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This paper deals with a parabolic-elliptic chemotaxis system with nonlineardiffusion. It was proved that there exists a solution of a Cahn-Hilliard systemas an approximation of a nonlinear diffusion equation by applying an abstracttheory by Colli-Visintin [Comm. Partial Differential Equations 15 (1990),737-756] for a doubly nonlinear evolution inclusion with some bounded monotoneoperator and subdifferential operator of a proper lower semicontinuous convexfunction (cf. Colli-Fukao [J. Math. Anal. Appl. 429 (2015), 1190-1213]).Moreover, Colli-Fukao [J. Differential Equations 260 (2016), 6930-6959]established existence of solutions to the nonlinear diffusion equation bypassing to the limit in the Cahn-Hilliard equation. However, Cahn-Hilliardapproaches to chemotaxis systems with nonlinear diffusions seem not to bestudied yet. This paper will try to derive existence of solutions to aparabolic-elliptic chemotaxis system with nonlinear diffusion by passing to thelimit in a Cahn-Hilliard type chemotaxis system.

Convergence of a Cahn-Hilliard type system to a parabolic-elliptic chemotaxis system with nonlinear diffusion