# American Institute of Mathematical Sciences

April  2021, 20(4): 1545-1557. doi: 10.3934/cpaa.2021032

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

 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 Published  April 2021 Early access  March 2021

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.

Citation: Lu Li. On a coupled Cahn–Hilliard/Cahn–Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1545-1557. doi: 10.3934/cpaa.2021032
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