Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis

TL;DR

This paper establishes the well-posedness of a tumor growth model involving a Cahn--Hilliard equation with Dirichlet boundary conditions, extending previous results to higher polynomial and singular potentials, and analyzes a quasi-static nutrient variant.

Contribution

It introduces the analysis of a tumor growth model with Dirichlet boundary conditions, including higher polynomial and singular potentials, and studies the quasi-static nutrient limit.

Findings

Well-posedness for regular potentials with higher polynomial growth

Well-posedness for singular potentials

Analysis of the quasi-static nutrient limit

Abstract

We consider a diffuse interface model for tumor growth consisting of a Cahn--Hilliard equation with source terms coupled to a reaction-diffusion equation, which models a tumor growing in the presence of a nutrient species and surrounded by healthy tissue. The well-posedness of the system equipped with Neumann boundary conditions was found to require regular potentials with quadratic growth. In this work, Dirichlet boundary conditions are considered, and we establish the well-posedness of the system for regular potentials with higher polynomial growth and also for singular potentials. New difficulties are encountered due to the higher polynomial growth, but for regular potentials, we retain the continuous dependence on initial and boundary data for the chemical potential and for the order parameter in strong norms as established in the previous work. Furthermore, we deduce the well-posedness of a variant of the model with quasi-static nutrient by rigorously passing to the limit where the ratio of the nutrient diffusion time-scale to the tumor doubling time-scale is small.