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Analysis of the Laplacian and spectral operators on the Vicsek set
The existence of weak solutions for a generalized Camassa-Holm equation
1. | Department of Mathematics,Sichuan Normal University, Chengdu, Department of Mathematics and Statistics,Curtin University of Technology, Perth, China |
2. | Department of Applied Mathematics, Southwestern University of Finance and Economics, 610074, Chengdu, China, China, China |
References:
[1] |
R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: doi:10.1103/PhysRevLett.71.1661. |
[2] |
Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.
doi: doi:10.1006/jdeq.1999.3683. |
[3] |
Z. H. Guo, M. Jiang, Z. Wang and G. F. Zheng, Global weak solutions to the Camassa-Holm equation, Discrete and Continuous Dynamical Systems, 21 (2008), 883-906.
doi: doi:10.3934/dcds.2008.21.883. |
[4] |
A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.
doi: doi:10.1007/s002200050801. |
[5] |
R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444.
doi: doi:10.1016/S0022-0396(03)00096-2. |
[6] |
A. M. Wazwaz, A class of nonlinear fourth order variant of a generalized Camassa-Holm equation with compact and noncompact solutions, Appl. Math. Comput., 165 (2005), 485-501.
doi: doi:10.1016/j.amc.2004.04.029. |
[7] |
A. M. Wazwaz, The tanh-coth and the sine-cosine methods for kinks, solitons and periodic solutions for the Pochhammer-Chree equations, Appl. Math. Comput., 195 (2008), 24-33.
doi: doi:10.1016/j.amc.2007.04.066. |
[8] |
L. X. Tian and X. Y. Song, New peaked solitary wave solutions of the generalized Camassa-Holm equation, Chaos, Solitons and Fractals, 19 (2004), 621-637.
doi: doi:10.1016/S0960-0779(03)00192-9. |
[9] |
Y. Zheng and S. Y. Lai, Peakons, solitary patterns and periodic solutions for generalized Camassa-Holm equations, Phys. Lett. A, 372 (2008), 4141-4143.
doi: doi:10.1016/j.physleta.2007.03.096. |
[10] |
S. Hakkaev and K. Kirchev, Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation, Communications in Partial Differential Equations, 30 (2005), 761-781.
doi: doi:10.1081/PDE-200059284. |
[11] |
T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equation, Commun. Pure Appl. Math., 41 (1988), 891-907.
doi: doi:10.1002/cpa.3160410704. |
[12] |
J. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation, Phil. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601.
doi: doi:10.1098/rsta.1975.0035. |
[13] |
A. Moameni, Soliton solutions for quasilinear Schr$\ddoto$dinger equations involving supercritical exponent in $R^N$, Communications on Pure and Applied Analysis, 7 (2008), 89-105.
doi: doi:10.3934/cpaa.2008.7.89. |
[14] |
R. M. Colombo and G. Guerra, Hyperbolic balance laws with a dissipative non local source, Communications on Pure and Applied Analysis, 7 (2008), 1077-1090.
doi: doi:10.3934/cpaa.2008.7.1077. |
[15] |
D. Pilod, Sharp well-posedness results for the Kuramoto-Velarde equation, Communications on Pure and Applied Analysis, 7 (2008), 867-881.
doi: doi:10.3934/cpaa.2008.7.867. |
[16] |
K. Y. Wang, Global well-posedness for a transport equation with non-local velocity and critical diffusion, Communications on Pure and Applied Analysis, 7 (2008), 1203-1210.
doi: doi:10.3934/cpaa.2008.7.1203. |
[17] |
C. L. He, D. X. Kong and K. F. Liu, Hyperbolic mean curvature flow, J. Differential Equations, 246 (2009), 373-390.
doi: doi:10.1016/j.jde.2008.06.026. |
show all references
References:
[1] |
R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
doi: doi:10.1103/PhysRevLett.71.1661. |
[2] |
Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.
doi: doi:10.1006/jdeq.1999.3683. |
[3] |
Z. H. Guo, M. Jiang, Z. Wang and G. F. Zheng, Global weak solutions to the Camassa-Holm equation, Discrete and Continuous Dynamical Systems, 21 (2008), 883-906.
doi: doi:10.3934/dcds.2008.21.883. |
[4] |
A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61.
doi: doi:10.1007/s002200050801. |
[5] |
R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444.
doi: doi:10.1016/S0022-0396(03)00096-2. |
[6] |
A. M. Wazwaz, A class of nonlinear fourth order variant of a generalized Camassa-Holm equation with compact and noncompact solutions, Appl. Math. Comput., 165 (2005), 485-501.
doi: doi:10.1016/j.amc.2004.04.029. |
[7] |
A. M. Wazwaz, The tanh-coth and the sine-cosine methods for kinks, solitons and periodic solutions for the Pochhammer-Chree equations, Appl. Math. Comput., 195 (2008), 24-33.
doi: doi:10.1016/j.amc.2007.04.066. |
[8] |
L. X. Tian and X. Y. Song, New peaked solitary wave solutions of the generalized Camassa-Holm equation, Chaos, Solitons and Fractals, 19 (2004), 621-637.
doi: doi:10.1016/S0960-0779(03)00192-9. |
[9] |
Y. Zheng and S. Y. Lai, Peakons, solitary patterns and periodic solutions for generalized Camassa-Holm equations, Phys. Lett. A, 372 (2008), 4141-4143.
doi: doi:10.1016/j.physleta.2007.03.096. |
[10] |
S. Hakkaev and K. Kirchev, Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation, Communications in Partial Differential Equations, 30 (2005), 761-781.
doi: doi:10.1081/PDE-200059284. |
[11] |
T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equation, Commun. Pure Appl. Math., 41 (1988), 891-907.
doi: doi:10.1002/cpa.3160410704. |
[12] |
J. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation, Phil. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601.
doi: doi:10.1098/rsta.1975.0035. |
[13] |
A. Moameni, Soliton solutions for quasilinear Schr$\ddoto$dinger equations involving supercritical exponent in $R^N$, Communications on Pure and Applied Analysis, 7 (2008), 89-105.
doi: doi:10.3934/cpaa.2008.7.89. |
[14] |
R. M. Colombo and G. Guerra, Hyperbolic balance laws with a dissipative non local source, Communications on Pure and Applied Analysis, 7 (2008), 1077-1090.
doi: doi:10.3934/cpaa.2008.7.1077. |
[15] |
D. Pilod, Sharp well-posedness results for the Kuramoto-Velarde equation, Communications on Pure and Applied Analysis, 7 (2008), 867-881.
doi: doi:10.3934/cpaa.2008.7.867. |
[16] |
K. Y. Wang, Global well-posedness for a transport equation with non-local velocity and critical diffusion, Communications on Pure and Applied Analysis, 7 (2008), 1203-1210.
doi: doi:10.3934/cpaa.2008.7.1203. |
[17] |
C. L. He, D. X. Kong and K. F. Liu, Hyperbolic mean curvature flow, J. Differential Equations, 246 (2009), 373-390.
doi: doi:10.1016/j.jde.2008.06.026. |
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