Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR IN A THREE-DIMENSIONAL TWO-SPECIES CHEMOTAXIS-STOKES SYSTEM WITH TENSOR-VALUED SENSITIVITY

  • Liu, Bin (School of Mathematics and Statistics Huazhong University of Science and Technology) ;
  • Ren, Guoqiang (School of Mathematics and Statistics Huazhong University of Science and Technology)
  • Received : 2019.01.09
  • Accepted : 2019.08.14
  • Published : 2019.12.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we deal with a two-species chemotaxis-Stokes system with Lotka-Volterra competitive kinetics under homogeneous Neumann boundary conditions in a general three-dimensional bounded domain with smooth boundary. Under appropriate regularity assumptions on the initial data, by some Lp-estimate techniques, we show that the system possesses at least one global and bounded weak solution, in addition to discussing the asymptotic behavior of the solutions. Our results generalizes and improves partial previously known ones.

Keywords

Acknowledgement

Supported by : NNSF of China

The authors express their gratitude to the anonymous reviewers and editors for their valuable comments and suggestions which led to the improvement of the original manuscript.

References

  1. X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J. 65 (2016), no. 2, 553-583. https://doi.org/10.1512/iumj.2016.65.5776
  2. T. Black, Global very weak solutions to a chemotaxis-fluid system with nonlinear diffusion, SIAM J. Math. Anal. 50 (2018), no. 4, 4087-4116. https://doi.org/10.1137/17M1159488
  3. N. Bellomo, A. Bellouquid, Y. Tao, and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci. 25 (2015), no. 9, 1663-1763. https://doi.org/10.1142/S021820251550044X
  4. X. Cao, Global classical solutions in chemotaxis(-Navier)-Stokes system with rotational flux term, J. Differential Equations 261 (2016), no. 12, 6883-6914. https://doi.org/10.1016/j.jde.2016.09.007
  5. X. Cao, S. Kurima, and M. Mizukami, Global existence an asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics, Math. Methods Appl. Sci. 41 (2018), no. 8, 3138-3154. https://doi.org/10.1002/mma.4807
  6. X. Cao, S. Kurima, and M. Mizukami, Global existence and asymptotic behavior of classical solutions for a 3D two-species Keller-Segel-Stokes system with competitive kinetics, preprint.
  7. D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, 1981.
  8. M. A. Herrero and J. J. L. Velazquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), no. 4, 633-683.
  9. M. Hirata, S. Kurima, M. Mizukami, and T. Yokota, Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, J. Differential Equations 263 (2017), no. 1, 470-490. https://doi.org/10.1016/j.jde.2017.02.045
  10. M. Hirata, S. Kurima, M. Mizukami, and T. Yokota, Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, Proceedings of EQUADIFF 2017 Conference, (2017) 11-20.
  11. D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations 215 (2005), no. 1, 52-107. https://doi.org/10.1016/j.jde.2004.10.022
  12. S. Ishida, K. Seki, and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations 256 (2014), no. 8, 2993-3010. https://doi.org/10.1016/j.jde.2014.01.028
  13. 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
  14. O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type (Russian), Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, 1968.
  15. H. Li and Y. Tao, Boundedness in a chemotaxis system with indirect signal production and generalized logistic source, Appl. Math. Lett. 77 (2018), 108-113. https://doi.org/10.1016/j.aml.2017.10.006
  16. X. Li, Global classical solutions in a Keller-Segel(-Navier)-Stokes system modeling coral fertilization, J. Differential Equations, In press.
  17. K. Lin, C. Mu, and L. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci. 38 (2015), no. 18, 5085-5096. https://doi.org/10.1002/mma.3429
  18. K. Lin, C. Mu, and H. Zhong, A new approach toward stabilization in a two-species chemotaxis model with logistic source, Comput. Math. Appl. 75 (2018), no. 3, 837-849. https://doi.org/10.1016/j.camwa.2017.10.007
  19. J. Liu and Y. Wang, Boundedness and decay property in a three-dimensional Keller- Segel-Stokes system involving tensor-valued sensitivity with saturation, J. Differential Equations 261 (2016), no. 2, 967-999. https://doi.org/10.1016/j.jde.2016.03.030
  20. J. Liu and Y. Wang, Global existence and boundedness in a Keller-Segel-(Navier-)Stokes system with signal-dependent sensitivity, J. Math. Anal. Appl. 447 (2017), no. 1, 499-528. https://doi.org/10.1016/j.jmaa.2016.10.028
  21. J. Liu and Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations 262 (2017), no. 10, 5271-5305. https://doi.org/10.1016/j.jde.2017.01.024
  22. M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), no. 6, 2301-2319. https://doi.org/10.3934/dcdsb.2017097
  23. M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), no. 6, 2301-2319. https://doi.org/10.3934/dcdsb.2017097
  24. M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion, J. Differential Equations 261 (2016), no. 5, 2650-2669. https://doi.org/10.1016/j.jde.2016.05.008
  25. T. Nagai, T. Senba, and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac. 40 (1997), no. 3, 411-433.
  26. M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal. 46 (2014), no. 6, 3761-3781. https://doi.org/10.1137/140971853
  27. M. Negreanu and J. I. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations 258 (2015), no. 5, 1592-1617. https://doi.org/10.1016/j.jde.2014.11.009
  28. G. Ren and B. Liu, Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals, Nonlinear Anal. Real World Appl. 48 (2019), 288-325. https://doi.org/10.1016/j.nonrwa.2019.01.017
  29. G. Ren and B. Liu, Global existence of bounded solutions for a quasilinear chemotaxis system with logistic source, Nonlinear Anal. Real World Appl. 46 (2019), 545-582. https://doi.org/10.1016/j.nonrwa.2018.09.020
  30. G. Ren and B. Liu, Boundedness of solutions for a quasilinear chemotaxis-haptotaxis model, Hakkaido Mathematical Journal, Accepted.
  31. H. Sohr, The Navier-Stokes equations, Birkhauser Advanced Texts: Basler Lehrbucher., Birkhauser Verlag, Basel, 2001.
  32. C. Stinner, J. I. Tello, and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol. 68 (2014), no. 7, 1607-1626. https://doi.org/10.1007/s00285-013-0681-7
  33. Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Ann. Inst. H. Poincare Anal. Non Lineaire 30 (2013), no. 1, 157-178. https://doi.org/10.1016/j.anihpc.2012.07.002
  34. Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys. 66 (2015), no. 5, 2555-2573. https://doi.org/10.1007/s00033-015-0541-y
  35. Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, Z. Angew. Math. Phys. 67 (2016), no. 6, Art. 138, 23 pp. https://doi.org/10.1007/s00033-016-0732-1
  36. I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler, and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. 102 (2005), 2277-2282. https://doi.org/10.1073/pnas.0406724102
  37. Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes sys- tem with subcritical sensitivity, Math. Models Methods Appl. Sci. 27 (2017), no. 14, 2745-2780. https://doi.org/10.1142/S0218202517500579
  38. Y. Wang, M. Winkler, and Z. Xiang, Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), no. 2, 421-466.
  39. Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations 259 (2015), no. 12, 7578-7609. https://doi.org/10.1016/j.jde.2015.08.027
  40. Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: the 3D case, J. Differential Equations 261 (2016), no. 9, 4944-4973. https://doi.org/10.1016/j.jde.2016.07.010
  41. M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations 248 (2010), no. 12, 2889-2905. https://doi.org/10.1016/j.jde.2010.02.008
  42. M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations 37 (2012), no. 2, 319-351. https://doi.org/10.1080/03605302.2011.591865
  43. M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations 54 (2015), no. 4, 3789-3828. https://doi.org/10.1007/s00526-015-0922-2
  44. M. Winkler, How far do chemotaxis-driven forces in uence regularity in the Navier-Stokes system?, Trans. Amer. Math. Soc. 369 (2017), no. 5, 3067-3125. https://doi.org/10.1090/tran/6733
  45. M. Winkler, Does fluid interaction aect regularity in the three-dimensional Keller-Segel system with saturated sensitivity?, J. Math. Flfluid Mech. 20 (2018), no. 4, 1889-1909. https://doi.org/10.1007/s00021-018-0395-0
  46. M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differential Equations 264 (2018), no. 10, 6109-6151. https://doi.org/10.1016/j.jde.2018.01.027
  47. Q. Zhang and Y. Li, Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys. 66 (2015), no. 1, 83-93. https://doi.org/10.1007/s00033-013-0383-4