Chemotaxis model for tumour growth

We discuss analytic results for a new diffuse interface model describing the evolution of a tumour mass under the effects of a chemical substance (e.g., a nutrient). The process is described by utilising an order parameter representing the local proportion of tumour cells, and a variable describing the concentration of the chemical. The order parameter is assumed to satisfy a suitable form of the Cahn–Hilliard equation with mass source and logarithmic potential of Flory–Huggins type, whereas the chemical concentration satisfies a reaction-diffusion equation where the cross-diffusion term has the same expression as in the celebrated
Keller–Segel model. Weak well-posedness, regularity, and continuous dependence results are presented.
This is a joint work with E. Rocca (University of Pavia) and G. Schimperna (University of Pavia).