×

On the Cauchy problem for the \(b\)-family equations with a strong dispersive term. (English) Zbl 1228.35206

Summary: We consider \(b\)-family equations with a strong dispersive term. First, we present a criterion on blow-up. Then global existence and persistence property of the solution are also established. Finally, we discuss infinite propagation speed of this equation.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B44 Blow-up in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] D. D. Holm and M. F. Staley, “Nonlinear balance and exchange of stability of dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary PDE,” Physics Letters A, vol. 308, no. 5-6, pp. 437-444, 2003. · Zbl 1010.35066
[2] Y. Zhou, “On solutions to the Holm-Staley b-family of equations,” Nonlinearity, vol. 23, no. 2, pp. 369-381, 2010. · Zbl 1189.37083
[3] R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661-1664, 1993. · Zbl 0972.35521
[4] A. Degasperis and M. Procesi, “Asymptotic integrability,” in Symmetry and Perturbation Theory, A. Degasperis and G. Gaeta, Eds., pp. 23-37, World Scientific, Singapore, 1999. · Zbl 0963.35167
[5] A. Constantin and J. Escher, “Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation,” Communications on Pure and Applied Mathematics, vol. 51, no. 5, pp. 475-504, 1998. · Zbl 0934.35153
[6] A. A. Himonas, G. Misiolek, G. Ponce, and Y. Zhou, “Persistence properties and unique continuation of solutions of the Camassa-Holm equation,” Communications in Mathematical Physics, vol. 271, no. 2, pp. 511-522, 2007. · Zbl 1142.35078
[7] H. P. McKean, “Breakdown of a shallow water equation,” Asian Journal of Mathematics, vol. 2, no. 4, pp. 867-874, 1998. · Zbl 0959.35140
[8] L. Molinet, “On well-posedness results for Camassa-Holm equation on the line: a survey,” Journal of Nonlinear Mathematical Physics, vol. 11, no. 4, pp. 521-533, 2004. · Zbl 1069.35076
[9] Y. Zhou, “Wave breaking for a shallow water equation,” Nonlinear Analysis, vol. 57, no. 1, pp. 137-152, 2004. · Zbl 1106.35070
[10] Y. Zhou, “Wave breaking for a periodic shallow water equation,” Journal of Mathematical Analysis and Applications, vol. 290, no. 2, pp. 591-604, 2004. · Zbl 1042.35060
[11] Y. Zhou, “Blow-up phenomenon for the integrable Degasperis-Procesi equation,” Physics Letters A, vol. 328, no. 2-3, pp. 157-162, 2004. · Zbl 1134.37361
[12] Y. Zhou, “Stability of solitary waves for a rod equation,” Chaos, Solitons and Fractals, vol. 21, no. 4, pp. 977-981, 2004. · Zbl 1046.35094
[13] Z. Jiang, L. Ni, and Y. Zhou, “Wave breaking for the Camassa-Holm equation,” In press. · Zbl 1247.35104
[14] L. Ni and Y. Zhou, “A new asymptotic behavior of solutions to the Camassa-Holm equation,” Proceedings of the American Mathematical Society. In press. · Zbl 1259.37046
[15] Z. Guo, “Blow up, global existence, and infinite propagation speed for the weakly dissipative Camassa-Holm equation,” Journal of Mathematical Physics, vol. 49, no. 3, Article ID 033516, 2008. · Zbl 1153.81368
[16] Z. Guo, “Some properties of solutions to the weakly dissipative Degasperis-Procesi equation,” Journal of Differential Equations, vol. 246, no. 11, pp. 4332-4344, 2009. · Zbl 1170.35083
[17] L. Ni and Y. Zhou, “Wave breaking and propagation speed for a class of nonlocal dispersive \theta -equations,” Nonlinear Analysis. Real World Applications, vol. 12, no. 1, pp. 592-600, 2011. · Zbl 1206.35221
[18] X. Li, “The infinite propagation speed and the limit behavior for the b-family equation with dispersive term,” International Journal of Nonlinear Science, vol. 7, no. 3, pp. 345-352, 2009. · Zbl 1177.35204
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.