July  2018, 38(7): 3595-3616. doi: 10.3934/dcds.2018155

Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity

Department of Mathematics, South China University of Technology, Guangzhou 510640, China

* Corresponding author: Hai-Yang Jin

Received  October 2017 Published  April 2018

Fund Project: The research of H.Y. Jin was supported by the NSF of China No. 11501218, and the Fundamental Research Funds for the Central Universities (No. 2017MS107).

This paper is concerned with the following Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity
$\begin{cases}\tag{*}n_t+u·\nabla n = \nabla ·(d(c)\nabla n)-\nabla ·(χ (c) n\nabla c)+a n-bn^2, &x∈ Ω, ~~t>0, \\ c_t+u·\nabla c = Δ c+ n-c,&x∈ Ω, ~~t>0, \\ u_t+ u·\nabla u = Δ u-\nabla P+n\nabla φ,&x∈ Ω, ~~t>0, \\\nabla · u = 0& x∈ Ω, \ t>0, \end{cases}$
in a bounded smooth domain
$Ω\subset \mathbb{R}^2$
with homogeneous Neumann boundary conditions, where
$a≥0$
and
$b>0$
are constants, and the functions
$d(c)$
and
$χ(c)$
satisfy the following assumptions:
$(d(c), χ (c))∈ [C^2([0, ∞))]^2$
with
$d(c), χ(c)>0$
for all
$c≥0$
,
$d'(c)<0$
and
$\lim\limits_{c\to∞}d(c) = 0$
.
$\lim\limits_{c\to∞} \frac{χ (c)}{d(c)}$
and
$\lim\limits_{c\to∞}\frac{d'(c)}{d(c)}$
exist.
The difficulty in analysis of system (*) is the possible degeneracy of diffusion due to the condition
$\lim\limits_{c\to∞}d(c) = 0$
. In this paper, we will use function
$d(c)$
as weight function and employ the method of energy estimate to establish the global existence of classical solutions of (*) with uniform-in-time bound. Furthermore, by constructing a Lyapunov functional, we show that the global classical solution
$(n, c, u)$
will converge to the constant state
$(\frac{a}{b}, \frac{a}{b}, 0)$
if
$b>\frac{K_0}{16}$
with
$K_0 = \max\limits_{0≤c ≤∞}\frac{|χ(c)|^2}{d(c)}$
.
Citation: Hai-Yang Jin. Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3595-3616. doi: 10.3934/dcds.2018155
References:
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N. BellomoA. BellouquidY. S. Tao and M. Winkler, Towards a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[2]

J. P. Bourguignon and H. Brezis, Remarks on Euler Equation, J. Functional Analysis, 15 (1974), 341-363.  doi: 10.1016/0022-1236(74)90027-5.

[3]

M. ChaeK. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297.  doi: 10.3934/dcds.2013.33.2271.

[4]

M. ChaeK. Kang and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differential Equations, 39 (2014), 1205-1235.  doi: 10.1080/03605302.2013.852224.

[5]

A. ChertockK. FellnerA. KurganovA. Lorz and P. A. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: A high-resolution numerical approach, J. Fluid Mech., 694 (2012), 155-190.  doi: 10.1017/jfm.2011.534.

[6]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.  doi: 10.1016/0025-5564(81)90055-9.

[7]

T. Ciéslak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.

[8]

T. Ciéslak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004.

[9]

M. DiFrancescoA. Lorz and P. A. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453.  doi: 10.3934/dcds.2010.28.1437.

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R. J. DuanA. Lorz and P. A. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.  doi: 10.1080/03605302.2010.497199.

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R. J. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Not. IMRN, 7 (2014), 1833-1852.  doi: 10.1093/imrn/rns270.

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M. Eisenbach, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion. Chemotaxis, Imperial College Press, London, 2004.

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E. Espejo and T. Suzuki, Reaction terms avoiding aggregation in slow fluids, Nonlinear Anal. Real World Appl., 21 (2015), 110-126.  doi: 10.1016/j.nonrwa.2014.07.001.

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X. FuH. TangC. LiuJ. D. HuangT. Hwa and P. Lenz, Stripe formation in bacterial system with density-suppressed motility, Phys. Rev. Lett., 108 (2012), 198102.  doi: 10.1103/PhysRevLett.108.198102.

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Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solution of the Navier-Stokes system, J. Differential Equations, 61 (1986), 186-212.  doi: 10.1016/0022-0396(86)90096-3.

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Y. Giga and H. Sohr, Abstract Lp estimate for the Cauchy problem with application to the Navier-Stokes system in exterior domains, J. Funct. Anal., 102 (1991), 72-94.  doi: 10.1016/0022-1236(91)90136-S.

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T. Hillen and K. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

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T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783-1801.  doi: 10.1002/mma.569.

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D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ, Jahresber. Deutsch. Math.-Verien., 105 (2003), 103-165. 

[21]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅱ, Jahresber. Deutsch. Math.-Verien., 106 (2004), 51-69. 

[22]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[23]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.

[24]

H. Y. Jin, Y. J. Kim and Z. A. Wang, Boundedness, stabilization and pattern formation driven by density-suppressed motility, SIAM J. Appl. Math., 2018, to appear.

[25]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. 

[26]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.  doi: 10.1016/j.jmaa.2008.01.005.

[27]

O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968.

[28]

J. G. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652.  doi: 10.1016/j.anihpc.2011.04.005.

[29]

A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.  doi: 10.1142/S0218202510004507.

[30]

A. Lorz, A coupled Keller-Segel-Stokes model: Global existence for small initial data and blow-up delay, Commun. Math. Sci., 10 (2012), 555-574.  doi: 10.4310/CMS.2012.v10.n2.a7.

[31]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. 

[32]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. 

[33]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 

[34]

K. Painter and J. A. Sherratt, Modeling the movement of interacting cell populations, J. Theor. Biol., 225 (2003), 327-339.  doi: 10.1016/S0022-5193(03)00258-3.

[35]

M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045.

[36]

T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.  doi: 10.4310/MAA.2001.v8.n2.a9.

[37]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.

[38]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002.

[39]

Y. S. Tao and M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115.

[40]

Y. S. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system Z. Angew. Math. Phys., 67 (2016), Art. 138, 23 pp. doi: 10.1007/s00033-016-0732-1.

[41]

Y. S. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2016), 2555-2573.  doi: 10.1007/s00033-015-0541-y.

[42]

Y. Tao and M. Winkler, Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Meth. Appl. Sci., 27 (2017), 1645-1683.  doi: 10.1142/S0218202517500282.

[43]

I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Nat. Acad. Sci., USA, 102 (2005), 2277–2282 doi: 10.1073/pnas.0406724102.

[44]

M. Winkler, Does a "volume-filling effect" always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.

[45]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.

[46]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.

[47]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.

[48]

M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Rational Mech. Anal., 211 (2014), 455-487.  doi: 10.1007/s00205-013-0678-9.

[49]

M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.

[50]

M. Winkler, Global existence and slow grow-up in a quasilinear Keller-Segel system with exponentially decaying diffusivity, Nonlinearity, 30 (2017), 735-764.  doi: 10.1088/1361-6544/aa565b.

[51]

C. Yoon and Y. J. Kim, Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Application Mathematics, 149 (2017), 101-123.  doi: 10.1007/s10440-016-0089-7.

show all references

References:
[1]

N. BellomoA. BellouquidY. S. Tao and M. Winkler, Towards a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[2]

J. P. Bourguignon and H. Brezis, Remarks on Euler Equation, J. Functional Analysis, 15 (1974), 341-363.  doi: 10.1016/0022-1236(74)90027-5.

[3]

M. ChaeK. Kang and J. Lee, Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297.  doi: 10.3934/dcds.2013.33.2271.

[4]

M. ChaeK. Kang and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differential Equations, 39 (2014), 1205-1235.  doi: 10.1080/03605302.2013.852224.

[5]

A. ChertockK. FellnerA. KurganovA. Lorz and P. A. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: A high-resolution numerical approach, J. Fluid Mech., 694 (2012), 155-190.  doi: 10.1017/jfm.2011.534.

[6]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.  doi: 10.1016/0025-5564(81)90055-9.

[7]

T. Ciéslak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.  doi: 10.1016/j.jde.2012.01.045.

[8]

T. Ciéslak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004.

[9]

M. DiFrancescoA. Lorz and P. A. Markowich, Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453.  doi: 10.3934/dcds.2010.28.1437.

[10]

R. J. DuanA. Lorz and P. A. Markowich, Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.  doi: 10.1080/03605302.2010.497199.

[11]

R. J. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Not. IMRN, 7 (2014), 1833-1852.  doi: 10.1093/imrn/rns270.

[12]

M. Eisenbach, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion. Chemotaxis, Imperial College Press, London, 2004.

[13]

E. Espejo and T. Suzuki, Reaction terms avoiding aggregation in slow fluids, Nonlinear Anal. Real World Appl., 21 (2015), 110-126.  doi: 10.1016/j.nonrwa.2014.07.001.

[14]

X. FuH. TangC. LiuJ. D. HuangT. Hwa and P. Lenz, Stripe formation in bacterial system with density-suppressed motility, Phys. Rev. Lett., 108 (2012), 198102.  doi: 10.1103/PhysRevLett.108.198102.

[15]

Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solution of the Navier-Stokes system, J. Differential Equations, 61 (1986), 186-212.  doi: 10.1016/0022-0396(86)90096-3.

[16]

Y. Giga and H. Sohr, Abstract Lp estimate for the Cauchy problem with application to the Navier-Stokes system in exterior domains, J. Funct. Anal., 102 (1991), 72-94.  doi: 10.1016/0022-1236(91)90136-S.

[17]

M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683. 

[18]

T. Hillen and K. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[19]

T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783-1801.  doi: 10.1002/mma.569.

[20]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ, Jahresber. Deutsch. Math.-Verien., 105 (2003), 103-165. 

[21]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅱ, Jahresber. Deutsch. Math.-Verien., 106 (2004), 51-69. 

[22]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022.

[23]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.

[24]

H. Y. Jin, Y. J. Kim and Z. A. Wang, Boundedness, stabilization and pattern formation driven by density-suppressed motility, SIAM J. Appl. Math., 2018, to appear.

[25]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. 

[26]

R. Kowalczyk and Z. Szymańska, On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.  doi: 10.1016/j.jmaa.2008.01.005.

[27]

O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968.

[28]

J. G. Liu and A. Lorz, A coupled chemotaxis-fluid model: Global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652.  doi: 10.1016/j.anihpc.2011.04.005.

[29]

A. Lorz, Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.  doi: 10.1142/S0218202510004507.

[30]

A. Lorz, A coupled Keller-Segel-Stokes model: Global existence for small initial data and blow-up delay, Commun. Math. Sci., 10 (2012), 555-574.  doi: 10.4310/CMS.2012.v10.n2.a7.

[31]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. 

[32]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. 

[33]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 

[34]

K. Painter and J. A. Sherratt, Modeling the movement of interacting cell populations, J. Theor. Biol., 225 (2003), 327-339.  doi: 10.1016/S0022-5193(03)00258-3.

[35]

M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.  doi: 10.1006/jdeq.1993.1045.

[36]

T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.  doi: 10.4310/MAA.2001.v8.n2.a9.

[37]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.

[38]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.  doi: 10.1016/j.anihpc.2012.07.002.

[39]

Y. S. Tao and M. Winkler, Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115.

[40]

Y. S. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system Z. Angew. Math. Phys., 67 (2016), Art. 138, 23 pp. doi: 10.1007/s00033-016-0732-1.

[41]

Y. S. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2016), 2555-2573.  doi: 10.1007/s00033-015-0541-y.

[42]

Y. Tao and M. Winkler, Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Meth. Appl. Sci., 27 (2017), 1645-1683.  doi: 10.1142/S0218202517500282.

[43]

I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Nat. Acad. Sci., USA, 102 (2005), 2277–2282 doi: 10.1073/pnas.0406724102.

[44]

M. Winkler, Does a "volume-filling effect" always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.

[45]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008.

[46]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865.

[47]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020.

[48]

M. Winkler, Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Rational Mech. Anal., 211 (2014), 455-487.  doi: 10.1007/s00205-013-0678-9.

[49]

M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.  doi: 10.1016/j.anihpc.2015.05.002.

[50]

M. Winkler, Global existence and slow grow-up in a quasilinear Keller-Segel system with exponentially decaying diffusivity, Nonlinearity, 30 (2017), 735-764.  doi: 10.1088/1361-6544/aa565b.

[51]

C. Yoon and Y. J. Kim, Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Application Mathematics, 149 (2017), 101-123.  doi: 10.1007/s10440-016-0089-7.

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