A Cahn–Hilliard model with a proliferation term for the proliferative-to-invasive transition of hypoxic glioma cells

Alain Miranville (School of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan, China; and Laboratoire I3M et Laboratoire de Mathématiques et Applications, Université de Poitiers, France)

Rémy Guillevin (Laboratoire I3M et Laboratoire de Mathématiques et Applications, Université de Poitiers, France; and CHU de Poitiers, France)

Abstract

Our aim in this paper is to prove the existence of global-in-time solutions for a model for the proliferative-to-invasive transition of hypoxic glioma cells. The equations consist of the coupling of a Cahn–Hilliard type equation for the tumor density and of a reaction-diffusion equation for the oxygen concentration. The main difficulty is to prove the existence of a biologically relevant solution. Note indeed that, because of the proliferation term, one cannot exclude the possibility of blow up in finite time when considering an approximated scheme. Our goal is achieved by considering a modified equation and taking a logarithmic nonlinear term in the Cahn–Hilliard equation. We also study permanence of the tumor. We finally give some numerical simulations.

Keywords

hypoxic glioma cells, proliferative-to-invasive transition, Cahn–Hilliard equation, proliferation term, logarithmic nonlinear term, existence of solutions, permanence, simulations

2010 Mathematics Subject Classification

35B45, 35K55, 35Q92

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Received 24 July 2020

Accepted 21 January 2021

Published 2 August 2021