Contraction in $L^1$ and large time behavior for a system arising in   chemical reactions and molecular motors
  We prove a contraction in $L^1$ property for the solutions of a nonlinear reaction--diffusion system whose special cases include intercellular transport as well as reversible chemical reactions. Assuming the existence of stationary solutions we show that the solutions stabilize as $t$ tends to infinity. Moreover, in the special case of linear reaction terms, we prove the existence and the uniqueness (up to a multiplicative constant) of the stationary solution. 
M. Chipot, D. Hilhorst, D. Kinderlehrer and M. Olech