Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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LARGE TIME CONVERGENCE FOR A CHEMOTAXIS MODEL WITH DEGENERATE LOCAL SENSING AND CONSUMPTION

  • Philippe Laurencot (Laboratoire de Mathematiques (LAMA) UMR 5127 Universite Savoie Mont Blanc, CNRS)
  • Received : 2023.03.24
  • Accepted : 2023.05.09
  • Published : 2024.03.31
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Abstract

Convergence to a steady state in the long term limit is established for global weak solutions to a chemotaxis model with degenerate local sensing and consumption, when the motility function is C1-smooth on [0, ∞), vanishes at zero, and is positive on (0, ∞). A condition excluding that the large time limit is spatially homogeneous is also provided. These results extend previous ones derived for motility functions vanishing algebraically at zero and rely on a completely different approach.

Keywords

Acknowledgement

Enlightening (electronic) discussions with Michael Winkler on the topic studied in this paper are gratefully acknowledged. I also thank the referee for helpful remarks. Part of this work was done while enjoying the kind hospitality of the Department of Mathematics, Indian Institute of Technology Roorkee.

References

  1. M. Burger, Ph. Lauren,cot, and A. Trescases, Delayed blow-up for chemotaxis models with local sensing, J. Lond. Math. Soc. (2) 103 (2021), no. 4, 1596-1617. https://doi.org/10.1112/jlms.12420 
  2. L. Desvillettes, Y. Kim, A. Trescases, and C. Yoon, A logarithmic chemotaxis model featuring global existence and aggregation, Nonlinear Anal. Real World Appl. 50 (2019), 562-582. https://doi.org/10.1016/j.nonrwa.2019.05.010 
  3. L. Desvillettes, Ph. Lauren,cot, A. Trescases, and M. Winkler, Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing, Nonlinear Anal. 226 (2023), Paper No. 113153, 26 pp. https://doi.org/10.1016/j.na.2022.113153 
  4. K. Fujie and J. Jiang, Boundedness of classical solutions to a degenerate Keller-Segel type model with signal-dependent motilities, Acta Appl. Math. 176 (2021), Paper No. 3, 36 pp. https://doi.org/10.1007/s10440-021-00450-1 
  5. K. Fujie and J. Jiang, Comparison methods for a Keller-Segel-type model of pattern formations with density-suppressed motilities, Calc. Var. Partial Differential Equations 60 (2021), no. 3, Paper No. 92, 37 pp. https://doi.org/10.1007/s00526-021-01943-5 
  6. K. Fujie and J. Jiang, A note on construction of nonnegative initial data inducing unbounded solutions to some two-dimensional Keller-Segel systems, Math. Eng. 4 (2022), no. 6, Paper No. 045, 12 pp. https://doi.org/10.3934/mine.2022045 
  7. K. Fujie and T. Senba, Global boundedness of solutions to a parabolic-parabolic chemotaxis system with local sensing in higher dimensions, Nonlinearity 35 (2022), no. 7, 3777-3811. https://doi.org/10.1088/1361-6544/ac6659 
  8. K. Fujie and T. Senba, Global existence and infinite time blow-up of classical solutions to chemotaxis systems of local sensing in higher dimensions, Nonlinear Anal. 222 (2022), Paper No. 112987, 7 pp. https://doi.org/10.1016/j.na.2022.112987 
  9. J. Jiang, Ph. Lauren,cot, and Y. Zhang, Global existence, uniform boundedness, and stabilization in a chemotaxis system with density-suppressed motility and nutrient consumption, Comm. Partial Differential Equations 47 (2022), no. 5, 1024-1069. https://doi.org/10.1080/03605302.2021.2021422 
  10. H.-Y. Jin and Z.-A. Wang, Critical mass on the Keller-Segel system with signal-dependent motility, Proc. Amer. Math. Soc. 148 (2020), no. 11, 4855-4873. https://doi.org/10.1090/proc/15124 
  11. E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (1970), no. 3, 399-415. https://doi.org/10.1016/0022-5193(70)90092-5 
  12. E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: a theoretical analysis, J. Theoret. Biol. 30 (1971), 235-248. https://doi.org/10.1016/0022-5193(71)90051-8 
  13. Ph. Lauren,cot, Long term spatial homogeneity for a chemotaxis model with local sensing and consumption, Commun. Math. Sci. 21 (2023), no. 6, 1743-1750. https://dx.doi.org/10.4310/CMS.2023.v21.n6.a14 
  14. H. Li and J. Jiang, Global existence of weak solutions to a signal-dependent Keller-Segel model for local sensing chemotaxis, Nonlinear Anal. Real World Appl. 61 (2021), Paper No. 103338, 14 pp. https://doi.org/10.1016/j.nonrwa.2021.103338 
  15. G. Li and M. Winkler, Relaxation in a Keller-Segel-consumption system involving signal-dependent motilities, Commun. Math. Sci. 21 (2023), no. 2, 299-322. https://dx.doi.org/10.4310/CMS.2023.v21.n2.a1 
  16. G. Li and M. Winkler, Refined regularity analysis for a Keller-Segel-consumption system involving signal-dependent motilities, Applicable Anal. 103 (2024), no. 1, 45-64. https://doi.org/10.1080/00036811.2023.2173183 
  17. D. Li and J. Zhao, Global boundedness and large time behavior of solutions to a chemotaxis-consumption system with signal-dependent motility, Z. Angew. Math. Phys. 72 (2021), no. 2, Paper No. 57, 20 pp. https://doi.org/10.1007/s00033-021-01493-y 
  18. C.-S. Lin, W.-M. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988), no. 1, 1-27. https://doi.org/10.1016/0022-0396(88)90147-7 
  19. Y. Tao and M. Winkler, Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Methods Appl. Sci. 27 (2017), no. 9, 1645-1683. https://doi.org/10.1142/S0218202517500282 
  20. Z.-A. Wang and X. Xu, Steady states and pattern formation of the density-suppressed motility model, IMA J. Appl. Math. 86 (2021), no. 3, 577-603. https://doi.org/10.1093/imamat/hxab006 
  21. M. Winkler, Application of the Moser-Trudinger inequality in the construction of global solutions to a strongly degenerate migration model, Bull. Math. Sci. 13 (2023), no. 2, Paper No. 2250012, 16pp. https://dx.doi.org/10.1142/S1664360722500126 
  22. M. Winkler, A quantitative strong parabolic maximum principle and application to a taxis-type migration-consumption model involving signal-dependent degenerate diffusion, Ann. Inst. H. Poincare Anal. Non Lineaire 41 (2024), no. 1, 95-127. https://doi.org/10.4171/aihpc/73 
  23. M. Winkler, Global generalized solvability in a strongly degenerate taxis-type parabolic system modeling migration-consumption interaction, Z. Angew. Math. Phys. 74 (2023), no. 1, Paper No. 32, 20 pp. https://doi.org/10.1007/s00033-022-01925-3 
  24. M. Winkler, Stabilization despite pervasive strong cross-degeneracies in a nonlinear diffusion model for migration-consumption interaction, Nonlinearity 36 (2023), no. 8, 4438-4469. https://doi.org/10.1088/1361-6544/ace22e 
  25. Y. Xiao and J. Jiang, Global existence and uniform boundedness in a fully parabolic Keller-Segel system with non-monotonic signal-dependent motility, J. Differential Equations 354 (2023), 403-429. https://doi.org/10.1016/j.jde.2023.02.028 
  26. C. Yoon and Y.-J. Kim, Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Appl. Math. 149 (2017), 101-123. https://doi.org/10.1007/s10440-016-0089-7