Acta Applicandae Mathematicae

, Volume 153, Issue 1, pp 197–220 | Cite as

Blow up and Bounded Solutions in a Two-Species Chemotaxis System in Two Dimensional Domains

Article
  • 89 Downloads

Abstract

In this paper, we consider the initial-boundary value problem of the two-species chemotaxis Keller-Segel model
$$\begin{aligned} \textstyle\begin{cases} u_{t}=\Delta u-\chi_{1}\nabla \cdot (u\nabla w), &x\in \varOmega , \ t>0, \\ v_{t}=\Delta v-\chi_{2}\nabla \cdot (v\nabla w), &x\in \varOmega , \ t>0, \\ 0=\Delta w-\gamma w+\alpha_{1}u+\alpha_{2}v, &x\in \varOmega , \ t>0, \end{cases}\displaystyle \end{aligned}$$
where the parameters \(\chi_{1}\), \(\chi_{2}\), \(\alpha_{1}\), \(\alpha_{2}\), \(\gamma \) are positive constants, \(\varOmega \subset \mathbb{R}^{2}\) is a bounded domain with smooth boundary. We obtain the results for finite time blow-up and global bounded as follows: (1) For any fixed \(x_{0}\in \varOmega \), if \(\chi_{1}\alpha_{2}= \chi_{2}\alpha_{1}\), \(\int_{\varOmega }(u_{0}+v_{0})|x-x_{0}|^{2}dx\) is sufficiently small, and \(\int_{\varOmega }(u_{0}+v_{0})dx>\frac{8\pi ( \chi_{1}\alpha_{1}+\chi_{2}\alpha_{2})}{\chi_{1}\alpha_{1}\chi_{2} \alpha_{2}}\), then the nonradial solution of the two-species Keller-Segel model blows up in finite time. Moreover, if \(\varOmega \) is a convex domain, we find a lower bound for the blow-up time; (2) If \(\|u_{0}\|_{L^{1}(\varOmega )}\) and \(\|v_{0}\|_{L^{1}( \varOmega )}\) lie below some thresholds, respectively, then the solution exists globally and remains bounded.

Keywords

Chemotaxis Blow-up Boundedness 

Mathematics Subject Classification

92C17 35B44 39A22 

Notes

Acknowledgements

The authors are very grateful to the anonymous reviewers for their careful reading and valuable comments which greatly improved this work. This work is supported by NSFC (Grant No. 11371384 and No. 11571062) and the Basic and Advanced Research Project of CQC-STC (Grant No. cstc2015jcyjBX0007).

References

  1. 1.
    Biler, P., Guerra, I.: Blowup and self-similar solutions for two-component drift-diffusion systems. Nonlinear Anal. 75, 5186–5193 (2012) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Biler, P., Espejo, E.E., Guerra, I.: Blowup in higher dimensional two species chemotactic systems. Commun. Pure Appl. Anal. 12, 89–98 (2013) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Conca, C., Espejo, E., Vilches, K.: Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in \(R^{2}\). Eur. J. Appl. Math. 22, 553–580 (2011) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cieślak, T., Stinner, C.: Finite-time blowup and global-in-time unbounded solutions to a parabolicparabolic quasilinear Keller-Segel system in higher dimensions. J. Differ. Equ. 252, 5832–5851 (2012) CrossRefMATHGoogle Scholar
  5. 5.
    Espejo Arenas, E.E., Stevens, A., Velázquez, J.J.L.: Simultaneous finite time blow-up in a two-species model for chemotaxis. Analysis (Munich) 29, 317–338 (2009) MathSciNetMATHGoogle Scholar
  6. 6.
    Espejo, E.E., Stevens, A., Suzuki, T.: Simultaneous blowup and mass separation during collapse in an interacting system of chemotactic species. Differ. Integral Equ. 25, 251–288 (2012) MathSciNetMATHGoogle Scholar
  7. 7.
    Espejo, E., Suzuki, T.: Global existence and blow-up for a system describing the aggregation of microglia. Appl. Math. Lett. 35, 29–34 (2014) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Espejo, E.E., Stevens, A., Velázquez, J.J.L.: A note on non-simultaneous blow-up for a drift-diffusion model. Differ. Integral Equ. 23(5–6), 451–462 (2010) MathSciNetMATHGoogle Scholar
  9. 9.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics, vol. 224. Springer, New York (2001) MATHGoogle Scholar
  10. 10.
    Hillen, T., Painter, K.J.: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58, 183–217 (2009) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Horstmann, D.: From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I. Jahresber. Dtsch. Math.-Ver. 105, 103–165 (2003) MathSciNetMATHGoogle Scholar
  12. 12.
    Horstmann, D.: From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, II. Jahresber. Dtsch. Math.-Ver. 106, 51–69 (2004) MathSciNetMATHGoogle Scholar
  13. 13.
    Horstmann, D., Winkler, M.: Boundedness vs. blow-up in a chemotaxis system. J. Differ. Equ. 215, 52–107 (2005) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Horstmann, D., Wang, G.: Blow-up in a chemotaxis model without symmetry assumptions. Eur. J. Appl. Math. 12, 159–177 (2001) MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Itō, S.: Diffusion Equations, vol. 114. Springer, New York (1992) MATHGoogle Scholar
  16. 16.
    Jin, H.Y., Wang, Z.A.: Boundedness, blowup and critical mass phenomenon incompeting chemotaxis. J. Differ. Equ. 260, 162–196 (2016) CrossRefMATHGoogle Scholar
  17. 17.
    Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970) CrossRefMATHGoogle Scholar
  18. 18.
    Keller, E.F., Segel, L.A.: Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971) CrossRefMATHGoogle Scholar
  19. 19.
    Keller, E.F., Segel, L.A.: Traveling bands of chemotactic bacteria: a theoretical analysis. J. Theor. Biol. 30, 235–248 (1971) CrossRefMATHGoogle Scholar
  20. 20.
    Li, Y., Li, Y.X.: Blow-up of nonradial solutions to attraction-repulsion chemotaxis system in two dimensions. Nonlinear Anal., Real World Appl. 30, 170–183 (2016) MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Li, Y., Li, Y.X.: Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species. Nonlinear Anal. 109, 72–84 (2014) MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Li, Y.: Global bounded solutions and their asymptotic properties under small initial data condition in a two-dimensional chemotaxis system for two species. J. Math. Anal. Appl. 429, 1291–1304 (2015) MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Marras, M., Viglialoro, G.: Blow-up time of a general Keller-Segel system with source and damping terms. C. R. Acad. Bulgare Sci. 6, 687–696 (2016) MathSciNetMATHGoogle Scholar
  24. 24.
    Marras, M., Vernier-Piro, S., Viglialoro, G.: Blow-up phenomena in chemotaxis systems with a source term. Math. Methods Appl. Sci. 11, 2787–2798 (2016) MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Nagai, T.: Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains. J. Inequal. Appl. 6, 37–55 (2001) MathSciNetMATHGoogle Scholar
  26. 26.
    Nagai, T.: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5, 581–601 (1995) MathSciNetMATHGoogle Scholar
  27. 27.
    Nagai, T., Senba, T., Yoshida, K.: Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis. Funkc. Ekvacioj 40, 411–433 (1997) MathSciNetMATHGoogle Scholar
  28. 28.
    Payne, L.E., Song, J.C.: Lower bounds for blow-up in a model of chemotaxis. J. Math. Anal. Appl. 385, 672–676 (2012) MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Payne, L.E., Schaefer, P.W.: Lower bounds for blow-up time in parabolic problems under Neumann conditions. Appl. Anal. 85, 1301–1311 (2006) MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Tao, Y.S., Wang, Z.A.: Competing effects of attraction vs. repulsion in chemotaxis. Math. Models Methods Appl. Sci. 23, 1–36 (2013) MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Tao, Y.S., Winkler, M.: Dominance of chemotaxis in a chemotaxis-haptotaxis model. Nonlinearity 27, 1225–1239 (2014) MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Tao, Y.S., Winkler, M.: Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subscritical sensitivity. J. Differ. Equ. 252, 692–715 (2012) CrossRefMATHGoogle Scholar
  33. 33.
    Tao, Y.S., Winkler, M.: Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals. Discrete Contin. Dyn. Syst., Ser. B 20, 3165–3183 (2015) MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Tao, Y.S.: Boundedness in a chemotaxis model with oxygen consumption by bacteria. J. Math. Anal. Appl. 381, 521–529 (2011) MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Viglialoro, G.: Blow-up time of a Keller-Segel-type system with neumann and robin boundary conditions. Differ. Integral Equ. 3–4, 359–376 (2016) MathSciNetMATHGoogle Scholar
  36. 36.
    Winkler, M.: Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source. Commun. Partial Differ. Equ. 35, 1516–1537 (2010) MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Winkler, M.: Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system. J. Math. Pures Appl. 100(9), 748–767 (2013) MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Winkler, M.: Does a ‘volume-filling effect’ always prevent chemotactic collapse? Math. Methods Appl. Sci. 33, 12–24 (2010) MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  • Jie Zhao
    • 1
  • Chunlai Mu
    • 1
  • Liangchen Wang
    • 1
  • Deqin Zhou
    • 1
  1. 1.College of Mathematics and StatisticsChongqing UniversityChongqingP.R. China

Personalised recommendations