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

Asymptotics in a two-species chemotaxis system with logistic source

Wenji Zhang and  Pengcheng Niu ,

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.

Keywords: Chemotaxis system, logistic source, weak solution, asymptotic behavior.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35D30, 35K55.
Citation: Wenji Zhang, Pengcheng Niu. Asymptotics in a two-species chemotaxis system with logistic source. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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.  doi: 10.1142/S021820251550044X.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

X. Chen, A. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[18]

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

[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.   Google Scholar

[20]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

L. Wang, Y. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

C. Yang, X. Cao, Z. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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.  doi: 10.1142/S021820251550044X.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[8]

X. Chen, A. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.   Google Scholar

[20]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

L. Wang, Y. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

C. Yang, X. Cao, Z. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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