Permeation with plasticization.
It may be the case of solvent molecules in vitreous polymers. When dowstream P2 is nil, the membrane is swelled in its upstream part and completly dry in its downstream part. The concentration profile drops strongly from upstream to downstream. In the upstream swelled part of the membrane the concentration profile is flat. The front of swelling rases very slowly to the downstream face and one must wait for a complete swelling of the membrane in order to observe an increase of the quantity of throwed molecules. The time-lag can not give any estimation of some diffusion coefficient by Fick law but it is rather proportional to the inverse of plasticization (relaxation) speed.
Example 4, case of very low (also nil) dry diffusion speed, very low speed of plasticization (relaxation) speed and long (or infinite) lifetime of plasticized sites :