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We study a bulk-surface coupled system that describes the processes oflipid-phase separation and lipid-cholesterol interaction on cell membranes, inwhich cholesterol exchange between cytosol and cell membrane is alsoincorporated. The PDE system consists of a surface Cahn-Hilliard equation forthe relative concentration of saturated/unsaturated lipids and a surfacediffusion-reaction equation for the cholesterol concentration on the membrane,together with a diffusion equation for the cytosolic cholesterol concentrationin the bulk. The detailed coupling between bulk and surface evolutions ischaracterized by a mass exchange term $q$. For the system with a physicallyrelevant singular potential, we first prove the existence, uniqueness andregularity of global weak solutions to the full bulk-surface coupled systemunder suitable assumptions on the initial data and the mass exchange term $q$.Next, we investigate the large cytosolic diffusion limit that gives a reductionof the full bulk-surface coupled system to a system of surface equations withnon-local contributions. Afterwards, we study the long-time behavior of globalsolutions in two categories, i.e., the equilibrium and non-equilibrium modelsaccording to different choices of the mass exchange term $q$. For the fullbulk-surface coupled system with a decreasing total free energy, we prove thatevery 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, weestablish the existence of a global attractor.

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