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The Kardar-Parisi-Zhang (KPZ) equation for surface growth has been analyzedfor over three decades. Some experiments indicated the power law for theinterface width, $w(t)\sim t^\beta$, remains the same as in growth on planarsurfaces. 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. {\bf102}, 139601, 2009) argued, however, that the dynamics of the interface mustremain unchanged with a change in the geometry. Other studies indicated thatfor radial growth the exponent $\beta$ should remain the same as that of theplanar case, regardless of whether the growth is linear or nonlinear, but thatthe saturation regime will not be reached anymore. We present the results ofextensive numerical simulations in (1+1)-dimensions of the radial KPZ equation,starting from an initial circular substrate. We find that unlike the KPZequation for flat substrates, the transition from linear to nonlinearuniversality classes is not sharp. Moreover, in the long-time limit theinterface width exhibits logarithmic growth with the time, instead ofsaturation. We also find that evaporation dominates the growth process when thecoefficient of the nonlinear term in the KPZ equation is small, and that theaverage radius of the interface decreases with time and reaches a minimum butnot zero value.

Numerical Simulation and the Universality Class of the KPZ Equation for Curved Substrates