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Applicable Analysis
An International Journal
Volume 98, 2019 - Issue 3
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Articles

Infinite propagation speed and asymptotic behavior for a generalized fifth-order Camassa–Holm equation

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Pages 536-552 | Received 15 Sep 2017, Accepted 19 Oct 2017, Published online: 28 Nov 2017

References

  • Holm D , Staley MF . Wave structures and nonlinear balances in a family of 1+1 evolutionary PDEs. Siam J Appl Dyn Syst. 2002;2:2003.
  • Camassa R , Holm D . An integrable shallow water equation with peaked solitons. Phys Rev Lett. 1993;71:1661–1664.
  • Constantin A , Lannes D . The hydrodynamical relevance of the Camassa--Holm and Degasperis--Procesi equations. Arch Ration Mech Anal. 2009;192:165–186.
  • Constantin A . The Hamiltonian structure of the Camassa--Holm equation. Expos Math. 1997;15:53–85.
  • Constantin A , Strauss WA . Stability of the Camassa--Holm solitons. J Nonlinear Sci. 2002;12:415–422.
  • Constantin A . On the scattering problem for the Camassa--Holm equation. Proc R Soc London A. 2001;457:953–970.
  • Constantin A , Mckean HP . A shallow water equation on the circle. Commun Pure Appl Math. 1999;52:949–982.
  • Beals R , Sattinger DH , Szmigielski J . Multipeakons and the classical moment problem. Adv Math. 2000;154:229–257.
  • Bressan A , Constantin A . Global conservative solutions of the Camassa--Holm equation. Arch Ration Mech Anal. 2007;183:215–239.
  • Holden H , Raynaud X . Dissipative solutions for the Camassa--Holm equation. Discrete Contin Dyn Syst. 2009;24:1047–1112.
  • Danchin R . A few remarks on the Camassa--Holm equation. Differ Integral Equ. 2000;14:953–988.
  • Constantin A , Escher J . Global existence and blow-up for a shallow water equation. Ann Sc Norm Super Pisa Cl Sci. 1998;26:303–328.
  • Constantin A , Escher J . Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 1998;181:229–243.
  • Constantin A . Finite propagation speed for the Camassa--Holm equation. J Math Phys. 2005;46:L1–L4.
  • Ni L , Zhou Y . A new asymptotic behavior of solutions to the Camassa--Holm equation. P Am Math Soc. 2011;140:607–614.
  • Mi Y , Guo B , Mu C . Lower order regularity for the generalized Camassa--Holm equation. Appl Anal. 2016;1–12.
  • 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.
  • Mi Y , Mu C . Well-posedness and analyticity for the Cauchy problem for the generalized Camassa--Holm equation. J Math Anal Appl. 2013;405:173–182.
  • Henry D . Infinite propagation speed for the Degasperis--Procesi equation. J Math Anal Appl. 2005;311:755–759.
  • Degasperis A , Holm DD , Hone ANW . A new integrable equation with peakon solutions. Theor Math Phys. 2002;133:1463–1474.
  • Coclite GM , Karlsen KH . On the well-posedness of the Degasperis--Procesi equation. J Funct Anal. 2006;233:60–91.
  • Liu Y , Yin Z . Global existence and blow-up phenomena for the Degasperis--Procesi equation. Commun Math Phys. 2006;267:801–820.
  • Yin Z . Global weak solutions for a new periodic integrable equation with peakon solutions. J Funct Anal. 2004;212:182–194.
  • Constantin A , Ivanov RI , Lenells J . Inverse scattering transform for the Degasperis--Procesi equation. Nonlinearity. 2010;23:2559–2575.
  • Inci H . On the well-posedness of the Holm--Staley b-family of equations. J Nonlinear Math Phys. 2015;23:4821–4838.
  • Zhou Y . On solutions to the Holm--Staley b-family of equations. Nonlinearity. 2010;23:369–381.
  • Singh K , Gupta RK , Kumar S . Exact solutions of b-family equation: classical Lie approach and direct method. Int J Nonlinear Sci. 2011;11:59–67.
  • Lv GY , Wang MX . Blow-up solutions of the general b-equation. J Math Phys. 2010;51:215–237.
  • Holm D , Hone ANW . Nonintegrability of a fifth-order equation with integrable two-body dynamics. Theoret Math Phys. 2003;137:1459–1471.
  • Coclite GM , Holden H , Karlsen KH . Well-posedness of higher-order Camassa--Holm equations. J Differ Equ. 2009;246:929–963.
  • Constantin A . The trajectories of particles in Stokes waves. Invent Math. 2006;166:523–535.
  • Constantin A . Particle trajectories in extreme Stokes waves. IMA J Appl Math. 2012;77:293–307.
  • Constantin A , Escher J . Analyticity of periodic traveling free surface water waves with vorticity. Ann Math. 2011;173:559–568.
  • Constantin A , Strauss WA . Stability of peakons. Commun Pure Appl Math. 2000;53:603–610.
  • Lenells J . A Variational approach to the stability of periodic Peakons. J Nonlinear Math Phys. 2004;11:151–163.
  • Liu Y , Lin Z . Stability of peakons for the Degasperis--Procesi equation. Commun Pure Appl Math. 2009;62:125–146.
  • Tian L , Zhang P , Xia L . Global existence for the higher-order Camassa--Holm shallow water equation. Nonlinear Anal Theor. 2011;74:2468–2474.
  • Ding D . Traveling solutions and evolution properties of the higher order Camassa--Holm equation. Nonlinear Anal Theor. 2017;152:1–11.
  • Coclite GM , Ruvo LD . A note on the convergence of the solutions of the Camassa--Holm equation to the entropy ones of a scalar conservation law. Discrete Contin Dyn Syst. 2015;36:2981–2990.
  • Ding D , Lv P . Conservative solutions for higher-order Camassa--Holm equations. J Math Phys. 2010;51:929.
  • Tang H , Liu ZR . Well-posedness of the modified Camassa--Holm equation in Besov spaces. Z Angew Math Phys. 2015;66:1559–1580.
  • Wu X , Guo B . The exponential decay of solutions and traveling wave solutions for a modified Camassa--Holm equation with cubic nonlinearity. J Math Phys. 2014;55:081504.
  • Strichartz RS . A guide to distribution theory and Fourier transforms. Boca Raton (FL): CRC Press; 1994.
  • Yan K , Yin Z . Infinite propagation speed and asymptotic behavior for a two-component Degasperis--Procesi system. Monatsh Math. 2014;266:1–18.

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