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December  2013, 18(10): 2627-2646. doi: 10.3934/dcdsb.2013.18.2627

Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation

Etsushi Nakaguchi 1, and  Koichi Osaki 2,

1. 

College of Liberal Arts and Sciences, Tokyo Medical and Dental University, 2-8-30 Kohnodai, Ichikawa, Chiba 272-0827, Japan
2. 

Department of Mathematical Sciences, School of Science and Technology, Kwansei Gakuin University, 2-1 Gakuen, Sanda 669-1337, Japan

Received  December 2012 Revised  July 2013 Published  October 2013

We construct the global bounded solutions and the attractors of a parabolic-parabolic chemotaxis-growth system in two- and three-dimensional smooth bounded domains. We derive new $L_p$ and $H^2$ uniform estimates for these solutions. We then construct the absorbing sets and the global attractors for the dynamical systems generated by the solutions. We also show the existence of exponential attractors by applying the existence theorem of Eden-Foias-Nicolaenko-Temam.
Keywords: Chemotaxis, attractor, global existence, infinite-dimensional dynamical system..
Mathematics Subject Classification: 37L30, 35B41, 35K51, 35K57, 35K92, 92C1.
Citation: Etsushi Nakaguchi, Koichi Osaki. Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2627-2646. doi: 10.3934/dcdsb.2013.18.2627
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show all references

References:
[1]

J. London Math. Soc. (2), 74 (2006), 453-474. doi: 10.1112/S0024610706023015.  Google Scholar

[2]

Math. Biosci., 56 (1981), 217-237. doi: 10.1016/0025-5564(81)90055-9.  Google Scholar

[3]

With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon. Translated from the French by Alan Craig. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1.  Google Scholar

[4]

C. R. Acad. Sci. Paris, 310 (1990), 559-562.  Google Scholar

[5]

Research in Applied Mathematics, vol. 37, John-Wiley and Sons, Chichester, 1994.  Google Scholar

[6]

Ann. Scoula Norm. Sup. Pisa Cl. Sci. IV, 24 (1997), 633-683.  Google Scholar

[7]

J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[8]

Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.  Google Scholar

[9]

Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69.  Google Scholar

[10]

European J. Appl. Math., 12 (2001), 159-177. doi: 10.1017/S0956792501004363.  Google Scholar

[11]

Trans. Amer. Math. Soc., 329 (1992), 819-824. doi: 10.1090/S0002-9947-1992-1046835-6.  Google Scholar

[12]

J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[13]

GAKUTO Internat. Ser. Math. Sci. Appl., 29 (2008), 265-278.  Google Scholar

[14]

Physica D, 241 (2012), 1629-1639. doi: 10.1016/j.physd.2012.06.009.  Google Scholar

[15]

Physica A, 230 (1996), 499-543. doi: 10.1016/0378-4371(96)00051-9.  Google Scholar

[16]

Springer-Verlag, New York, 2003.  Google Scholar

[17]

Nonlinear Anal., 74 (2011), 286-297. doi: 10.1016/j.na.2010.08.044.  Google Scholar

[18]

Adv. Math. Sci. Appl., 5 (1995), 581-601.  Google Scholar

[19]

J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042.  Google Scholar

[20]

Nonlinear Anal., 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[21]

Funkcial. Ekvac., 44 (2001), 441-469.  Google Scholar

[22]

Adv. Math. Sci. Appl., 12 (2002), 587-606.  Google Scholar

[23]

Physica D: Nonlinear Phenomena, 240 (2011), 363-375. doi: 10.1016/j.physd.2010.09.011.  Google Scholar

[24]

Progress in Nonlinear Differential Equations and their Applications, 62. Birkhäuser Boston, Inc., Boston, MA, 2005. doi: 10.1007/0-8176-4436-9.  Google Scholar

[25]

Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003.  Google Scholar

[26]

Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997.  Google Scholar

[27]

Bull. Math. Biol., 70 (2008), 1570-1607. doi: 10.1007/s11538-008-9322-5.  Google Scholar

[28]

Johann Ambrosius Barth Verlag, Heidelberg/Leipzig, 1995.  Google Scholar

[29]

J. Math. Anal. Appl., 348 (2008), 708-729. doi: 10.1016/j.jmaa.2008.07.071.  Google Scholar

[30]

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[31]

J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[32]

Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar

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