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Asymptotic stability in a chemotaxis-competition system with indirect signal production
College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China |
$ \begin{eqnarray*} \label{1a} \left\{ \begin{split}{} &u_{t} = \text{div}(d_{u}\nabla u+\chi u\nabla w)+\mu_{1}u(1-u-a_{1}v), &(x,t)\in \Omega\times (0,\infty), \\ &v_{t} = d_{v}\Delta v+\mu_{2}v(1-v-a_{2}u), &(x,t)\in \Omega\times (0,\infty), \\ & w_{t} = d_{w}\Delta w-\lambda w+\alpha v, &(x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*} $ |
$ \Omega\subset \mathbb{R}^{N} $ |
$ N\geq 1 $ |
$ d_{u}>0, d_{v}>0 $ |
$ d_{w}>0 $ |
$ \chi\in \mathbb{R} $ |
$ \mu_{1}>0 $ |
$ \mu_{2}>0 $ |
$ a_{1}>0, a_{2}>0 $ |
$ \lambda $ |
$ \alpha $ |
$ \chi>0 $ |
References:
[1] |
X. Bai and M. Winkler,
Equilibration in a fully parabolic two-spescies chemotaxis system with competitive kinetics, Indiana University Math. J., 65 (2016), 553-583.
doi: 10.1512/iumj.2016.65.5776. |
[2] |
T. Black,
Global existence and asymptotic behavior in a competition two-species chemotaxis system with two signals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1253-1272.
doi: 10.3934/dcdsb.2017061. |
[3] |
T. Black, J. Lankeit and M. Mizukami,
On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876.
doi: 10.1093/imamat/hxw036. |
[4] |
X. Cao,
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.
doi: 10.3934/dcds.2015.35.1891. |
[5] |
X. Cao, S. Kurima and M. Mizukami,
Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics, Math. Meth. Appl. Sci., 41 (2018), 3138-3154.
doi: 10.1002/mma.4807. |
[6] |
X. Cao, S. Kurima and M. Mizukami, Global existence and asymptotic behavior of classical solutions for a 3D two-species Keller-Segel-Stokes system with competitive kinetics, preprint, arXiv: 1706.07910v1. |
[7] |
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. |
[8] |
M. Fuest,
Analysis of a chemotaxis model with indirect signal absorbtion, J. Differential Equations, 267 (2019), 4778-4806.
doi: 10.1016/j.jde.2019.05.015. |
[9] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[10] |
M. Hirata, S. Kurima, M. Mizukami and T. Yokota,
Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, J. Differential Equations, 263 (2017), 470-490.
doi: 10.1016/j.jde.2017.02.045. |
[11] |
M. Hirata, S. Kurima, M. Mizukami and T. Yokota, Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, preprint, arXiv: 1710.00957v1. |
[12] |
S. B. Hsu,
Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.
doi: 10.1137/0134064. |
[13] |
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. |
[14] |
H.-Y. Jin and T. Xiang,
Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes with competition kinetics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1919-1942.
doi: 10.3934/dcdsb.2018249. |
[15] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[16] |
E. F. Keller and L. A. Segel,
Traveling bans of chemotactic bacteria: a theoretical analysis, J. Theor. Biol., 30 (1971), 377-380.
|
[17] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Izdat. Nauka, Moscow, 1967. |
[18] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
doi: 10.1142/3302. |
[19] |
K. Lin, C. Mu and L. Wang,
Boundedness in a two-species chemotaxis system, Math. Meth. Appl. Sci., 38 (2015), 5085-5096.
doi: 10.1002/mma.3429. |
[20] |
A. J. Lotka, Elements of Physical Biology, Williams and Wilkins Co., Baltimore, 1925. |
[21] |
J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993.
doi: 10.1007/b98869. |
[22] |
C. S. Patlak,
Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[23] |
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. |
[24] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[25] |
Y. Tao and M. Winkler, Boundedness and competitive exclusion in a population model with cross-diffusion for one species, preprint. |
[26] |
Y. Tao and M. Winkler,
Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc. (JEMS), 19 (2017), 3641-3678.
doi: 10.4171/JEMS/749. |
[27] |
J. I. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[28] |
J. I. Tello and D. Wrzosek,
Inter-species competition and chemorepulsion, J. Math. Anal. Appl., 459 (2018), 1233-1250.
doi: 10.1016/j.jmaa.2017.11.021. |
[29] |
V. Volterra, Variazioni e Fluttuazioni del Numero d'individui in Specie Animali Conviventi, Mem. R. Accad. Naz. Dei Lincei. Ser. VI, 1926. |
[30] |
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. |
[31] |
Q. Wang, L. Zhang, J. Yang and J. Hu,
Global existence and steady states of a two competing species Keller-Segel chemotaxis model, Kinet. Relat. Models, 8 (2015), 777-807.
doi: 10.3934/krm.2015.8.777. |
[32] |
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. |
[33] |
J. Xing and P. Zheng, Global boundedness and long-time behavior for a two-dimensional quasilinear chemotaxis system with indirect signal consumption, preprint. |
[34] |
P. Zheng and C. Mu,
Global boundedness in a two-competing-species chemotaxis system with two chemicals, Acta Appl. Math., 148 (2017), 157-177.
doi: 10.1007/s10440-016-0083-0. |
[35] |
P. Zheng, C. Mu and X. Hu,
Global dynamics for an attraction-repulsion chemotaxis-(Navier)-Stokes system with logistic source, Nonl. Anal. Real World Appl., 45 (2019), 557-580.
doi: 10.1016/j.nonrwa.2018.07.028. |
[36] |
P. Zheng, C. Mu and Y. Mi,
Global stability in a two-competing-species chemotaxis system with two chemicals, Diff. Inte. Equa., 31 (2018), 547-558.
|
[37] |
P. Zheng, C. Mu, R. Willie and X. Hu,
Global asymptotic stability of steady states in a chemotaxis-growth system with singular sensitivity, Comput. Math. Appl., 75 (2018), 1667-1675.
doi: 10.1016/j.camwa.2017.11.032. |
[38] |
P. Zheng, R. Willie and C. Mu,
Global boundedness and stabilization in a two-competing-species chemotaxis-fluid system with two chemicals, J. Dyn. Differential Equations, 32 (2020), 1371-1399.
doi: 10.1007/s10884-019-09797-4. |
show all references
References:
[1] |
X. Bai and M. Winkler,
Equilibration in a fully parabolic two-spescies chemotaxis system with competitive kinetics, Indiana University Math. J., 65 (2016), 553-583.
doi: 10.1512/iumj.2016.65.5776. |
[2] |
T. Black,
Global existence and asymptotic behavior in a competition two-species chemotaxis system with two signals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1253-1272.
doi: 10.3934/dcdsb.2017061. |
[3] |
T. Black, J. Lankeit and M. Mizukami,
On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876.
doi: 10.1093/imamat/hxw036. |
[4] |
X. Cao,
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.
doi: 10.3934/dcds.2015.35.1891. |
[5] |
X. Cao, S. Kurima and M. Mizukami,
Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics, Math. Meth. Appl. Sci., 41 (2018), 3138-3154.
doi: 10.1002/mma.4807. |
[6] |
X. Cao, S. Kurima and M. Mizukami, Global existence and asymptotic behavior of classical solutions for a 3D two-species Keller-Segel-Stokes system with competitive kinetics, preprint, arXiv: 1706.07910v1. |
[7] |
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. |
[8] |
M. Fuest,
Analysis of a chemotaxis model with indirect signal absorbtion, J. Differential Equations, 267 (2019), 4778-4806.
doi: 10.1016/j.jde.2019.05.015. |
[9] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[10] |
M. Hirata, S. Kurima, M. Mizukami and T. Yokota,
Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, J. Differential Equations, 263 (2017), 470-490.
doi: 10.1016/j.jde.2017.02.045. |
[11] |
M. Hirata, S. Kurima, M. Mizukami and T. Yokota, Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, preprint, arXiv: 1710.00957v1. |
[12] |
S. B. Hsu,
Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), 760-763.
doi: 10.1137/0134064. |
[13] |
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. |
[14] |
H.-Y. Jin and T. Xiang,
Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes with competition kinetics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1919-1942.
doi: 10.3934/dcdsb.2018249. |
[15] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[16] |
E. F. Keller and L. A. Segel,
Traveling bans of chemotactic bacteria: a theoretical analysis, J. Theor. Biol., 30 (1971), 377-380.
|
[17] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Izdat. Nauka, Moscow, 1967. |
[18] |
G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
doi: 10.1142/3302. |
[19] |
K. Lin, C. Mu and L. Wang,
Boundedness in a two-species chemotaxis system, Math. Meth. Appl. Sci., 38 (2015), 5085-5096.
doi: 10.1002/mma.3429. |
[20] |
A. J. Lotka, Elements of Physical Biology, Williams and Wilkins Co., Baltimore, 1925. |
[21] |
J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993.
doi: 10.1007/b98869. |
[22] |
C. S. Patlak,
Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[23] |
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. |
[24] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[25] |
Y. Tao and M. Winkler, Boundedness and competitive exclusion in a population model with cross-diffusion for one species, preprint. |
[26] |
Y. Tao and M. Winkler,
Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production, J. Eur. Math. Soc. (JEMS), 19 (2017), 3641-3678.
doi: 10.4171/JEMS/749. |
[27] |
J. I. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[28] |
J. I. Tello and D. Wrzosek,
Inter-species competition and chemorepulsion, J. Math. Anal. Appl., 459 (2018), 1233-1250.
doi: 10.1016/j.jmaa.2017.11.021. |
[29] |
V. Volterra, Variazioni e Fluttuazioni del Numero d'individui in Specie Animali Conviventi, Mem. R. Accad. Naz. Dei Lincei. Ser. VI, 1926. |
[30] |
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. |
[31] |
Q. Wang, L. Zhang, J. Yang and J. Hu,
Global existence and steady states of a two competing species Keller-Segel chemotaxis model, Kinet. Relat. Models, 8 (2015), 777-807.
doi: 10.3934/krm.2015.8.777. |
[32] |
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. |
[33] |
J. Xing and P. Zheng, Global boundedness and long-time behavior for a two-dimensional quasilinear chemotaxis system with indirect signal consumption, preprint. |
[34] |
P. Zheng and C. Mu,
Global boundedness in a two-competing-species chemotaxis system with two chemicals, Acta Appl. Math., 148 (2017), 157-177.
doi: 10.1007/s10440-016-0083-0. |
[35] |
P. Zheng, C. Mu and X. Hu,
Global dynamics for an attraction-repulsion chemotaxis-(Navier)-Stokes system with logistic source, Nonl. Anal. Real World Appl., 45 (2019), 557-580.
doi: 10.1016/j.nonrwa.2018.07.028. |
[36] |
P. Zheng, C. Mu and Y. Mi,
Global stability in a two-competing-species chemotaxis system with two chemicals, Diff. Inte. Equa., 31 (2018), 547-558.
|
[37] |
P. Zheng, C. Mu, R. Willie and X. Hu,
Global asymptotic stability of steady states in a chemotaxis-growth system with singular sensitivity, Comput. Math. Appl., 75 (2018), 1667-1675.
doi: 10.1016/j.camwa.2017.11.032. |
[38] |
P. Zheng, R. Willie and C. Mu,
Global boundedness and stabilization in a two-competing-species chemotaxis-fluid system with two chemicals, J. Dyn. Differential Equations, 32 (2020), 1371-1399.
doi: 10.1007/s10884-019-09797-4. |
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