The Cauchy problem for the integrable Novikov equation

Yan, Wei; Li, Yongsheng; Zhang, Yimin

doi:10.1016/j.jde.2012.03.015

Cited by 102 publications

(42 citation statements)

References 14 publications

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“…The explicit formulas for multipeakon solutions of Novikov equation were derived in [50]. Recently, by the transport equations theory and the classical Friedrichs regularization method, the authors proved that the Cauchy problem for the Novikov equation is locally well-posed in the Besov spaces B s p,r (with 1 ≤ p, r ≤ +∞ and s > max{1+1/p, 3/2} in [72,80], and with the critical index s = 3/2, p = 2, r = 1 in [66]). It is also shown in [66] that the Novikov equation associated with the initial value is locally well-posed in Sobolev space H s with s > 3/2 by using the abstract Kato Theorem.…”

Section: Shouming Zhou Chunlai Mu and Liangchen Wangmentioning

confidence: 99%

Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation

Zhou¹,

Mu²,

Wang³

2013

DCDS-A

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This paper deals with the Cauchy problem for a weakly dissipative shallow water equation with high-order nonlinearities yt + u m+1 yx + bu m uxy + λy = 0, where λ, b are constants and m ∈ N, the notation y :which includes the famous b-equation and Novikov equations as special cases. The local well-posedness of solutions for the Cauchy problem in Besov space B s p,r with 1 ≤ p, r ≤ +∞ and s > max{1 + 1 p , 3 2 } is obtained. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and conditions that lead to the development of singularities in finite time for the solutions are acquired, moreover, the propagation behaviors of compactly supported solutions are also established. Finally, the weak solution and analytic solution for the equation are considered. 2010 Mathematics Subject Classification. Primary: 35G25, 35L05; Secondary: 35Q50.

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Section: Shouming Zhou Chunlai Mu and Liangchen Wangmentioning

confidence: 99%

Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation

Zhou¹,

Mu²,

Wang³

2013

DCDS-A

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“…As for the Novikov equation, the local well‐posedness of the Cauchy problem was investigated in . The global existence and the blow‐up phenomena of the strong solutions to the Novikov equation were studied in .…”

Section: Introductionmentioning

confidence: 99%

“…Coclite and Karlsen [4] obtained global existence results for entropy weak solutions in the class of 1 (ℝ) ∩ (ℝ) and the class of 2 (ℝ) ∩ 4 (ℝ). As for the Novikov equation, the local well-posedness of the Cauchy problem was investigated in [39,40,42,43]. The global existence and the blow-up phenomena of the strong solutions to the Novikov equation were studied in [39,43].…”

Section: Introductionmentioning

confidence: 99%

Global weak solutions for a generalized Camassa–Holm equation

Yin

2018

Mathematische Nachrichten

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In this paper we mainly investigate the Cauchy problem of a generalized Camassa–Holm equation. We first derive two global existence results and two blow‐up results from the relationship between the Degasperis–Procesi equation and the generalized Camassa–Holm equation. We then prove the existence and uniqueness of global weak solutions under some certain sign conditions on the initial data.

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“…The local well-posedness for the Novikov equation in Sobolev spaces and Besov spaces was studied in [49,50,52,53]. The global existence of strong solutions were established in [49] under some sign conditions and the blow-up phenomena of the strong solutions were shown in [53].…”

Section: Introductionmentioning

confidence: 99%