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Boundedness and Exponential Stabilization in a Parabolic-Elliptic Keller—Segel Model with Signal-Dependent Motilities for Local Sensing Chemotaxis

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Abstract

In this paper we consider the initial Neumann boundary value problem for a degenerate Keller—Segel model which features a signal-dependent non-increasing motility function. The main obstacle of analysis comes from the possible degeneracy when the signal concentration becomes unbounded. In the current work, we are interested in the boundedness and exponential stability of the classical solution in higher dimensions. With the aid of a Lyapunov functional and a delicate Alikakos—Moser type iteration, we are able to establish a time-independent upper bound of the concentration provided that the motility function decreases algebraically. Then we further prove the uniform-in-time boundedness of the solution by constructing an estimation involving a weighted energy. Finally, thanks to the Lyapunov functional again, we prove the exponential stabilization toward the spatially homogeneous steady states. Our boundedness result improves those in [1] and the exponential stabilization is obtained for the first time.

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References

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

    Article  MathSciNet  Google Scholar 

  2. Keller E F, Segel L A. Model for chemotaxis[J]. J Theoret Biol, 1971, 30: 225–234

    Article  Google Scholar 

  3. Fu X, Huang L H, Liu C, et al. Stripe formation in bacterial systems with density-suppressed motility[J]. Phys Rev Lett, 2012, 108: 198102

    Article  Google Scholar 

  4. Liu C L, Fu X F, Liu L Z, et al. Sequential establishment of stripe patterns in an expanding cell population[J]. Science, 2011, 334: 238

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Lv W, Yuan Q. Global existence for a class of chemotaxis systems with signal-dependent motility, indirect signal production and generalized logistic source[J]. Z Angew Math Phys, 2020, 71: 53

    Article  MathSciNet  Google Scholar 

  7. Wang J, Wang M. Boundedness in the higher-dimensional Keller-Segel model with signal-dependent motility and logistic growth[J]. J Math Phys, 2019, 60: 011507

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Burger M, Laurençot Ph, Trescases A. Delayed blow-up for chemotaxis models with local sensing[J]. J London Math Soc, 2020. doi:https://doi.org/10.1112/jlms.12420

  11. Fujie K, Jiang J. Global existence for a kinetic model of pattern formation with density-suppressed motilities[J]. J Differential Equations, 2020, 269: 5338–5778

    Article  MathSciNet  Google Scholar 

  12. Fujie K, Jiang J. Comparison methods for a Keller—Segel model of pattern formations with signal-dependent motilities[J]. Calc Var Partial Differential Equations, 2021, 60: 92

    Article  Google Scholar 

  13. Fujie K, Jiang J. Boundedness of Classical Solutions to a Degenerate Keller—Segel Type Model with Signal-dependent Motilities[J]. Acta Applicandae Mathematicae, 2021, 176: 3

    Article  MathSciNet  Google Scholar 

  14. Li H, Jiang J. Global Existence of Weak Solutions to a Signal-dependent Keller-Segel Model for Local Sensing Chemotaxis[J]. Nonlinear Analysis: Real World Applications, 2021, 61: 103338

    Article  MathSciNet  Google Scholar 

  15. Jin H Y, Wang Z A. Critical mass on the Keller—Segel system with signal-dependent motility[J]. Proc Amer Math Soc, 2020, 148: 4855–4873

    Article  MathSciNet  Google Scholar 

  16. Jin H Y, Wang Z A. The Keller-Segel system with logistic growth and signal-dependent motility[J]. Discrete Contin Dyn Syst Ser B, 2021, 26: 3023–3041

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Ma M, Peng R, Wang Z. Stationary and non-stationary patterns of the density-suppressed motility model[J]. Physica D, 2020, 402: 132259

    Article  MathSciNet  Google Scholar 

  19. Wang Z A. On the parabolic-elliptic Keller-Segel system with signal-dependent motilities: a paradigm for global boundedness[J]. Math Meth Appl Sci, 2021, 44: 10881–10898

    Article  MathSciNet  Google Scholar 

  20. Zheng J, Wang Z. Global Boundedness of the Fully Parabolic Keller-Segel System with Signal-Dependent Motilities[J]. Acta Appl Math, 2021, 171: 25

    Article  MathSciNet  Google Scholar 

  21. Nagai T, Senba T. Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis[J]. Adv Math Sci Appl, 1998, 8: 145–156

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Stinner C, Winkler M. Global weak solutions in a chemotaxis system with large singular sensitivity[J]. Nonlinear Anal, 2011, 12: 3727–3740

    MathSciNet  MATH  Google Scholar 

  24. Winkler M, Yokota T. Stabilization in the logarithmic Keller-Segel system[J]. Nonlinear Anal Theor Meth Appl, 2018, 170: 123–141

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Fujie K, Senba T. A sufficient condition of sensitivity functions for boundedness of solutions to a parabolic-parabolic chemotaxis system[J]. Nonlinearity, 2018, 31: 1639–1672

    Article  MathSciNet  Google Scholar 

  27. Lankeit L, Winkler M. A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: global solvability for large nonradial data[J]. NoDEA Nonlinear Differential Equations Appl, 2017, 24: 49

    Article  MathSciNet  Google Scholar 

  28. Cao X. Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces[J]. Discrete Contin Dynam Syst Ser A, 2015, 35: 1891–1904

    Article  MathSciNet  Google Scholar 

  29. Winkler M. Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model[J]. J Different Equ, 2010, 248: 2889–2905

    Article  MathSciNet  Google Scholar 

  30. Alikakos N D. An application of the invariance principle to reaction-diffusion equations[J]. J Diff Equ, 1979, 33: 201–225

    Article  MathSciNet  Google Scholar 

  31. Black T. Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity[J]. Discrete Contin Dyn Syst Ser S, 2020, 13: 119–137

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Innovation Academy for Precision Measurement Science and Technology, CAS, Wuhan, 430071, China

    Jie Jiang  (江杰)

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  1. Jie Jiang  (江杰)
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Correspondence to Jie Jiang  (江杰).

Additional information

This work was supported by Hubei Provincial Natural Science Foundation (2020CFB602).

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Jiang, J. Boundedness and Exponential Stabilization in a Parabolic-Elliptic Keller—Segel Model with Signal-Dependent Motilities for Local Sensing Chemotaxis. Acta Math Sci 42, 825–846 (2022). https://doi.org/10.1007/s10473-022-0301-y

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  • DOI: https://doi.org/10.1007/s10473-022-0301-y

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