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July  2013, 33(7): 3211-3223. doi: 10.3934/dcds.2013.33.3211

Persistence properties and infinite propagation for the modified 2-component Camassa--Holm equation

Xinglong Wu 1, and  Boling Guo 2,

1. 

Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China
2. 

Institute of Applied Physics & Computational Math., Beijing 100088

Received  April 2012 Revised  July 2012 Published  January 2013

In this paper, we mainly study persistence properties and infinite propagation for the modified 2-component Camassa--Holm equation. We first prove that persistence properties of the solution to the equation provided the initial potential satisfies a certain sign condition. Finally, we get the infinite propagation if the initial datas satisfy certain compact conditions, while the solution to system (1.1) instantly loses compactly supported, the solution has exponential decay as $|x|$ goes to infinity.
Keywords: exponential decay, infinite propagation speed., compact support, The modified 2-component Camassa--Holm equation, persistence properties.
Mathematics Subject Classification: 35G25, 35L0.
Citation: Xinglong Wu, Boling Guo. Persistence properties and infinite propagation for the modified 2-component Camassa--Holm equation. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3211-3223. doi: 10.3934/dcds.2013.33.3211
References:
[1]

R. Beals, D. Sattinger and J. Szmigielski, Multipeakons and a theorem of Stieltjes, Inverse Problems, 15 (1999), 1-4. doi: 10.1088/0266-5611/15/1/001.  Google Scholar

[2]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa- Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.  Google Scholar

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27. doi: 10.1142/S0219530507000857.  Google Scholar

[4]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[5]

R. Camassa, D. Holm and J. Hyman, An integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. Google Scholar

[6]

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701.  Google Scholar

[7]

A. Constantin, Global existence and breaking waves for a shallow water equation: A geometric approch, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.  Google Scholar

[8]

A. Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 46 (2005), 023506. doi: 10.1063/1.1845603.  Google Scholar

[9]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91. doi: 10.1007/PL00004793.  Google Scholar

[10]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.3.CO;2-C.  Google Scholar

[11]

H. H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mechanica, 127 (1998), 193-207. doi: 10.1007/BF01170373.  Google Scholar

[12]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[13]

A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246. Google Scholar

[14]

C. Guan and Z. Yin, Well-podness and blow-up pheonmena for a modified two-component Camassa- Holm equation, in "Nonlinear Partial Differential Equations and HyperbolicWave Phenomena" Contemporary Mathematics, Amer. Math. Soc., Providence, RI., 526 (2010), 199-220. Google Scholar

[15]

D. Henry, Compactly supported solutions of the Camassa-Holm equation, J. Nonlinear Math. Phys., 12 (2005), 342-347. doi: 10.2991/jnmp.2005.12.3.3.  Google Scholar

[16]

A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys., 271 (2007), 511-522. doi: 10.1007/s00220-006-0172-4.  Google Scholar

[17]

D. Holm, J. Marsden and T. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.  Google Scholar

[18]

D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified two-component Camassa-Holm equation, Phy. Rev. E., 79 (2009), 1-13. doi: 10.1103/PhysRevE.79.016601.  Google Scholar

[19]

T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99. Google Scholar

[20]

T. Kato, On the Cauchy problem for the generalized Korteweg-de Vries equation, in "Studies in Applied Mathematics" in "Adv. Math. Suppl. Stu.", 8, Academic Press, New York, (1983), 93-128.  Google Scholar

[21]

J. Liu and Z. Yin, Global existence and blow-up phenomena for a periodic modified two-component Camassa-Holm equation,, IMA J. Appl. Math., 34 (): 1.   Google Scholar

[22]

W. Tan and Z. Yin, Global periodic conservative solutions of a periodic modified two-component Camassa-Holm equation, J. Funct. Anal., 261 (2011), 1204-1226. doi: 10.1016/j.jfa.2011.04.015.  Google Scholar

show all references

References:
[1]

R. Beals, D. Sattinger and J. Szmigielski, Multipeakons and a theorem of Stieltjes, Inverse Problems, 15 (1999), 1-4. doi: 10.1088/0266-5611/15/1/001.  Google Scholar

[2]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa- Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.  Google Scholar

[3]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27. doi: 10.1142/S0219530507000857.  Google Scholar

[4]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[5]

R. Camassa, D. Holm and J. Hyman, An integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. Google Scholar

[6]

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701.  Google Scholar

[7]

A. Constantin, Global existence and breaking waves for a shallow water equation: A geometric approch, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.  Google Scholar

[8]

A. Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 46 (2005), 023506. doi: 10.1063/1.1845603.  Google Scholar

[9]

A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91. doi: 10.1007/PL00004793.  Google Scholar

[10]

A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.3.CO;2-C.  Google Scholar

[11]

H. H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mechanica, 127 (1998), 193-207. doi: 10.1007/BF01170373.  Google Scholar

[12]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[13]

A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid Mech., 78 (1976), 237-246. Google Scholar

[14]

C. Guan and Z. Yin, Well-podness and blow-up pheonmena for a modified two-component Camassa- Holm equation, in "Nonlinear Partial Differential Equations and HyperbolicWave Phenomena" Contemporary Mathematics, Amer. Math. Soc., Providence, RI., 526 (2010), 199-220. Google Scholar

[15]

D. Henry, Compactly supported solutions of the Camassa-Holm equation, J. Nonlinear Math. Phys., 12 (2005), 342-347. doi: 10.2991/jnmp.2005.12.3.3.  Google Scholar

[16]

A. Himonas, G. Misiolek, G. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa-Holm equation, Comm. Math. Phys., 271 (2007), 511-522. doi: 10.1007/s00220-006-0172-4.  Google Scholar

[17]

D. Holm, J. Marsden and T. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.  Google Scholar

[18]

D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified two-component Camassa-Holm equation, Phy. Rev. E., 79 (2009), 1-13. doi: 10.1103/PhysRevE.79.016601.  Google Scholar

[19]

T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99. Google Scholar

[20]

T. Kato, On the Cauchy problem for the generalized Korteweg-de Vries equation, in "Studies in Applied Mathematics" in "Adv. Math. Suppl. Stu.", 8, Academic Press, New York, (1983), 93-128.  Google Scholar

[21]

J. Liu and Z. Yin, Global existence and blow-up phenomena for a periodic modified two-component Camassa-Holm equation,, IMA J. Appl. Math., 34 (): 1.   Google Scholar

[22]

W. Tan and Z. Yin, Global periodic conservative solutions of a periodic modified two-component Camassa-Holm equation, J. Funct. Anal., 261 (2011), 1204-1226. doi: 10.1016/j.jfa.2011.04.015.  Google Scholar

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