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Recent experimental works on one-dimensional (1D) circularKardar-Parisi-Zhang (KPZ) systems whose radii decrease in time have reportedcontroversial conclusions about the statistics of their interfaces. Motivatedby this, we investigate here several 1D KPZ models on substrates whose sizechanges in time as $L(t)=L_0 + \omega t$, focusing on the case $\omega<0$. Fromextensive numerical simulations, we show that for $L_0 \gg 1$ there exists atransient regime in which the statistics is consistent with that of flat KPZsystems (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 betweeningrowing ($\omega<0$) and outgrowing ($\omega>0$) systems arises only at longtimes ($t \gtrsim t_c=L_0/|\omega|$), when the expanding surfaces cross over tothe statistics of curved KPZ systems, whereas the shrinking ones becomecompletely correlated. A generalization of the Family-Vicsek scaling for theroughness of ingrowing interfaces is presented. Our results demonstrate that atransient flat statistics is a general feature of systems starting with largeinitial sizes, regardless their curvature. This is consistent with their recentobservation in ingrowing turbulent liquid crystal interfaces, but it is incontrast with the apparent observation of curved statistics in colloidaldeposition at the edge of evaporating drops. A possible explanation for thislast result, as a consequence of the very small number of monolayers analyzedin this experiment, is given. This is illustrated in a competitive growth modelpresenting a few-monolayer transient and an asymptotic behavior consistent,respectively, with the curved and flat statistics.

Kardar-Parisi-Zhang growth on one-dimensional decreasing substrates