A new blow-up criterion for the <i>N</i> – <i>abc</i> family of Camassa-Holm type equation with both dissipation and dispersion

Zhang, Zaiyun; Li, Limei; Fang, Chunhua; He, Fan; Huang, Chuangxia; Wen, Zhu

doi:10.1515/math-2020-0012

Cited by 8 publications

(6 citation statements)

References 45 publications

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“…en, the propagation speed with compact supported initial datum was investigated, which improves early results [40][41][42]. On the other hand, the authors [39] constructed blowup threshold for a positive solution in finite time under certain condition of initial datum, that complements early works [36,37]. Now, we introduce the structure of equation ( 1).…”

Section: Introductionmentioning

confidence: 52%

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Global Dissipative Solution for an Extended Dullin–Gottwald–Holm Equation

Wang

Yan

2022

Journal of Mathematics

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The current paper devotes to the continuation of the solutions for the Dullin–Gottwald–Holm equation with forcing. Due to forcing, this equation loses the conservation of energy and cannot be regarded as the Hamiltonian system. Thus, through employing the characteristic method and exploiting the balance law, we prove the existence of a semigroup of a global dissipative solution, which is defined for each initial value u 0 ∈ H 1 ℝ and depends continuity on it.

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Section: Introductionmentioning

confidence: 52%

“…Lately, the authors discussed the blowup rate of the wave-breaking solution, which complement the result of [37]. Except D-G-H equation, Zhang [38,39] also studied the Cauchy problem for the N − abc family of the C-H type equation with dispersion and dissipation…”

Section: Introductionmentioning

confidence: 94%

Global Dissipative Solution for an Extended Dullin–Gottwald–Holm Equation

Wang

Yan

2022

Journal of Mathematics

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“…In this section, we turn to our attention to establish a semi-linear system for smooth solutions. For that purpose, it is crucial to introduce the characteristic equation [55,56]…”

Section: Semi-linear System For Smooth Solutionsmentioning

confidence: 99%

“…where N ∈ Z + , N ≥ 1, a, b, c are positive constants and a = b + c. More details are present in [55,56].…”

Section: Introductionmentioning

confidence: 99%

Global Energy Conservation Solution for the N − abc Family of Camassa-Holm Type Equation

ZHANG

Liu

Deng

2022

Preprint

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In this paper, we investigate the N − abc family of Camassa-Holm type equation with (N + 1)-order nonlinearities. Using both the lower order and the higher order energy conservation laws, as well as the characteristic method, we establish the global existence and uniqueness of the Hölder continuous energy weak solution to the N − abc family of Camassa-Holm type equation in the energy space H 1 (R) × W 1,2N (R). Moreover, we show that a very natural and interesting problem is to study how the regularity of solution changes with respect to N. More precisely, we establish Hölder continuous energy weak solutions with the exponent 1 − 1 2N. This result precisely shows how the regularity of solution changes with respect to the the power of nonlinear wave speed N, and it reveals an intrinsic relation between Camassa-Holm equation (linear wave speed u 1), Novikov equation (quadratic wave speed u 2) and the N − abc family of Camassa-Holm type equation (wave speed u N). This result improves earlier ones in the literatures, such as [1], [8], [45] and [50].

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“…e long-time behavior for the support of momentum density of the Camassa-Holm equation was discussed in [25]. Mathematical studies for the related models can been found in [26][27][28].…”

Section: Introductionmentioning

confidence: 99%