Theoretical and numerical analysis for a hybrid tumor model with diffusion depending on vasculature

TL;DR

This paper provides a comprehensive analysis of a PDE-ODE glioblastoma model with vasculature-dependent diffusion, establishing existence, stability, and a finite element scheme that preserves key properties.

Contribution

It introduces a novel analysis of a hybrid tumor model with anisotropic nonlinear diffusion depending on vasculature, including existence, stability, and numerical discretization.

Findings

Existence of global weak-strong solutions proven.

Stability results for critical points established.

A finite element scheme preserving key estimates developed.

Abstract

In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes an anisotropic nonlinear diffusion term with a diffusion velocity increasing with respect to vasculature. First, we prove the existence of global in time weak-strong solutions using a regularization technique via an artificial diffusion in the ODE-system and a fixed point argument. In addition, stability results of the critical points are given under some constraints on parameters. Finally, we design a fully discrete finite element scheme for the model which preserves the pointwise and energy estimates of the continuous problem.