Skip to main content
SpringerLink
Log in

Global boundedness to a chemotaxis system with singular sensitivity and logistic source

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

We consider the parabolic-parabolic Keller–Segel system with singular sensitivity and logistic source: \( u_t=\Delta u-\chi \nabla \cdot (\frac{u}{v}\nabla v) +ru-\mu u^2\), \(v_t=\Delta v-v+u\) under the homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset \mathbb {R}^2\), \(\chi ,\mu >0\) and \(r\in \mathbb {R}\). It is proved that the system exists globally bounded classical solutions if \(r>\frac{\chi ^2}{4}\) for \(0<\chi \le 2\), or \(r>\chi -1\) for \(\chi >2\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aida, M., Osaka, K., Tsujikawa, T., Yagi, A., Mimura, M.: Chemotaxis and growth system with sigular sensitivity function. Nonliear Anal. Real World Appl. 6, 323–336 (2005)

    Article  MATH  Google Scholar 

  2. Bai, X.L., Winkler, M.: Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics. Indiana Univ. Math. J. 65, 553–583 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Biler, P.: Global solutions to some parabolic-elliptic systems of chemotaxis. Adv. Math. Sci. Appl. 9, 347–359 (1999)

    MathSciNet  MATH  Google Scholar 

  4. Bellomo, N., Bellouquid, A., Tao, Y., Winkler, M.: Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues. Math. Mod. Meth. Appl. Sci. 9, 1663–1763 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cao, X.R.: Boundedness in a quasilinear parabolic-parabolic Keller–Segel system with logistic source. J. Math. Anal. Appl. 412, 181–188 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cao, X.R., Zheng, S.N.: Boundedness of solutions to a quasilinear parabolic-elliptic Keller–Segel system with logistic source. Math. Meth. Appl. Sci. 37, 2326–2330 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fujie, K.: Boundedness in a fully parabolic chemotaxis system with singular sensitivity. J. Math. Anal. Appl. 424, 675–684 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fujie, K., Senba, T.: Global existence and boundedness in a parabolic-elliptic Keller–Segel system with general sensitivity. Discrete Contin. Dyn. Syst. Ser. B 21, 81–102 (2016)

    MathSciNet  MATH  Google Scholar 

  9. Fujie, K., Winkler, M., Yokota, T.: Boundedness of solutions to parabolic-elliptic Keller–Segel systems with signal-dependent sensitivity. Math. Methods Appl. Sci. 38, 1212–1224 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fujie, K., Winkler, M., Yokota, T.: Blow-up prevention by logistic sources in a parabolic-elliptic Keller–Segel system with singular sensitivity. Nonliear Anal. 109, 56–71 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)

    Article  MATH  Google Scholar 

  12. Keller, E.F., Segel, L.A.: Traveling bans of chemotactic bacteria: a theoretical analysis. J. Theor. Biol. 30, 377–380 (1971)

    MATH  Google Scholar 

  13. Hillen, T., Painter, K.J.: A use’s guidence to PDE models for chemotaxis. J. Math. Biol. 58, 183–217 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lankeit, J.: A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity. Math. Methods Appl. Sci. 39, 394–404 (2015)

  15. Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-linear Equations of Parabolic Tpye. AMS, Providence (1968)

    Google Scholar 

  16. Nagai, T., Senba, T.: Behavior of radially symmetric solutions of a system related to chemotaxis. Nonlinear Anal. 30, 3837–3842 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nagai, T., Senba, T., Yoshida, K.: Global existence of solutions to the parabolic systems of chemotaxis. RIMS Kokyuroku 1009, 22–28 (1997)

    MathSciNet  MATH  Google Scholar 

  18. Nagai, T., Senba, T., Yoshida, K.: Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis. Funkc. Ekvac. 40(3), 411–433 (1997)

    MathSciNet  MATH  Google Scholar 

  19. Osaki, K., Yagi, A.: Finite dimensional attractors for one-dimensional Keller-Segel equations. Funkc. Ekvac. 44, 441–469 (2001)

    MathSciNet  MATH  Google Scholar 

  20. Osaki, K., Tsujikawa, T., Yagi, A., Mimura, M.: Exponential attractor for a chemotaxis-growth system of equations. Nonlinear Anal. 51, 119–144 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tello, J.I., Winkler, M.: A chemotaxis system with logistic source. Comm. Partial Differ. Equ. 32, 849–877 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Winkler, M.: Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source. Comm. Partial Differ. Equ. 35, 1516–1537 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Winkler, M.: Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model. J. Differ. Equ. 248, 2889–2905 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Winkler, M.: Global solutions in a fully parabolic chemotaxis system with singular sensitivity. Math. Methods Appl. Sci. 34, 176–190 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang, Y.L., Zheng, S.N.: Global boundedness of solutions to a quasilinear parabolic-parabolic Keller–Segel system with logistic source. Appl. Math. Lett. 52, 15–20 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dalian University of Technology, Dalian, China

    Xiangdong Zhao & Sining Zheng

Authors
  1. Xiangdong Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sining Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sining Zheng.

Additional information

Supported by the National Natural Science Foundation of China (11171048).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, X., Zheng, S. Global boundedness to a chemotaxis system with singular sensitivity and logistic source. Z. Angew. Math. Phys. 68, 2 (2017). https://doi.org/10.1007/s00033-016-0749-5

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-016-0749-5

Mathematics Subject Classification

Keywords

Search

Navigation