August  2013, 33(8): 3407-3441. doi: 10.3934/dcds.2013.33.3407

On the Cauchy problem for the two-component Dullin-Gottwald-Holm system

1. 

School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

2. 

Jiangsu Key Laboratory for NSLSCS and School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, China

3. 

Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019-0408

Received  May 2012 Revised  October 2012 Published  January 2013

Considered herein is the initial-value problem for a two-component Dullin-Gottwald-Holm system. The local well-posedness in the Sobolev space $H^{s}(\mathbb{R})$ with $s>3/2$ is established by using the bi-linear estimate technique to the approximate solutions. Then the wave-breaking criteria and global solutions are determined in $H^s(\mathbb{R}), s > 3/2.$ Finally, existence of the solitary-wave solutions is demonstrated.
Citation: Yong Chen, Hongjun Gao, Yue Liu. On the Cauchy problem for the two-component Dullin-Gottwald-Holm system. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3407-3441. doi: 10.3934/dcds.2013.33.3407
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show all references

References:
[1]

Comm. Partial Differential Equations, 28 (2003), 1943-1974. doi: 10.1081/PDE-120025491.  Google Scholar

[2]

J. Math. Pures Appl., 76 (1997), 377-430. doi: 10.1016/S0021-7824(97)89957-6.  Google Scholar

[3]

Phil. Trans. Roy. Soc. London A, 278 (1975), 555-601.  Google Scholar

[4]

Geom. Funct. Anal., 2 (1993), 107-156. doi: 10.1007/BF01896020.  Google Scholar

[5]

Geom. Funct. Anal., 3 (1993), 209-262. doi: 10.1007/BF01895688.  Google Scholar

[6]

Ph.D thesis, University of Notre Dame, 2003.  Google Scholar

[7]

Math. Proc. Cambridge Philos. Soc., 49 (1953), 695-706.  Google Scholar

[8]

Electron. J. Differential Equations, 26 (2001), 1-7.  Google Scholar

[9]

J. Amer. Math. Soc., 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[10]

SIAM J. Math. Anal., 33 (2001), 649-666. doi: 10.1137/S0036141001384387.  Google Scholar

[11]

Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362.  Google Scholar

[12]

Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.  Google Scholar

[13]

Math. Z., 233 (2000), 75-91. doi: 10.1007/PL00004793.  Google Scholar

[14]

Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303-328.  Google Scholar

[15]

Phys. Lett. A, 372 (2008), 7129-7132. doi: 10.1016/j.physleta.2008.10.050.  Google Scholar

[16]

Comm. Pure Appl. Math.,57 (2004), 481-527. doi: 10.1002/cpa.3046.  Google Scholar

[17]

J. Nonlinear Sci., 12 (2002), 415-22. doi: 10.1007/s00332-002-0517-x.  Google Scholar

[18]

J. Funct. Anal., 169 (1999), 532-558. doi: 10.1006/jfan.1999.3484.  Google Scholar

[19]

Phys. Rev. Lett., 87 (2001), 4501-4504. Google Scholar

[20]

Discrete Contin. Dynam. Systems, 19 (2007), 493-513. doi: 10.3934/dcds.2007.19.493.  Google Scholar

[21]

Math. Ann., 348 (2010), 415-448. doi: 10.1007/s00208-010-0483-9.  Google Scholar

[22]

J. Differential Equations, 248 (2010), 2003-2014. doi: 10.1016/j.jde.2009.08.002.  Google Scholar

[23]

J. Funct. Anal., 260 (2011), 1132-1154. doi: 10.1016/j.jfa.2010.11.015.  Google Scholar

[24]

Ann. I. H. Poincare-AN, 28 (2011), 623-641. doi: 10.1016/j.anihpc.2011.04.003.  Google Scholar

[25]

J. Funct. Anal., 258 (2010), 4251-4278. doi: 10.1016/j.jfa.2010.02.008.  Google Scholar

[26]

Math. Z., 268 (2011), 45-66. doi: 10.1007/s00209-009-0660-2.  Google Scholar

[27]

F. Guo, H. J. Gao and Y. Liu, Blow-up mechanism and global solutions for the two-component Dullin-Gottwald-Holm system,, Accepted by J. of the London Math. Soc., ().   Google Scholar

[28]

Comm. Partial Differential Equations, 23 (1998), 123-139. doi: 10.1080/03605309808821340.  Google Scholar

[29]

J. Differential Equations, 161 (2000), 479-495. doi: 10.1006/jdeq.1999.3695.  Google Scholar

[30]

in "Analysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis Contemp. Math." 251, Amer. Math. Soc., Providence, RI, (2000), 309-320. doi: 10.1090/conm/251/03878.  Google Scholar

[31]

Wave Motion, 46 (2009), 389-396. doi: 10.1016/j.wavemoti.2009.06.012.  Google Scholar

[32]

J. Differential Equations, 246 (2009), 2448-2467. doi: 10.1016/j.jde.2008.10.027.  Google Scholar

[33]

Fluid Dynam. Res., 33 (2003), 97-111. doi: 10.1016/S0169-5983(03)00036-4.  Google Scholar

[34]

Manuscripta Math., 28 (1979), 89-99. Google Scholar

[35]

Commun. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.  Google Scholar

[36]

Indiana Univ. Math. J., 40 (1991), 33-69. doi: 10.1512/iumj.1991.40.40003.  Google Scholar

[37]

Duke Math. J., 71 (1993), 1-21. doi: 10.1215/S0012-7094-93-07101-3.  Google Scholar

[38]

J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.  Google Scholar

[39]

J. Differential Equations, 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683.  Google Scholar

[40]

Acta Math. Sin. (Engl. Ser.), 30 (2004), 1-16. doi: 10.1007/s10114-004-0420-5.  Google Scholar

[41]

Math. Ann., 324 (2002), 255-275. doi: 10.1007/s00208-002-0338-0.  Google Scholar

[42]

Phys. Rev. E., 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900.  Google Scholar

[43]

Amer J. Math., 123 (2001), 839-908.  Google Scholar

[44]

Comm. Math. Phys., 257 (2005), 667-701. doi: 10.1007/s00220-005-1356-z.  Google Scholar

[45]

J. Differential Equations, 230 (2006), 600-613. doi: 10.1016/j.jde.2006.04.008.  Google Scholar

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J. Differential Equations, 248 (2010), 1458-1472. doi: 10.1016/j.jde.2010.01.004.  Google Scholar

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Int. Math. Res. Not. IMRN, 11 (2010), 1981-2021. doi: 10.1093/imrn/rnp211.  Google Scholar

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