Abstract
In this paper, we focus on the large time behavior of compact support of the potential for a Camassa–Holm-type equation with nonlinearities of degree \(k+1\) if the compactly supported initial potential keeps its sign. Moreover, persistence property in weighted Sobolev spaces is also investigated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Private discussion with Professor Tudor Stefan Ratiu.
References
Aldroubi, A., Gröchenig, K.: Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev. 43, 585–620 (2001)
Anco, S.C., da Silva, P.L., Freire, I.L.: A family of wave-breaking equations generalizing the Camassa–Holm and Novikov equations. J. Math. Phys. 56(9), 091506 (2015). (21pp)
Anco, S.C., Recio, E., Gandarias, M.L., Bruzón, M.S.: A nonlinear generalization of the Camassa–Holm equation with peakon solutions. In: Proceedings of the 10th AIMS International Conference Dynamical Systems, Differential Equations and Applications, Madrid, Spain, pp. 29–37 (2015)
Brandolese, L.: Breakdown for the Camassa–Holm equation using decay criteria and persistence in weighted spaces. Int. Math. Res. Not. 22, 5161–5181 (2012)
Barostichi, R., Himonas, A., Petronilho, G.: Global analyticity for a generalized Camassa–Holm equation and decay of the radius of spatial analyticity. J. Differ. Equ. 263, 732–764 (2017)
Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)
Constantin, A., Escher, J.: Analyticity of periodic travelling free surface water waves with vorticity. Ann. Math. 173, 559–568 (2011)
Constantin, A., Strauss, W.: Stability of peakons. Commun. Pure Appl. Math. 53, 603–610 (2000)
Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192, 165–186 (2009)
Degasperis, A., Procesi, M.: Asymptotic integrability. In: Degasperis, A., Gaeta, G. (eds.) Symmetry and Perturbation Theory, pp. 23–37. World Scientific, Singapore (1999)
Fokas, A., Fuchssteiner, B.: Symplectic structures, Bälund transformations and hereditary symmetries. Physica D 4(1), 47–66 (1981)
Gröchenig, K.: Weight functions in time–frequency analysis, pseudodifferential operators: partial differential equations and time–frequency analysis. Fields Institute Communications, vol. 52, American Mathematical Society, Providence, RI, pp. 343–366 (2007)
Grayshan, K., Himonas, A.: Equations with peakon traveling wave solutions. Adv. Dyn. Syst. Appl. 8, 217–232 (2013)
Grayshan, K.: Peakon solutions of the Novikov equation and properties of the data-to-solution map. J. Math. Anal. Appl. 397, 515–521 (2013)
Himonas, A., Holliman, C.: The Cauchy problem for the Novikov equation. Nonlinearity 25, 449–479 (2012)
Himonas, A., Holliman, C.: The Cauchy problem for a generalized Camassa–Holm equation. Adv. Differ. Equ. 19, 161–200 (2014)
Hone, A., Lafortune, S.: Stability of stationary solutions for nonintegrable peakon equations. Physica D 269, 28–36 (2014)
Himonas, A., Misiolek, G.: The Cauchy problem for an integrable shallow water equation. Differ. Integr. Equ. 14, 821–831 (2001)
Himonas, A., Thompson, R.: Persistence properties and unique continuation for a generalized Camassa–Holm equation. J. Math. Phys. 55, 091503 (2014)
Himonas, A., Misiolek, G., Ponce, G., Zhou, Y.: Persistence properties and unique continuation of solutions of Camassa–Holm equation. Commun. Math. Phys. 271, 511–522 (2007)
Lai, S.: Global weak solutions to the Novikov equation. J. Funct. Anal. 265, 520–544 (2013)
Liu, X., Liu, Y., Qu, C.: Stability of peakons for the Novikov equation. J. Math. Pures Appl. 101, 172–187 (2014)
McKean, H.P.: Breakdown of the Camassa–Holm equation. Commun. Pure Appl. Math. 57, 416–418 (2004)
Novikov, V.S.: Generalizations of the Camassa–Holm equation. J. Phys. A 42, 342002 (2009)
Ni, L., Zhou, Y.: A new asymptotic behavior for solutions of the Camassa–Holm equation. Proc. Am. Math. Soc. 140, 607–614 (2012)
Ni, L., Zhou, Y.: Well-posedness and persistence properties for the Novikov equation. J. Differ. Equ. 250, 3002–3021 (2011)
da Silva, P.L., Freire, I.L.: An equation unifying both Camassa–Holm and Novikov equations. In: Proceedings of the 10th AIMS International Conference (2015). https://doi.org/10.3934/proc.2015.0304
Tiglay, F.: The periodic Cauchy problem for Novikov’s equation. Int. Math. Res. Not. 20, 4633–4648 (2011)
Wei, L., Qiao, Z., Wang, Y., Zhou, S.: Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete Contin. Dyn. Syst. 37, 1733–1748 (2017)
Wu, X., Yin, Z.: Global weak solutions for the Novikov equation. J. Phys. A Math. Theor. 44, 055202 (2011). (17pp)
Yan, W., Li, Y., Zhang, Y.: The Cauchy problem for the Novikov equation. Nonlinear Differ. Equ. Appl. 20, 1157–1169 (2013)
Acknowledgements
The authors thank Professor Tudor Stefan Ratiu for his fruitful discussions and are very grateful to the anonymous reviewers for their careful reading and useful suggestions which greatly improved the presentation of the paper. This work was partially supported by National Natural Science Foundation of China, under Grant No. 11301394 and China Postdoctoral Science Foundation, under Grant No. 2017M620149.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Melvin Leok.
Rights and permissions
About this article
Cite this article
Guo, Z., Li, X. & Yu, C. Some Properties of Solutions to the Camassa–Holm-Type Equation with Higher-Order Nonlinearities. J Nonlinear Sci 28, 1901–1914 (2018). https://doi.org/10.1007/s00332-018-9469-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-018-9469-7