On a Cahn--Hilliard--Darcy system for tumour growth with solution dependent source terms

TL;DR

This paper investigates the mathematical existence of weak solutions for a complex tumour growth model combining Cahn--Hilliard--Darcy equations with a reaction-diffusion component, accounting for solution-dependent source terms.

Contribution

It establishes the existence of weak solutions for a coupled tumour growth system with solution-dependent sources and various boundary conditions.

Findings

Proves existence of weak solutions under specified conditions

Handles both Dirichlet and Robin boundary conditions

Models tumour growth with coupled PDEs including source dependencies

Abstract

We study the existence of weak solutions to a mixture model for tumour growth that consists of a Cahn--Hilliard--Darcy system coupled with an elliptic reaction-diffusion equation. The Darcy law gives rise to an elliptic equation for the pressure that is coupled to the convective Cahn--Hilliard equation through convective and source terms. Both Dirichlet and Robin boundary conditions are considered for the pressure variable, which allows for the source terms to be dependent on the solution variables.