Abstract
In this paper, we establish a new blow-up criterion for the two-component Dullin–Gottwald–Holm system. We show that strong solutions must blowup in finite time if the integral concerning initial momentum changes sign at some point \(x_0\in {\mathbb {R}}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen, Y., Gao, H., Liu, Y.: On the Cauchy problem for the two-component Dullin–Gottwald–Holm system. Discret. Contin. Dyn. Syst. 33, 3407–3441 (2013)
Constantin, A.: The trajectories of particles in stokes waves. Invent. Math. 166, 523–535 (2006)
Dullin, R., Gottwald, G., Holm, D.: An integrable shallow water equation with linear and nonlinear dispersion. Phys. Rev. Lett. 87, 4501–4504 (2001)
Guo, F., Gao, H., Liu, Y.: On the wave-breaking phenomena for the two-component Dullin–Gottwald–Holm system. J. Lond. Math. Soc. 86, 810–834 (2012)
Guo, F., Wang, R.: On the persistence and unique continuation properties for an integrable two-component Dullin–Gottwald–Holm system. Nonlinear Anal. 96, 38–46 (2014)
Han, Y., Guo, F., Gao, H.: On solitary waves and wave-breaking phenomena for a generalized two-component integrable Dullin–Gottwald–Holm system. J. Nonlinear Sci. 23, 617–656 (2013)
Hoang, D.: The local criteria for blowup of the Dullin–Gottwald–Holm equation and the two-component Dullin–Gottwald–Holm system. Ann. Fac. Sci. Toulouse Math. 6(25), 995–1012 (2016)
Liu, X., Yin, Z.: Local well-posedness and stability of solitary waves for the two-component Dullin–Gottwald–Holm system. Nonlinear Anal. 88, 1–15 (2013)
Tian, L., Gui, G., Liu, Y.: On the well-posedness problem and the scattering problem for the Dullin–Gottwald–Holm equation. Commun. Math. Phys. 257, 667–701 (2005)
Wang, F., Chen, Q., Mei, J.: Global existence of weak solutions to a weakly dissipative modified two-component Dullin–Gottwald–Holm system. Math. Methods Appl. Sci. 38, 2349–2362 (2015)
Zhou, Y.: On solutions to the Holm–Staley b-family of equations. Nonlinearity 23, 369–381 (2010)
Zhou, Y.: Blow-up of solutions to the DGH equation. J. Funct. Anal. 250, 227–248 (2007)
Zhu, M., Jin, L., Jiang, Z.: A new blow-up criterion for the DGH equation. Abstr. Appl. Anal. (2012). https://doi.org/10.1155/2012/515948
Acknowledgements
The authors are indebted to anonymous referees for their helpful comments. This work was partially supported by ZJNSF (Grant No. LQ17A010006) and NSFC (Grant Nos. 11771183 and 11301394).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Yong Zhou.
Rights and permissions
About this article
Cite this article
Guo, Z., Cao, Y. & Zhu, M. Blowup of Solutions to the Two-Component Dullin–Gottwald–Holm System. Bull. Malays. Math. Sci. Soc. 43, 201–209 (2020). https://doi.org/10.1007/s40840-018-0672-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-018-0672-3