  The Kardar-Parisi-Zhang (KPZ) equation for surface growth has been analyzed
for over three decades. Some experiments indicated the power law for the
interface width, $w(t)\sim t^\beta$, remains the same as in growth on planar
surfaces. Escudero (Phys. Rev. Lett. {\bf 100}, 116101, 2008) argued, however,
that for the radial KPZ equations in (1+1)-dimension $w(t)$ should increase as
$w(t)\sim [\ln(t)]^{1/2}$ in the long-time limit. Krug (Phys. Rev. Lett. {\bf
102}, 139601, 2009) argued, however, that the dynamics of the interface must
remain unchanged with a change in the geometry. Other studies indicated that
for radial growth the exponent $\beta$ should remain the same as that of the
planar case, regardless of whether the growth is linear or nonlinear, but that
the saturation regime will not be reached anymore. We present the results of
extensive numerical simulations in (1+1)-dimensions of the radial KPZ equation,
starting from an initial circular substrate. We find that unlike the KPZ
equation for flat substrates, the transition from linear to nonlinear
universality classes is not sharp. Moreover, in the long-time limit the
interface width exhibits logarithmic growth with the time, instead of
saturation. We also find that evaporation dominates the growth process when the
coefficient of the nonlinear term in the KPZ equation is small, and that the
average radius of the interface decreases with time and reaches a minimum but
not zero value.
