Global attractor for a Cahn-Hilliard-chemotaxis model with logistic degradation

We consider a mathematical model coupling the Cahn-Hilliard system for phase separation with an additional equation describing the diffusion process of a chemical quantity whose concentration influences the physical process. The main application of the model refers to tumor progression, where the phase variable phi denotes the local proportion of active cancer cells and the chemical concentration sigma may refer to a nutrient transported by the blood flow or to a drug administered to the patient. The resulting system is characterized by cross-diffusion effects similar to those appearing in the Keller-Segel model for chemotaxis; in particular, the nutrient tends to be attracted toward the regions where more active tumor cells are present (and consume it in a quickier way). Complementing various recent results on related models, we investigate here the long-time behavior of solutions under the perspective of infinite-dimensional dynamical systems. To this aim, we first identify a regularity setting in which the system is well-posed and generates a closed semigroup according to the terminology introduced by Pata and Zelik. Then, partly based on the approach introduced by Rocca and the first author for the Cahn-Hilliard system with singular potential, we prove that the semigroup is strongly dissipative and asymptotically compact, so guaranteeing the existence of the global attractor in a suitable phase space. Finally, we discuss the sign properties of sigma and show that, if the initial datum for the nutrient is a.e. strictly positive in the reference domain then it becomes separated from 0 in the uniform norm for strictly positive times. It is not clear, however, whether this property is uniform with respect to large values of the time variable.