Open AccessA coupled Cahn–Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells
Author(s) -
Lü Li,
Alain Miranville,
Rémy Guillevin
Publication year - 2020
Publication title -
quarterly of applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.603
H-Index - 41
eISSN - 1552-4485
pISSN - 0033-569X
DOI - 10.1090/qam/1585
Subject(s) - cahn–hilliard equation , logarithm , chemical equation , nonlinear system , glioma , coupling (piping) , diffusion equation , mathematics , mathematical analysis , physics , chemistry , partial differential equation , materials science , medicine , quantum mechanics , cancer research , economy , economics , metallurgy , service (business)
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 reaction-diffusion equation for the tumor density and of 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 a modified equation and taking a logarithmic nonlinear term in the Cahn–Hilliard equation.