The Kardar-Parisi-Zhang (KPZ) equation for surface growth has been analyzed for over three decades , and its properties and universality classes are well established. The vast majority of all the past studies were, however, concerned with surface growth that started from flat substrates. In several natural phenomena, as well as technological processes, interface growth occurs on curved surfaces. Examples include tumour and bacterial growth, as well as the interface between two fluid phases during injection of a fluid into a porous medium in which it moves radially. Since in growth on flat substrates the linear size of the system remains constant, whereas it increases in the caseof growth on curved substrates, the universality class of{\bf  [  in the long-time limit. Krug (Phys. Rev. Lett. }{\bf   the resulting growth process has remained controversial . We present the results of extensive numerical simulations in (1+1)-dimensions of the  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, the interface width exhibits logarithmic growth with the time, instead of saturation , in the long-time limit. Furthermore, we find that evaporation dominates the growth process when the coefficient of the nonlinear term in the KPZ equation is small, and that  average radius of  interface decreases with time and reaches a minimum but not zero value.
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.
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, and its properties and universality classes are well established. The vast majority of all the past studies were, however, concerned with surface growth that started from flat substrates. In several natural phenomena, as well as technological processes, interface growth occurs on curved surfaces. Examples include tumour and bacterial growth, as well as the interface between two fluid phases during injection of a fluid into a porous medium in which it moves radially. Since in growth on flat substrates the linear size of the system remains constant, whereas it increases in the caseof growth on curved substrates, the universality class of
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