Formation of a boundary-layer solution in a problem for a system of reaction-diffusion equations in a limited volume

We consider a system of equations that describes a two-component chemical reaction in a limited volume. The reaction is assumed to occur in a solution, the concentration of reaction products increases in time, then becomes maximal possible under the given conditions (i.e., saturation occurs), and then the reaction terminates. A similar formulation can be used for describing microscopic processes occurring when CO$_2$ is injected into a rock, which is a porous medium with pores filled with water. For a system of two equations of the “reaction-diffusion” type on a segment, we show that in a finite time, a solution close to a stationary distribution corresponding to the concentration of a saturated solution under given conditions is formed from a given initial function.