Global existence and boundedness in a higher-dimensional quasilinear chemotaxis system

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• 作者：Yilong Wang ; Zhaoyin Xiang
• 关键词：35K57 ; 35Q92 ; 35A01 ; 92C17 ; Chemotaxis ; Boundedness ; Global existence
• 刊名：Zeitschrift f篓鹿r angewandte Mathematik und Physik
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：66
• 期：6
• 页码：3159-3179
• 全文大小：675 KB
• 参考文献：1.Chae M., Kang K., Lee J.: Existence of smooth solutions to coupled chemotaxis-fluid equations. Discrete Contin. Dyn. Syst. A 33, 2271-297 (2013)MATH MathSciNet CrossRef
  2.Chae M., Kang K., Lee J.: Global existence and temporal decay in Keller–Segel models coupled to fluid equations. Commun. Partial Differ. Equ. 39, 1205-235 (2014)MATH MathSciNet CrossRef
  3.Difrancesco M., Lorz A., Markowich P.A.: Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior. Discrete Contin. Dyn. Syst. A 28, 1437-453 (2010)MathSciNet CrossRef
  4.Duan R., Lorz A., Markowich P.A.: Global solutions to the coupled chemotaxis-fluid equations. Commun. Partial Differ. Equ. 35, 1635-673 (2010)MATH MathSciNet CrossRef
  5.Duan R., Xiang Z.: A note on global existence for the chemotaxis Stokes model with nonlinear diffusion. Int. Math. Res. Not. 2014, 1833-852 (2014)MathSciNet
  6.Horstmann D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math. Verien 105, 103-65 (2003)MATH MathSciNet
  7.Horstmann D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. II. Jahresber. Deutsch. Math. Verien 106, 51-9 (2004)MATH MathSciNet
  8.Hillen T., Painter K.: A users guide to PDE models for chemotaxis. J. Math. Biol. 58, 183-17 (2009)MATH MathSciNet CrossRef
  9.Herrero M.A., Velázquez J.L.L.: A blow-up mechanism for a chemotaxis model. Ann. Scuola Norm. Super. Pisa Cl. Sci. 24, 633-83 (1997)MATH
  10.Horstmann D., Winkler M.: Boundedness vs. blow-up in a chemotaxis system. J. Differ. Equ. 215, 52-07 (2005)MATH MathSciNet CrossRef
  11.Ishida S., Seki K., Yokota T.: Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains. J. Differ. Equ. 256, 2993-010 (2014)MATH MathSciNet CrossRef
  12.J?ger W., Luckhaus S.: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Am. Math. Soc. 329, 819-24 (1992)MATH CrossRef
  13.Keller E.F., Segel L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399-15 (1970)MATH CrossRef
  14.Li X., Xiang Z.: Boundedness in quasilinear Keller–Segel equations with nonlinear sensitivity and logistic source. Discrete Contin. Dyn. Syst. A 35, 3503-531 (2015)MATH MathSciNet CrossRef
  15.Li T., Suen A., Winkler M., Xue C.: Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms. Math. Models Methods Appl. Sci. 25, 721-46 (2015)MathSciNet CrossRef
  16.Liu J., Lorz A.: A coupled chemotaxis-fluid model: global existence. Ann. I. H. Poincaré Anal. 28, 643-52 (2011)MATH MathSciNet CrossRef
  17.Lorz A.: Coupled chemotaxis fluid equations. Math. Models Methods Appl. Sci. 20, 987-004 (2010)MATH MathSciNet CrossRef
  18.Nagai T.: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5, 581-01 (1995)MATH MathSciNet
  19.Mizoguchi, N., Winkler, M.: Is aggregation a generic phenomenon in the two dimensional Keller–Segel system?. Preprint
  20.Nagai T., Senba T., Yoshida K.: Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis. Funkc. Ekvacioj Ser. Int. 40, 411-33 (1997)MATH MathSciNet
  21.Osaki K., Yagi A.: Finite dimensional attractors for one-dimensional Keller–Segel equations. Funkcial. Ekvacioj 44, 441-69 (2001)MATH MathSciNet
  22.Tuval I., Cisneros L., Dombrowski C., Wolgemuth C.W., Kessler J.O., Goldstein R.E.: Bacterial swimming and oxygen transport near contact lines. Proc. Natl. Acad. Sci. USA 102, 2277-282 (2005)MATH CrossRef
  23.Tao Y.: Boundedness in a chemotaxis model with oxygen consumption by bacteria. J. Math. Anal. Appl. 381, 521-29 (2011)MATH MathSciNet CrossRef
  24.Tao Y., Winkler M.: Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity. J. Differ. Equ. 252, 692-15 (2012)MATH MathSciNet CrossRef
  25.Tao Y., Winkler M.: Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion. Ann. I. H. Poincaré Anal. 30, 157-78 (2013)MATH MathSciNet CrossRef
  26.Tao Y., Winkler M.: Global existence and boundedness in a Keller–Segel–Stokes model with arbitrary porous medium diffusion. Discrete Contin. Dyn. Syst. A 32, 1901-914 (2012)MATH MathSciNet CrossRef
  27.Tao Y., Winkler M.: Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant. J. Differ. Equ. 252, 2520-543 (2012)MATH MathSciNet CrossRef
  28.Vorotnikov D.: Weak solutions for a bioconvection model related to Bacillus subtilis. Commun. Math. Sci. 12, 545-63 (2014)MATH MathSciNet CrossRef
  29.Wang L., Mu C., Zhou S.: Boundedness in a parabolic-parabolic chemotaxis system with nonlinear diffusion. Z. Angew. Math. Phys. 65, 1137-152
• 作者单位：Yilong Wang (1) (2)
  Zhaoyin Xiang (1)
  
  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China
  2. School of Sciences, Southwest Petroleum University, Chengdu, 610500, China
  
• 刊物主题：Theoretical and Applied Mechanics; Mathematical Methods in Physics;
• 出版者：Springer Basel
• ISSN：1420-9039

文摘

This paper deals with the boundedness of global solutions to the quasilinear Keller–Segel system $$\left\{\begin{array}{ll}u_t=\nabla\cdot\big(D(u)\nabla u-u\nabla v\big), &\quad x\in\Omega,\,\, t>0,\\ v_t=\Delta v-uf(v),&\quad x\in\Omega, \,\,t>0,\\ \nabla u\cdot \nu=0,\,\, \nabla v\cdot\nu=0,&\quad x\in \partial\Omega,\,\, t>0\end{array}\right.$$