Phenotypic heterogeneity in a model of tumor growth: existence of solutions and incompressible limit

TL;DR

This paper investigates a mathematical model of tumor growth considering phenotypic heterogeneity, proving existence of solutions and analyzing the behavior as pressure becomes stiff, leading to a Hele-Shaw type free boundary problem.

Contribution

It extends methods from two-species cross-diffusion systems to a tumor growth model, establishing existence of solutions and analyzing the incompressible limit.

Findings

Existence of weak solutions for the tumor growth model.

Derivation of the incompressible limit as pressure stiffens.

Identification of a free boundary problem of Hele-Shaw type in the limit.

Abstract

We consider a (degenerate) cross-diffusion model of tumor growth structured by phenotypic trait. We prove the existence of weak solutions and the incompressible limit as the pressure becomes stiff extending methods recently introduced in the context of two-species cross-diffusion systems. In the stiff-pressure limit, the compressible model generates a free boundary problem of Hele-Shaw type. Moreover, we prove a new L4-bound on the pressure gradient.