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 $ \varphi $ 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 $ \sigma_0 $ is a.e. strictly positive in the reference domain $ \Omega $ with $ \ln \sigma_0\in L^1(\Omega) $, then, for every $ 00 $ a.e. in $ \Omega\times(\tau, T) $. It is not clear, however, whether this property is uniform in $ T $; actually, we cannot exclude that $ \delta $ could degenerate to $ 0 $ as $ T $ goes to infinity.