id="M11">\begin{document}$ m $\end{document}</tex-math></inline-formula> in space-time Radon measure space) to the gmCH equation by some space-time BV compactness arguments.</p>

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On the well-posedness of the Cauchy problem for the two-component peakon system in $$C^k\cap W^{k,1}$$

Karlsen,

Rybalko

2024

Z. Angew. Math. Phys.

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This study focuses on the Cauchy problem associated with the two-component peakon system featuring a cubic nonlinearity, constrained to the class $$(m,n)\in C^{k}(\mathbb {R}) \cap W^{k,1}(\mathbb {R})$$ ( m , n ) ∈ C k ( R ) ∩ W k , 1 ( R ) with $$k\in \mathbb {N}\cup \{0\}$$ k ∈ N ∪ { 0 } . This system extends the celebrated Fokas–Olver–Rosenau–Qiao equation and the following nonlocal (two-place) counterpart proposed by Lou and Qiao: $$\begin{aligned} \partial _t m(t,x)= \partial _x[m(t,x)(u(t,x)-\partial _xu(t,x)) (u(-t,-x)+\partial _x(u(-t,-x)))], \end{aligned}$$ ∂ t m ( t , x ) = ∂ x [ m ( t , x ) ( u ( t , x ) - ∂ x u ( t , x ) ) ( u ( - t , - x ) + ∂ x ( u ( - t , - x ) ) ) ] , where $$m(t,x)=\left( 1-\partial _{x}^2\right) u(t,x)$$ m ( t , x ) = 1 - ∂ x 2 u ( t , x ) . Employing an approach based on Lagrangian coordinates, we establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map in the class $$C^k\cap W^{k,1}$$ C k ∩ W k , 1 . Moreover, we derive criteria for blow-up of the local solution in this class.

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Blow-up phenomena for the generalized FORQ/MCH equation

Cited by 6 publications

References 37 publications

Orbital Stability of Peakons and Multi-peakons for a Generalized Cubic–Quintic Camassa–Holm Type Equation

Orbital Stability of Peakons and Multi-peakons for a Generalized Cubic–Quintic Camassa–Holm Type Equation

Global weak solutions to the generalized mCH equation via characteristics

On the well-posedness of the Cauchy problem for the two-component peakon system in $$C^k\cap W^{k,1}$$

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