We prove a contraction in L1 property for the solutions of a nonlinear reaction–
diffusion system whose special cases include a system related to intracellular transport as well
as reversible chemical reactions. We then consider the special case of the linear molecular motor
problem and prove the existence and uniqueness of the stationary solution up to a multiplicative
constant, extending to arbitrary space dimension results which were already known in the one
dimensional case; this in turn implies the convergence to stationary solutions of the solutions of
the time evolution linear molecular motor problem.