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We propose a new type of diffuse interface model describing the evolution ofa tumor mass under the effects of a chemical substance (e.g., a nutrient or adrug). The process is described by utilizing the variables $\varphi$, an orderparameter representing the local proportion of tumor cells, and $\sigma$,representing the concentration of the chemical. The order parameter $\varphi$is assumed to satisfy a suitable form of the Cahn-Hilliard equation with masssource and logarithmic potential of Flory-Huggins type (or generalizations ofit). The chemical concentration $\sigma$ satisfies a reaction-diffusionequation where the cross-diffusion term has the same expression as in thecelebrated Keller-Segel model. In this respect, the model we propose representsa new coupling between the Cahn-Hilliard equation and a subsystem of theKeller-Segel model. We believe that, compared to other models, this choice ismore effective in capturing the chemotactic effects that may occur in tumorgrowth dynamics (chemically induced tumor evolution and consumption ofnutrient/drug by tumor cells). Note that, in order to prevent finite timeblowup of $\sigma$, we assume a chemical source term of logistic type. Our mainmathematical result is devoted to proving existence of weak solutions in arather general setting that covers both the two- and three- dimensional cases.Under more restrictive assumptions on coefficients and data, and in some caseson the spatial dimension, we prove various regularity results. Finally, in aproper class of smooth solutions we show uniqueness and continuous dependenceon the initial data in a number of significant cases.

On a Cahn-Hilliard-Keller-Segel model with generalized logistic source describing tumor growth