Blow-up criteria for modified two-component generalization of hyper-elastic rod equation

References

• Dai H-H. Model equations for nonlinear dispersive waves in a compressible Mooney–Rivlin rod. Acta Mech. 1998;127(1–4):193–207.
   Web of Science ®Google Scholar
• Dai H-H, Huo Y. Solitary shock waves and other travelling waves in a general compressible hyperelastic rod. R Soc Lond Proc Ser A Math Phys Eng Sci. 2000;456:331–363.
   Web of Science ®Google Scholar
• Camassa R, Holm D. An integrable shallow water equation with peaked solitons. Phys Rev Lett. 1993;71:1661–1664.
   PubMed Web of Science ®Google Scholar
• Fokas A, Fuchssteiner B. Symplectic structures, their backlund transformation and hereditary symmetries. Phys D. 1981;4:47–66.
   Web of Science ®Google Scholar
• Beals R, Sattinger D, Szmigielski J. Acoustic scattering and the extended Korteweg de Vries hierarchy. Adv Math. 1998;140:190–206.
   Web of Science ®Google Scholar
• Boutet de Monvel A, Kostenko A, Shepelsky D, et al. Long-time asymptotic for the Camassa–Holm equation. SIAM J Math Anal. 2009;41:1559–1588.
   Web of Science ®Google Scholar
• Camassa R, Holm D, Hyman J. A new integrable shallow water equation. Adv Appl Mech. 1994;31:1–33.
   Google Scholar
• Constantin A, McKean HP. A shallow water equation on the circle. Comm Pure Appl Math. 1999;52:949–982.
   Web of Science ®Google Scholar
• Constantin A. On the scattering problem for the Camassa–Holm equation. Proc Roy Soc London A. 2001;457:953–970.
   Web of Science ®Google Scholar
• Constantin A, Lannes D. The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch Ration Mech Anal. 2009;192:165–186.
   Web of Science ®Google Scholar
• Bressan A, Constantin A. Global conservative solutions of the Camassa–Holm equation. Arch Ration Mech Anal. 2007;183:215–239.
   Web of Science ®Google Scholar
• Bressan A, Constantin A. Global dissipative solutions of the Camassa–Holm equation. Anal Appl. 2007;5:1–27.
   Web of Science ®Google Scholar
• Constantin A, Escher J. Wave breaking for nonlinear nonlocal shallow water equation. Acta Math. 1998;181:229–243.
   Web of Science ®Google Scholar
• Jiang Z, Ni L, Zhou Y. Wave breaking of the Camassa–Holm equation. J Nonlinear Sci. 2012;22(2):235–245.
   Web of Science ®Google Scholar
• Li Y, Olver P. Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J Differ Equ. 2000;162:27–63.
   Web of Science ®Google Scholar
• Constantin A. Finite propagation speed for the Camassa–Holm equation. J Math Phys. 2005;46(2):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 A, Misiolek G, Ponce G, et al. Persistence properties and unique continuation of solutions of the Camassa–Holm equation. Comm Math Phys. 2007;271(2):511–522.
   Web of Science ®Google Scholar
• Novruzov E, Hagverdiyev A. On the behavior of the solution of the dissipative Camassa–Holm equation with the arbitrary dispersion coefficient. J Differ Equ. 2014;257:4525–4541.
   Web of Science ®Google Scholar
• Zhou Y, Chen H. Wave breaking and propagation speed for the Camassa–Holm equation with k≠0. Nonlinear Anal Real World Appl. 2011;12:1875–1882.
   Web of Science ®Google Scholar
• Holm D, Ó Náraigh L, Tronci C. Singular solutions of a modified two-component Camassa–Holm equation. Phys Rev E(3). 2009;79:016601, 13 pp.
   Web of Science ®Google Scholar
• Holm D, Marsden J, Ratiu T. The Euler–Poincar e equations and semidirect products with applications to continuum theories. Adv Math. 1998;137:1–81.
   Web of Science ®Google Scholar
• Guo Z, Zhu M. Wave breaking for a modified two-component Camassa–Holm system. J Differ Equ. 2012;252:2759–2770.
   Web of Science ®Google Scholar
• Guan C, Karlsen KH, Yin Z. Well-posedness and blow-up phenomena for a modified two-component Camassa–Holm equation. Proceedings of the 2008–2009 Special Year in Nonlinear Partial Differential Equations; 2010, p. 199–220. (Contemp. Math., Amer. Math. Soc).
   Google Scholar
• Guo Z, Zhu M, Ni L. Blow-up criteria of solutions to a modified two-component Camassa–Holm system. Nonlinear Anal Real World Appl. 2011;12:3531–3540.
   Web of Science ®Google Scholar
• Olver P, Rosenau P. Tri-hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys Rev E. 1996;53:1900–1906.
   PubMed Web of Science ®Google Scholar
• Constantin A, Ivanov RI. On an integrable two-component Camassa–Holm shallow water system. Phys Lett A. 2008;372:7129–7132.
   Web of Science ®Google Scholar
• Ivanov R. Two-component integrable systems modelling shallow water waves: the constant vorticity case. Wave Motion. 2009;46:389–396.
   Web of Science ®Google Scholar
• Hone ANW, Novikov V, Wang JP. Two-component generalizations of the Camassa–Holm equation equation. Nonlinearity. 2017;30(2):622–658.
   Web of Science ®Google Scholar
• Escher J, Lechtenfeld O, Yin ZY. Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation. Discrete Contin Dyn Syst. 2007;19:493–513.
   Web of Science ®Google Scholar
• Kato T. Quasi-linear equation of evolution, with applications to partial differential equations. Spectral theory and differential equations;1975. p. 25–70. (Lecture Notes in Math.; 448).
   Google Scholar
• Gui G, Liu Y. On the Cauchy problem for the two-component Camassa–Holm system. Math Z. 2010;268:45–66.
   Web of Science ®Google Scholar
• Liu Y, Zhang P. Stability of solitary waves and wave-breaking phenomena for the two-component Camassa–Holm system. Int Math Res Not. 2009;211:1981–2021.
   Google Scholar
• Aratyn H, Gomes JF, Zimerman AH. On a negative flow of the AKNS hierarchy and its relation to a two-component Camassa–Holm equation. Symm Integr Geom Methods Appl. 2006;2:070. 12pp.
   Web of Science ®Google Scholar
• Chen RM, Liu Y. Wave breaking and global existence for a generalized two-component Camassa–Holm system. Int Math Res Not. 2011;6:1381–1418.
   Google Scholar
• Chen RM, Liu Y, Qiao Z. Stability of solitary waves and global existence of a generalized two-component Camassa–Holm system. Comm Partial Differ Equ. 2011;36:2162–2188.
   Web of Science ®Google Scholar
• Guan CX, Yin ZY. Global weak solutions for a two-component Camassa–Holm shallow water systems. J Funct Anal. 2011;260:1132–1154.
   Web of Science ®Google Scholar
• Gui G, Liu Y. On the global existence and wave-breaking criteria for the two-component Camassa–Holm system. J Funct Anal. 2011;258:4251–4278.
   Web of Science ®Google Scholar
• Yang S, Xu T. Local-in-space blow-up and symmetric waves for a generalized two-component Camassa–Holm system. Appl Math Comput. 2019;347:514–521.
   Web of Science ®Google Scholar
• Zhu M, Xu J. On the wave-breaking phenomena for the periodic two-component Dullin–Gottwald–Holm system. J Math Anal Appl. 2012;391:415–428.
   Web of Science ®Google Scholar
• Bayrak V, Novruzov E, Ozkol I. Local-in-space blow-up criteria for two-component nonlinear dispersive wave system. DCDS. 0000;39(10):6023–6037.
   Google Scholar
• Yin Z. On the blow-up scenario for the generalized Camassa–Holm equation. Commun Partial Differ Equ. 2004;29(5 & 6):867–877.
   Web of Science ®Google Scholar
• Brandolese L. Local-in-space criteria for blowup in shallow water and dispersive rod equations. Comm Math Phys. 2014;330:401–414.
   Web of Science ®Google Scholar
• Tian C, Yan W, Zhang H. The Cauchy problem for the generalized hyperelastic rod equation. Math Nachr. 2014;287(17–18):2116–2137.
   Web of Science ®Google Scholar
• Yin Z. Well-posedness, global solutions and blow-up phenomena for a nonlinearly dispersive wave equation. J Evo Eqns. 2004;4:391–419.
   Web of Science ®Google Scholar
• Zhou Y. Local well-posedness and blow-up criteria of solutions for a rod equation. Math Nachr. 2005;278(14):1726–1739.
   Web of Science ®Google Scholar
• Brandolese L, Cortez MF. Blowup issues for a class of nonlinear dispersive wave equations. J Differ Equ. 2014;256:3981–3998.
   Web of Science ®Google Scholar
• Brandolese L, Cortez MF. On permanent and breaking waves in hyperelastic rods and rings. J Funct Anal. 2014;266:6954–6987.
   Web of Science ®Google Scholar
• Novruzov E, Yazar B. On blow-up criteria for a class of nonlinear dispersive wave equations with dissipation. Monatshefte Für Mathematik. 2019;188(1):163–181.
   Web of Science ®Google Scholar
• Novruzov E. Local-in-space blow-up criteria for a class of nonlinear dispersive wave equations. J Differ Equ. 2017;263:5773–5786.
   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
• McKean HP. Breakdown of a shallow water equation. Asian J Math. 1998;2(4):867–874.
   Google Scholar