Infinite propagation speed and asymptotic behavior for a generalized Camassa-Holm equation with cubic nonlinearity. (English) Zbl 1380.35063
Summary: This paper is devoted to the study of the infinite propagation speed and the asymptotic behavior for a generalized Camassa-Holm equation with cubic nonlinearity. First, we get the infinite propagation speed in the sense that the corresponding solution with compactly supported initial data does not have compact support any longer in its lifespan. Then, the asymptotic behavior of the solution at infinity is investigated. Especially, we prove that the solution decays algebraically with the same exponent as that of the initial data.
MSC:
| 35G25 | Initial value problems for nonlinear higher-order PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
References:
| [1] | Novikov, V., Generalizations of the Camassa-Holm equation, J. Phys. A, 42, 342002 (2009) · Zbl 1181.37100 |
| [2] | Li, M.; Yin, Z. Y., Blow-up phenomena and local well-posedness for a generalized Camassa-Holm equation with cubic nonlinearity, Nonlinear Anal.-Theor., 151, 208-226 (2017) · Zbl 1356.35060 |
| [3] | Fu, Y., A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity, Discrete Contin. Dyn. Syst., 35, 2011-2039 (2017) · Zbl 1310.35073 |
| [4] | Mi, Y. S.; Mu, C. L., On the cauchy problem for the modified novikov equation with peakon solutions, J. Differential Equations, 254, 961-982 (2013) · Zbl 1258.35065 |
| [5] | Hone, A. N.W.; Wang, J. P., Integrable peakon equations with cubic nonlinearity, J. Phys. A, 41, 4359-4380 (2008) · Zbl 1153.35075 |
| [6] | Wu, X. L.; Yin, Z. Y., Global weak solutions for the Novikov equation, J. Phys. A, 44, 055202 (2011) · Zbl 1210.35217 |
| [7] | Ni, L. D.; Zhou, Y., Well-posedness and persistence properties for the Novikov equation, J. Differential Equations, 250, 3002-3021 (2011) · Zbl 1215.37051 |
| [8] | Wu, X. L.; Yin, Z. Y., A note on the cauchy problem of the Novikov equation, Appl. Anal., 92, 1116-1137 (2013) · Zbl 1273.35097 |
| [9] | Fokas, A. S., On a class of physically important integrable equations, Physica D, 87, 145-150 (1995) · Zbl 1194.35363 |
| [10] | Fuchssteiner, B., Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation, Physica D, 95, 229-243 (1996) · Zbl 0900.35345 |
| [11] | Olver, P. J.; Rosenau, P., Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53, 1900-1906 (1996) |
| [12] | Qiao, Z. J., A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47, 1661-1664 (2006) · Zbl 1112.37063 |
| [13] | Hu, H. C.; Yin, W.; Wu, H. X., Bilinear equations and new multi-soliton solution for the modified Camassa-Holm equation, Appl. Math. Lett., 59, 18-23 (2016) · Zbl 1338.35099 |
| [14] | Qu, C. Z.; Liu, X. C.; Liu, Y., Stability of peakons for an integrable modified Camassa-Holm equation with cubic nonlinearity, Comm. Math. Phys., 322, 967-997 (2013) · Zbl 1307.35264 |
| [15] | Wu, X. L.; Guo, B. L., The exponential decay of solutions and traveling wave solutions for a modified Camassa-Holm equation with cubic nonlinearity, J. Math. Phys., 55, 081504 (2014) · Zbl 1366.35162 |
| [16] | Ni, L. D.; Zhou, Y., A new asymptotic behavior of solutions to the Camassa-Holm equation, Proc. Amer. Math. Soc., 140, 607-614 (2011) · Zbl 1259.37046 |
| [17] | Bahouri, H.; Chemin, J. Y.; Danchin, R., Fourier Analysis and Nonlinear Partial Differential Equations (2011), Springer Berlin Heidelberg · Zbl 1227.35004 |
| [18] | Danchin, R., (Fourier Analysis Methods for PDE’S. Fourier Analysis Methods for PDE’S, Lecture Notes (2005)), 14 November |
| [19] | Strichartz, R. S., A Guide To Distribution Theory and Fourier Transforms (1994), CRC Press: CRC Press Boca Raton · Zbl 0854.46035 |
| [20] | Yan, K.; Yin, Z. Y., Infinite propagation speed and asymptotic behavior for a two-component degasperis-procesi system, Monatsh. Math., 181, 217-234 (2016) · Zbl 1367.35068 |
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