This thesis aims to explore the applications of Cahn-Hilliard-type models in biology and image processing. In the first part, we initially study a Cahn-Hilliard model for glial cells, proving the existence of a biologically relevant solution and a strict separation of pure states in 1D and 2D. We then consider a Cahn-Hilliard-Oono model and draw similar conclusions. Furthermore, we study a coupled model for the transition from proliferative to invasive states in hypoxic glioma cells; we consider Cahn-Hilliard-type equations in three cases and prove, in particular, the existence of global solutions in time. We specifically study tumor persistence and provide some numerical simulations in certain cases. In the second part, we study a Cahn-Hilliard model for image segmentation. The well-posedness issue has been addressed; since the solution could be unbounded as time tends to infinity, we consider a Cahn-Hilliard-Oono model to perform numerical simulations that illustrate the theoretical results. We then study the asymptotic behavior of Cahn-Hilliard-Oono-type models with cubic nonlinear terms and logarithmic nonlinear terms; more specifically, the existence of attractors of finite dimension.
Author : Li Lu
Management team
- A.Miranville
- R.Guillevin
Thesis defended on March 10, 2025
Accepted publications :
- L.Li, A. Miranville and R. Guillevin, « Cahn-Hilliard models for glial cells » . A paraître dans Applied Mathematics and Optimization
- L.Li, A. Miranville and R. Guillevin, « A coupled Cahn-Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells » . A paraître dans Quarterly of Applied Mathematics