May  2014, 13(3): 1283-1304. doi: 10.3934/cpaa.2014.13.1283

On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity

1. 

Department of mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China

2. 

Department of Mathematics, University of Texas-Pan American, Edinburg, Texas 78541, United States

3. 

Department of Mathematics, Zhongshan University, 510275 Guangzhou, China

Received  June 2013 Revised  October 2013 Published  December 2013

In this paper, we study the Cauchy problem for a generalized integrable Camassa-Holm equation with both quadratic and cubic nonlinearity. By overcoming the difficulties caused by the complicated mixed nonlinear structure, we firstly establish the local well-posedness result in Besov spaces, and then present a precise blow-up scenario for strong solutions. Furthermore, we show the existence of single peakon by the method of analysis.
Citation: Xingxing Liu, Zhijun Qiao, Zhaoyang Yin. On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1283-1304. doi: 10.3934/cpaa.2014.13.1283
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show all references

References:
[1]

Springer-Verlag, Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

J. Math. Phys., 53 (2012), 073710. doi: 10.1063/1.4736845.  Google Scholar

[3]

Phys. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[4]

Adv. Appl. Mech., 31 (1994), 1-33. Google Scholar

[5]

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

[6]

Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar

[7]

IMA J. Appl. Math., 77 (2012), 293-307. doi: 10.1093/imamat/hxs033.  Google Scholar

[8]

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

[9]

Comm. Pure Appl. Math., 51 (1998), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.  Google Scholar

[10]

Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[11]

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.  Google Scholar

[12]

Phys. Lett. A, 270 (2000), 140-148. doi: 10.1016/S0375-9601(00)00255-3.  Google Scholar

[13]

J. Nonlinear. Sci., 12 (2002), 415-422. doi: 10.1007/s00332-002-0517-x.  Google Scholar

[14]

Diff. Int. Eq., 14 (2001), 953-988.  Google Scholar

[15]

Acta Appl. Math., 39 (1995), 295-305. doi: 10.1007/BF00994638.  Google Scholar

[16]

Physica D, 87 (1995), 145-150. doi: 10.1016/0167-2789(95)00133-O.  Google Scholar

[17]

Physica D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X.  Google Scholar

[18]

J. Differential Equations, 255 (2013), 1905-1938. doi: 10.1016/j.jde.2013.05.024.  Google Scholar

[19]

Physica D, 95 (1996), 229-243. doi: 10.1016/0167-2789(96)00048-6.  Google Scholar

[20]

Comm. Math. Phys., 319 (2013), 731-759. doi: 10.1007/s00220-012-1566-0.  Google Scholar

[21]

Diff. Int. Eq., 14 (2001), 821-831.  Google Scholar

[22]

J. Math. Phys., 53 (2012), 123701. Google Scholar

[23]

Nonlinear Anal. Real World Appl., 10 (2009), 1797-1802. doi: 10.1016/j.nonrwa.2008.02.016.  Google Scholar

[24]

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

[25]

Nonlinear Anal. TMA, 74 (2011), 2497-2507. doi: 10.1016/j.na.2010.12.005.  Google Scholar

[26]

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

[27]

J. Math. Phys., 47 (2006), 112701. doi: 10.1063/1.2365758.  Google Scholar

[28]

Theor. Math. Phys., 267 (2011), 584-589. Google Scholar

[29]

Z. Qiao, B. Xia and J. B. Li, Integrable system with peakon, weak kink, and kink-peakon interactional solutions,, preprint, ().   Google Scholar

[30]

Comm. Math. Phys., 322 (2013), 967-997. doi: 10.1007/s00220-013-1749-3.  Google Scholar

[31]

Nonlinear Anal. TMA, 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X.  Google Scholar

[32]

J. Math. Phys., 52 (2011), 023509. doi: 10.1063/1.3548837.  Google Scholar

[33]

Topol. Methods Nonlinear Anal., 7 (1996), 1-48.  Google Scholar

[34]

Illinois. J. Math., 47 (2003), 649-666.  Google Scholar

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