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Mathematical Research for Models Which is Related to Chemotaxis System

  • Jiashan Zheng
Chapter
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Abstract

This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the Chemotaxis system and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow-up of solutions. The presentation is organized in six parts. The first part focuses on background of the models which is related to Chemotaxis system and its development. The second–five part are devoted to the qualitative analysis of the (quasilinear) Keller–Segel model, the (quasilinear) chemotaxis–haptotaxis model, the (quasilinear) chemotaxis system with consumption of chemoattractant, and the (quasilinear) Keller–Segel–Navier–Stokes system. Finally, an overview of the entire contents leads to suggestions for future research activities.

Keywords

Boundedness Navier–Stokes system Keller–Segel model Chemotaxis models Chemotaxis-haptotaxis model Global existence Nonlinear diffusion 

Notes

Acknowledgements

This work is partially supported by the Shandong Provincial Science Foundation for Outstanding Youth (No. ZR2018JL005), the National Natural Science Foundation of China (No. 11601215), the Natural Science Foundation of Shandong Province of China (No. ZR2016AQ17), and the Doctor Start-up Funding of Ludong University (No. LA2016006).

References

  1. 1.
    N.D. Alikakos, L p bounds of solutions of reaction-diffusion equations. Commun. Partial Differ. Equ. 4, 827–868 (1979)MathSciNetzbMATHGoogle Scholar
  2. 2.
    X. Bai, M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics. Ind. Univ. Math. J. 65, 553–583 (2016)MathSciNetzbMATHGoogle Scholar
  3. 3.
    N. Bellomo, A. Belloquid, Y. Tao, M. Winkler, Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. 25(9), 1663–1763 (2015)MathSciNetzbMATHGoogle Scholar
  4. 4.
    T. Black, Sublinear signal production in a two-dimensional Keller–Segel–Stokes system. Nonlinear Anal. Real World Appl. 31, 593–609 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    M. Burger, M. Di Francesco, Y. Dolak-Struss, The Keller–Segel model for chemotaxis with prevention of overcrowding: linear vs nonlinear diffusion. SIAM J. Math. Anal. 38, 1288–1315 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    V. Calvez, J.A. Carrillo, Volume effects in the Keller–Segel model: energy estimates preventing blow-up. J. Math. Pures Appl. 9(86), 155–175 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    X. Cao, Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with logistic source. J. Math. Anal. Appl. 412, 181–188 (2014)MathSciNetzbMATHGoogle Scholar
  8. 8.
    X. Cao, Boundedness in a three-dimensional chemotaxis–haptotaxis model. Z. Angew. Math. Phys. 67(1), 1–13 (2016)MathSciNetzbMATHGoogle Scholar
  9. 9.
    X. Cao, S. Zheng, Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source. Math. Methods Appl. Sci. 37, 2326–2330 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    M. Chae, K. Kang, J. Lee, Global existence and temporal decay in Keller–Segel models coupled to fluid equations. Commun. Partial Differ. Equ. 39, 1205–1235 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    M.A.J. Chaplain, A.R.A. Anderson, Mathematical modelling of tissue invasion, in Cancer Modelling and Simulation, ed. by L. Preziosi (Chapman Hall/CRC, Boca Raton, 2003), pp. 269–297Google Scholar
  12. 12.
    M.A.J. Chaplain, G. Lolas, Mathematical modelling of cancer invasion of tissue: the role of the urokinase plasminogen activation system. Math. Models Methods Appl. Sci. 11, 1685–1734 (2005)MathSciNetzbMATHGoogle Scholar
  13. 13.
    M.A.J. Chaplain, G. Lolas, Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity. Netw. Heterog. Media 1, 399–439 (2006)MathSciNetzbMATHGoogle Scholar
  14. 14.
    T. Cieślak, P. Laurençot, Finite time blow-up for a one-dimensional quasilinear parabolic–parabolic chemotaxis system. Ann. Inst. Henri Poincaré Non Linear Anal. 27, 437–446 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    T. Cieślak, C. Stinner, 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)MathSciNetzbMATHGoogle Scholar
  16. 16.
    T. Cieślak, C. Stinner, Finite-time blow-up in a supercritical quasilinear parabolic–parabolic Keller–Segel system in dimension 2. Acta Appl. Math. 129, 135–146 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    T. Cieślak, C. Stinner, New critical exponents in a fully parabolic quasilinear Keller–Segel system and applications to volume filling models. J. Differ. Equ. 258, 2080–2113 (2015)MathSciNetzbMATHGoogle Scholar
  18. 18.
    T. Cieślak, M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis. Nonlinearity 21, 1057–1076 (2008)MathSciNetzbMATHGoogle Scholar
  19. 19.
    L. Corrias, B. Perthame, H. Zaag, A chemotaxis model motivated by angiogenesis. C. R. Math. 336, 141–146 (2003)MathSciNetzbMATHGoogle Scholar
  20. 20.
    L. Corrias, B. Perthame, H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Milan J. Math. 72, 1–28 (2004)MathSciNetzbMATHGoogle Scholar
  21. 21.
    M. Di Francesco, A. Lorz, P. Markowich, Chemotaxis–fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior. Discrete Contin. Dyn. Syst. 28, 1437–1453 (2010)MathSciNetzbMATHGoogle Scholar
  22. 22.
    C. Dombrowski, L. Cisneros, S. Chatkaew, R.E. Goldstein, J.O. Kessler, Self-concentration and large-scale coherence in bacterial dynamics. Phys. Rev. Lett. 93(9), 098103 (2004)Google Scholar
  23. 23.
    R. Duan, Z.Y. Xiang, A note on global existence for the chemotaxis–Stokes model with nonlinear diffusion. Int. Math. Res. Not. 7, 1833–1852 (2014)MathSciNetzbMATHGoogle Scholar
  24. 24.
    R. Duan, A. Lorz, Markowich, P.A.: Global solutions to the coupled chemotaxis-fluid equations. Commun. Partial Differ. Equ. 35, 1635–1673 (2010)Google Scholar
  25. 25.
    H. Gajewski, K. Zacharias, Global behavior of a reaction-diffusion system modeling chemotaxis. Math. Nachr. 195, 177–194 (1998)zbMATHGoogle Scholar
  26. 26.
    Y. Giga, Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier–Stokes system. J. Differ. Equ. 61, 186–212 (1986)MathSciNetzbMATHGoogle Scholar
  27. 27.
    D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edn. (Springer, Berlin, 1983)zbMATHGoogle Scholar
  28. 28.
    H. Hajaiej, L. Molinet, T. Ozawa, B. Wang, Necessary and sufficient conditions for the fractional Gagliardo–Nirenberg inequalities and applications to Navier–Stokes and generalized boson equations, in Harmonic Analysis and Nonlinear Partial Differential Equations, RIMS Kôkyûroku Bessatsu, vol. B26 (Research Institute for Mathematical Sciences (RIMS), Kyoto, 2011), pp. 159–175Google Scholar
  29. 29.
    D.D. Haroske, H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations (European Mathematical Society, Zurich, 2008)Google Scholar
  30. 30.
    M.A. Herrero, J.J.L. Velázquez, Singularity patterns in a chemotaxis model. Math. Ann. 306, 583–623 (1996)MathSciNetzbMATHGoogle Scholar
  31. 31.
    M. Herrero, J. Velázquez, Chemotactic collapse for the Keller–Segel model. J. Math. Biol. 35, 177–194 (1996)MathSciNetzbMATHGoogle Scholar
  32. 32.
    M. Herrero, J. Velázquez, A blow-up mechanism for a chemotaxis model. Ann. Scuola Norm. Super. Pisa Cl. Sci. 24(4), 633–683 (1997)MathSciNetzbMATHGoogle Scholar
  33. 33.
    M. Hieber, J. Prüss, Heat kernels and maximal L p-L q estimate for parabolic evolution equations. Commun. Partial Differ. Equ. 22, 1647–1669 (1997)MathSciNetzbMATHGoogle Scholar
  34. 34.
    T. Hillen, K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding. Adv. Appl. Math. 26, 281–301 (2001)MathSciNetzbMATHGoogle Scholar
  35. 35.
    T. Hillen, K.J. Painter, A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58, 183–217 (2009)MathSciNetzbMATHGoogle Scholar
  36. 36.
    T. Hillen, K.J. Painter, M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model. Math. Models Methods Appl. Sci. 23, 165–198 (2013)MathSciNetzbMATHGoogle Scholar
  37. 37.
    D. Horstmann, From 1970 until present: the Keller–Segel model in chemotaxis and its consequences, I. Jahresberichte der Deutschen Mathematiker-Vereinigung 105, 103–165 (2003)MathSciNetzbMATHGoogle Scholar
  38. 38.
    D. Horstmann, G. Wang, Blow-up in a chemotaxis model without symmetry assumptions. Eur. J. Appl. Math. 12, 159–177 (2001)MathSciNetzbMATHGoogle Scholar
  39. 39.
    D. Horstmann, M. Winkler, Boundedness vs. blow-up in a chemotaxis system. J. Differ. Equ. 215, 52–107 (2005)Google Scholar
  40. 40.
    S. Ishida, Global existence and boundedness for chemotaxis–Navier–Stokes system with position-dependent sensitivity in 2d bounded domains. Discrete Contin. Dyn. Syst. Ser. A 32, 3463–3482 (2015)MathSciNetzbMATHGoogle Scholar
  41. 41.
    S. Ishida, K. Seki, T, Yokota, Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains. J. Differ. Equ. 256, 2993–3010 (2014)Google Scholar
  42. 42.
    W. Jäger, S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Am. Math. Soc. 329, 819–824 (1992)MathSciNetzbMATHGoogle Scholar
  43. 43.
    E. Keller, L. Segel, Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1970)zbMATHGoogle Scholar
  44. 44.
    E. Keller, L. Segel, Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)MathSciNetzbMATHGoogle Scholar
  45. 45.
    E. Keller, L. Segel, Traveling bands of chemotactic bacteria: a theoretical analysis. J. Theor. Biol. 30, 377–380 (1971)zbMATHGoogle Scholar
  46. 46.
    A. Kiselev, L. Ryzhik, Biomixing by chemotaxis and enhancement of biological reactions. Commun. Partial Differ Equ. 37(2), 298–318 (2012)MathSciNetzbMATHGoogle Scholar
  47. 47.
    R. Kowalczyk, Preventing blow-up in a chemotaxis model. J. Math. Anal. Appl. 305, 566–585 (2005)MathSciNetzbMATHGoogle Scholar
  48. 48.
    O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’eva, Linear and Quasi-Linear Equations of Parabolic Type. American Mathematical Society Translation, vol. 23 (AMS, Providence, 1968)Google Scholar
  49. 49.
    J. Lankeit, Eventual smoothness and asymptotics in a three dimensional chemotaxis system with logistic source. J. Differ. Equ. 258(4), 1158–1191 (2015)MathSciNetzbMATHGoogle Scholar
  50. 50.
    T. Li, A. Suen, C. Xue, M. Winkler, Global small-data solutions of a two-dimensional chemotaxis system with rotational ux term. Math. Models Methods Appl. Sci. 25, 721–746 (2015)MathSciNetzbMATHGoogle Scholar
  51. 51.
    X. Li, Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete Contin. Dyn. Syst. 35, 3503–3531 (2015)MathSciNetzbMATHGoogle Scholar
  52. 52.
    X. Li, Y. Wang, Z. Xiang, Global existence and boundedness in a 2D Keller–Segel-Stokes system with nonlinear diffusion and rotational flux. Commun. Math. Sci. 14, 1889–1910 (2016)MathSciNetzbMATHGoogle Scholar
  53. 53.
    G. Liţanu, C. Morales-Rodrigo, Asymptotic behavior of global solutions to a model of cell invasion. Math. Models Methods Appl. Sci. 20, 1721–1758 (2010)MathSciNetzbMATHGoogle Scholar
  54. 54.
    J.-G. Liu, A. Lorz, A coupled chemotaxis–fluid model: global existence. Ann. Inst. Henri Poincaré Anal. Non Linéaire 28(5), 643–652 (2011)MathSciNetzbMATHGoogle Scholar
  55. 55.
    J. Liu, J. Zheng, Y. Wang, Boundedness in a quasilinear chemotaxis-haptotaxis system with logistic source. Z. Angew. Math. Phys. 67(2), 1–33 (2016)MathSciNetzbMATHGoogle Scholar
  56. 56.
    A. Lorz, Coupled chemotaxis fluid model. Math. Models Methods Appl. Sci. 20, 987–1004 (2010)MathSciNetzbMATHGoogle Scholar
  57. 57.
    A. Marciniak-Czochra, M. Ptashnyk, Boundedness of solutions of a haptotaxis model. Math. Models Methods Appl. Sci. 20, 449–476 (2010)MathSciNetzbMATHGoogle Scholar
  58. 58.
    G. Meral, C. Stinner, C. Surulescu, On a multiscale model involving cell contractivity and its effects on tumor invasion. Discrete Contin. Dyn. Syst. Ser. B 20, 189–213 (2015)MathSciNetzbMATHGoogle Scholar
  59. 59.
    M. Mizukami, Remarks on smallness of chemotactic effect for asymptotic stability in a two-species chemotaxis system. AIMS Math. 1(3), 156–164 (2016)zbMATHGoogle Scholar
  60. 60.
    N. Mizoguchi, P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller–Segel system. Ann. Inst. Henri Poincaré Non Linéaire 31(4), 851–875 (2014)MathSciNetzbMATHGoogle Scholar
  61. 61.
    M. Mizukami, T. Yokota, Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion. J. Differ. Equ. 261, 2650–2669 (2016)MathSciNetzbMATHGoogle Scholar
  62. 62.
    J. Murray, Mathematical Biology: II. Spatial Models and Biomedical Applications, 3rd edn. (Springer, New York, 2003)Google Scholar
  63. 63.
    T. Nagai, T. Senba, K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis. Funkc. Ekvacioj Ser. Int. 40, 411–433 (1997)MathSciNetzbMATHGoogle Scholar
  64. 64.
    K. Osaki, A. Yagi, Finite dimensional attractor for one-dimensional Keller–Segel equations. Funkcial. Ekvac. 44, 441–469 (2001)MathSciNetzbMATHGoogle Scholar
  65. 65.
    K. Osaki, T. Tsujikawa, A. Yagi, M. Mimura, Exponential attractor for a chemotaxis growth system of equations. Nonlinear Anal. Theory Methods Appl. 51, 119–144 (2002)MathSciNetzbMATHGoogle Scholar
  66. 66.
    K.J. Painter, T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement. Can. Appl. Math. Q. 10, 501–543 (2002)MathSciNetzbMATHGoogle Scholar
  67. 67.
    C.S. Patlak, Random walk with persistence and external bias. Bull. Math. Biophy. 15, 311–338 (1953)MathSciNetzbMATHGoogle Scholar
  68. 68.
    B. Perthame, Transport Equations in Biology (Birkhäuser Verlag, Basel, 2007)zbMATHGoogle Scholar
  69. 69.
    M. Rascle, C. Ziti, Finite time blow-up in some models of chemotaxis. J. Math. Biol. 33, 388–414 (1995)MathSciNetzbMATHGoogle Scholar
  70. 70.
    J. Simon, Compact sets in the space L p(O, T; B). Annali di Matematica Pura ed Applicata 146(1), 65–96 (1986)Google Scholar
  71. 71.
    H. Sohr, The Navier–Stokes Equations, An Elementary Functional Analytic Approach (Birkhäuser Verlag, Basel, 2001)zbMATHGoogle Scholar
  72. 72.
    C. Stinner, C. Surulescu, M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion. SIAM J. Math. Anal. 46, 1969–2007 (2014)MathSciNetzbMATHGoogle Scholar
  73. 73.
    Y. Tao, Global existence of classical solutions to a combined chemotaxis–haptotaxis model with logistic source. J. Math. Anal. Appl. 354, 60–69 (2009)MathSciNetzbMATHGoogle Scholar
  74. 74.
    Y. Tao, Boundedness in a two-dimensional chemotaxis–haptotaxis system. J. Oceanogr. 70(70), 165–174 (2014)Google Scholar
  75. 75.
    Y. Tao, M. Wang, Global solution for a chemotactic–haptotactic model of cancer invasion. Nonlinearity 21, 2221–2238 (2014)MathSciNetzbMATHGoogle Scholar
  76. 76.
    J.I. Tello, M. Winkler, A chemotaxis system with logistic source. Commun. Partial Differ. Equ. 32, 849–877 (2007)MathSciNetzbMATHGoogle Scholar
  77. 77.
    Y. Tao, M. Winkler, A chemotaxis-haptotaxis model: the roles of nonlinear diffusion and logisticsource. SIAM J. Math. Anal. 43, 685–704 (2011)MathSciNetzbMATHGoogle Scholar
  78. 78.
    Y. Tao, M. Winkler, Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity. J. Differ. Equ. 252, 692–715 (2012)MathSciNetzbMATHGoogle Scholar
  79. 79.
    Y. Tao, M. Winkler, Global existence and boundedness in a Keller–Segel–Stokes model with arbitrary porous medium diffusion. Discrete Contin. Dyn. Syst. Ser. A 32, 1901–1914 (2012)MathSciNetzbMATHGoogle Scholar
  80. 80.
    Y. Tao, M. Wiklner, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant. J. Differ. Equ. 252, 2520–2543 (2012)MathSciNetzbMATHGoogle Scholar
  81. 81.
    Y. Tao, M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion. Ann. Inst. Henri Poincaré Anal. Non Linéaire 30, 157–178 (2013)MathSciNetzbMATHGoogle Scholar
  82. 82.
    Y. Tao, M. Winkler, Boundedness and stabilization in a multi-dimensional chemotaxis–haptotaxis model. Proc. R. Soc. Edinb. 144, 1067–1084 (2014)MathSciNetzbMATHGoogle Scholar
  83. 83.
    Y. Tao, M. Winkler, Dominance of chemotaxis in a chemotaxis–haptotaxis model. Nonlinearity 27, 1225–1239 (2014)MathSciNetzbMATHGoogle Scholar
  84. 84.
    Y. Tao, M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractant. J. Differ. Equ. 257, 784–815 (2014)MathSciNetzbMATHGoogle Scholar
  85. 85.
    Y. Tao, M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis–fluid system. Z. Angew. Math. Phys. 66, 2555–2573 (2015)MathSciNetzbMATHGoogle Scholar
  86. 86.
    R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, 2nd edn. (Springer, New York, 1997)Google Scholar
  87. 87.
    I. Tuval, L. Cisneros, C. Dombrowski, et al., Bacterial swimming and oxygen transport near contact lines. Proc. Natl. Acad. Sci. USA 102, 2277–2282 (2005)zbMATHGoogle Scholar
  88. 88.
    G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source. J. Math. Anal. Appl. 439(1), 197–212 (2016)MathSciNetzbMATHGoogle Scholar
  89. 89.
    C. Walker, G.F. Webb, Global existence of classical solutions for a haptotaxis model. SIAM J. Math. Anal. 38, 1694–1713 (2007)MathSciNetzbMATHGoogle Scholar
  90. 90.
    Y. Wang, Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion. J. Differ. Equ. 260(2), 1975–1989 (2016)MathSciNetzbMATHGoogle Scholar
  91. 91.
    Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity. Math. Models Methods Appl. Sci. 27(14), 2745–2780 (2017)MathSciNetzbMATHGoogle Scholar
  92. 92.
    Y. Wang, Y. Ke, Large time behavior of solution to a fully parabolic chemotaxis-haptotaxis model in higher dimensions. J. Differ. Equ. 260(9), 6960–6988 (2016)MathSciNetzbMATHGoogle Scholar
  93. 93.
    Y. Wang, Z. Xiang, Global existence and boundedness in a higher-dimensional quasilinear chemotaxis system. Z. Angew. Math. Phys. 66(6), 3159–3179 (2015)MathSciNetzbMATHGoogle Scholar
  94. 94.
    Y. Wang, Z. Xiang, Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation. J. Differ. Equ. 259, 7578–7609 (2015)MathSciNetzbMATHGoogle Scholar
  95. 95.
    Y. Wang, Z. Xiang, Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation: the 3D case. J. Differ. Equ. 261, 4944–4973 (2016)MathSciNetzbMATHGoogle Scholar
  96. 96.
    Y. Wang, J. Yin, Coexistence periodic solutions of a doubly nonlinear parabolic system with Neumann boundary conditions. J. Math. Anal. Appl. 396, 704–714 (2012)MathSciNetzbMATHGoogle Scholar
  97. 97.
    Z. Wang, M. Winkler, D. Wrzosek, Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume-filling effect and degenerate diffusion. SIAM J. Math. Anal. 44, 3502–3525 (2012)Google Scholar
  98. 98.
    L. Wang, Y. Li, C. Mu, Boundedness in a parabolic–parabolic quasilinear chemotaxis system with logistic source. Discrete Contin. Dyn. Syst. Ser. A. 34, 789–802 (2014)MathSciNetzbMATHGoogle Scholar
  99. 99.
    L. Wang, C. Mu, S. Zhou, Boundedness in a parabolic–parabolic chemotaxis system with nonlinear diffusion. Z. Angew. Math. Phys. 65, 1137–1152 (2014)MathSciNetzbMATHGoogle Scholar
  100. 100.
    L. Wang, C. Mu, P. Zheng, On a quasilinear parabolic–elliptic chemotaxis system with logistic source. J. Differ. Equ. 256, 1847–1872 (2014)MathSciNetzbMATHGoogle Scholar
  101. 101.
    L. Wang, C. Mu, K. Lin, J. Zhao, Global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant. Z. Angew. Math. Phys. 66(4), 1–16 (2015)MathSciNetzbMATHGoogle Scholar
  102. 102.
    M. Winkler, Chemotaxis with logistic source: very weak global solutions and their boundedness properties. J. Math. Anal. Appl. 348, 708–729 (2008)MathSciNetzbMATHGoogle Scholar
  103. 103.
    M. Winkler, Does a volume-filling effect always prevent chemotactic collapse. Math. Methods Appl. Sci. 33, 12–24 (2010)MathSciNetzbMATHGoogle Scholar
  104. 104.
    M. Winkler, Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source. Commun. Partial Differ. Equ. 35, 1516–1537 (2010)MathSciNetzbMATHGoogle Scholar
  105. 105.
    M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model. J. Differ. Equ. 248, 2889–2905 (2010)Google Scholar
  106. 106.
    M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction. J. Math. Anal. Appl. 384, 261–272 (2011)MathSciNetzbMATHGoogle Scholar
  107. 107.
    M. Winkler, 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)MathSciNetzbMATHGoogle Scholar
  108. 108.
    M. Winkler, Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system. J. Math. Pures Appl. 100, 748–767 (2013)MathSciNetzbMATHGoogle Scholar
  109. 109.
    M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities? J. Nonlinear Sci. 24, 809–855 (2014)MathSciNetzbMATHGoogle Scholar
  110. 110.
    M. Winkler, Stabilization in a two-dimensional chemotaxis–Navier–Stokes system. Arch. Ration. Mech. Anal. 211, 455–487 (2014)MathSciNetzbMATHGoogle Scholar
  111. 111.
    M. Winkler, Global asymptotic stability of constant equilibriain a fully parabolic chemotaxis system with strong logistic dampening. J. Differ. Equ. 257, 1056–1077 (2014)zbMATHGoogle Scholar
  112. 112.
    M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion and general sensitivity. Cal. Var. Partial Differ. Equ. 54, 3789–3828 (2015)MathSciNetzbMATHGoogle Scholar
  113. 113.
    M. Winkler, Global weak solutions in a three-dimensional chemotaxis–Navier–Stokes system. Ann. Inst. Henri Poincaré Anal. Non Linéaire 33(5), 1329–1352 (2016)MathSciNetzbMATHGoogle Scholar
  114. 114.
    M. Winkler, K.C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect. Nonlinear Anal. Theory Methods Appl. 72, 1044–1064 (2010)MathSciNetzbMATHGoogle Scholar
  115. 115.
    T. Xiang, Boundedness and global existence in the higher-dimensional parabolic–parabolic chemotaxis system with/without growth source. J. Differ. Equ. 258, 4275–4323 (2015)MathSciNetzbMATHGoogle Scholar
  116. 116.
    C. Xue, H.G. Othmer, Multiscale models of taxis-driven patterning in bacterial population. SIAM J. Appl. Math. 70, 133–167 (2009)MathSciNetzbMATHGoogle Scholar
  117. 117.
    Q. Zhang, Y. Li, Global boundedness of solutions to a two-species chemotaxis system. Z. Angew. Math. Phys. 66(1), 83–93 (2015)MathSciNetzbMATHGoogle Scholar
  118. 118.
    Q. Zhang, X. Zheng, Global well-posedness for the two-dimensional incompressible chemotaxis–Navier–Stokes equations. SIAM J. Math. Anal. 46, 3078–3105 (2014)MathSciNetzbMATHGoogle Scholar
  119. 119.
    J. Zheng, A new approach toward locally bounded global solutions to a 3D chemotaxis-stokes system with nonlinear diffusion and rotation. arXiv:1701.01334Google Scholar
  120. 120.
    J. Zheng, Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source. J. Differ. Equ. 259(1), 120–140 (2015)MathSciNetzbMATHGoogle Scholar
  121. 121.
    J. Zheng, Boundedness of solutions to a quasilinear parabolic–parabolic Keller–Segel system with logistic source. J. Math. Anal. Appl. 431(2), 867–888 (2015)MathSciNetzbMATHGoogle Scholar
  122. 122.
    J. Zheng, Boundedness in a three-dimensional chemotaxis–fluid system involving tensor-valued sensitivity with saturation. J. Math. Anal. Appl. 442(1), 353–375 (2016)MathSciNetzbMATHGoogle Scholar
  123. 123.
    J. Zheng, Boundedness and global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with nonlinear a logistic source. J. Math. Anal. Appl. 450, 104–1061 (2017)MathSciNetGoogle Scholar
  124. 124.
    J. Zheng, A note on boundedness of solutions to a higher-dimensional quasi–linear chemotaxis system with logistic source. Zeitschriftfür Angewandte Mathematik und Mechanik 97(4), 414–421 (2017)MathSciNetGoogle Scholar
  125. 125.
    J. Zheng, Boundedness in a two-species quasi-linear chemotaxis system with two chemicals. Topol. Methods Nonlinear Anal. 49(2), 463–480 (2017)MathSciNetzbMATHGoogle Scholar
  126. 126.
    J. Zheng, Boundedness of solution of a higher-dimensional parabolic–ODE–parabolic chemotaxis–haptotaxis model with generalized logistic source. Nonlinearity 30, 1987–2009 (2017)MathSciNetzbMATHGoogle Scholar
  127. 127.
    J. Zheng, Boundedness of solutions to a quasilinear higher-dimensional chemotaxis–haptotaxis model with nonlinear diffusion. Discrete Contin. Dyn. Syst. 37(1), 627–643 (2017)MathSciNetzbMATHGoogle Scholar
  128. 128.
    J. Zheng, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion. J. Differ. Equ. 263, 2606–2629 (2017)MathSciNetzbMATHGoogle Scholar
  129. 129.
    J. Zheng, Y. Wang, Boundedness and decay behavior in a higher-dimensional quasilinear chemotaxis system with nonlinear logistic source. Comput. Math. Appl. 72(10), 2604–2619 (2016)MathSciNetzbMATHGoogle Scholar
  130. 130.
    J. Zheng, Y. Wang, Boundedness of solutions to a quasilinear chemotaxis–haptotaxis model. Comput. Math. Appl. 71, 1898–1909 (2016)MathSciNetzbMATHGoogle Scholar
  131. 131.
    J. Zheng, Y. Wang, A note on global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant. Discrete Contin. Dyn. Syst. Ser. B 22(2), 669–686 (2017)MathSciNetzbMATHGoogle Scholar
  132. 132.
    P. Zheng, C. Mu, X. Song, On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion. Discrete Cont. Dyn. Syst. 36(3), 1737–1757 (2015)MathSciNetzbMATHGoogle Scholar
  133. 133.
    P. Zheng, C. Mu, X. Hu, 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)MathSciNetzbMATHGoogle Scholar
  134. 134.
    J. Zheng, Y. Li, G. Bao, X. Zou, A new result for global existence and boundedness of solutions to a parabolic–parabolic Keller–Segel system with logistic source. J. Math. Anal. Appl. 462, 1–25 (2018)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Jiashan Zheng
    • 1
  1. 1.School of Mathematics and Statistics ScienceLudong UniversityYantaiP.R. China

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