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

Abstract

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 denotes the local proportion of active cancer cells and the chemical concentration 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 towards 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 the nutrient and prove that, under additional assumptions on the initial data, its concentration is uniformly larger than some strictly positive constant at least on finite time intervals.