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doi: 10.3934/dcdsb.2022096
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Emergence of lager densities in chemotaxis system with indirect signal production and non-radial symmetry case

College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua 321004, China

* Corresponding author: Guangyu Xu

Received  November 2021 Revised  January 2022 Early access May 2022

Fund Project: This work is supported by the Scientific Research Fund (YS304221937, 2021ZS0802) and the Young Doctor Program of Zhejiang Normal University (ZZ323205020520013068)

This paper deals with the classical solution of the following chemotaxis system with generalized logistic growth and indirect signal production
$ \begin{eqnarray} \left\{ \begin{array}{llll} u_t = \epsilon\Delta u-\nabla\cdot(u\nabla v)+ru-\mu u^\theta, &\\ 0 = d_1\Delta v-\beta v+\alpha w, &\\ 0 = d_2\Delta w-\delta w+\gamma u, & \end{array} \right. \end{eqnarray} \quad\quad\quad\quad(1)$
and the so-called strong
$ W^{1, q}( \Omega) $
-solution of hyperbolic-elliptic-elliptic model
$ \begin{eqnarray} \left\{ \begin{array}{llll} u_t = -\nabla\cdot(u\nabla v)+ru-\mu u^\theta, &\\ 0 = d_1\Delta v-\beta v+\alpha w, &\\ 0 = d_2\Delta w-\delta w+\gamma u, & \end{array} \right. \end{eqnarray} \quad\quad\quad\quad(2)$
in arbitrary bounded domain
$ \Omega\subset\mathbb{R}^n $
,
$ n\geq1 $
, where
$ r, \mu, d_1, d_2, \alpha, \beta, \gamma, \delta>0 $
and
$ \theta>1 $
. Via applying the viscosity vanishing method, we first prove that the classical solution of (1) will converge to the strong
$ W^{1, q}( \Omega) $
-solution of (2) as
$ \epsilon\rightarrow0 $
. After structuring the local well-pose of (2), we find that the strong
$ W^{1, q}( \Omega) $
-solution will blow up in finite time with non-radial symmetry setting if
$ \Omega $
is a bounded convex domain,
$ \theta\in(1, 2] $
, and the initial data is suitable large. Moreover, for any positive constant
$ M $
and the classical solution of (1), if we add another hypothesis that there exists positive constant
$ \epsilon_0(M) $
with
$ \epsilon\in(0,\ \epsilon_0(M)) $
, then the classical solution of (1) can exceed arbitrarily large finite value in the sense: one can find some points
$ \left(\tilde{x}, \tilde{t}\right) $
such that
$ u(\tilde{x}, \tilde{t})>M $
.
Citation: Guangyu Xu. Emergence of lager densities in chemotaxis system with indirect signal production and non-radial symmetry case. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022096
References:
[1]

X. Bai and S. Liu, A new criterion to a two-chemical substances chemotaxis system with critical dimension, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3717-3721.  doi: 10.3934/dcdsb.2018074.

[2]

N. BellomoA. BellouquidY. 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.

[3]

R. DillonP. K. Maini and H. G. Othmer, Pattern formation in generalised turing systems I. Steady-state patterns in systems with mixed boundary conditions, J. Math. Biol., 32 (1994), 345-393.  doi: 10.1007/BF00160165.

[4]

M. Ding and W. Wang, Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4665-4684.  doi: 10.3934/dcdsb.2018328.

[5]

Y. Dong and Y. Peng, Global boundedness in the higher-dimensional chemotaxis system with indirect signal production and rotational flux, Appl. Math. Lett., 112 (2021), 106700, 8 pp. doi: 10.1016/j.aml.2020.106700.

[6]

A. Friedman, Partial Differential Equations, Dover Books on Mathematics Series, Dover Publications, Incorporated, 2008.

[7]

M. Fuest, Analysis of a chemotaxis model with indirect signal absorption, J. Differential Equations, 267 (2019), 4778-4806.  doi: 10.1016/j.jde.2019.05.015.

[8]

K. Fujie and T. Senba, Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differential Equations, 263 (2017), 88-148.  doi: 10.1016/j.jde.2017.02.031.

[9]

K. Fujie and T. Senba, Blowup of solutions to a two-chemical substances chemotaxis system in the critical dimension, J. Differential Equations, 266 (2019), 942-976.  doi: 10.1016/j.jde.2018.07.068.

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.

[11]

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

[12]

D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller-Segel model, J. Math. Biol., 44 (2002), 463-478.  doi: 10.1007/s002850100134.

[13]

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

[14]

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.

[15]

B. Hu and Y. Tao, To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128.  doi: 10.1142/S0218202516400091.

[16]

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.

[17]

H.-Y. Jin and Z.-A. Wang, Global stabilization of the full attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst., 40 (2020), 3509-3527.  doi: 10.3934/dcds.2020027.

[18]

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.

[19]

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.

[20]

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.

[21]

K. Kang and A. Stevens, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal. Theor. Methods Appl., 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017.

[22]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[23]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.

[24]

H. Kozono and Y. Taniuchi, Limiting case of the sobolev inequality in BMO with application to the Euler equations, Commun. Math. Phys., 214 (2000), 191-200.  doi: 10.1007/s002200000267.

[25]

J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499.

[26]

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.

[27]

P. Laurençot, Global bounded and unbounded solutions to a chemotaxis system with indirect signal production, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6419-6444. 

[28]

H. Li and Y. Tao, Boundedness in a chemotaxis system with indirect signal production and generalized logistic source, Appl. Math. Lett., 77 (2018), 108-113.  doi: 10.1016/j.aml.2017.10.006.

[29]

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.

[30]

Y. Li, Emergence of large densities and simultaneous blow-up in a two-species chemotaxis system with competitive kinetics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5461-5480.  doi: 10.3934/dcdsb.2019066.

[31]

Y. Li and W. Wang, Boundedness in a four-dimensional attraction-repulsion chemotaxis system with logistic source, Math. Methods Appl. Sci., 41 (2018), 4936-4942.  doi: 10.1002/mma.4942.

[32]

K. Lin and C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.  doi: 10.3934/dcds.2016018.

[33]

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.

[34]

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

[35]

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.

[36]

Y. LiuZ. Li and J. Huang, Global boundedness and large time behavior of achemotaxis system with indirect signal absorption, J. Differential Equations, 269 (2020), 6365-6399.  doi: 10.1016/j.jde.2020.05.008.

[37]

M. LucaA. Chavez-RossL. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bulletin Math. Biol., 65 (2003), 693-730.  doi: 10.1016/S0092-8240(03)00030-2.

[38]

W. Lv and Q. Wang, Global existence for a class of chemotaxis systems with signal-dependent motility, indirect signal production and generalized logistic source, Z. Angew. Math. Phys., 71 (2020), Article number: 53. doi: 10.1007/s00033-020-1276-y.

[39]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.

[40]

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

[41]

E. NakaguchiK. NodaK. Osaki and K. Uemichi, Global attractor for a two-dimensional chemotaxis system with linear degradation and indirect signal production, Japan J. Indust. Appl. Math., 37 (2020), 49-80.  doi: 10.1007/s13160-019-00376-0.

[42]

K. Noda and K. Osaki, Global attractor and Lyapunov function for one-dimensional Deneubourg chemotaxis system, Hiroshima Math. J., 49 (2019), 251-271.  doi: 10.32917/hmj/1564106547.

[43]

T. Ogawa and Y. Taniuchi, On blow-up criteria of smooth solutions to the 3-D Euler equations in a bounded domain, J. Differential Equations, 190 (2003), 39-63.  doi: 10.1016/S0022-0396(03)00013-5.

[44]

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

[45]

K. J. PainterP. K. Maini and H. G. Othmer, Development and applications of a model for cellular response to multiple chemotactic cues, J. Math. Biol., 41 (2000), 285-314.  doi: 10.1007/s002850000035.

[46]

S. QiuC. Mu and Y. Li, Boundedness and stability in a chemotaxis-growth model with indirect attractant production and signal-dependent sensitivity, Acta Appl. Math., 169 (2020), 341-360.  doi: 10.1007/s10440-019-00301-0.

[47]

S. QiuC. Mu and L. Wang, Boundedness in the higher-dimensional quasilinear chemotaxis-growth system with indirect attractant production, Comput. Math. Appl., 75 (2018), 3213-3223.  doi: 10.1016/j.camwa.2018.01.042.

[48]

G. Q. Ren and B. Liu, Boundedness in a chemotaxis system under a critical parameter condition, Bull. Braz. Math. Soc. (N.S.), 52 (2021), 281-289.  doi: 10.1007/s00574-020-00202-z.

[49]

S. StrohmR. C. Tyson and J. A. Powell, Pattern formation in a model for mountain pine beetle dispersal: linking model predictions to data, Bull. Math. Biol., 75 (2013), 1778-1797.  doi: 10.1007/s11538-013-9868-8.

[50]

C. Surulescu and M. Winkler, Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis-haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more), European J. Appl. Math., 32 (2021), 618–651. arXiv: 1904.11210. doi: 10.1017/S0956792520000236.

[51]

Q. TangQ. Xin and C. Mu, Boundedness of the higher-dimensional quasilinear chemotaxis system with generalized logistic source, Acta Math. Sci. Ser. B (Engl. Ed.), 40 (2020), 713-722.  doi: 10.1007/s10473-020-0309-0.

[52]

Y. 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.

[53]

Y. Tao and M. Winkler, Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc., 19 (2017), 3641-3678.  doi: 10.4171/JEMS/749.

[54]

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

[55]

J. I. Tello and D. Wrzosek, Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.  doi: 10.1142/S0218202516400108.

[56]

M. TianL. Hong and S. Zheng, A hyperbolic-elliptic-elliptic system of an attraction-repulsion chemotaxis model with nonlinear productions, J. Evolution Equations, 18 (2018), 973-1001.  doi: 10.1007/s00028-018-0428-4.

[57]

X. TuC.-L. Tang and S. Qiu, The phenomenon of large population densities in a chemotaxis competition system with loop, J. Evolution Equ., 21 (2021), 1717-1754.  doi: 10.1007/s00028-020-00650-6.

[58]

I. TuvalL. CisnerosC. DombrowskiC. WolgemuthJ. Kessler and R. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Nat. Acad. Sci. USA, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.

[59]

W. Wang, A quasilinear fully parabolic chemotaxis system with indirect signal production and logistic source, J. Math. Anal. Appl., 477 (2019), 488-522.  doi: 10.1016/j.jmaa.2019.04.043.

[60]

W. WangM. Zhuang and S. Zheng, Positive effects of repulsion on boundedness in a fully parabolic attraction-repulsion chemotaxis system with logistic source, J. Differential Equations, 264 (2018), 2011-2027.  doi: 10.1016/j.jde.2017.10.011.

[61]

Y. Wang, M. Winkler and Z. Xiang, The fast signal diffusion limit in Keller-Segel(-fluid) systems, Calc. Var. Part. Differ. Eq., 58 (2019), Paper No. 196, 40 pp. doi: 10.1007/s00526-019-1656-3.

[62]

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.

[63]

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.

[64]

M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.  doi: 10.1007/s00332-014-9205-x.

[65]

M. Winkler, Emergence of large population densities despite logistic restricitiions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793.  doi: 10.3934/dcdsb.2017135.

[66]

M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phy., 69 (2018), Paper No. 69, 40 pp. doi: 10.1007/s00033-018-0935-8.

[67]

D. Wrzosek, Long-time behaviour of solutions to a chemotaxis model with volume-filling effect, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 431-444.  doi: 10.1017/S0308210500004649.

[68]

Z. Q. Wu, J. X. Yin and C. P. Wang, Elliptic and Parabolic Equations, World Scientific Publishing, Co. Pte. Ltd., Hackensack, NJ, 2006.

[69]

T. Xiang, How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?, J. Math. Anal. Appl., 459 (2018), 1172-1200.  doi: 10.1016/j.jmaa.2017.11.022.

[70]

T. Xiang, Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion, Comm. Pure Appl. Anal., 18 (2019), 255-284.  doi: 10.3934/cpaa.2019014.

[71]

G. Xu, The carrying capacity analyses to chemotaxis system with two species and competitive kinetics in $N$ dimensions, Z. Angew. Math. Phy., 71 (2020), Paper No. 133, 28 pp. doi: 10.1007/s00033-020-01363-z.

[72]

H. Yang, X. Tu and C. Mu, Property of the large densities in a two-species and two-stimuli chemotaxis system with competitive kinetics, J. Math. Anal. Appl., 499 (2021), Paper No. 125066, 23 pp. doi: 10.1016/j.jmaa.2021.125066.

[73]

W. Zhang, S. Liu and P. Niu, Asymptotic behavior in a quasilinear chemotaxis-growth system with indirect signal production, J. Math. Anal. Appl., 486 (2020), 123855, 13 pp. doi: 10.1016/j.jmaa.2020.123855.

[74]

W. ZhangP. Niu and S. Liu, Large time behavior in a chemotaxis model with logistic growth and indirect signal production, Nonlinear Anal.: Real World Appl., 50 (2019), 484-497.  doi: 10.1016/j.nonrwa.2019.05.002.

[75]

P. Zheng, Asymptotic stability in a chemotaxis-competition system with indirect signal production, Discrete Contin. Dyn. Syst. Ser., 41 (2021), 1207-1223.  doi: 10.3934/dcds.2020315.

show all references

References:
[1]

X. Bai and S. Liu, A new criterion to a two-chemical substances chemotaxis system with critical dimension, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3717-3721.  doi: 10.3934/dcdsb.2018074.

[2]

N. BellomoA. BellouquidY. 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.

[3]

R. DillonP. K. Maini and H. G. Othmer, Pattern formation in generalised turing systems I. Steady-state patterns in systems with mixed boundary conditions, J. Math. Biol., 32 (1994), 345-393.  doi: 10.1007/BF00160165.

[4]

M. Ding and W. Wang, Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4665-4684.  doi: 10.3934/dcdsb.2018328.

[5]

Y. Dong and Y. Peng, Global boundedness in the higher-dimensional chemotaxis system with indirect signal production and rotational flux, Appl. Math. Lett., 112 (2021), 106700, 8 pp. doi: 10.1016/j.aml.2020.106700.

[6]

A. Friedman, Partial Differential Equations, Dover Books on Mathematics Series, Dover Publications, Incorporated, 2008.

[7]

M. Fuest, Analysis of a chemotaxis model with indirect signal absorption, J. Differential Equations, 267 (2019), 4778-4806.  doi: 10.1016/j.jde.2019.05.015.

[8]

K. Fujie and T. Senba, Application of an Adams type inequality to a two-chemical substances chemotaxis system, J. Differential Equations, 263 (2017), 88-148.  doi: 10.1016/j.jde.2017.02.031.

[9]

K. Fujie and T. Senba, Blowup of solutions to a two-chemical substances chemotaxis system in the critical dimension, J. Differential Equations, 266 (2019), 942-976.  doi: 10.1016/j.jde.2018.07.068.

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.

[11]

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

[12]

D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller-Segel model, J. Math. Biol., 44 (2002), 463-478.  doi: 10.1007/s002850100134.

[13]

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

[14]

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.

[15]

B. Hu and Y. Tao, To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128.  doi: 10.1142/S0218202516400091.

[16]

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.

[17]

H.-Y. Jin and Z.-A. Wang, Global stabilization of the full attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst., 40 (2020), 3509-3527.  doi: 10.3934/dcds.2020027.

[18]

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.

[19]

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.

[20]

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.

[21]

K. Kang and A. Stevens, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal. Theor. Methods Appl., 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017.

[22]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[23]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.

[24]

H. Kozono and Y. Taniuchi, Limiting case of the sobolev inequality in BMO with application to the Euler equations, Commun. Math. Phys., 214 (2000), 191-200.  doi: 10.1007/s002200000267.

[25]

J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499.

[26]

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.

[27]

P. Laurençot, Global bounded and unbounded solutions to a chemotaxis system with indirect signal production, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 6419-6444. 

[28]

H. Li and Y. Tao, Boundedness in a chemotaxis system with indirect signal production and generalized logistic source, Appl. Math. Lett., 77 (2018), 108-113.  doi: 10.1016/j.aml.2017.10.006.

[29]

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.

[30]

Y. Li, Emergence of large densities and simultaneous blow-up in a two-species chemotaxis system with competitive kinetics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5461-5480.  doi: 10.3934/dcdsb.2019066.

[31]

Y. Li and W. Wang, Boundedness in a four-dimensional attraction-repulsion chemotaxis system with logistic source, Math. Methods Appl. Sci., 41 (2018), 4936-4942.  doi: 10.1002/mma.4942.

[32]

K. Lin and C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046.  doi: 10.3934/dcds.2016018.

[33]

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.

[34]

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

[35]

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.

[36]

Y. LiuZ. Li and J. Huang, Global boundedness and large time behavior of achemotaxis system with indirect signal absorption, J. Differential Equations, 269 (2020), 6365-6399.  doi: 10.1016/j.jde.2020.05.008.

[37]

M. LucaA. Chavez-RossL. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bulletin Math. Biol., 65 (2003), 693-730.  doi: 10.1016/S0092-8240(03)00030-2.

[38]

W. Lv and Q. Wang, Global existence for a class of chemotaxis systems with signal-dependent motility, indirect signal production and generalized logistic source, Z. Angew. Math. Phys., 71 (2020), Article number: 53. doi: 10.1007/s00033-020-1276-y.

[39]

T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042.

[40]

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

[41]

E. NakaguchiK. NodaK. Osaki and K. Uemichi, Global attractor for a two-dimensional chemotaxis system with linear degradation and indirect signal production, Japan J. Indust. Appl. Math., 37 (2020), 49-80.  doi: 10.1007/s13160-019-00376-0.

[42]

K. Noda and K. Osaki, Global attractor and Lyapunov function for one-dimensional Deneubourg chemotaxis system, Hiroshima Math. J., 49 (2019), 251-271.  doi: 10.32917/hmj/1564106547.

[43]

T. Ogawa and Y. Taniuchi, On blow-up criteria of smooth solutions to the 3-D Euler equations in a bounded domain, J. Differential Equations, 190 (2003), 39-63.  doi: 10.1016/S0022-0396(03)00013-5.

[44]

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

[45]

K. J. PainterP. K. Maini and H. G. Othmer, Development and applications of a model for cellular response to multiple chemotactic cues, J. Math. Biol., 41 (2000), 285-314.  doi: 10.1007/s002850000035.

[46]

S. QiuC. Mu and Y. Li, Boundedness and stability in a chemotaxis-growth model with indirect attractant production and signal-dependent sensitivity, Acta Appl. Math., 169 (2020), 341-360.  doi: 10.1007/s10440-019-00301-0.

[47]

S. QiuC. Mu and L. Wang, Boundedness in the higher-dimensional quasilinear chemotaxis-growth system with indirect attractant production, Comput. Math. Appl., 75 (2018), 3213-3223.  doi: 10.1016/j.camwa.2018.01.042.

[48]

G. Q. Ren and B. Liu, Boundedness in a chemotaxis system under a critical parameter condition, Bull. Braz. Math. Soc. (N.S.), 52 (2021), 281-289.  doi: 10.1007/s00574-020-00202-z.

[49]

S. StrohmR. C. Tyson and J. A. Powell, Pattern formation in a model for mountain pine beetle dispersal: linking model predictions to data, Bull. Math. Biol., 75 (2013), 1778-1797.  doi: 10.1007/s11538-013-9868-8.

[50]

C. Surulescu and M. Winkler, Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis-haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more), European J. Appl. Math., 32 (2021), 618–651. arXiv: 1904.11210. doi: 10.1017/S0956792520000236.

[51]

Q. TangQ. Xin and C. Mu, Boundedness of the higher-dimensional quasilinear chemotaxis system with generalized logistic source, Acta Math. Sci. Ser. B (Engl. Ed.), 40 (2020), 713-722.  doi: 10.1007/s10473-020-0309-0.

[52]

Y. 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.

[53]

Y. Tao and M. Winkler, Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc., 19 (2017), 3641-3678.  doi: 10.4171/JEMS/749.

[54]

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

[55]

J. I. Tello and D. Wrzosek, Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.  doi: 10.1142/S0218202516400108.

[56]

M. TianL. Hong and S. Zheng, A hyperbolic-elliptic-elliptic system of an attraction-repulsion chemotaxis model with nonlinear productions, J. Evolution Equations, 18 (2018), 973-1001.  doi: 10.1007/s00028-018-0428-4.

[57]

X. TuC.-L. Tang and S. Qiu, The phenomenon of large population densities in a chemotaxis competition system with loop, J. Evolution Equ., 21 (2021), 1717-1754.  doi: 10.1007/s00028-020-00650-6.

[58]

I. TuvalL. CisnerosC. DombrowskiC. WolgemuthJ. Kessler and R. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Nat. Acad. Sci. USA, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102.

[59]

W. Wang, A quasilinear fully parabolic chemotaxis system with indirect signal production and logistic source, J. Math. Anal. Appl., 477 (2019), 488-522.  doi: 10.1016/j.jmaa.2019.04.043.

[60]

W. WangM. Zhuang and S. Zheng, Positive effects of repulsion on boundedness in a fully parabolic attraction-repulsion chemotaxis system with logistic source, J. Differential Equations, 264 (2018), 2011-2027.  doi: 10.1016/j.jde.2017.10.011.

[61]

Y. Wang, M. Winkler and Z. Xiang, The fast signal diffusion limit in Keller-Segel(-fluid) systems, Calc. Var. Part. Differ. Eq., 58 (2019), Paper No. 196, 40 pp. doi: 10.1007/s00526-019-1656-3.

[62]

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.

[63]

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.

[64]

M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.  doi: 10.1007/s00332-014-9205-x.

[65]

M. Winkler, Emergence of large population densities despite logistic restricitiions in fully parabolic chemotaxis systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2777-2793.  doi: 10.3934/dcdsb.2017135.

[66]

M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phy., 69 (2018), Paper No. 69, 40 pp. doi: 10.1007/s00033-018-0935-8.

[67]

D. Wrzosek, Long-time behaviour of solutions to a chemotaxis model with volume-filling effect, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 431-444.  doi: 10.1017/S0308210500004649.

[68]

Z. Q. Wu, J. X. Yin and C. P. Wang, Elliptic and Parabolic Equations, World Scientific Publishing, Co. Pte. Ltd., Hackensack, NJ, 2006.

[69]

T. Xiang, How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?, J. Math. Anal. Appl., 459 (2018), 1172-1200.  doi: 10.1016/j.jmaa.2017.11.022.

[70]

T. Xiang, Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion, Comm. Pure Appl. Anal., 18 (2019), 255-284.  doi: 10.3934/cpaa.2019014.

[71]

G. Xu, The carrying capacity analyses to chemotaxis system with two species and competitive kinetics in $N$ dimensions, Z. Angew. Math. Phy., 71 (2020), Paper No. 133, 28 pp. doi: 10.1007/s00033-020-01363-z.

[72]

H. Yang, X. Tu and C. Mu, Property of the large densities in a two-species and two-stimuli chemotaxis system with competitive kinetics, J. Math. Anal. Appl., 499 (2021), Paper No. 125066, 23 pp. doi: 10.1016/j.jmaa.2021.125066.

[73]

W. Zhang, S. Liu and P. Niu, Asymptotic behavior in a quasilinear chemotaxis-growth system with indirect signal production, J. Math. Anal. Appl., 486 (2020), 123855, 13 pp. doi: 10.1016/j.jmaa.2020.123855.

[74]

W. ZhangP. Niu and S. Liu, Large time behavior in a chemotaxis model with logistic growth and indirect signal production, Nonlinear Anal.: Real World Appl., 50 (2019), 484-497.  doi: 10.1016/j.nonrwa.2019.05.002.

[75]

P. Zheng, Asymptotic stability in a chemotaxis-competition system with indirect signal production, Discrete Contin. Dyn. Syst. Ser., 41 (2021), 1207-1223.  doi: 10.3934/dcds.2020315.

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