On a coupled Cahn–Hilliard/Cahn–Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells

Lu Li

doi:10.3934/cpaa.2021032

Article Contents

2021, Volume 20, Issue 4: 1545-1557. Doi: 10.3934/cpaa.2021032

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On a coupled Cahn–Hilliard/Cahn–Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells

Lu Li^a,b,

a.
Université de Poitiers, Laboratoire I3M et Laboratoire de Mathématiques et Applications, UMR CNRS 7348, Equipe DACTIM-MIS, 11 Boulevard Marie et Pierre Curie-Bâtiment H3-TSA 61125, 86073 Poitiers Cedex 9, France
b.
CHU de Poitiers, 2 rue de la Milétrie, 86000 Poitiers, France

Received: October 2020

Revised: January 2021

Early access: March 2021

Published: April 2021

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Abstract

Our aim in this paper is to prove the existence of solutions for a model for the proliferative-to-invasive transition of hypoxic glioma cells. The equations consist of the coupling of a Cahn–Hilliard equation for the tumor density and a Cahn–Hilliard type equation for the oxygen concentration. The main difficulty is to prove the existence of a biologically relevant solution. This is achieved by considering modified equations and taking logarithmic nonlinear terms in the Cahn–Hilliard equations. After that we show a local in time weak solution which is conditionally global in time.

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Mathematics Subject Classification: Primary: 35K55, 35B45; Secondary: 35Q92.

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