×

Conserved quantities, global existence and blow-up for a generalized CH equation. (English) Zbl 1364.35053

Summary: In this paper, we study conserved quantities, blow-up criterions, and global existence of solutions for a generalized CH equation. We investigate the classification of self-adjointness, conserved quantities for this equation from the viewpoint of Lie symmetry analysis. Then, based on these conserved quantities, blow-up criterions and global existence of solutions are presented.

MSC:

35B44 Blow-up in context of PDEs
35G25 Initial value problems for nonlinear higher-order PDEs
35L05 Wave equation
35B06 Symmetries, invariants, etc. in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. Bressan, Global conservative solutions of the Camassa-Holm equation,, Arch. Rat. Mech. Anal., 183, 215 (2007) · Zbl 1105.76013 · doi:10.1007/s00205-006-0010-z
[2] A. Bressan, Global dissipative solutions of the Camassa-Holm equation,, Anal. Appl., 5, 1 (2007) · Zbl 1139.35378 · doi:10.1142/S0219530507000857
[3] V. Busuioc, On second grade fluids with vanishing viscosity,, C. R. Acad. Sci. Paris Sér. I Math., 328, 1241 (1999) · Zbl 0935.76004 · doi:10.1016/S0764-4442(99)80447-9
[4] R. Camassa, An integrable shallow water equation with peaked solitons,, Phys. Rev. Letters, 71, 1661 (1993) · Zbl 0972.35521 · doi:10.1103/PhysRevLett.71.1661
[5] A. Constantin, On the inverse spectral problem for the Camassa-Holm equation,, J. Funct. Anal., 155, 352 (1998) · Zbl 0907.35009 · doi:10.1006/jfan.1997.3231
[6] A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50, 321 (2000) · Zbl 0944.00010 · doi:10.5802/aif.1757
[7] A. Constantin, On the scattering problem for the Camassa-Holm equation,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457, 953 (2001) · Zbl 0999.35065 · doi:10.1098/rspa.2000.0701
[8] A. Constantin, Well-posedness, global existence, and blow-up phenomena for a periodic quasi-linear hyperbolic equation,, Comm. Pure Appl. Math., 51, 475 (1998) · Zbl 0934.35153 · doi:10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5
[9] A. Constantin, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Super. Pisa Cl. Sci., 26, 303 (1998) · Zbl 0918.35005
[10] A. Constantin, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181, 229 (1998) · Zbl 0923.76025 · doi:10.1007/BF02392586
[11] A. Constantin, Inverse scattering transform for the Degasperis-Procesi equation,, Nonlinearity, 23, 2559 (2010) · Zbl 1211.37081 · doi:10.1088/0951-7715/23/10/012
[12] A. Constantin, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192, 165 (2009) · Zbl 1169.76010 · doi:10.1007/s00205-008-0128-2
[13] A. Constantin, A shallow water equation on the circle,, Comm. Pure Appl. Math., 52, 949 (1999) · Zbl 0940.35177 · doi:10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D
[14] H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod,, Acta Mech., 127, 193 (1998) · Zbl 0910.73036 · doi:10.1007/BF01170373
[15] A. Degasperis, A new integrable equation with peakon solutions,, Theoret. Math. Phys., 133, 1463 (2002) · doi:10.1023/A:1021186408422
[16] A. Degasperis, Integral and non-integrable equations with peakons,, in Nonlinear Physics: Theory and Experiment, 37 (2003) · Zbl 1053.37039 · doi:10.1142/9789812704467_0005
[17] A. Degasperis, Asymptotic integrability,, in Symmetry and Perturbation Theory (Rome, 23 (1999) · Zbl 0963.35167
[18] H. Dullin, On asymptotically equivalent shallow water wave equations,, Phys. D., 190, 1 (2004) · Zbl 1050.76008 · doi:10.1016/j.physd.2003.11.004
[19] H. Dullin, An integrable shallow water equation with linear and nonlinear dispersion,, Phys. Rev. Lett., 87 (2001) · doi:10.1103/PhysRevLett.87.194501
[20] J. Escher, Well-posedness, blow-up phenomena, and global solutions for the \(b\)-equation,, J. Reine Angew. Math., 624, 51 (2008) · Zbl 1159.35060 · doi:10.1515/CRELLE.2008.080
[21] B. Fuchssteiner, Symplectic structures, their backlund transformation and hereditary symmetries,, Phys. D, 4, 47 (1981) · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X
[22] M. Gandarias, Weak self-adjoint differential equations,, J. Phys. A: Math. Theor., 44 (2011) · Zbl 1223.35203
[23] M. Gandarias, Some weak self-adjoint Hamilton Jacobi Bellman equations arising in financial mathematics,, Nonlinear Anal.: RWA, 13, 340 (2012) · Zbl 1238.35048 · doi:10.1016/j.nonrwa.2011.07.041
[24] K. Grayshan, Equations with peakon traveling wave solutions,, Adv. Dyn. Syst. Appl., 8, 217 (2013) · Zbl 1256.35109
[25] G. Gui, Global existence and blow-up phenomena for the peakon \(b\)-family of equations,, Indiana Univ. Math. J., 57, 1209 (2008) · Zbl 1170.35369 · doi:10.1512/iumj.2008.57.3213
[26] A. Himonas, The Cauchy problem for a generalized Camassa-Holm equation,, Adv. Differ. Equations, 19, 161 (2014) · Zbl 1285.35093
[27] A. Himonas, Persistence properties and unique continuation for a generalized Camassa-Holm equation,, J. Math. Phys., 55 (2014) · Zbl 1366.35157 · doi:10.1063/1.4895572
[28] H. Holden, Dissipative solutions for the Camassa-Holm equation,, Discrete Contin. Dyn. Syst., 24, 1047 (2009) · Zbl 1178.65099 · doi:10.3934/dcds.2009.24.1047
[29] D. Holm, Wave structure and nonlinear balances in a family of evolutionary PDEs,, SIAM J. Appl. Dyn. Syst., 2, 323 (2003) · Zbl 1088.76531 · doi:10.1137/S1111111102410943
[30] J. Holmes, Continuity properties of the data-to-solution map for the generalized Camassa-Holm equation,, J. Math. Anal. Appl., 417, 635 (2014) · Zbl 1304.35610 · doi:10.1016/j.jmaa.2014.03.033
[31] A. Hone, Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable Camassa-Holm equation,, Dyn. Partial Differential Eqns., 6, 253 (2009) · Zbl 1179.37092 · doi:10.4310/DPDE.2009.v6.n3.a3
[32] A. Hone, Integrable peakon equations with cubic nonlinearity,, J. Phys. A, 41 (2008) · Zbl 1153.35075 · doi:10.1088/1751-8113/41/37/372002
[33] Y. Hou, Algebro-geometric solutions for the Degasperis-Procesi hierarchy,, SIAM J. Math. Anal., 45, 1216 (2013) · Zbl 1291.35286 · doi:10.1137/12089689X
[34] N. Ibragimov, A new conservation theorem,, J. Math. Anal. Appl., 333, 311 (2007) · Zbl 1160.35008 · doi:10.1016/j.jmaa.2006.10.078
[35] N. Ibragimov, Quasi-self-adjoint differential equations,, Archives of ALGA., 4, 55 (2007)
[36] N. Ibragimov, Nonlinear self-adjointness and conservation laws,, J. Phys. A: Math. Theor., 44 (2011) · Zbl 1270.35031 · doi:10.1088/1751-8113/44/43/432002
[37] N. Ibragimov, Self-adjointness and conservation laws of a generalized Burgers equation,, J. Phys. A: Math. Theor., 44 (2011) · Zbl 1216.35115 · doi:10.1088/1751-8113/44/14/145201
[38] D. Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation,, J. Nonlinear Math. Phys., 14, 303 (2007) · Zbl 1157.76005 · doi:10.2991/jnmp.2007.14.3.1
[39] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455, 63 (2002) · Zbl 1037.76006 · doi:10.1017/S0022112001007224
[40] R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow,, Fluid Dynam. Res., 33, 97 (2003) · Zbl 1032.76519 · doi:10.1016/S0169-5983(03)00036-4
[41] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, Spectral theory and differential equations, 25 (1974)
[42] H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation,, J. Nonlinear Sci., 17, 169 (2007) · Zbl 1185.35194 · doi:10.1007/s00332-006-0803-3
[43] H. McKean, Breakdown of a shallow water equation,, Asian J. Math., 2, 867 (1998) · Zbl 0959.35140 · doi:10.4310/AJM.1998.v2.n4.a10
[44] A. Mikhailov, Perturbative symmetry approach,, J. Phys. A, 35, 4775 (2002) · Zbl 1039.35008 · doi:10.1088/0305-4470/35/22/309
[45] L. Ni, Well-posedness and persistence properties for the Novikov equation,, J. Differential Equations, 250, 3002 (2011) · Zbl 1215.37051 · doi:10.1016/j.jde.2011.01.030
[46] W. Niu, Blow-up phenomena and global existence for the nonuniform weakly dissipative b-equation,, J. Math. Anal. Appl., 374, 166 (2011) · Zbl 1203.35050 · doi:10.1016/j.jmaa.2010.08.002
[47] V. Novikov, Generalizations of the Camassa-Holm equation,, J. Phys. A, 42 (2009) · Zbl 1181.37100 · doi:10.1088/1751-8113/42/34/342002
[48] P. Olver, <em>Applications of Lie Groups to Differential Equations</em>,, New York: Springer-Verlag (1993) · Zbl 0785.58003 · doi:10.1007/978-1-4612-4350-2
[49] Z. Qiao, The Camassa-Holm hierarchy, \(N\)-dimensional integrable systems, and algebrogeometric solution on a symplectic submanifold,, Commu. Math. Phys., 239, 309 (2003) · Zbl 1020.37046 · doi:10.1007/s00220-003-0880-y
[50] Z. Qiao, Integrable hierarchy, \(3\times 3\) constrained systems, and parametric and stationary solutions,, Acta Appl. Math., 83, 199 (2004) · Zbl 1078.37045 · doi:10.1023/B:ACAP.0000038872.88367.dd
[51] L. Wei, Conservation laws for a modified lubrication equation,, Nonlinear Analysis: RWA, 26, 44 (2015) · Zbl 1331.35024 · doi:10.1016/j.nonrwa.2015.04.005
[52] L. Wei, Self-adjointness and conservation laws for Kadomtsev-Petviashvili-Burgers equation,, Nonlinear Analysis: RWA, 23, 123 (2015) · Zbl 1316.35013 · doi:10.1016/j.nonrwa.2014.11.008
[53] X. Wu, Well-posedness and global existence for the Novikov equation,, Ann. Sc. Norm. Sup. Pisa CI. Sci., 11, 707 (2012) · Zbl 1261.35041
[54] Z. Xin, On the weak solution to a shallow water equation,, Comm. Pure Appl. Math., 53, 1411 (2000) · Zbl 1048.35092 · doi:10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5
[55] W. Yan, The Cauchy problem for the integrable Novikov equation,, J. Differential Equations, 253, 298 (2012) · Zbl 1248.35051 · doi:10.1016/j.jde.2012.03.015
[56] Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions,, Illinois J. Math., 47, 649 (2003) · Zbl 1061.35142
[57] S. Zhou, The properties of solutions for a generalized \(b\)-family equation with peakons,, J. Nonlinear Sci., 23, 863 (2013) · Zbl 1277.35134 · doi:10.1007/s00332-013-9171-8
[58] S. Zhou, The Cauchy problem for a generalized \(b\)-equation with higher-order nonlinearities in critical Besov spaces and weighted spaces,, Discrete Contin. Dyn. Syst., 34, 4967 (2014) · Zbl 1312.35045 · doi:10.3934/dcds.2014.34.4967
[59] S. Zhou, Well-posedness, blow-up phenomena and global existence for the generalized \(b\)-equation with higher-order nonlinearities and weak dissipation,, Discrete Contin. Dyn. Syst., 34, 843 (2014) · Zbl 1277.35135 · doi:10.3934/dcds.2014.34.843
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.