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Classification and evolution of bifurcation curves for a one-dimensional Dirichlet-Neumann problem with a specific cubic nonlinearity
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 |
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.
References:
[1] |
N. Bellomo, A. Bellouquid, Y. 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.
|
[2] |
G. Bunting, Y.-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. |
[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.
|
[4] |
J. I. Diaz, T. 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. |
[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. |
[6] |
E. Galakhov, O. 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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
K. Lin, C. 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. |
[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. |
[17] |
M. Luca, A. Chavez-Ross, L. 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. |
[18] |
T. Nagai, T. Senba and K. Yoshida,
Application of the trudinger-moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[26] |
L. Wang, C. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
P. Zheng, C. Mu, X. 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. |
[37] |
P. Zheng, C. Mu, X. 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. |
show all references
References:
[1] |
N. Bellomo, A. Bellouquid, Y. 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.
|
[2] |
G. Bunting, Y.-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. |
[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.
|
[4] |
J. I. Diaz, T. 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. |
[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. |
[6] |
E. Galakhov, O. 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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
K. Lin, C. 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. |
[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. |
[17] |
M. Luca, A. Chavez-Ross, L. 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. |
[18] |
T. Nagai, T. Senba and K. Yoshida,
Application of the trudinger-moser inequality to a parabolic system of chemotaxis, Funkcialaj Ekvacioj, 40 (1997), 411-433.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[26] |
L. Wang, C. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[36] |
P. Zheng, C. Mu, X. 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. |
[37] |
P. Zheng, C. Mu, X. 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. |
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