Cahn-Hilliard-Brinkman model for tumor growth with possibly singular potentials

TL;DR

This paper investigates a complex tumor growth model combining Cahn-Hilliard and Brinkman equations, proving the existence of weak solutions for various nonlinear potentials and boundary conditions, advancing mathematical understanding of tumor dynamics.

Contribution

It establishes the existence of weak solutions for a tumor growth model with diverse nonlinear potentials and boundary conditions, including Dirichlet conditions for the chemical potential.

Findings

Existence of weak solutions for regular, logarithmic, and double obstacle potentials.

Inclusion of Dirichlet boundary conditions for the chemical potential.

Abstract growth conditions for source terms depending on solution variables.

Abstract

We analyze a phase field model for tumor growth consisting of a Cahn-Hilliard-Brinkman system, ruling the evolution of the tumor mass, coupled with an advection-reaction-diffusion equation for a chemical species acting as a nutrient. The main novelty of the paper concerns the discussion of the existence of weak solutions to the system covering all the meaningful cases for the nonlinear potentials; in particular,the typical choices given by the regular, the logarithmic, and the double obstacle potentials are admitted in our treatise. Compared to previous results related to similar models, we suggest, instead of the classical no-flux condition, a Dirichlet boundary condition for the chemical potential appearing in the Cahn-Hilliard-type equation. Besides, abstract growth conditions for the source terms that may depend on the solution variables are postulated.