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On the solutions of the Dullin-Gottwald-Holm equation in Besov spaces. (English) Zbl 1253.35129

Summary: This paper is concerned with the Cauchy problem for the Dullin-Gottwald-Holm equation. First, the local well-posedness for this system in Besov spaces is established. Second, the blow-up criterion for solutions to the equation is derived. Then, the existence and uniqueness of global solutions to the equation are investigated. Finally, the sharp estimate from below and lower semicontinuity for the existence time of solutions to this equation are presented.

MSC:

35Q35 PDEs in connection with fluid mechanics
35B44 Blow-up in context of PDEs
35C08 Soliton solutions
30H25 Besov spaces and \(Q_p\)-spaces
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[1] Dullin, H. R.; Gottwald, G. A.; Holm, D. D., An integral shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett., 87, 1945-1948 (2001)
[2] Drazin, P. G.; Johnson, R. S., Solitons: An Introduction (1989), Cambridge University Press: Cambridge University Press Cambridge, New York · Zbl 0661.35001
[3] Lopes, O., A linearized instability result for solitary waves, Discrete Contin. Dynam. Systems, 8, 115-119 (2002) · Zbl 1001.35108
[4] Bona, J. L.; Scott, R., Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces, Duke Math. J., 43, 87-99 (1976) · Zbl 0335.35032
[5] Kato, T., On the Korteweg-de Vries equation, Manuscripta Math., 23, 89-99 (1979) · Zbl 0415.35070
[6] Kenig, C.; Ponce, G.; Vega, L., Well-posedness and scattering results for the generalized Korteweg-de Veris equation via the contraction principle, Comm. Pure Appl. Math., 46, 527-620 (1993) · Zbl 0808.35128
[7] Camassa, R.; Holm, D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661-1664 (1993) · Zbl 0972.35521
[8] Constantin, A.; Lannes, D., The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 1992, 165-186 (2009) · Zbl 1169.76010
[9] Johnson, R., Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 457, 63-82 (2002) · Zbl 1037.76006
[10] Dai, H., Model equations for nolinear dispersive waves in compressible Mooney-Rivlin rod, Acta Mech., 127, 193-207 (1998) · Zbl 0910.73036
[11] Constantin, A., The Hamiltonian structure of the Camassa-Holm equation, Exposition. Math., 15, 53-85 (1997) · Zbl 0881.35094
[12] Fokas, A.; Fuchssteiner, B., Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4, 47-66 (1981) · Zbl 1194.37114
[13] Constantin, A., On the scattering problem for the Camassa-Holm equation, Proc. Roy. Soc. London A, 457, 953-970 (2001) · Zbl 0999.35065
[14] Camassa, R.; Holm, D.; Hyman, J., A new integrable shallow water equation, Adv. Appl. Mech., 31, 1-33 (1994) · Zbl 0808.76011
[15] Constantin, A.; Escher, J., Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44, 423-431 (2007) · Zbl 1126.76012
[16] Wen, Z.; Liu, Z., Bifurcation of peakons and periodic cusp waves for the generalization of the Camassa-Holm equation, Nonlinear Analysis: Real World Applications, 12, 1698-1707 (2011) · Zbl 1218.35026
[17] Constantin, A.; Strauss, W., Stability of peakons, Comm. Pure Appl. Math., 53, 603-610 (2000) · Zbl 1049.35149
[18] Beals, R.; Scattinger, D.; Szmigielski, J., Acoustic scatting and the extended Korteweg-de Vries hierarchy, Adv. Math., 140, 190-206 (1998) · Zbl 0919.35118
[19] Constantin, A.; Escher, J., Global existence and blow-up for a shallow water equation, Annali Sc. Norm. Sup. Pisa, 26, 303-328 (1998) · Zbl 0918.35005
[20] Constantin, A.; Escher, J., Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51, 475-504 (1998) · Zbl 0934.35153
[21] Dachin, R., A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14, 953-988 (2001) · Zbl 1161.35329
[22] Dachin, R., A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192, 429-444 (2003) · Zbl 1048.35076
[23] Escher, J.; Yin, Z., Initial boundary value problems of the Camassa-Holm equation, Comm. Partial Differential Equations, 33, 377-395 (2008) · Zbl 1145.35031
[24] Escher, J.; Yin, Z., Initial boundary value problems for nonlinear dispersive wave equations, J. Funct. Anal., 256, 479-508 (2009) · Zbl 1193.35108
[25] Li, Y.; Oliver, P., Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation, J. Differential Equations, 162, 27-63 (2000) · Zbl 0958.35119
[26] Rodriguez-Blanco, G., On the Cauchy problem for the Camassa-Holm equation, Nonlinear Analysis TMA, 46, 309-327 (2001) · Zbl 0980.35150
[27] Yin, Z., Well-posedness, blowup, and global existence for an integrable shallow water equation, Discrete Contin. Dyn. Syst., 11, 393-411 (2004) · Zbl 1061.35123
[28] Lai, S.; Wu, Y., A model containing both the Camassa-Holm and Degasperis-Procesi equations, J. Math. Anal. Appl., 374, 458-469 (2011) · Zbl 1202.35231
[29] Constantin, A., Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble), 50, 321-362 (2000) · Zbl 0944.35062
[30] Constantin, A.; Escher, J., Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181, 229-243 (1998) · Zbl 0923.76025
[31] Bressan, A.; Constantin, A., Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183, 215-239 (2007) · Zbl 1105.76013
[32] Bressan, A.; Constantin, A., Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.), 5, 1-27 (2007) · Zbl 1139.35378
[33] Coclite, G. M.; Holden, H.; Karlsen, K. H., Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37, 1044-1069 (2006) · Zbl 1100.35106
[34] Constantin, A.; Molinet, L., Global weak solutions for a shallow water equation, Comm. Math. Phys., 211, 45-61 (2000) · Zbl 1002.35101
[35] Holden, H.; Raynaud, X., Global conservative solutions of the Camassa-Holm equation — a Lagrangian point of view, Comm. Partial Differential Equations, 32, 1511-1549 (2007) · Zbl 1136.35080
[36] Xin, Z.; Zhang, P., On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53, 1411-1433 (2000) · Zbl 1048.35092
[37] Whitham, G. B., Linear and Nonlinear Waves (1980), J. Wiley and Sons: J. Wiley and Sons New York · Zbl 0373.76001
[38] Shen, C.; Tian, L.; Gao, A., Optimal control of the viscous Dullin-Gottwalld-Holm equation, Nonlinear Analysis: Real World Applications, 11, 480-491 (2010) · Zbl 1181.35200
[39] Ai, X.; Gui, G., On the inverse scattering problem and the low regularity solutions for the Dullin-Gottwald-Holm equation, Nonlinear Analysis: Real World Applications, 11, 888-894 (2010) · Zbl 1186.35235
[40] Guo, Z.; Ni, L., Wave breaking for the periodic weakly dissipative Dullin-Gottwald-Holm equation, Nonlinear Analysis TMA, 74, 965-973 (2011) · Zbl 1202.37109
[41] Meng, Q.; He, B.; Long, Y.; Li, Z., New exact periodic wave solutions for the Dullin-Gottwald-Holm equation, Appl. Math. Comput., 218, 4533-4537 (2011) · Zbl 1239.35133
[42] Sun, B., Maximum principle for optimal distributed control of the viscous Dullin-Gottwald-Holm equation, Nonlinear Analysis: Real World Applications, 13, 325-332 (2012) · Zbl 1238.49034
[43] Liu, Y., Global existence and blow-up solutions for a nonlinear shallow water equation, Math. Ann., 335, 717-735 (2006) · Zbl 1102.35021
[44] Tian, L.; Gui, G.; Liu, Y., On the Cauchy problem and the scattering problem for the Dullin-Gottwald-Holm equation, Comm. Math. Phys., 257, 667-701 (2005) · Zbl 1080.76016
[45] Yin, Z., Global existence and blow-up for a periodic integrable shallow water equation with linear and nonlinear dispersion, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 12, 87-101 (2005) · Zbl 1096.35118
[46] Zhang, S.; Yin, Z., On the blow-up phenomena of the periodic Dullin-Gottwald-Holm equation, J. Math. Phys., 49, 1-16 (2008) · Zbl 1159.81344
[47] Zhang, S.; Yin, Z., Global weak solutions for the Dullin-Gottwald-Holm equation, Nonlinear Analysis TMA, 72, 1690-1700 (2010) · Zbl 1180.35170
[48] Mustafa, O. G., Global conservative solutions of the Dullin-Gottwald-Holm equation, Discrete Contin. Dyn. Syst., 19, 575-594 (2007) · Zbl 1142.35074
[49] Yan, K.; Yin, Z., Analytic solutions of the Cauchy problem for two-component shallow water systems, Math. Z., 269, 1113-1127 (2011) · Zbl 1252.35012
[50] Gui, G.; Liu, Y., On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258, 4251-4278 (2010) · Zbl 1189.35254
[51] Dachin, R., (Fourier Analysis Methods for PDEs. Fourier Analysis Methods for PDEs, Lecture Notes (2003)), 14 November
[52] Escher, J.; Yin, Z., Well-posedness, blow-up phenomena and global solutions for the \(b\)-equation, J. Reine Angew. Math., 624, 51-80 (2008) · Zbl 1159.35060
[53] Gui, G.; Liu, Y., On the Cauchy problem for the Degasperis-Procesi equation, Quart. Appl. Math., 69, 445-464 (2011) · Zbl 1229.35243
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