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Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models
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 |
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. |
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. |
[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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