Global boundedness and large time behavior of solutions to a chemotaxis–consumption system with signal-dependent motility

Abstract

This paper deals with the following chemotaxis–consumption system with signal-dependent motility

$$\begin{aligned} \left\{ \begin{array}{llll} u_{t}=\Delta (\gamma (v)u),\,\,\, &{}x\in \Omega ,\,\,\, t>0,\\ v_{t}=\Delta v-uv, \,\,\,&{}x\in \Omega ,\,\,\, t>0,\\ \end{array} \right. \end{aligned}$$

under no-flux boundary conditions in a smoothly bounded domain \(\Omega \subset {\mathbb {R}}^{n}\). \(\gamma (s)\) is the motility function. For the case of positive motility function, it is shown that the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded. Moreover, it is asserted that the solution to the system exponentially converges to constant equilibria in the large time. Finally, if the motility function is zero at some point, we obtain the existence of the weak solution.

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

References

  1. 1.

    Ahn, J., Yoon, C.: Global well-posedness and stability of constant equilibria in parabolic–elliptic chemotaxis systems without gradient sensing. Nonlinearity 32, 1327–1351 (2019)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Cieálak, T., Stinner, C.: Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions. J. Differ. Equ. 252, 5832–5851 (2012)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Fujie, K., Jiang, J.: Global existence for a kinetic model of pattern formation with density-suppressed motilities. J. Differ. Equ. 269, 5338–5378 (2020)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Friedman, A., Tello, J.I.: Stability of solutions of chemotaxis equations in reinforced random walks. J. Math. Anal. Appl. 272, 138–163 (2002)

    MathSciNet  Article  Google Scholar 

  5. 5.

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

    MathSciNet  Article  Google Scholar 

  6. 6.

    Ishida, S., Seki, K., Yokota, T.: Boundedness in quasilinear Keller–Segel system of parabolic-parabolic type on nonconvex bounded domains. J. Differ. Equ. 256, 2993–3010 (2014)

    Article  Google Scholar 

  7. 7.

    Jin, H.Y., Kim, Y.J., Wang, Z.A.: Boundedness, stabilization, and pattern formation driven by density-suppressed motility. SIAM J. Appl. Math. 78(3), 1632–1657 (2018)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Jin, H.Y., Shi, S.J., Wang, Z.A.: Boundedness and asymptotics of a reaction–diffusion system with density-dependent motility. J. Differ. Equ. 269, 6758–6793 (2020)

    MathSciNet  Article  Google Scholar 

  9. 9.

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

    MathSciNet  Article  Google Scholar 

  10. 10.

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

    MathSciNet  Article  Google Scholar 

  11. 11.

    Lv, W.B., Wang, Q.Y.: Global existence for a class of chemotaxis systems with signal dependent motility, indirect signal production and generalized logistic source. Z. Angew. Math. Phys. 71, 53 (2020)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Lv, W.B., Wang, Q.Y.: An n-dimensional chemotaxis system with signal-dependent motility and generalized logistic source: global existence and asymptotic stabilization. Proc. R. Soc. Edinb. A (2020). https://doi.org/10.1017/prm.2020.38

    Article  MATH  Google Scholar 

  13. 13.

    Lv, W.B., Wang, Q.Y.: Global existence for a class of Keller–Segel model with signal dependent motility and general logistic term. Evol. Equ. Control Theory (2019). https://doi.org/10.3934/eect.2020040

    Article  Google Scholar 

  14. 14.

    Lv, W.B., Wang, Q.: A chemotaxis system with signal-dependent motility, indirect signal production and generalized logistic source: global existence and asymptotic stabilization. J. Math. Anal. Appl. 488(2), 124108 (2020)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Ladyzenskaja, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type, vol. 23. American Mathematical Society, Providence (1968)

    Google Scholar 

  16. 16.

    Nagai, T.: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5, 581–601 (1995)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Nagai, T.: Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains. J. Inequal. Appl. 6, 37–55 (2001)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    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  MATH  Google Scholar 

  19. 19.

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

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Porzio, M.M., Vespri, V.: Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations. J. Differ. Equ. 103, 146–178 (1993)

    Article  Google Scholar 

  21. 21.

    Tao, Y.S., Wang, Z.A.: Competing effects of attraction vs. repulsion in chemotaxis. Math. Models Methods Appl. Sci. 23, 1–36 (2013)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Tao, Y.S., Winkler, M.: Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subscritical sensitivity. J. Differ. Equ. 252, 692–715 (2012)

    Article  Google Scholar 

  23. 23.

    Tao, Y.S., Winkler, M.: Effects of signal-dependent motilities in a Keller–Segel-type reaction–diffusion system. Math. Models Methods Appl. Sci. 27(9), 1645–1683 (2017)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Tao, Y.S.: Boundedness in a chemotaxis model with oxygen consumption by bacteria. J. Math. Anal. Appl. 381(2), 521–529 (2011)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Tao, Y.S., Winkler, M.: Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant. J. Differ. Equ. 252(3), 2520–2543 (2012)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Temam, R.: Navier–Stokes Equations. Theory and Numerical Analysis. Studies in Applied Mathematics, vol. 2. North-Holland, Amsterdam (1977)

    Google Scholar 

  27. 27.

    Wang, J.P., Wang, M.X.: Boundedness in the higher-dimensional Keller–Segel model with signal-dependent motility and logistic growth. J. Math. Phys. 60(1), 011507 (2019)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Wang, Y., Winkler, M., Xiang, Z.: The small-convection limit in a two-dimensional chemotaxis-Navier–Stokes system. Math. Z. 289, 71–108 (2018)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Winkler, M.: Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system. J. Math. Pures Appl. 100(9), 748–767 (2013)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Winkler, M.: Asymptotic homogenization in a three-dimensional nutrient taxis system involving food-supported proliferation. J. Differ. Equ. 263(8), 4826–4869 (2017)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Winkler, M.: Renormalized radial large-data solutions to the higher-dimensional Keller–Segel system with singular sensitivity and signal absorption. J. Differ. Equ. 264(3), 2310–2350 (2018)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Winkler, M.: Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops. Commun. Partial Differ. Equ. 37(2), 319–351 (2012)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Winkler, M.: Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement. J. Differ. Equ. 264(10), 6109–6151 (2018)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Winkler, M.: A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: global weak solutions and asymptotic stabilization. J. Funct. Anal. 276(5), 1339–1401 (2019)

    MathSciNet  Article  Google Scholar 

  35. 35.

    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  Article  Google Scholar 

  36. 36.

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

    MathSciNet  Article  Google Scholar 

  37. 37.

    Zhao, J.: Large time behavior of solution to quasilinear chemotaxis system with logistic source. Discrete Contin. Dyn. Syst. Ser. A 40, 1737–1755 (2020)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

This work is supported by Doctoral Start-up Fund of CWNU (No. 18Q058) and (No. 19B044). Research and innovation Team of China West Normal University (CXTD2020-5).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jie Zhao.

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

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, 57 (2021). https://doi.org/10.1007/s00033-021-01493-y

Download citation

Keywords

  • Chemotaxis
  • Boundedness
  • Signal-dependent motility

Mathematics Subject Classification

  • 92C17
  • 35K55
  • 35Q92