Weak solutions and weak-strong uniqueness for a Cahn-Hilliard type model with chemotaxis

We prove existence of weak solutions and weak-strong uniqueness for a mathematical model which 
couples the evolution of a phase-parameter φ satisfying a Cahn-Hilliard type relation with
the one of an additional variable σ influencing the phase separation process. The main application
of the model refers to cancer growth processes, where σ may represent the concentration of a
chemical substance affecting the evolution of the tumor, and is governed by a nonlinear parabolic
equation characterized by a cross-diffusion term alike that occurring in the Keller-Segel model
for chemotaxis. This term is also responsible for the most relevant difficulties in the mathematical
analysis of the system. Complementing previous results on the model, we prove here global in
time existence for a very weak notion of solution to which a suitable energy imbalance and a
logarithmic inequality for the nutrient are added. Noting that the system also admits local in time
“strong” solutions, we can also exhibit a weak-strong uniqueness result whose proof exploits in
an essential way the entropy-type inequality satisfied by weak solutions.