Skip to main content
SpringerLink
Log in

Global boundedness and large time behavior in a signal-dependent motility system with nonlinear signal consumption

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

Abstract

In this paper, we deal with the following system with nonlinear signal consumption

$$\begin{aligned} \left\{ \begin{array}{llll} {u_t} = \Delta \left( {\gamma \left( v \right) u} \right) + ru - \mu {u^\alpha },\quad &{}x\in \Omega ,\quad &{}t>0,\\ {v_t} = \Delta v - {u^\beta }v,\quad &{}x\in \Omega ,\quad &{}t>0,\\ \end{array} \right. \end{aligned}$$

under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \in {\mathbb {R}^n} \left( {n \ge 2} \right) \). It shown that whenever \(r> 0,\mu> 0,\alpha> 2, \beta> 0 \text { and } \frac{\alpha }{\beta } > \frac{{n + 2}}{2}\), then the original system will produce a global classical solution and the solution converges to equilibrium

$$\begin{aligned} \left( {{{\left( {\frac{r}{\mu }} \right) }^{\frac{1}{{\alpha - 1}}}},0} \right) \quad \text { as } t \rightarrow \infty . \end{aligned}$$

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. Alikakos, N.: \({L^p}\) bounds of solutions of reaction–diffusion equations. Commun. Part. Diff. Equ. 4, 827–868 (1979)

    Article  Google Scholar 

  2. Amann, H.: Dynamic theory of quasilinear parabolic equations, II: reaction–diffusion systems. Differ. Integral Equ. 3(1), 13–75 (1990)

    MathSciNet  Google Scholar 

  3. Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Function Spaces, Differential Operators and Nonlinear Analysis, Friedrichroda, 1992, Teubner-Texte Mathematics, vol. 133, Teubner, Stuttgart (1993)

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

    Article  MathSciNet  Google Scholar 

  5. Cao, X.: Global bounded solutions of the higher-dimensional Keller–Segel system under smallness conditions in optimal spaces. Discrete Cont. Dyn. A. 35, 1891–1904 (2015)

    Article  MathSciNet  Google Scholar 

  6. Fujie, K., Jiang, J.: Boundedness of classical solutions to a degenerate Keller–Segel type model with signal-dependent motilities. Acta Appl. Math. 176, 1–36 (2021)

    Article  MathSciNet  Google Scholar 

  7. Horstmann, D., Wang, G.: Blow-up in a chemotaxis model without symmetry assumptions. Eur. J. Appl. Math. 12, 159–177 (2001)

    Article  MathSciNet  CAS  Google Scholar 

  8. Jin, H., Wang, Z.: Critical mass on the Keller–Segel system with signal-dependent motility. Proc. Am. Math. Soc. 148, 4855–4873 (2020)

    Article  MathSciNet  Google Scholar 

  9. Jin, H., Wang, Z., Kim, Y.: Boundedness, stabilization, and pattern formation driven by density-suppressed motility. SIAM J. Appl. Math. 78, 1632–1657 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  11. Lankeit, J., Wang, Y.: Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption. Discrete Cont. Dyn. A 37, 6099–6121 (2017)

    Article  MathSciNet  Google Scholar 

  12. Lady’zhenskaya, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear Equations of Parabolic Type. AMS, Providence (1968)

    Book  Google Scholar 

  13. Li, D., Zhao, J.: Global boundedness and large time behavior of solutions to a chemotaxis-consumption system with signal-dependent motility. Z. Angew. Math. Phys. 72, 1–20 (2021)

    MathSciNet  Google Scholar 

  14. Li, G., Wang, L.: Boundedness in a taxis-consumption system involving signal-dependent motilities and concurrent enhancement of density-determined diffusion and cross-diffusion. Z. Angew. Math. Phys. 74, 92 (2023)

    Article  MathSciNet  Google Scholar 

  15. Li, G., Winkler, M.: Refined regularity analysis for a Keller–Segel-consumption system involving signal-dependent motilities. Appl. Anal. 8, 1–20 (2023)

    Google Scholar 

  16. Li, G., Winkler, M.: Relaxation in a Keller–Segel-consumption system involving signal-dependent motilities. Commun. Math. Sci. 21(2), 299–322 (2023)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  17. Li, X., Wang, L., Pan, X.: Boundedness and stabilization in the chemotaxis consumption model with signal-dependent motility. Z. Angew. Math. Phys. 72, 1–18 (2021)

    MathSciNet  Google Scholar 

  18. Liu, Z., Xu, J.: Large time behavior of solutions for density-suppressed motility system in higher dimensions. J. Math. Anal. Appl. 475, 1596–1613 (2019)

    Article  MathSciNet  Google Scholar 

  19. Lv, W.: Global existence for a class of chemotaxis-consumption systems with signal-dependent motility and generalized logistic source. Nonlinear Anal. Real World Appl. 56, 103160 (2020)

    Article  MathSciNet  Google Scholar 

  20. Lv, W., Wang, Q.: An n-dimensional chemotaxis system with signal-dependent motility and generalized logistic source: global existence and asymptotic stabilization. Proc. Roy. Soc. Edinb. A 151, 821–841 (2021)

    Article  MathSciNet  Google Scholar 

  21. Lv, W., Wang, Q.: Global existence for a class of Keller–Segel models with signal-dependent motility and general logistic term. Evol. Equ. Control Theory 10, 25 (2021)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  24. Tao, Y.: Boundedness in a chemotaxis model with oxygen consumption by bacteria. J. Math. Anal. Appl. 381, 521–529 (2011)

    Article  MathSciNet  Google Scholar 

  25. Tao, X., Fang, Z.: Global boundedness and stability in a density-suppressed motility model with generalized logistic source and nonlinear signal production. Z. Angew. Math. Phys. 73, 1–19 (2022)

    Article  MathSciNet  Google Scholar 

  26. Tao, Y., Winkler, M.: Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant. J. Differ. Equ. 252, 2520–2543 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  27. Tao, Y., Winkler, M.: Effects of signal-dependent motilities in a Keller–Segel-type reaction-diffusion system. Math. Mod. Meth. Appl. Sci. 27, 1645–1683 (2017)

    Article  MathSciNet  CAS  Google Scholar 

  28. Tao, Y., Winkler, M.: Global solutions to a Keller–Segel-consumption system involving singularly signal-dependent motilities in domains of arbitrary dimension. J. Differ. Equ. 343, 390–418 (2023)

    Article  ADS  MathSciNet  Google Scholar 

  29. Tello, J., Winkler, M.: A chemotaxis system with logistic source. Commun. Part. Diff. Equ. 32, 849–877 (2007)

    Article  MathSciNet  Google Scholar 

  30. Wang, L.: Improvement of conditions for boundedness in a chemotaxis consumption system with density-dependent motility. Appl. Math. Lett. 125, 107724 (2022)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  32. Winkler, M.: Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source. Commun. Part. Diff. Equ. 35, 1516–1537 (2010)

    Article  MathSciNet  Google Scholar 

  33. Winkler, M.: Global asympototic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening. J. Differ. Equ. 257, 1056–1077 (2014)

    Article  ADS  Google Scholar 

  34. Winkler, M.: Application of the Moser–Trudinger inequality in the construction of global solutions to a strongly degenerate migration model. Bull. Math. Sci. (2023). https://doi.org/10.1142/S1664360722500126

    Article  MathSciNet  Google Scholar 

  35. Winkler, M.: Global generalized solvability in a strongly degenerate taxis-type parabolic system modeling migration-consumption interaction. Z. Angew. Math. Phys. 74, 32 (2023)

    Article  MathSciNet  Google Scholar 

  36. Winkler, M.: Can simultaneous density-determined enhancement of diffusion and cross-diffusion foster boundedness in Keller–Segel type systems involving signal-dependent motilities? Nonlinearity 33, 6590–6623 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  37. Yoon, C., Kim, Y.: Global existence and aggregation in a Keller–Segel model with Fokker–Planck diffusion. Acta Appl. Math. 149, 101–123 (2017)

    Article  MathSciNet  Google Scholar 

  38. Zhang, Q., Li, Y.: Stabilization and convergence rate in a chemotaxis system with consumption of chemoattractant. J. Math. Phys. 56, 081506 (2015)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the editors and reviewers for their helpful and constructive comments.

Author information

Authors and Affiliations

  1. School of Science, Chongqing University of Posts and Telecommunications, Chongqing, 400065, People’s Republic of China

    Ya Tian & Guoqing Xie

Authors
  1. Ya Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guoqing Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

YT provided the main idea of this paper, gave the solution of this paper and made continuous modifications in the subsequent process. GX completed the main manuscript text. All authors reviewed the manuscript.

Corresponding author

Correspondence to Ya Tian.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tian, Y., Xie, G. Global boundedness and large time behavior in a signal-dependent motility system with nonlinear signal consumption. Z. Angew. Math. Phys. 75, 7 (2024). https://doi.org/10.1007/s00033-023-02149-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-023-02149-9

Keywords

Mathematics Subject Classification

Search

Navigation