  We prove existence, uniqueness, and regularity for a reaction-diffusion
system of coupled bulk-surface equations on a moving domain modelling
receptor-ligand dynamics in cells. The nonlinear coupling between the three
unknowns is through the Robin boundary condition for the bulk quantity and the
right hand sides of the two surface equations. Our results are new even in the
non-moving setting, and in this case we also show exponential convergence to a
steady state. The primary complications in the analysis are indeed the
nonlinear coupling and the Robin boundary condition. For the well posedness and
essential boundedness of solutions we use several De Giorgi-type arguments, and
we also develop some useful estimates to allow us to apply a Steklov averaging
technique for time-dependent operators to prove that solutions are strong. Some
of these auxiliary results presented in this paper are of independent interest
by themselves.
