The Cauchy problem and continuation of periodic solutions for a generalized Camassa–Holm equation

References

• Degasperis A, Holm DD, Hone ANW. A new integrable equation with peakon solutions. Theor Math Phys. 2002;133:1463–1474.
   Web of Science ®Google Scholar
• Dullin HR, Gottwald GA, Holm DD. Camassa–Holm, Korteweg–de Vries-5 and other asymptotically equivalent equations for shallow water waves. Fluid Dyn Res. 2003;33:73–95.
   Web of Science ®Google Scholar
• Dullin HR, Gottwald GA, Holm DD. On asymptotically equivalent shallow water wave equations. PhysD. 2004;190:1–14.
   Google Scholar
• Holm DD, Staley MF. Wave structure and nonlinear balances in a family of evolutionary PDEs. SIAM J Appl Dyn Syst. 2003;2:323–380.
   Web of Science ®Google Scholar
• Holm D, Staley M. Nonlinear balance and exchange of stability in dynamics of solitons, Peakons, Ramp/Cliffs and Leftons in 1+1 nonlinear evolutionary PDE. Phys Lett. 2003;308:437–444.
   Web of Science ®Google Scholar
• Degasperis A, Procesi M. Asymptotic integrability. In: Symmetry and perturbation theory II, SPT98. Singapore: World Scientific; 1999. p. 23–37.
   Google Scholar
• Camassa R, Holm DD. An integrable shallow water equation with peaked solitons. Phys Rev Lett. 1993;71:1661–1664.
   PubMed Web of Science ®Google Scholar
• da Silva PL, Freire IL. An equation unifying both Camassa-Holm and Novikov equations. In: Proceedings of the 10th AIMS International Conference; 2015. DOI:10.3934/proc.2015.0304
   Google Scholar
• Anco S, da Silva PL, Freire IL. A family of wave-breaking equations generalizing the Camassa–Holm and Novikov equations. J Math Phys. 2015;56:Article ID 091506.
   PubMed Web of Science ®Google Scholar
• Himonas AA, Holliman C. The Cauchy problem for a generalized Camassa-Holm equations. Adv Differ Equ. 2014;19:161–200.
   Web of Science ®Google Scholar
• Novikov VS. Generalizations of the Camassa-Holm equation. J Phys A Math Theor. 2009;42:342002.
   Web of Science ®Google Scholar
• Freire IL. A look on some results about Camassa–Holm type equations. Commun Math. 2021;29:115–130.
   Google Scholar
• Himonas AA, Thompson RC. Persistence properties and unique continuation for a generalized Camassa–Holm equation. J Math Phys. 2014;55:Article ID 091503.
   Web of Science ®Google Scholar
• Zhou Y. On solutions to the Holm-Staley b-family of equations. Nonlinearity. 2010;23:369–381.
   Web of Science ®Google Scholar
• Christov O, Hakkaev S. On the Cauchy problem for the periodic b-family of equations and of the non-uniform continuity of Degasperis–Procesi equation. J Math Anal. 2009;360:47–56.
   Web of Science ®Google Scholar
• Yan K. Wave breaking and global existence for a family of peakon equations with high order nonlinearity. Nonlinear Anal Real World Appl. 2019;45:721–735.
   Web of Science ®Google Scholar
• Lai S, Wu Y. Global solutions and blow-up phenomena to a shallow water equation. J Differ Equ. 2010;249:693–706.
   Web of Science ®Google Scholar
• Wei L, Qiao Z, Wang Y, et al. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete Contin Dyn Syst Ser A. 2017;37:1733–1748.
   Web of Science ®Google Scholar
• Chen L, Guan C. Global solutions for the generalized Camassa–Holm equation. Nonlinear Anal Real World Appl. 2021;58:Article ID 103227.
   Web of Science ®Google Scholar
• da Silva PL, Freire IL. Existence, persistence, and continuation of solutions for a generalized 0-Holm-Staley equation. J Differ Equ. 2022;320:371–398. https://doi.org/10.1016/j.jde.2022.02.058
   Web of Science ®Google Scholar
• Constantin A. Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann Inst Fourier. 2000;50:321–362.
   Web of Science ®Google Scholar
• Constantin A, Escher J. On the blow-up rate and the blow-up set of breaking waves for a shallow water equation. Math Z. 2000;233:75–91.
   Web of Science ®Google Scholar
• Constantin A. Finite propagation speed for the Camassa–Holm equation. J Math Phys. 2005;46:Article ID 023506.
   Web of Science ®Google Scholar
• Henry D. Compactly supported solutions of the Camassa–Holm equation. J Nonlinear Math Phys. 2005;12:342–347.
   Web of Science ®Google Scholar
• Himonas AA, Misiolek G, Ponce G, et al. Persistence properties and unique continuation of solutions of the Camassa-Holm equation. Commun Math Phys. 2007;271:511–522.
   Web of Science ®Google Scholar
• Freire IL. Conserved quantities, continuation and compactly supported solutions of some shallow water models. J Phys A Math Theor. 2021;54:Article ID 015207.
   Web of Science ®Google Scholar
• Freire IL. Corrigendum to “Conserved quantities, continuation and compactly supported solutions of some shallow water models”. J Phys A Math Theor. 2021;54:Article ID 015207.
   Web of Science ®Google Scholar
• Linares F, Ponce G. Unique continuation properties for solutions to the Camassa–Holm equation and related models. Proc Amer Math Soc. 2020;148:3871–3879.
   Web of Science ®Google Scholar