Well-posedness and persistence property for a shallow water wave equation for waves of large amplitude

References

• Korteweg DJ , de Vries G . On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philos Mag. 1895;39:422–443.
   Google Scholar
• Drazin PG , Johnson RS . Solitons: an introduction. Cambridge: Cambridge University Press; 1989.
   Google Scholar
• Kato T . On the Korteweg-de Vries equation. Manuscripta Math. 1979;28:89–99.
   Web of Science ®Google Scholar
• Dodd RK , Morris HC , Eilbeck JC , et al . Soliton and nonlinear wave equations. London: Academic Press; 1982. p. 1.
   Google Scholar
• Constantin A , Escher J . Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 1998;181:229–243.
   Web of Science ®Google Scholar
• Whitham GB . Linear and nonlinear waves. New York (NY): Wiley; 1980.
   Google Scholar
• Fokas A , Fuchssteiner B . Symplectic structures, their Bäklund transformation and hereditary symmetries. Physica D. 1981;4:47–66.
   Web of Science ®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
• Constantin A , McKean HP . A shallow water equation on the circle. Commun Pure Appl Math. 1999;52:949–982.
   Web of Science ®Google Scholar
• Constantin A . Nonlinear water waves with applications to wave-current interactions and tsunamis. Vol. 81, CBMS-NSF regional conference series in applied mathematics. Philadephia: SIAM; 2011.
   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 . Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation. Commun Pure Appl Math. 1998;51(5):475–504.
   Web of Science ®Google Scholar
• Constantin A . Finite propagation speed for the Camassa--Holm equation. J Math Phys. 2005;46. Art. Nr. 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
• Danchin R . A few remarks on the Camassa--Holm equation. Differ Integral Equ. 2001;14:953–988.
   Google Scholar
• Danchin R . A note on well-posedness for Camassa--Holm equation. J Differ Equ. 2003;192:429–444.
   Web of Science ®Google Scholar
• Rodriguez-Blanco G . On the Cauchy problem for the Camassa--Holm equation. Nonlinear Anal. 2001;46:309–327.
   Web of Science ®Google Scholar
• Li J , Yin Z . Remarks on the well-posedness of Camassa--Holm type equations in Besov spaces. J Differ Equ. 2016;261:6125–6143.
   Web of Science ®Google Scholar
• Constantin A , Strauss W . Stability of the Camassa--Holm solitons. J Nonlinear Sci. 2002;12:415–422.
   Web of Science ®Google Scholar
• Constantin A , Strauss W . Stability of peakons. Comm Pure Appl Math. 2000;53:603–610.
   Web of Science ®Google Scholar
• Guan C , Yin Z . Global weak solutions for a two-component Camassa--Holm shallow water system. J Funct Anal. 2011;260:1132–1154.
   Web of Science ®Google Scholar
• Guan C , Yin Z . Global existence and blow-up phenomena for an integrable two-component Camassa--Holm shallow water system. J Differ Equ. 2010;248:2003–2014.
   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. 2010;258:4251–4278.
   Web of Science ®Google Scholar
• Himonas A , Kenig C , Misiolek G . Non-uniform dependence for the periodic Camassa--Holm equation. Commun Partial Differ Equ. 2010;35:1145–1162.
   Web of Science ®Google Scholar
• Himonas A , Misiolek G , Ponce G , Zhou Y . Persistence properties and unique continuation of solutions of the Camassa--Holm equation. Commun Math Phys. 2007;271:511–512.
   Web of Science ®Google Scholar
• Ni L , Zhou Y . A new asymptotic behavior of solutions to the Camassa--Holm equation. Proc Amer Math Soc. 2012;140:607–614.
   Web of Science ®Google Scholar
• Holden H , Raynaud X . Global conservative solutions of the Camassa--Holm equations-a Lagrangian point of view. Commun Partial Differ Equ. 2007;32:1511–1549.
   Web of Science ®Google Scholar
• Quirchmayr Ronald . A new highly nonlinear shallow water wave equation. J Evol Equ. 2016;16:539–567.
   Web of Science ®Google Scholar
• Kato T . Quasi-linear equations of evolution, with applications to partial differential equations. Spectral theory and differential equations. Vol. 448, Lecture notes in mathematics. Berlin: Springer-Verlag; 1975. p. 25–70.
   Google Scholar
• Kato T . On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Stud Appl Math. 1983;8:93–126.
   Google Scholar
• Pazy A . Semigroups of linear operators and applications to partial differential equations. Vol. 44, Applied mathematical sciences. New York (NY): Springer-Verlag; 1983. viii+279 p.
   Google Scholar