A parabolic–elliptic chemotaxis system with nonlinear diffusion approached from a Cahn–Hilliard-type system


This paper deals with a parabolic–elliptic chemotaxis system with nonlinear diffusion. It was proved that there exists a solution of a Cahn–Hilliard system as an approximation of a nonlinear diffusion equation by applying an abstract theory by Colli–Visintin (Commun Partial Differ Equ 15:737–756, 1990) for a doubly nonlinear evolution inclusion with some bounded monotone operator and subdifferential operator of a proper lower semicontinuous convex function (cf. Colli–Fukao in J Math Anal Appl 429, 2015:1190–1213). Moreover, Colli–Fukao (J Differ Equ 260:6930–6959, 2016) established existence of solutions to the nonlinear diffusion equation by passing to the limit in the Cahn–Hilliard equation. However, Cahn–Hilliard approaches to chemotaxis systems with nonlinear diffusions seem to be not studied yet. This paper will try to derive existence of solutions to a parabolic–elliptic chemotaxis system with nonlinear diffusion by passing to the limit in a Cahn–Hilliard-type chemotaxis system. Although in this paper, we employ a time discretization scheme to prove existence for the Cahn–Hilliard-type chemotaxis system in reference to Colli–Kurima (Nonlinear Anal 190:111613, 2020), please note that this reference does not deal with chemotaxis terms.

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  1. 1.

    G. Akagi, G. Schimperna, A. Segatti, Fractional Cahn–Hilliard, Allen–Cahn and porous medium equations, J. Differential Equations 261 (2016), 2935–2985.

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    V. Barbu, “Nonlinear semigroups and differential equations in Banach spaces”, Translated from the Romanian, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976.

  3. 3.

    V. Barbu, “Nonlinear Differential Equations of Monotone Types in Banach Spaces”, Springer, London, 2010.

    Google Scholar 

  4. 4.

    H. Brézis, “Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Especes de Hilbert”, North-Holland, Amsterdam, 1973.

    Google Scholar 

  5. 5.

    P. Colli, T. Fukao, Cahn–Hilliard equation with dynamic boundary conditions and mass constraint on the boundary, J. Math. Anal. Appl. 429 (2015), 1190–1213.

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    P. Colli, T. Fukao, Nonlinear diffusion equations as asymptotic limits of Cahn–Hilliard systems, J. Differential Equations 260 (2016), 6930–6959.

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    P. Colli, S. Kurima, Time discretization of a nonlinear phase field system in general domains, Comm. Pure Appl. Anal. 18 (2019), 3161–3179.

    MathSciNet  Article  Google Scholar 

  8. 8.

    P. Colli, S. Kurima, Global existence for a phase separation system deduced from the entropy balance, Nonlinear Anal. 190 (2020) 111613, 31 pp.

  9. 9.

    P. Colli, A. Visintin, On a class of doubly nonlinear evolution equations, Comm. Partial Differential Equations 15 (1990), 737–756.

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    A. Friedman, “Partial differential equations”, Holt, Rinehart and Winston Inc., New York, 1969.

  11. 11.

    G. Gilardi, A. Miranville, G. Schimperna, On the Cahn–Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881–912.

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    J.W. Jerome, “Approximations of Nonlinear Evolution Systems”, Mathematics in Science and Engineering 164, Academic Press Inc., Orlando, 1983.

  13. 13.

    G. Marinoschi, Well-posedness of singular diffusion equations in porous media with homogeneous Neumann boundary conditions, Nonlinear Anal. 72 (2010), 3491–3514.

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    G. Marinoschi, Well-posedness for chemotaxis dynamics with nonlinear cell diffusion, J. Math. Anal. Appl. 402 (2013), 415–439.

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    N. Okazawa, An application of the perturbation theorem for m-accretive operators, Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), 88–90.

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

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

    MathSciNet  MATH  Google Scholar 

  17. 17.

    R. E. Showalter, “Monotone Operators in Banach Space and Nonlinear Partial Differential Equations”, Mathematical Surveys and Monographs, 49, American Mathematical Society, Providence, RI, 1997.

  18. 18.

    J. Simon, Compact sets in the space \(L^p(0, T; B)\), Ann. Mat. Pura Appl. (4) 146 (1987), 65–96.

  19. 19.

    J. L. Vázquez, “The Porous Medium Equation”, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.

  20. 20.

    H. -M. Yin, On a degenerate parabolic system, J. Differential Equations 245 (2008), 722–736.

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    T. Yokota, N. Yoshino, Local and global existence of solutions to a quasilinear degenerate chemotaxis system with unbounded initial data, Math. Methods Appl. Sci. 39 (2016), 3361–3380.

    MathSciNet  Article  MATH  Google Scholar 

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The author would like to thank the anonymous referee for careful reading and helpful comments. The author is supported by JSPS Research Fellowships for Young Scientists (No. 18J21006).

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Correspondence to Shunsuke Kurima.

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Kurima, S. A parabolic–elliptic chemotaxis system with nonlinear diffusion approached from a Cahn–Hilliard-type system. J. Evol. Equ. (2020). https://doi.org/10.1007/s00028-020-00651-5

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  • Cahn–Hilliard approaches
  • Nonlinear diffusions
  • Parabolic–elliptic chemotaxis systems
  • Existence
  • Time discretizations

Mathematics Subject Classification

  • 35G31
  • 35B25
  • 35D30
  • 35A40