Boundedness and Stability in a Chemotaxis-Growth Model with Indirect Attractant Production and Signal-Dependent Sensitivity

Abstract

We study the chemotaxis-growth system with signal-dependent sensitivity function and logistic source

$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} u_{t}=\Delta u-\nabla \cdot \bigl(u\chi (v)\nabla v\bigr)+\mu u(1-u), &x \in \varOmega ,\ t>0, \\ v_{t}=d\Delta v+h(v,w), &x\in \varOmega ,\ t>0, \\ \tau w_{t}=-\delta w+u, &x\in \varOmega ,\ t>0, \\ \end{array}\displaystyle \right . \end{aligned}$$

under homogeneous Neumann boundary conditions in a smooth bounded domain \(\varOmega \subset \mathbb{R}^{n}\ (n\geq 1)\), where the parameters \(\mu , \tau , \delta >0\) and \(d\geq 0\), the functions \(\chi (v)\), \(h(v,w)\) satisfying some conditions represent the chemotactic sensitivity and the balance between the production and degradation of the chemical signal which relies explicitly on the living organisms, respectively. In the case that \(\chi (v)\equiv 1\), \(d=1\) and \(h(v,w)=-v+w\), Hu and Tao (Math. Models Methods Appl. Sci. 26:2111–2128, 2016) asserted global existence of bounded solutions for arbitrary \(\mu >0\) and established asymptotic behavior of solutions to the mentioned system under the condition \(\mu >\frac{1}{8\delta ^{2}}\) in the three dimensional space. The purpose of the present paper is to investigate the global existence and boundedness of classical solutions and to improve the condition assumed in Hu and Tao (Math. Models Methods Appl. Sci. 26:2111–2128, 2016) by extending the previous method for obtaining asymptotic stability. Consequently, the range of \(\mu \) is extended in the present paper.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Ahn, J.: Global well-posedness and asymptotic stabilization for chemotaxis system with signal-dependent sensitivity. J. Differ. Equ. 266(10), 6866–6904 (2019). https://doi.org/10.1016/j.jde.2018.11.015

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Bellomo, N., Belloquid, A., Tao, Y., Winkler, M.: Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. 25, 1663–1763 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

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

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Cao, J., Wang, W., Yu, H.: Asymptotic behavior of solutions to two-dimensional chemotaxis system with logistic source and singular sensitivity. J. Math. Anal. Appl. 436, 382–392 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Chaplain, M.A.J., Lolas, G.: Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity. Netw. Heterog. Media 1, 399–439 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Fontelos, M.A., Friedman, A., Hu, B.: Mathematical analysis of a model for the initiation of angiogenesis. SIAM J. Math. Anal. 33, 1330–1355 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

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

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Fujie, K.: Study of Reaction-Diffusion Systems Modeling Chemotaxis. Doctoral thesis (2016)

  9. 9.

    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 

  10. 10.

    Fujie, K., Senba, T.: Global existence and boundedness of radial solutions to a two dimensional fully parabolic chemotaxis system with general sensitivity. Nonlinearity 29, 2417–2450 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Fujie, K., Yokota, T.: Boundedness in a fully parabolic chemotaxis system with strongly singular sensitivity. Appl. Math. Lett. 38, 140–143 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    He, X., Zheng, S.: Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source. J. Math. Anal. Appl. 436, 970–982 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathe-Matics, vol. 840. Springer, Berlin-New York (1981)

    MATH  Book  Google Scholar 

  14. 14.

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

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Horstmann, D., Winkler, M.: Boundedness vs. blow-up in a chemotaxis system. J. Differ. Equ. 215, 52–107 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Hu, B., Tao, Y.: To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production. Math. Models Methods Appl. Sci. 26, 2111–2128 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Jin, H.Y., Xiang, T.: Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes system with competitive kinetics. Discrete Contin. Dyn. Syst., Ser. B 24(4), 1919–1942 (2019). https://doi.org/10.3934/dcdsb.2018249

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

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

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Lankeit, J.: A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity. Math. Methods Appl. Sci. 39, 394–404 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Li, H., Tao, Y.: Boundedness in a chemotaxis system with indirect signal production and generalized logistic source. Appl. Math. Lett. 77, 108–113 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Maini, P.K., Myerscough, M.R., Winters, K.H., Murray, J.D.: Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation. Bull. Math. Biol. 53, 701–719 (1991)

    MATH  Article  Google Scholar 

  22. 22.

    Mizukami, M.: Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete Contin. Dyn. Syst., Ser. B 22, 2301–2319 (2017)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Mizukami, M., Yokota, T.: Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion. J. Differ. Equ. 261, 2650–2669 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Mizukami, M., Yokota, T.: A unified method for boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity. Math. Nachr. 290, 2648–2660 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Negreanu, M., Tello, J.I.: Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant. J. Differ. Equ. 258, 1592–1617 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

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

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Qiu, S., Mu, C.L., Wang, L.: Boundedness in the higher-dimensional quasilinear chemotaxis-growth system with indirect attractant production. Comput. Appl. Math. 75, 3213–3223 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Stinner, C., Winkler, M.: Global weak solutions in a chemotaxis system with large singular sensitivity. Nonlinear Anal., Real World Appl. 12, 3727–3740 (2011)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Stinner, C., Surulescu, C., Winkler, M.: Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion. SIAM J. Math. Anal. 46, 1969–2007 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Strohm, S., Tyson, R.C., Powell, J.A.: Pattern formation in a model for mountain pine beetle dispersal: linking model predictions to data. Bull. Math. Biol. 75, 1778–1797 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Tao, Y., Winker, M.: Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production (2017). arXiv:1608.07622v2

  32. 32.

    Tao, Y., Winkler, M.: Large time behavior in a multi-dimensional chemotaxis-haptotaxis model with slow signal diffusion. SIAM J. Math. Anal. 47, 4229–4250 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Temam, R.: Infnite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Applied Mathematical Sciences, vol. 68. Springer, New York (1997)

    MATH  Book  Google Scholar 

  34. 34.

    Tindall, M.J., Maini, P.K., Porter, S.L., Armitage, J.P.: Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations. Bull. Math. Biol. 70, 1570–1607 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Winkler, M.: Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity. Math. Nachr. 283, 1664–1673 (2010)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

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

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

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

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Winkler, M.: Chemotactic cross-diffusion in complex frameworks. Math. Models Methods Appl. Sci. 26, 2035–2040 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Winkler, M., Yokota, T.: Stabilization in the logarithmic Keller-Segel system. Nonlinear Anal. 170, 123–141 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Zhao, X., Zheng, S.: Global boundedness to a chemotaxis system with singular and logistic source. Z. Angew. Math. Phys. 68(2), 826–865 (2019)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Zheng, P., Mu, C.L., Hu, X.: Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source. Discrete Contin. Dyn. Syst., Ser. A 35, 2299–2323 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Zheng, P., Mu, C.L., Wang, L., Li, L.: Boundedness and asymptotic behavior in a fully parabolic chemotaxis-growth system with signal-dependent sensitivity. J. Evol. Equ. 17, 909–929 (2017)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

The second author is supported by NSFC (Grant Nos. 11571062 and 11771062), the Fundamental Research Funds for the Central Universities (Grant Nos. 106112016CDJXZ238826 and 2019CDJCYJ001).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Shuyan Qiu.

Additional information

Publisher’s Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Qiu, S., Mu, C. & Li, Y. Boundedness and Stability in a Chemotaxis-Growth Model with Indirect Attractant Production and Signal-Dependent Sensitivity. Acta Appl Math 169, 341–360 (2020). https://doi.org/10.1007/s10440-019-00301-0

Download citation

Keywords

  • Chemotaxis
  • Boundedness
  • Indirect attractant production
  • Logistic growth
  • Asymptotic behavior

Mathematics Subject Classification

  • 35A01
  • 92C17
  • 35B45
  • 35B40
  • 35K57
  • 35Q92