WELL-POSEDNESS AND LONG-TIME BEHAVIOR OF A BULK-SURFACE COUPLED CAHN-HILLIARD-DIFFUSION SYSTEM WITH SINGULAR POTENTIAL FOR LIPID RAFT FORMATION

Abstract

We study a bulk-surface coupled system that describes the processes of lipid-phase separation and lipid-cholesterol interaction on cell membrane, in which cholesterol exchange between cytosol and cell membrane is also incorporated. The PDE system consists of a surface Cahn-Hilliard equation for the relative concentration of saturated/unsaturated lipids and a surface diffusion-reaction equation for the cholesterol concentration on the membrane, together with a diffusion equation for the cytosolic cholesterol concentration in the bulk. The detailed coupling between bulk and surface evolutions is characterized by a mass exchange term q. Working with a physically relevant singular potential, we first prove the existence, uniqueness and regularity of global weak solutions to the full bulk-surface coupled system under suitable assumptions on the initial data and the mass exchange term q. Next, we investigate the large cytosolic diffusion limit that gives a reduction of the full bulk-surface coupled system to a system of surface equations with non-local contributions. Afterwards, we study the long-time behavior of global solutions in two categories, i.e., the equilibrium and non-equilibrium models according to different choices of the mass exchange term q. For the full bulk-surface coupled system with a decreasing total free energy, we prove that every global weak solution converges to a single equilibrium as t → +∞. For the reduced surface system with a mass exchange term of reaction type, we establish the existence of a global attractor.