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

TL;DR

This paper introduces a mathematical model for tumor growth using a phase-field system with a degenerate mobility and singular potential, proving existence of solutions through a relaxation approach.

Contribution

It develops a relaxed Cahn-Hilliard tumor model with a single-well potential and degenerate mobility, establishing existence of weak solutions and their convergence as the relaxation parameter tends to zero.

Findings

Existence of weak solutions for the relaxed system.

Convergence of solutions as the relaxation parameter approaches zero.

Mathematical validation of the tumor growth model.

Abstract

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.