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Kinetic models for traffic flow resulting in a reduced space of microscopic velocities
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 |
$\begin{cases} u_{t}=Δ u-χ\nabla·(u\nabla v)+ξ\nabla·(u\nabla w)+f(u), &x∈Ω,\ t>0,\\ v_{t}=Δ v+α u-β v, &x∈Ω,\ t>0,\\ w_{t}=Δ w+γ u-δ w, &x∈Ω,\ t>0,\\\end{cases}\;\;\;\;(*)$ |
$Ω \subset \mathbb{R}^n(n≥ 1)$ |
$(u_0,v_0,w_0)$ |
$χ≥ 0,ξ≥ 0,α, β, γ, δ>0$ |
$f$ |
$f(0)≥ 0$ |
$f(u)≤ a-bu^θ, \ \ u≥ 0,\ \ \mathrm{with~some} \ \ a≥ 0,b>0,θ≥1.$ |
$χα=ξγ$ |
$θ$ |
$n$ |
$\begin{cases} θ > \max\{1,3-\frac6n\}, &\text{when }\ \ 1≤ n≤ 5,\\ θ≥ 2, &\text{when }\ \ 6≤ n≤ 9,\\ θ>1+\frac{2(n-4)}{n+2}, &\text{when} \ \ \ n≥10.\\\end{cases}$ |
$f(u)=μ u(1-u)$ |
$μ>\frac{χ^2α^2(β-δ)^2}{8δβ^2}$ |
$(u,v,w)$ |
$(1,\frac{α}{β},\frac{γ}{δ})$ |
$1≤ n≤ 9$ |
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. |
[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. |
[3] |
N. Bellomo, A. Bellouquid, Y. 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. |
[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. |
[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. |
[6] |
K. Fujie, A. Ito, M. 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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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.
|
[13] |
S. Ishida, K. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
K. Lin, C. 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. |
[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. |
[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. |
[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. |
[28] |
P. Liu, J. 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. |
[29] |
M. Luca, A. Chavez-Ross, L. 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.
|
[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. |
[31] |
N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint. |
[32] |
L. Nirenberg,
An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 733-737.
|
[33] |
K. Osaki, T. Tsujikawa, A. 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. |
[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.
|
[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] |
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. |
[37] |
P. Souplet and P. Quittner, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel/Boston/Berlin, 2007. |
[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. |
[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 Z. A. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[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. |
[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. |
[43] |
M. Winkler,
A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.
doi: 10.1002/mma.319. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[3] |
N. Bellomo, A. Bellouquid, Y. 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. |
[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. |
[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. |
[6] |
K. Fujie, A. Ito, M. 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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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.
|
[13] |
S. Ishida, K. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
K. Lin, C. 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. |
[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. |
[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. |
[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. |
[28] |
P. Liu, J. 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. |
[29] |
M. Luca, A. Chavez-Ross, L. 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.
|
[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. |
[31] |
N. Mizoguchi and M. Winkler, Finite-time blow-up in the two-dimensional Keller-Segel system, preprint. |
[32] |
L. Nirenberg,
An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 733-737.
|
[33] |
K. Osaki, T. Tsujikawa, A. 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. |
[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.
|
[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] |
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. |
[37] |
P. Souplet and P. Quittner, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel/Boston/Berlin, 2007. |
[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. |
[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 Z. A. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[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. |
[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. |
[43] |
M. Winkler,
A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925.
doi: 10.1002/mma.319. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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