Abstract
This paper deals with the boundedness of global solutions to the quasilinear Keller–Segel system
in a bounded domain \({\Omega\subset \mathbb{R}^{n}(n\geq 3)}\) with smooth boundary, where D(u) is supposed to satisfy D(u) ≥ D 0 u m-1 with some positive constant D 0. It is proved that when \({m>2-\frac{n+2}{2n}}\), the system possesses global bounded weak solutions for any sufficiently smooth nonnegative initial data. In particular, we improved the recent result by Wang et al. (Z Angew Math Phys, 2015. doi:10.1007/s00033-014-0491-9) in the sense that we established the global boundedness of weak solutions. We also removed the convexity assumption on the domain used by Wang et al. (Z Angew Math Phys 65:1137–1152, 2014, 2015).
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References
Chae M., Kang K., Lee J.: Existence of smooth solutions to coupled chemotaxis-fluid equations. Discrete Contin. Dyn. Syst. A 33, 2271–2297 (2013)
Chae M., Kang K., Lee J.: Global existence and temporal decay in Keller–Segel models coupled to fluid equations. Commun. Partial Differ. Equ. 39, 1205–1235 (2014)
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, 1437–1453 (2010)
Duan R., Lorz A., Markowich P.A.: Global solutions to the coupled chemotaxis-fluid equations. Commun. Partial Differ. Equ. 35, 1635–1673 (2010)
Duan R., Xiang Z.: A note on global existence for the chemotaxis Stokes model with nonlinear diffusion. Int. Math. Res. Not. 2014, 1833–1852 (2014)
Horstmann D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math. Verien 105, 103–165 (2003)
Horstmann D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. II. Jahresber. Deutsch. Math. Verien 106, 51–69 (2004)
Hillen T., Painter K.: A users guide to PDE models for chemotaxis. J. Math. Biol. 58, 183–217 (2009)
Herrero M.A., Velázquez J.L.L.: A blow-up mechanism for a chemotaxis model. Ann. Scuola Norm. Super. Pisa Cl. Sci. 24, 633–683 (1997)
Horstmann D., Winkler M.: Boundedness vs. blow-up in a chemotaxis system. J. Differ. Equ. 215, 52–107 (2005)
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, 2993–3010 (2014)
Jäger W., Luckhaus S.: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Am. Math. Soc. 329, 819–824 (1992)
Keller E.F., Segel L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)
Li X., Xiang Z.: Boundedness in quasilinear Keller–Segel equations with nonlinear sensitivity and logistic source. Discrete Contin. Dyn. Syst. A 35, 3503–3531 (2015)
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, 721–746 (2015)
Liu J., Lorz A.: A coupled chemotaxis-fluid model: global existence. Ann. I. H. Poincaré Anal. 28, 643–652 (2011)
Lorz A.: Coupled chemotaxis fluid equations. Math. Models Methods Appl. Sci. 20, 987–1004 (2010)
Nagai T.: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5, 581–601 (1995)
Mizoguchi, N., Winkler, M.: Is aggregation a generic phenomenon in the two dimensional Keller–Segel system?. Preprint
Nagai T., Senba T., Yoshida K.: Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis. Funkc. Ekvacioj Ser. Int. 40, 411–433 (1997)
Osaki K., Yagi A.: Finite dimensional attractors for one-dimensional Keller–Segel equations. Funkcial. Ekvacioj 44, 441–469 (2001)
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, 2277–2282 (2005)
Tao Y.: Boundedness in a chemotaxis model with oxygen consumption by bacteria. J. Math. Anal. Appl. 381, 521–529 (2011)
Tao Y., Winkler M.: Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity. J. Differ. Equ. 252, 692–715 (2012)
Tao Y., Winkler M.: Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion. Ann. I. H. Poincaré Anal. 30, 157–178 (2013)
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, 1901–1914 (2012)
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, 2520–2543 (2012)
Vorotnikov D.: Weak solutions for a bioconvection model related to Bacillus subtilis. Commun. Math. Sci. 12, 545–563 (2014)
Wang L., Mu C., Zhou S.: Boundedness in a parabolic-parabolic chemotaxis system with nonlinear diffusion. Z. Angew. Math. Phys. 65, 1137–1152 (2014)
Wang, L., Mu, C., Lin, K., Zhao, J.: Global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant. Z. Angew. Math. Phys. (2015). doi:10.1007/s00033-014-0491-9
Winkler M.: Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model. J. Differ. Equ. 248, 2889–2905 (2010)
Winkler M.: Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system. J. Math. Pures Appl. 100, 748–767 (2013)
Winkler M.: Global large-data solutions in a chemotaxis-(Navier–)Stokes system modeling cellular swimming in fluid drops. Commun. Partial Differ. Equ. 37, 319–351 (2012)
Winkler, M.: A two-dimensional chemotaxis-Stokes system with rotational flux: global solvability, eventual smoothness and stabilization. Preprint
Winkler M.: Global weak solutions in a three-dimensional chemotaxis-Navier–Stokes system. Ann. I. H. Poincaré Anal. (2015). doi:10.1016/j.anihpc.2015.05.002
Winkler M.: Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity. arXiv:1501.07059v1
Winkler M.: How far do chemotaxis-driven forces influence regularity in the Navier–Stokes system? Tran. Am. Math. Soc. (2015). doi:10.1090/tran/6733
Zhang Q., Zheng X.: Global well-posedness for the two-dimensional incompressible chemotaxis-Navier–Stokes equations. SIAM J. Math. Anal. 46, 3078–3105 (2014)
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Wang, Y., Xiang, Z. Global existence and boundedness in a higher-dimensional quasilinear chemotaxis system. Z. Angew. Math. Phys. 66, 3159–3179 (2015). https://doi.org/10.1007/s00033-015-0557-3
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DOI: https://doi.org/10.1007/s00033-015-0557-3