# American Institute of Mathematical Sciences

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

 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.
Citation: Xinglong Wu, Boling Guo. Persistence properties and infinite propagation for the modified 2-component Camassa--Holm equation. Discrete and 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. [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. [3] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27. doi: 10.1142/S0219530507000857. [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. [5] R. Camassa, D. Holm and J. Hyman, An integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. [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. [7] A. Constantin, Global existence and breaking waves for a shallow water equation: A geometric approch, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. [8] A. Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 46 (2005), 023506. doi: 10.1063/1.1845603. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99. [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. [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, 1-15. [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.

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##### 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. [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. [3] A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27. doi: 10.1142/S0219530507000857. [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. [5] R. Camassa, D. Holm and J. Hyman, An integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. [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. [7] A. Constantin, Global existence and breaking waves for a shallow water equation: A geometric approch, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. [8] A. Constantin, Finite propagation speed for the Camassa-Holm equation, J. Math. Phys., 46 (2005), 023506. doi: 10.1063/1.1845603. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] T. Kato, On the Korteweg-de Vries equation, Manuscripta Math., 28 (1979), 89-99. [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. [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, 1-15. [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.
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