Skip to main content
SpringerLink
Log in

Global existence and propagation speed for a generalized Camassa–Holm model with both dissipation and dispersion

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

In this paper, we study a generalized Camassa–Holm (gCH) model with both dissipation and dispersion, which has (\(N+1\))-order nonlinearities and includes the following three integrable equations: the Camassa–Holm, the Degasperis–Procesi, and the Novikov equations, as its reductions. We first present the local well-posedness and a precise blow-up scenario of the Cauchy problem for the gCH equation. Then, we provide several sufficient conditions that guarantee the global existence of the strong solutions to the gCH equation. Finally, we investigate the propagation speed for the gCH equation when the initial data are compactly supported.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. A. Bressan and A. Constantin, Global conservative solutions of the Camassa–Holm equation, Arch. Rat. Mech. Anal.,183 (2007), 215–239.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Bressan and A. Constantin, Global dissipative solutions of the Camassa–Holm equation, Anal. Appl.,5 (2007), 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  3. R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters,71 (1993), 1661–1664.

    Article  MathSciNet  MATH  Google Scholar 

  4. R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech.,31 (1994), 1–33.

    Article  MATH  Google Scholar 

  5. G. Coclite and K. Karlsen, On the well-posedness of the Degasperis–Procesi equation, J. Funct. Anal.,233 (2006), 60–91.

    Article  MathSciNet  MATH  Google Scholar 

  6. G. M. Coclite, H. Holden and K.H. Karlsen, Well-posedness of higher-order Camassa–Holm equations. J. Differential Equations,246 (2009) 929–963.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Constantin, The Hamiltonian structure of the Camassa–Holm equation, Exposition Math.,15 (1997), 53–85.

    MathSciNet  MATH  Google Scholar 

  8. A. Constantin, The Cauchy problem for the periodic Camassa–Holm equation, J. Differential Equations,141 (1997), 218–235.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Constantin, On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci.,10 (2000), 391–399.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier, Grenoble,50 (2000), 321–362.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica,181 (1998), 229–243.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math.,51 (1998), 475–504.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Comm. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze,Srie 4, Tome 26 (no. 2), 303–328.

  14. A. Constantin and J. Escher, On the structure of a family of quasilinear equations arising in a shallow water theory, Math. Ann.,312 (1998), 403–416.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Constantin and L. Molinet, The initial value problem for a generalized Boussinesq equation, Differential and Integral equations15 (2002) 1061–1072.

    MathSciNet  MATH  Google Scholar 

  16. A. Degasperis and M. Procesi, Asymptotic integrability symmetry and perturbation theory, (Rome, 1998), World Sci. Publ., (1999), 23–37.

  17. A. Degasperis, D. Holm and A. Hone, A new integrable equation with peakon solution, Theoret. and Math. Phys,133 (2002) 1463–1474.

    Article  MathSciNet  Google Scholar 

  18. H. Holden and X. Raynaud, Global conservative solutions of the Camassa–Holm equation—a Lagrangian point of view, Comm. Partial Differential Equations,32 (2007), 1511–1549.

    Article  MathSciNet  MATH  Google Scholar 

  19. H. Holden and X. Raynaud, Dissipative solutions for the Camassa–Holm equation, Discrete Contin. Dyn. Syst.,24 (2009), 1047–1112.

    Article  MathSciNet  MATH  Google Scholar 

  20. Q. Hu and Z. Yin, Blowup and blowup rate of solutions to a weakly dissipative periodic rod equation, J.Math. Phys.,50 083503 (2009), 1–16.

    MathSciNet  MATH  Google Scholar 

  21. Q. Hu, Global existence and blow-up phenomena for a weakly dissipative periodic 2-component Camassa–Holm system, J. Math. Phys.,52 103701 (2011), 1–13.

    MathSciNet  MATH  Google Scholar 

  22. J. Escher, Y. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis–Procesi equation, J. Funct. Anal., 241 (2006), 257–485.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. Escher, Y. Liu and Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis–Procesi equation, Indiana Univ. Math. J., 56 (2007), 87–117.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Fokas, On a class of physically important integrable equations, Physica D87 (1995), 145-150.

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D,4 (1981), 47–66.

    Article  MathSciNet  MATH  Google Scholar 

  26. J. Ghidaglia, Weakly damped forced Korteweg–de Vries equations behave as a finite dimensional dynamical system in the long time, J. Differential Equations,74 (1988), 369–390.

    Article  MathSciNet  MATH  Google Scholar 

  27. A. Himonas and C. Holliman, The Cauchy problem for a generalized Camassa–Holm equation, Advances in Differential Equations,19 (2013), 161–200.

    MathSciNet  MATH  Google Scholar 

  28. A. Himonas and C. Holliman, On well-posedness of the Degasperis–Procesi equation, Discrete Contin. Dyn. Syst.,31 (2011), 469–488.

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Himonas, C. Holliman and K. Grayshan, Norm inflation and ill-posedness for the Degasperis–Procesi equation, Comm. Partial Differential Equations39 (2014) 2198–2215.

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation, Nonlinearity,25 (2012), 449–479.

    Article  MathSciNet  MATH  Google Scholar 

  31. A. Himonas and R. Thompson, Persistence properties and unique continuation for a generzlized Camassa–Holm Equation, J. Math. Phys.,55 (2014), 091503.

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Hone and J. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A: Math. Theor.41 (2008), 372002–372010.

    Article  MathSciNet  MATH  Google Scholar 

  33. T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in: Spectral Theory and Differential Equations, Lecture Notes in Math., Springer Verlag, Berlin, 448 (1975), 25–70.

    Google Scholar 

  34. T. Kato and G. Ponce, Commutator estimates and Navier–Stokes equations, Comm. Pure Appl. Math.,41 (1988), 203–208.

    Article  MathSciNet  MATH  Google Scholar 

  35. Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations,162 (2000), 27–63.

    Article  MathSciNet  MATH  Google Scholar 

  36. Y. Li and P. Olver, Well-posedness and blow-up solutions for phenomona for the Degasperis–Procesi equation, Comm. Math. Phys.,267 (2006), 801–820.

    Article  MathSciNet  Google Scholar 

  37. H. McKean, Breakdown of a shallow water equation, Asian J. Math.,2(4) (1998) 867–874.

    Article  MathSciNet  MATH  Google Scholar 

  38. V. Novikov, Generalizations of the Camassa–Holm equation, J. Phys. A,42 (2009), 342002, 14 pages.

    Article  MathSciNet  MATH  Google Scholar 

  39. E. Novruzova and A. Hagverdiyevb, On the behavior of the solution of the dissipative Camassa–Holm equation with the arbitrary dispersion coefficient J. Differential Equations,257 (2014), 4525–4541.

    Article  MathSciNet  Google Scholar 

  40. P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E,53 (1996), 1900–1906.

    Article  MathSciNet  Google Scholar 

  41. E. Ott and R. Sudan, Damping of solitary waves, Phys. Fluids,13 (1970), 1432–1434.

    Article  Google Scholar 

  42. Z. Qiao, The Camassa-Holm hierarchy, related \(N\)-dimensional integrable systems and algebro-geometric solution on a symplectic submanifold, Commun. Math. Phys.239 (2003) 309 – 341.

    Article  MATH  Google Scholar 

  43. Z. Qiao, Integrable hierarchy (the DP hierarchy), \(3 \times 3 \) constrained systems, and parametric and stationary solutions, Acta Applicandae Mathematicae83 (2004), 199-220.

    Article  MathSciNet  MATH  Google Scholar 

  44. Z. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys.47 (2006), 112701 – 112710.

    Article  MathSciNet  MATH  Google Scholar 

  45. G. Rodriguez-Blanco, On the Cauchy problem for the Camassa–Holm equation, Nonlinear Anal.,46 (2001), 309–327.

    Article  MathSciNet  MATH  Google Scholar 

  46. F. Tiğlay, The periodic Cauchy problem for Novikov’s equation, Int. Math. Res. Not., (2010) rnq267, 1–16.

  47. S. Wu and Z. Yin, Blow-up, blow-up rate and decay of the solution of the weakly dissipative Camassa–Holm equation, J. Math. Phys.,47, 013504 (2006), 1–12.

    MathSciNet  MATH  Google Scholar 

  48. S. Wu and Z. Yin, Blow-up and decay of the solution of the weakly dissipative Degasperis–Procesi equation, SIAM J. Math. Anal.,40 (2008), 475–490.

    Article  MathSciNet  MATH  Google Scholar 

  49. X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation, Ann. Sc. Norm. Sup. Pisa CI. Sci.,5 (2012), 707–727.

    MathSciNet  MATH  Google Scholar 

  50. X. Wu and Z. Yin, Global weak solutions for the Novikov equation, J. Phys. A: Math. Theor.,44 (2011), 055202 1–17.

    MathSciNet  MATH  Google Scholar 

  51. Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math.,53 (2000), 1411–1433.

    Article  MathSciNet  MATH  Google Scholar 

  52. Z. Yin, Global existence for a new periodic integrable equation, J. Math. Anal. Appl.,283 (2003), 129–139.

    Article  MathSciNet  MATH  Google Scholar 

  53. Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math.,47 (2003), 649–666.

    Article  MathSciNet  MATH  Google Scholar 

  54. S. Zhou and C. Mu, The properties of solutions for a generalized b-family equation with peakons, J Nonlinear Sci,23 (2013), 863–889.

    Article  MathSciNet  MATH  Google Scholar 

  55. S. Zhou, C. Mu and L. Wang, Well-posedness, blow-up phenomena and global existence for the generalized b-equation with higher-order nonlinearities and weak dissipation, Discrete Contin. Dyn. Syst. Ser. A,34 (2014), 843–867.

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by NSF of China (No. 11401223), NSF of Guangdong (No. 2015A030313424), the Science and Technology Program of Guangzhou (Grant No. 201607010005), and China Scholarship Council. The first author would like to thank Professor Zhijun Qiao for his kind hospitality and encouragement during her visit in the University of Texas Rio Grande Valley, and the second author (Qiao) thanks the UT Presidents Endowed Professorship (Project # 450000123) for its partial support.

Author information

Authors and Affiliations

  1. Department of Mathematics, South China Agricultural University, Guangzhou, 510642, Guangdong, People’s Republic of China

    Qiaoyi Hu

  2. School of Mathematical and Statistical Science, University of Texas, Rio Grande Valley, Edinburg, TX, 78539, USA

    Qiaoyi Hu & Zhijun Qiao

Authors
  1. Qiaoyi Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhijun Qiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhijun Qiao.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, Q., Qiao, Z. Global existence and propagation speed for a generalized Camassa–Holm model with both dissipation and dispersion. J. Evol. Equ. 20, 403–419 (2020). https://doi.org/10.1007/s00028-019-00533-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00028-019-00533-5

Mathematics Subject Classification

Keywords

Search

Navigation