Well-posedness and Long-time Behavior of a Bulk-surface Coupled Cahn-Hilliard-diffusion System with Singular Potential for Lipid Raft Formation

TL;DR

This paper analyzes a complex bulk-surface PDE system modeling lipid raft formation, proving well-posedness, exploring large diffusion limits, and studying long-term behavior including convergence to equilibrium and existence of attractors.

Contribution

It establishes the existence, uniqueness, and regularity of solutions for a coupled lipid membrane model with singular potential, and investigates its asymptotic dynamics.

Findings

Global weak solutions exist and are unique.

Solutions converge to equilibrium over time.

Reduced models possess global attractors.

Abstract

We study a bulk-surface coupled system that describes the processes of lipid-phase separation and lipid-cholesterol interaction on cell membranes, 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 $q$. For the system 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 $t\to +\infty$. For the reduced surface system with a mass exchange term of reaction type, we establish the existence of a global attractor.