August  2021, 26(8): 4281-4298. doi: 10.3934/dcdsb.2020288

Asymptotics in a two-species chemotaxis system with logistic source

School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an, Shaanxi 710129, China

* Corresponding author: Pengcheng Niu

Received  May 2020 Revised  August 2020 Published  August 2021 Early access  October 2020

This paper deals with nonnegative solutions of a fully parabolic two-species chemotaxis system with competitive kinetics under homogeneous Neumann boundary conditions in a N-dimensional bounded smooth domain with reasonably smooth nonnegative initial data. In a previous paper of Bai & Winkler (2016), the equilibrium of the global bounded classical solution was shown in both coexistence and extinction cases. We extend this result to weak solutions and prove these solutions globally exist and finally converge to the same semi-trivial steady state in a certain sense.

Citation: Wenji Zhang, Pengcheng Niu. Asymptotics in a two-species chemotaxis system with logistic source. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4281-4298. doi: 10.3934/dcdsb.2020288
References:
[1]

S. Ahmad, Convergence and ultimate bounds of solutions of the nonautonomous Volterra-Lotka competition equations, J. Math. Anal. Appl., 127 (1987), 377-387.  doi: 10.1016/0022-247X(87)90116-8.

[2]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis (eds. Schmeisser, H. and Triebel, H.), Teubner-Texte Math., Teubner, Stuttgart, 133 (1993), 9–126. doi: 10.1007/978-3-663-11336-2_1.

[3]

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.

[4]

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.

[5]

X. Cao, An interpolation inequality and its application in Keller-Segel model, preprint, arXiv: 1707.09235.

[6]

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.

[7]

X. Cao, Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3369-3378.  doi: 10.3934/dcdsb.2017141.

[8]

X. ChenA. Jüngel and J.-G. Liu, A note on Aubin-Lions-Dubinskiĭ lemmas, Acta Appl. Math., 133 (2014), 33-43.  doi: 10.1007/s10440-013-9858-8.

[9]

L. Corrias and B. Perthame, Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces, Math. Comput. Model., 47 (2008), 755-764.  doi: 10.1016/j.mcm.2007.06.005.

[10]

X. He and S. Zheng, Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source, J. Math. Anal. Appl., 436 (2016), 970-982.  doi: 10.1016/j.jmaa.2015.12.058.

[11]

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

[12]

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.

[13]

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.

[14]

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.

[15]

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.

[16]

J. Lankeit and Y. Wang, Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption, Discrete Contin. Dyn. Syst., 37 (2017), 6099-6121.  doi: 10.3934/dcds.2017262.

[17]

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

[18]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 

[19]

K. Osaki and A. Yagi, Global existence of a chemotaxis-growth system in ${\Bbb R}^2$, Adv. Math. Sci. Appl., 12 (2002), 587-606. 

[20]

C. StinnerJ. 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.

[21]

Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.

[22]

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.

[23]

L. WangY. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802.  doi: 10.3934/dcds.2014.34.789.

[24]

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.

[25]

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.

[26]

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.

[27]

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.

[28]

M. Winkler, Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM. J. Math. Anal., 47 (2015), 3092-3115.  doi: 10.1137/140979708.

[29]

C. YangX. CaoZ. Jiang and S. Zheng, Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source, J. Math. Anal. Appl., 430 (2015), 585-591.  doi: 10.1016/j.jmaa.2015.04.093.

[30]

M. L. Zeeman, Extinction in competitive Lotka-Volterra systems, Proc. Amer. Math. Soc., 123 (1995), 87-96.  doi: 10.1090/S0002-9939-1995-1264833-2.

[31]

Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.  doi: 10.1007/s00033-013-0383-4.

show all references

References:
[1]

S. Ahmad, Convergence and ultimate bounds of solutions of the nonautonomous Volterra-Lotka competition equations, J. Math. Anal. Appl., 127 (1987), 377-387.  doi: 10.1016/0022-247X(87)90116-8.

[2]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function Spaces, Differential Operators and Nonlinear Analysis (eds. Schmeisser, H. and Triebel, H.), Teubner-Texte Math., Teubner, Stuttgart, 133 (1993), 9–126. doi: 10.1007/978-3-663-11336-2_1.

[3]

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.

[4]

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.

[5]

X. Cao, An interpolation inequality and its application in Keller-Segel model, preprint, arXiv: 1707.09235.

[6]

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.

[7]

X. Cao, Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3369-3378.  doi: 10.3934/dcdsb.2017141.

[8]

X. ChenA. Jüngel and J.-G. Liu, A note on Aubin-Lions-Dubinskiĭ lemmas, Acta Appl. Math., 133 (2014), 33-43.  doi: 10.1007/s10440-013-9858-8.

[9]

L. Corrias and B. Perthame, Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces, Math. Comput. Model., 47 (2008), 755-764.  doi: 10.1016/j.mcm.2007.06.005.

[10]

X. He and S. Zheng, Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source, J. Math. Anal. Appl., 436 (2016), 970-982.  doi: 10.1016/j.jmaa.2015.12.058.

[11]

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

[12]

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.

[13]

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.

[14]

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.

[15]

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.

[16]

J. Lankeit and Y. Wang, Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption, Discrete Contin. Dyn. Syst., 37 (2017), 6099-6121.  doi: 10.3934/dcds.2017262.

[17]

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

[18]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469. 

[19]

K. Osaki and A. Yagi, Global existence of a chemotaxis-growth system in ${\Bbb R}^2$, Adv. Math. Sci. Appl., 12 (2002), 587-606. 

[20]

C. StinnerJ. 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.

[21]

Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.

[22]

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.

[23]

L. WangY. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802.  doi: 10.3934/dcds.2014.34.789.

[24]

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.

[25]

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.

[26]

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.

[27]

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.

[28]

M. Winkler, Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM. J. Math. Anal., 47 (2015), 3092-3115.  doi: 10.1137/140979708.

[29]

C. YangX. CaoZ. Jiang and S. Zheng, Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source, J. Math. Anal. Appl., 430 (2015), 585-591.  doi: 10.1016/j.jmaa.2015.04.093.

[30]

M. L. Zeeman, Extinction in competitive Lotka-Volterra systems, Proc. Amer. Math. Soc., 123 (1995), 87-96.  doi: 10.1090/S0002-9939-1995-1264833-2.

[31]

Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.  doi: 10.1007/s00033-013-0383-4.

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