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We prove existence, uniqueness, and regularity for a reaction-diffusionsystem of coupled bulk-surface equations on a moving domain modellingreceptor-ligand dynamics in cells. The nonlinear coupling between the threeunknowns is through the Robin boundary condition for the bulk quantity and theright hand sides of the two surface equations. Our results are new even in thenon-moving setting, and in this case we also show exponential convergence to asteady state. The primary complications in the analysis are indeed thenonlinear coupling and the Robin boundary condition. For the well posedness andessential boundedness of solutions we use several De Giorgi-type arguments, andwe also develop some useful estimates to allow us to apply a Steklov averagingtechnique for time-dependent operators to prove that solutions are strong. Someof these auxiliary results presented in this paper are of independent interestby themselves.

A coupled ligand-receptor bulk-surface system on a moving domain: well posedness, regularity and convergence to equilibrium