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

Abstract

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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Acknowledgements

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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Keywords

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

Mathematics Subject Classification

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