On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials

TL;DR

This paper develops a mathematical model for tumor growth involving coupled Cahn-Hilliard-Darcy equations with singular potentials, proving the existence of weak solutions despite complex constraints and nonzero forcing terms.

Contribution

It introduces a novel tumor growth model with singular potentials ensuring physical constraints and establishes the existence of weak solutions under these conditions.

Findings

Successfully incorporates singular potentials in the model

Proves existence of weak solutions with nonzero forcing terms

Handles constraints on cell concentration variables

Abstract

We consider a model describing the evolution of a tumor inside a host tissue in terms of the parameters $\varphi_p$, $\varphi_d$ (proliferating and dead cells, respectively), $u$ (cell velocity) and $n$ (nutrient concentration). The variables $\varphi_p$, $\varphi_d$ satisfy a Cahn-Hilliard type system with nonzero forcing term (implying that their spatial means are not conserved in time), whereas $u$ obeys a form of the Darcy law and $n$ satisfies a quasistatic diffusion equation. The main novelty of the present work stands in the fact that we are able to consider a configuration potential of singular type implying that the concentration vector $(\varphi_p,\varphi_d)$ is constrained to remain in the range of physically admissible values. On the other hand, in view of the presence of nonzero forcing terms, this choice gives rise to a number of mathematical difficulties, especially related to the control of the mean values of $\varphi_p$ and $\varphi_d$. For the resulting mathematical problem, by imposing suitable initial-boundary conditions, our main result concerns the existence of weak solutions in a proper regularity class.