Well-posedness of a Cahn--Hilliard system modelling tumour growth with chemotaxis and active transport

TL;DR

This paper proves well-posedness for a complex tumour growth model involving coupled Cahn--Hilliard and reaction-diffusion equations with transport mechanisms, addressing mathematical challenges from source terms.

Contribution

It establishes the existence, uniqueness, and continuous dependence of solutions for a tumour growth model with chemotaxis and active transport, including a variant with quasi-static nutrient.

Findings

Proved well-posedness of the coupled PDE system.

Addressed difficulties from source terms in a priori estimates.

Showed continuous dependence on initial and boundary data.

Abstract

We consider a diffuse interface model for tumour growth consisting of a Cahn--Hilliard equation with source terms coupled to a reaction-diffusion equation. The coupled system of partial differential equations models a tumour growing in the presence of a nutrient species and surrounded by healthy tissue. The model also takes into account transport mechanisms such as chemotaxis and active transport. We establish well-posedness results for the tumour model and a variant with a quasi-static nutrient. It will turn out that the presence of the source terms in the Cahn--Hilliard equation leads to new difficulties when one aims to derive a priori estimates. However, we are able to prove continuous dependence on initial and boundary data for the chemical potential and for the order parameter in strong norms.