Qualitative analysis for the new shallow-water model with cubic nonlinearity

Mi, Yongsheng; Liu, Yue; Huang, Dajian; Guo, Boling

doi:10.1016/j.jde.2020.04.003

Cited by 14 publications

(16 citation statements)

References 47 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where a = (−3k 3 ± 9k 2 3 + 24ck 1 )/(4k 1 ) [66,76]; the mCH-Novikov equation for k 1 k 2 = 0, k 3 = 0, where a = ± 3c/(2k 1 + 3k 2 ) [70] (Notice that we here correct it); or the Novikov-CH equation for k…”

Section: Is Called a Global Weak Solution If T Can Be Taken Arbitrari...mentioning

confidence: 99%

“…Remark 1.10. When k 1 = k 2 = 0 and k 3 = 1, system (1.29) reduces to the dynamical system for the CH equation [9]; when k 1 = k 3 = 0 and k 2 = 1, system (1.29) becomes the dynamical system for the Novikov equation [56]; when k 2 = k 3 = 0 and k 1 = 1, system (1.29) reduces to the dynamical system for the mCH equation [47], when k 2 = 0, system (1.29) becomes the dynamical system for the mCH-CH equation [85]; when k 3 = 0, system (1.29) becomes the dynamical system for the mCH-Novikov equation [70]. System (1.29) was also given in [1] by using the distribution theory [42].…”

Section: Is Called a Global Weak Solution If T Can Be Taken Arbitrari...mentioning

confidence: 99%

“…We have investigated some properties of the Cauchy problem (1.1), including the local well-posedness of its strong solution, the Hölder continuity of its data-to-solution map, the blow-up criterion, the precise blow-up quantity and the wave-breaking phenomenon and so on. Some other properties associated with the Cauchy problem (1.1) can also be concerned with, for instance, the analyticity the persistence properties of its solution, which can be finished using similar methods as that of [52,53,70,72]. However, we will not consider these problems in the current paper in detail.…”

Section: Is Called a Global Weak Solution If T Can Be Taken Arbitrari...mentioning

confidence: 99%

“…Remark 1.4. The result obtained in Theorem 1.7 is inspired by [47,70]. However, we need the additional condition u 0 (x 1 ) + 1 ≤ C 2 m 0 (x 1 ) as stated in Theorem 1.7.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Mi et al [70] obtained the local well-posedness of Eq. (1.7) in noncritical or critical Besov spaces and provided the blow-up criterion in Sobolev spaces as well as the wave breaking condition, the persistence property and the analyticity of the solution.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The Cauchy problem and multi-peakons for the mCH-Novikov-CH equation with quadratic and cubic nonlinearities

Qina¹,

Yan²,

Guo³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

This paper investigates the Cauchy problem of a generalized Camassa-Holm equation with quadratic and cubic nonlinearities (alias the mCH-Novikov-CH equation), which is a generalization of some special equations such as the Camassa-Holm (CH) equation, the modified CH (mCH) equation ((alias the Fokas-Olver-Rosenau-Qiao equation), the Novikov equation, the CH-mCH equation, the mCH-Novikov equation, and the CH-Novikov equation. We first show the local well-posedness for the strong solutions of the mCH-Novikov-CH equation in Besov spaces by means of the Littlewood-Paley theory and the transport equations theory.Then, the H ölder continuity of the data-to-solution map to this equation are exhibited in some Sobolev spaces. After providing the blow-up criterion and the precise blow-up quantity in light of the Moser-type estimate in the Sobolev spaces, we then trace a portion and the whole of the precise blow-up quantity, respectively, along the characteristics associated with this equation, and obtain two kinds of sufficient conditions on the gradient of the initial data to guarantee the occurance of the wave-breaking phenomenon. Finally, the non-periodic and periodic peakon and multi-peakon solutions for this equation are also explored.

show abstract

Section: Is Called a Global Weak Solution If T Can Be Taken Arbitrari...mentioning

confidence: 99%

Section: Is Called a Global Weak Solution If T Can Be Taken Arbitrari...mentioning

confidence: 99%

Section: Is Called a Global Weak Solution If T Can Be Taken Arbitrari...mentioning

confidence: 99%

“…Remark 1.4. The result obtained in Theorem 1.7 is inspired by [47,70]. However, we need the additional condition u 0 (x 1 ) + 1 ≤ C 2 m 0 (x 1 ) as stated in Theorem 1.7.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations