On a relaxed Cahn-Hilliard tumour growth model with single-well potential and degenerate mobility

We consider a phase-field system modelling solid tumour growth.

This system consists of a Cahn-Hilliard equation coupled with a nutrient equation.

The former is characterised by a degenerate mobility and a singular potential.

Both equations are subject to suitable reaction terms which model proliferation and nutrient consumption.

Chemotactic effects are also taken into account.

Adding an elliptic regularisation, depending on a relaxation parameter $\delta>0$, in the equation for the chemical potential, we prove the existence of a weak solution to an initial and boundary value problem for the relaxed system.

Then, we let $\delta$ go to zero, and we recover the existence of a weak solution to the original system.