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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  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, doi: 10.3934/cpaa.2021032
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