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November  2014, 34(11): 4565-4576. doi: 10.3934/dcds.2014.34.4565

On ill-posedness for the generalized BBM equation

Xavier Carvajal 1, and  Mahendra Panthee 2,

1. 

Instituto de Matemática - UFRJ Av. Horácio Macedo, Centro de Tecnologia, Cidade Universitária, Ilha do Fundão, 21941-972 Rio de Janeiro, RJ, Brazil
2. 

Departamento de Matemática, Universidade Estadual de Campinas (UNICAMP), Rua Sergio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brazil

Received  October 2013 Revised  December 2013 Published  May 2014

We consider the Cauchy problem associated to the generalized Benjamin-Bona-Mahony (BBM) equation for given data in the $L^2$-based Sobolev spaces. Depending on the order of nonlinearity and dispersion, we prove that the Cauchy problem is ill-posed for data with lower order Sobolev regularity. We also prove that, in certain range of the Sobolev regularity, even if the solution exists globally in time, it fails to be smooth.
Keywords: Initial value problem, ill-posedness., well-posedness, BBM equation.
Mathematics Subject Classification: Primary: 35Q53; Secondary: 35A0.
Citation: Xavier Carvajal, Mahendra Panthee. On ill-posedness for the generalized BBM equation. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4565-4576. doi: 10.3934/dcds.2014.34.4565
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show all references

References:
[1]

Adv. Differential Equations, 11 (2006), 121-166.  Google Scholar

[2]

Discrete Contin. Dyn. Syst., 30 (2011), 851-871. doi: 10.3934/dcds.2011.30.851.  Google Scholar

[3]

Phil. Trans. Royal Soc. London, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032.  Google Scholar

[4]

Philos. Trans. Royal Soc. London Series A, 302 (1981), 457-510. doi: 10.1098/rsta.1981.0178.  Google Scholar

[5]

Discrete and Continuous Dynamical Systems, 23 (2009), 1241-1252. doi: 10.3934/dcds.2009.23.1241.  Google Scholar

[6]

Disc. Cont. Dynamical Systems, 23 (2009), 1253-1275. doi: 10.3934/dcds.2009.23.1253.  Google Scholar

[7]

Selecta Math. New Ser., 3 (1997), 115-159. doi: 10.1007/s000290050008.  Google Scholar

[8]

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

[9]

J. Diff. Equations, 240 (2007), 125-144. doi: 10.1016/j.jde.2007.05.030.  Google Scholar

[10]

Amer. J. Math., 125 (2003), 1235-1293. doi: 10.1353/ajm.2003.0040.  Google Scholar

[11]

arXiv:1110.2352v1, Proc. Amer. Math. Soc., 141 (2013), 2793-2798. doi: 10.1090/S0002-9939-2013-11693-X.  Google Scholar

[12]

Int. Math. Research Notices, (2002), 1979-2005. doi: 10.1155/S1073792802112104.  Google Scholar

[13]

Proceedings of the Royal Society of Edinburgh, 132 (2002), 1407-1416.  Google Scholar

[14]

SIAM. J. Math. Anal., 33 (2001), 982-988. doi: 10.1137/S0036141001385307.  Google Scholar

[15]

Discrete Contin. Dyn. Syst., 30 (2011), 253-259. doi: 10.3934/dcds.2011.30.253.  Google Scholar

[16]

C. R. Acad. Sci. Paris Ser. I Math., 329 (1999), 1043-1047. doi: 10.1016/S0764-4442(00)88471-2.  Google Scholar

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