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February  2020, 40(2): 1107-1130. doi: 10.3934/dcds.2020072

Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain

1. 

School of Mathematical Science, Zhejiang University, Hangzhou 310027, China

2. 

School of Mathematics, Jilin University, Changchun, Jilin 130012, China

3. 

Department of Mathematics and Statistics, Auburn University, AL 36849, USA

* Corresponding author: Lianzhang Bao

Received  May 2019 Revised  August 2019 Published  November 2019

Fund Project: The first author is supported by China Postdoctoral Science Foundation (183816). The second author is supported by NSF grant DMS–1645673

The current series of research papers is to investigate the asymptotic dynamics in logistic type chemotaxis models in one space dimension with a free boundary or an unbounded boundary. Such a model with a free boundary describes the spreading of a new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front. In this first part of the series, we investigate the dynamical behaviors of logistic type chemotaxis models on the half line $ \mathbb{R}^+ $, which are formally corresponding limit systems of the free boundary problems. In the second of the series, we will establish the spreading-vanishing dichotomy in chemoattraction-repulsion systems with a free boundary as well as with double free boundaries.

Citation: Lianzhang Bao, Wenxian Shen. Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1107-1130. doi: 10.3934/dcds.2020072
References:
[1]

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.   Google Scholar

[2]

G. BuntingY.-H. Du and K. Kratowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.  doi: 10.3934/nhm.2012.7.583.  Google Scholar

[3]

J. I. Diaz and T. Nagai, Symmetrization in a parabolic-elliptic system related to chemotaxis, Advances in Mathematical Science and Applications, 5 (1995), 659-680.   Google Scholar

[4]

J. I. DiazT. Nagai and J.-M. Rakotoson, Symmetrization techniques on unbounded domains: Application to a chemotaxis system on $\mathbb{R}^N$, J. Differential Equations, 145 (1998), 156-183.  doi: 10.1006/jdeq.1997.3389.  Google Scholar

[5]

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

[6]

E. GalakhovO. Salieva and J. I. Tello, On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.  doi: 10.1016/j.jde.2016.07.008.  Google Scholar

[7]

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

[8]

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

[9]

D. Horstmann, Generalizing the Keller–Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, Journal of Nonlinear Science, 21 (2011), 231-270.  doi: 10.1007/s00332-010-9082-x.  Google Scholar

[10]

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.  Google Scholar

[11]

H. Jin, Boundedness of the attraction-repulsion Keller-Segel system, Journal of Mathematical Analysis and Applications, 422 (2015), 1463-1478.  doi: 10.1016/j.jmaa.2014.09.049.  Google Scholar

[12]

K. Kanga and A. Steven, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017.  Google Scholar

[13]

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.  Google Scholar

[14]

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

[15]

K. LinC. Mu and Y. Gao, Boundedness and blow up in the higher-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion, Journal of Differential Equations, 261 (2016), 4524-4572.  doi: 10.1016/j.jde.2016.07.002.  Google Scholar

[16]

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

[17]

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

[18]

T. NagaiT. Senba and K. Yoshida, Application of the trudinger-moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433.   Google Scholar

[19]

R. B. Salako and W. Shen, Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, Discrete Contin. Dyn. Syst., 37 (2017), 6189-6225.  doi: 10.3934/dcds.2017268.  Google Scholar

[20]

R. B. Salako and W. Shen, Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^N$. I. Persistence and asymptotic spreading, Mathematical Models and Methods in Applied Sciences, 28 (2018), 2237-2273.  doi: 10.1142/S0218202518400146.  Google Scholar

[21]

R. B. Salako and W. Shen, Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^N$. II. Existence, uniqueness, and stability of strictly positive entire solutions, J. Math. Anal. Appl., 464 (2018), 883-910.  doi: 10.1016/j.jmaa.2018.04.034.  Google Scholar

[22]

R. B. Salako and W. Shen, Global classical solutions, stability of constant equilibria, and spreading speeds in attraction-repulsion chemotaxis systems with logistic source on $\mathbb{R}^N$, Journal of Dynamics and Differential Equations, 31 (2019), 1301-1325.  doi: 10.1007/s10884-017-9602-6.  Google Scholar

[23]

R. B. Salako and W. Shen, Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, J. Differential Equations, 262 (2017), 5635-5690.  doi: 10.1016/j.jde.2017.02.011.  Google Scholar

[24]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow up in super critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.   Google Scholar

[25]

Y. Sugiyama and H. Kunii, Global Existence and decay properties for a degenerate keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.  doi: 10.1016/j.jde.2006.03.003.  Google Scholar

[26]

L. WangC. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007.  Google Scholar

[27]

Y. Wang, Global bounded weak solutions to a degenerate quasilinear attraction-repulsion chemotaxis system with rotation, Computers and Mathematics with Applications, 72 (2016), 2226-2240.  doi: 10.1016/j.camwa.2016.08.024.  Google Scholar

[28]

Y. Wang and Z.-Y. Xiang, Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system, Discrete and Continuous Dynamical Systems-Series B, 21 (2016), 1953-1973.  doi: 10.3934/dcdsb.2016031.  Google Scholar

[29]

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

[30]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis and Applications, 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.  Google Scholar

[31]

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

[32]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.  Google Scholar

[33]

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.  Google Scholar

[34]

T. Yokota and N. Yoshino, Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst. Dynamical Systems, Differential Equations and Applications. 10th AIMS Conference. Suppl., 2015, 1125–1133. doi: 10.3934/proc.2015.1125.  Google Scholar

[35]

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

[36]

P. ZhengC. MuX. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), 509-522.  doi: 10.1016/j.jmaa.2014.11.031.  Google Scholar

[37]

P. ZhengC. MuX. Hu and Y. Tian, Boundedness in the higher dimensional attraction-repulsion chemotaxis-growth system, Computers and Mathematics with Applications, 72 (2016), 2194-2202.  doi: 10.1016/j.camwa.2016.08.028.  Google Scholar

show all references

References:
[1]

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.   Google Scholar

[2]

G. BuntingY.-H. Du and K. Kratowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.  doi: 10.3934/nhm.2012.7.583.  Google Scholar

[3]

J. I. Diaz and T. Nagai, Symmetrization in a parabolic-elliptic system related to chemotaxis, Advances in Mathematical Science and Applications, 5 (1995), 659-680.   Google Scholar

[4]

J. I. DiazT. Nagai and J.-M. Rakotoson, Symmetrization techniques on unbounded domains: Application to a chemotaxis system on $\mathbb{R}^N$, J. Differential Equations, 145 (1998), 156-183.  doi: 10.1006/jdeq.1997.3389.  Google Scholar

[5]

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

[6]

E. GalakhovO. Salieva and J. I. Tello, On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.  doi: 10.1016/j.jde.2016.07.008.  Google Scholar

[7]

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

[8]

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

[9]

D. Horstmann, Generalizing the Keller–Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, Journal of Nonlinear Science, 21 (2011), 231-270.  doi: 10.1007/s00332-010-9082-x.  Google Scholar

[10]

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.  Google Scholar

[11]

H. Jin, Boundedness of the attraction-repulsion Keller-Segel system, Journal of Mathematical Analysis and Applications, 422 (2015), 1463-1478.  doi: 10.1016/j.jmaa.2014.09.049.  Google Scholar

[12]

K. Kanga and A. Steven, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Analysis, 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017.  Google Scholar

[13]

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.  Google Scholar

[14]

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

[15]

K. LinC. Mu and Y. Gao, Boundedness and blow up in the higher-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion, Journal of Differential Equations, 261 (2016), 4524-4572.  doi: 10.1016/j.jde.2016.07.002.  Google Scholar

[16]

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

[17]

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

[18]

T. NagaiT. Senba and K. Yoshida, Application of the trudinger-moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433.   Google Scholar

[19]

R. B. Salako and W. Shen, Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, Discrete Contin. Dyn. Syst., 37 (2017), 6189-6225.  doi: 10.3934/dcds.2017268.  Google Scholar

[20]

R. B. Salako and W. Shen, Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^N$. I. Persistence and asymptotic spreading, Mathematical Models and Methods in Applied Sciences, 28 (2018), 2237-2273.  doi: 10.1142/S0218202518400146.  Google Scholar

[21]

R. B. Salako and W. Shen, Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^N$. II. Existence, uniqueness, and stability of strictly positive entire solutions, J. Math. Anal. Appl., 464 (2018), 883-910.  doi: 10.1016/j.jmaa.2018.04.034.  Google Scholar

[22]

R. B. Salako and W. Shen, Global classical solutions, stability of constant equilibria, and spreading speeds in attraction-repulsion chemotaxis systems with logistic source on $\mathbb{R}^N$, Journal of Dynamics and Differential Equations, 31 (2019), 1301-1325.  doi: 10.1007/s10884-017-9602-6.  Google Scholar

[23]

R. B. Salako and W. Shen, Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$, J. Differential Equations, 262 (2017), 5635-5690.  doi: 10.1016/j.jde.2017.02.011.  Google Scholar

[24]

Y. Sugiyama, Global existence in sub-critical cases and finite time blow up in super critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.   Google Scholar

[25]

Y. Sugiyama and H. Kunii, Global Existence and decay properties for a degenerate keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.  doi: 10.1016/j.jde.2006.03.003.  Google Scholar

[26]

L. WangC. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007.  Google Scholar

[27]

Y. Wang, Global bounded weak solutions to a degenerate quasilinear attraction-repulsion chemotaxis system with rotation, Computers and Mathematics with Applications, 72 (2016), 2226-2240.  doi: 10.1016/j.camwa.2016.08.024.  Google Scholar

[28]

Y. Wang and Z.-Y. Xiang, Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system, Discrete and Continuous Dynamical Systems-Series B, 21 (2016), 1953-1973.  doi: 10.3934/dcdsb.2016031.  Google Scholar

[29]

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

[30]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, Journal of Mathematical Analysis and Applications, 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.  Google Scholar

[31]

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

[32]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.  Google Scholar

[33]

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.  Google Scholar

[34]

T. Yokota and N. Yoshino, Existence of solutions to chemotaxis dynamics with logistic source, Discrete Contin. Dyn. Syst. Dynamical Systems, Differential Equations and Applications. 10th AIMS Conference. Suppl., 2015, 1125–1133. doi: 10.3934/proc.2015.1125.  Google Scholar

[35]

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

[36]

P. ZhengC. MuX. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), 509-522.  doi: 10.1016/j.jmaa.2014.11.031.  Google Scholar

[37]

P. ZhengC. MuX. Hu and Y. Tian, Boundedness in the higher dimensional attraction-repulsion chemotaxis-growth system, Computers and Mathematics with Applications, 72 (2016), 2194-2202.  doi: 10.1016/j.camwa.2016.08.028.  Google Scholar

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