  Recent experimental works on one-dimensional (1D) circular
Kardar-Parisi-Zhang (KPZ) systems whose radii decrease in time have reported
controversial conclusions about the statistics of their interfaces. Motivated
by this, we investigate here several 1D KPZ models on substrates whose size
changes in time as $L(t)=L_0 + \omega t$, focusing on the case $\omega<0$. From
extensive numerical simulations, we show that for $L_0 \gg 1$ there exists a
transient regime in which the statistics is consistent with that of flat KPZ
systems (the $\omega=0$ case), for both $\omega<0$ and $\omega>0$. Actually,
for a given model, $L_0$ and $|\omega|$, we observe that a difference between
ingrowing ($\omega<0$) and outgrowing ($\omega>0$) systems arises only at long
times ($t \gtrsim t_c=L_0/|\omega|$), when the expanding surfaces cross over to
the statistics of curved KPZ systems, whereas the shrinking ones become
completely correlated. A generalization of the Family-Vicsek scaling for the
roughness of ingrowing interfaces is presented. Our results demonstrate that a
transient flat statistics is a general feature of systems starting with large
initial sizes, regardless their curvature. This is consistent with their recent
observation in ingrowing turbulent liquid crystal interfaces, but it is in
contrast with the apparent observation of curved statistics in colloidal
deposition at the edge of evaporating drops. A possible explanation for this
last result, as a consequence of the very small number of monolayers analyzed
in this experiment, is given. This is illustrated in a competitive growth model
presenting a few-monolayer transient and an asymptotic behavior consistent,
respectively, with the curved and flat statistics.
