  We study dynamics of filling of an initially empty finite medium by diffusing
particles $A$ and $B$, which arise on the surface upon dissociation of $AB$
molecules, impinging on it with a fixed flux density $I$, and desorb from it by
the reaction $A + B\to AB\to 0$. We show that once the bulk diffusivities
differ ($p=D_{A}/D_{B}<1$), there exists a critical flux density $I_{c}(p)$,
above which the relaxation dynamics to the steady state is qualitatively
changed: on time dependencies of $c_{As}/c_{e}$ ($c_{e}$ being the steady state
concentration at $t\to \infty$) a maximum appears, the amplitude of which grows
both with $I$ and with $D_{B}/D_{A}$ ratio. In the diffusion-controlled limit
$I \gg I_{c}$ at $p \ll 1$ the reaction "selects" the {\it universal laws} for
the particles number growth ${\cal N}_{A}={\cal N}_{B}\propto t^{1/4}$ and the
evolution of the surface concentrations $c_{As}\propto t^{-1/4},c_{Bs}\propto
t^{1/4}$, which are approached by one of the {\it two characteristic regimes}
with the corresponding hierarchy of the intermediate power-law asymptotics. In
the first of these $c_{As}$ goes through a comparatively {\it sharp}
max$(c_{As}/c_{e})\propto I^{1/6}$, the amplitude of which is $p$-independent,
in the second one $c_{As}$ goes through a {\itplateau-like}
max$(c_{As}/c_{e})\propto p^{-1/4}$, the amplitude of which is $I$-independent.
We demonstrate that on the main filling stage the evolution of the ${\cal
N}(t)/{\cal N}_{e}, c_{As}(t)/c_{e},$ and $c_{Bs}(t)/c_{e}$ trajectories with
changing $p$ or $J$ between the limiting regimes is unambiguously defined by
the value of the scaling parameter ${\cal K}=p^{3/2}J$ ($J$ being the reduced
flux density) and is described by the set of {\it scaling laws}, which we study
in detail analytically and numerically.
