September  2017, 10(3): 855-878. doi: 10.3934/krm.2017034

Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source

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

* Corresponding author: Hai-Yang Jin

Received  June 2016 Revised  August 2016 Published  December 2016

In this paper, we study an attraction-repulsion Keller-Segel chemotaxis model with logistic source
$\begin{cases} u_{t}=Δ u-χ\nabla·(u\nabla v)+ξ\nabla·(u\nabla w)+f(u), &x∈Ω,\ t>0,\\ v_{t}=Δ v+α u-β v, &x∈Ω,\ t>0,\\ w_{t}=Δ w+γ u-δ w, &x∈Ω,\ t>0,\\\end{cases}\;\;\;\;(*)$
in a smooth bounded domain
$Ω \subset \mathbb{R}^n(n≥ 1)$
, with homogeneous Neumann boundary conditions and nonnegative initial data
$(u_0,v_0,w_0)$
satisfying suitable regularity, where
$χ≥ 0,ξ≥ 0,α, β, γ, δ>0$
and
$f$
is a smooth growth source satisfying
$f(0)≥ 0$
and
$f(u)≤ a-bu^θ, \ \ u≥ 0,\ \ \mathrm{with~some} \ \ a≥ 0,b>0,θ≥1.$
When
$χα=ξγ$
(i.e. repulsion cancels attraction), the boundedness of classical solution of system (*) is established if the dampening parameter
$θ$
and the space dimension
$n$
satisfy
$\begin{cases} θ > \max\{1,3-\frac6n\}, &\text{when }\ \ 1≤ n≤ 5,\\ θ≥ 2, &\text{when }\ \ 6≤ n≤ 9,\\ θ>1+\frac{2(n-4)}{n+2}, &\text{when} \ \ \ n≥10.\\\end{cases}$
Furthermore, when
$f(u)=μ u(1-u)$
and repulsion cancels attraction, by constructing appropriate Lyapunov functional, we show that if
$μ>\frac{χ^2α^2(β-δ)^2}{8δβ^2}$
, the solution
$(u,v,w)$
exponentially stabilizes to the constant stationary solution
$(1,\frac{α}{β},\frac{γ}{δ})$
in the case of
$1≤ n≤ 9$
. Our results implies that when repulsion cancels attraction the logistic source play an important role on the solution behavior of the attraction-repulsion chemotaxis system.
Citation: Shijie Shi, Zhengrong Liu, Hai-Yang Jin. Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source. Kinetic & Related Models, 2017, 10 (3) : 855-878. doi: 10.3934/krm.2017034
References:
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X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.  Google Scholar

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N. BellomoA. BellouquidY. S. Tao and M. Winkler, Toward 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.  Google Scholar

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

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E. Espejo and T. Suzuki, Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34.  doi: 10.1016/j.aml.2014.04.007.  Google Scholar

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K. FujieA. ItoM. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169.  doi: 10.3934/dcds.2016.36.151.  Google Scholar

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

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T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721.  Google Scholar

[9]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[10]

S. Hittmeir and A. Jüngel, Cross diffusion preventing blow-up in the two-dimensional Keller-Segel model, SIAM J. Math. Anal., 43 (2011), 997-1022.  doi: 10.1137/100813191.  Google Scholar

[11]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.  Google Scholar

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

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S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.  Google Scholar

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W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.2307/2153966.  Google Scholar

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H. Y. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.  doi: 10.1016/j.jmaa.2014.09.049.  Google Scholar

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H. Y. Jin and Z. A. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Methods Appl. Sci., 38 (2015), 444-457.  doi: 10.1002/mma.3080.  Google Scholar

[17]

H. Y. Jin and Z. Liu, Large time behavior of the full attraction-repulsion Keller-Segel system in the whole space, Appl. Math. Lett., 47 (2015), 13-20.  doi: 10.1016/j.aml.2015.03.004.  Google Scholar

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H. Y. Jin and Z. A. Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 260 (2016), 162-196.  doi: 10.1016/j.jde.2015.08.040.  Google Scholar

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O. A. Ladyžhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R. I., 1968.  Google Scholar

[20]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.  Google Scholar

[21]

X. Li, Boundedness in a two-dimensional attraction-repulsion system with nonlinear diffusion, Math. Methods Appl. Sci., 39 (2016), 289-301.  doi: 10.1002/mma.3477.  Google Scholar

[22]

X. Li and Z. Xiang, On an attraction-repulsion chemotaxis system with a logistic source, IMA J. Appl. Math., 81 (2016), 165-198.  doi: 10.1093/imamat/hxv033.  Google Scholar

[23]

Y. Li and Y. X. Li, Blow-up of nonradial solutions to attraction-repulsion chemotaxis system in two dimensions, Nonlinear Anal. Real World Appl., 30 (2016), 170-183.  doi: 10.1016/j.nonrwa.2015.12.003.  Google Scholar

[24]

K. LinC. Mu and L. Wang, Large-time behavior of an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124.  doi: 10.1016/j.jmaa.2014.12.052.  Google Scholar

[25]

K. Lin and C. Mu, Global existence and convergence to steady states for an attraction-repulsion chemotaxis system, Nonlinear Anal. Real World Appl., 31 (2016), 630-642.  doi: 10.1016/j.nonrwa.2016.03.012.  Google Scholar

[26]

D. Liu and Y. S. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546.  doi: 10.1002/mma.3240.  Google Scholar

[27]

J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41.  doi: 10.1080/17513758.2011.571722.  Google Scholar

[28]

P. LiuJ. P. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.  doi: 10.3934/dcdsb.2013.18.2597.  Google Scholar

[29]

M. LucaA. Chavez-RossL. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, Microglia, and alzheimer's disease senile plagues: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730.   Google Scholar

[30]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875.  doi: 10.1016/j.anihpc.2013.07.007.  Google Scholar

[31]

N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint. Google Scholar

[32]

L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 733-737.   Google Scholar

[33]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[34]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.   Google Scholar

[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.  Google Scholar

[36]

R. Shi and W. Wang, Well-posedness for a model derived from an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 423 (2015), 497-520.  doi: 10.1016/j.jmaa.2014.10.006.  Google Scholar

[37]

P. Souplet and P. Quittner, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel/Boston/Berlin, 2007.  Google Scholar

[38]

Y. S. 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.  Google Scholar

[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.  Google Scholar

[40]

Y. S. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.  Google Scholar

[41]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.  Google Scholar

[42]

Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13 pp.  doi: 10.1063/1.2766864.  Google Scholar

[43]

M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.  doi: 10.1002/mma.319.  Google Scholar

[44]

M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.  doi: 10.1016/j.jmaa.2008.07.071.  Google Scholar

[45]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

[46]

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.  Google Scholar

[47]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.  Google Scholar

[48]

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.  Google Scholar

[49]

T. Xiang, Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differential Equations, 258 (2015), 4275-4323.  doi: 10.1016/j.jde.2015.01.032.  Google Scholar

[50]

Q. S. Zhang and Y. X. Li, An attraction-repulsion chemotaxis system with logistic source, ZAMM Z. Angew. Math. Mech., 96 (2016), 570-584.  doi: 10.1002/zamm.201400311.  Google Scholar

[51]

Q. S. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.  doi: 10.1007/s00033-015-0532-z.  Google Scholar

show all references

References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.  doi: 10.1080/03605307908820113.  Google Scholar

[2]

X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.  Google Scholar

[3]

N. BellomoA. BellouquidY. S. Tao and M. Winkler, Toward 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.  Google Scholar

[4]

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

[5]

E. Espejo and T. Suzuki, Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34.  doi: 10.1016/j.aml.2014.04.007.  Google Scholar

[6]

K. FujieA. ItoM. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169.  doi: 10.3934/dcds.2016.36.151.  Google Scholar

[7]

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

[8]

T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721.  Google Scholar

[9]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[10]

S. Hittmeir and A. Jüngel, Cross diffusion preventing blow-up in the two-dimensional Keller-Segel model, SIAM J. Math. Anal., 43 (2011), 997-1022.  doi: 10.1137/100813191.  Google Scholar

[11]

D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363.  Google Scholar

[12]

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

[13]

S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.  Google Scholar

[14]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.2307/2153966.  Google Scholar

[15]

H. Y. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.  doi: 10.1016/j.jmaa.2014.09.049.  Google Scholar

[16]

H. Y. Jin and Z. A. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Methods Appl. Sci., 38 (2015), 444-457.  doi: 10.1002/mma.3080.  Google Scholar

[17]

H. Y. Jin and Z. Liu, Large time behavior of the full attraction-repulsion Keller-Segel system in the whole space, Appl. Math. Lett., 47 (2015), 13-20.  doi: 10.1016/j.aml.2015.03.004.  Google Scholar

[18]

H. Y. Jin and Z. A. Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 260 (2016), 162-196.  doi: 10.1016/j.jde.2015.08.040.  Google Scholar

[19]

O. A. Ladyžhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R. I., 1968.  Google Scholar

[20]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.  Google Scholar

[21]

X. Li, Boundedness in a two-dimensional attraction-repulsion system with nonlinear diffusion, Math. Methods Appl. Sci., 39 (2016), 289-301.  doi: 10.1002/mma.3477.  Google Scholar

[22]

X. Li and Z. Xiang, On an attraction-repulsion chemotaxis system with a logistic source, IMA J. Appl. Math., 81 (2016), 165-198.  doi: 10.1093/imamat/hxv033.  Google Scholar

[23]

Y. Li and Y. X. Li, Blow-up of nonradial solutions to attraction-repulsion chemotaxis system in two dimensions, Nonlinear Anal. Real World Appl., 30 (2016), 170-183.  doi: 10.1016/j.nonrwa.2015.12.003.  Google Scholar

[24]

K. LinC. Mu and L. Wang, Large-time behavior of an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124.  doi: 10.1016/j.jmaa.2014.12.052.  Google Scholar

[25]

K. Lin and C. Mu, Global existence and convergence to steady states for an attraction-repulsion chemotaxis system, Nonlinear Anal. Real World Appl., 31 (2016), 630-642.  doi: 10.1016/j.nonrwa.2016.03.012.  Google Scholar

[26]

D. Liu and Y. S. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546.  doi: 10.1002/mma.3240.  Google Scholar

[27]

J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 31-41.  doi: 10.1080/17513758.2011.571722.  Google Scholar

[28]

P. LiuJ. P. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625.  doi: 10.3934/dcdsb.2013.18.2597.  Google Scholar

[29]

M. LucaA. Chavez-RossL. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, Microglia, and alzheimer's disease senile plagues: Is there a connection?, Bull. Math. Biol., 65 (2003), 693-730.   Google Scholar

[30]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875.  doi: 10.1016/j.anihpc.2013.07.007.  Google Scholar

[31]

N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint. Google Scholar

[32]

L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 733-737.   Google Scholar

[33]

K. OsakiT. TsujikawaA. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X.  Google Scholar

[34]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.   Google Scholar

[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.  Google Scholar

[36]

R. Shi and W. Wang, Well-posedness for a model derived from an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 423 (2015), 497-520.  doi: 10.1016/j.jmaa.2014.10.006.  Google Scholar

[37]

P. Souplet and P. Quittner, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel/Boston/Berlin, 2007.  Google Scholar

[38]

Y. S. 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.  Google Scholar

[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.  Google Scholar

[40]

Y. S. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443.  Google Scholar

[41]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.  Google Scholar

[42]

Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13 pp.  doi: 10.1063/1.2766864.  Google Scholar

[43]

M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.  doi: 10.1002/mma.319.  Google Scholar

[44]

M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.  doi: 10.1016/j.jmaa.2008.07.071.  Google Scholar

[45]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.  doi: 10.1080/03605300903473426.  Google Scholar

[46]

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.  Google Scholar

[47]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.  Google Scholar

[48]

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.  Google Scholar

[49]

T. Xiang, Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differential Equations, 258 (2015), 4275-4323.  doi: 10.1016/j.jde.2015.01.032.  Google Scholar

[50]

Q. S. Zhang and Y. X. Li, An attraction-repulsion chemotaxis system with logistic source, ZAMM Z. Angew. Math. Mech., 96 (2016), 570-584.  doi: 10.1002/zamm.201400311.  Google Scholar

[51]

Q. S. Zhang and Y. Li, Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.  doi: 10.1007/s00033-015-0532-z.  Google Scholar

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