August  2015, 35(8): 3799-3825. doi: 10.3934/dcds.2015.35.3799

Global attractor for weakly damped gKdV equations in higher sobolev spaces

1. 

School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, China

Received  August 2014 Revised  December 2014 Published  February 2015

Long time behavior of solutions for weakly damped gKdV equations on the real line is studied. With some weak regularity assumptions on the force $f$, we prove the existence of global attractor in $H^s$ for any $s\geq 1$. The asymptotic compactness of solution semigroup is shown by Ball's energy method and Goubet's high-low frequency decomposition if $s$ is an integer and not an integer, respectively.
Citation: Ming Wang. Global attractor for weakly damped gKdV equations in higher sobolev spaces. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3799-3825. doi: 10.3934/dcds.2015.35.3799
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show all references

References:
[1]

International Mathematics Research Notices, 6 (1996), 277-304. doi: 10.1155/S1073792896000207.  Google Scholar

[2]

Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912.  Google Scholar

[3]

Journal of the American Mathematical Society, 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[4]

Discrete and Continuous Dynamical Systems, 6 (2000), 625-644.  Google Scholar

[5]

Journal of Differential Equations, 185 (2002), 25-53. doi: 10.1006/jdeq.2001.4163.  Google Scholar

[6]

Differential and Integral Equations, 18 (2005), 1333-1339.  Google Scholar

[7]

Providence, RI: American Mathematical Society, 1988.  Google Scholar

[8]

Journal of the American Mathematical Society, 4 (1991), 323-347. doi: 10.1090/S0894-0347-1991-1086966-0.  Google Scholar

[9]

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

[10]

Communications on Pure and Applied Mathematics, 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.  Google Scholar

[11]

Analysis & PDE, 5 (2012), 145-198. doi: 10.2140/apde.2012.5.145.  Google Scholar

[12]

Philos. Mag., 39 (1895), 422-443. Google Scholar

[13]

Springer, New York, 2009.  Google Scholar

[14]

Journal of Differential Equations, 170 (2001), 281-316. doi: 10.1006/jdeq.2000.3827.  Google Scholar

[15]

Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255.  Google Scholar

[16]

Journal of the American Mathematical Society, 15 (2002), 617-664. Google Scholar

[17]

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 141 (2011), 287-317. doi: 10.1017/S030821051000003X.  Google Scholar

[18]

Acta Mathematica, 212 (2014), 59-140. doi: 10.1007/s11511-014-0109-2.  Google Scholar

[19]

Communications in mathematical physics, 231 (2002), 347-373. Google Scholar

[20]

Advances in Differential Equations, 2 (1997), 257-296.  Google Scholar

[21]

Dynamics of Partial Differential Equations, 6 (2009), 15-34. doi: 10.4310/DPDE.2009.v6.n1.a2.  Google Scholar

[22]

VI Workshop on Partial Differential Equations, Part II (Rio de Janeiro, 1999). Mat. Contemp., 19 (2000), 129-152.  Google Scholar

[23]

Springer, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[24]

Indiana Univ. Math. J., 60 (2011), 1487-1516. doi: 10.1512/iumj.2011.60.4399.  Google Scholar

[25]

Duke Mathematical Journal, 86 (1997), 109-142. doi: 10.1215/S0012-7094-97-08604-X.  Google Scholar

[26]

Asymptotic Analysis, 59 (2008), 51-81. doi: 10.3233/ASY-2008-0886.  Google Scholar

[27]

Transactions of the American Mathematical Society, 363 (2011), 6085-6109. doi: 10.1090/S0002-9947-2011-05373-0.  Google Scholar

[28]

Nonlinear Anal.: Theory, Methods Appl., 63 (2005), 49-65. doi: 10.1016/j.na.2005.04.034.  Google Scholar

[29]

volume 106 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006.  Google Scholar

[30]

Journal of Differential Equations, 232 (2007), 623-651. doi: 10.1016/j.jde.2006.07.019.  Google Scholar

[31]

Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[32]

Journal of Physics A: Mathematical and General, 31 (1998), 7635-7645. doi: 10.1088/0305-4470/31/37/021.  Google Scholar

[33]

Journal of Mathematical Analysis and Applications, 390 (2012), 136-150. doi: 10.1016/j.jmaa.2012.01.031.  Google Scholar

[34]

Differential And Integral Equations, 23 (2010), 569-600.  Google Scholar

[35]

Nonlinear Analysis: Real World Applications, 11 (2010), 913-919. doi: 10.1016/j.nonrwa.2009.01.022.  Google Scholar

[36]

Journal of Mathematical Physics, 54 (2013), 092701, 11pp. doi: 10.1063/1.4818983.  Google Scholar

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